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abstract: 'Wiseman and co-workers (Phys. Rev. Lett. **98**, 140402, 2007) proposed a distinction between the nonlocality classes of Bell nonlocality, steering and entanglement based on whether or not an overseer *trusts* each party in a bipartite scenario where they are asked to demonstrate entanglement. Here we extend that concept to the multipartite case and derive inequalities that progressively test for those classes of nonlocality, with different thresholds for each level. This framework includes the three classes of nonlocality above in special cases and introduces a family of others.'
author:
- 'E. G. Cavalcanti,$^{\text{1}}$ Q. Y. He,$^{\text{2}}$ M. D. Reid,$^{\text{2}}$ and H. M. Wiseman$^{1,\text{3}}$'
title: Unified criteria for multipartite quantum nonlocality
---
The Einstein-Podolsky-Rosen (EPR) paradox [@einstein] revealed an incompatibility between local causality and the completeness of quantum mechanics. Bell later discovered that quantum mechanics (QM) can violate certain inequalities that are predicted by all Local Hidden Variable (LHV) theories [@Bell]. Schrödinger [@sch] introduced the term *steering* to describe the “spooky action at a distance” envisaged by EPR where an agent can apparently “steer” a distant quantum state, and the term *entanglement* for the nonfactorisable property possessed by all states that demonstrate this phenomenon.
Entanglement, EPR paradox, and Bell’s nonlocality were seen as requiring the same resources until Werner [@werner] discovered that not all entangled states can violate a Bell inequality. At a similar time, experiments by Ou *et al.* [@eprr-1] demonstrated the EPR paradox, for measurements which would admit a LHV model but based on criteria specific to the EPR paradox [@eprr-2], thus further suggesting that the three types of nonlocality embody different physics.
This idea wasn’t fully formalised until recently, when Wiseman, Jones, and Doherty (WJD) [@hw-1] presented a definition of steering as violation of what they termed a *Local Hidden State* (LHS) model. They showed that the states exhibiting steering are a strict *subset* of the entangled states, and a strict *superset* of the Bell-nonlocal states (those violating a LHV model). In Ref. [@cavalsteer], the EPR paradox and steering were shown to be equivalent concepts, and EPR-steering criteria were defined as any experimental consequences, usually in the form of inequalities, of the LHS model. Based on that work, Saunders et al. [@steerexp] experimentally demonstrated EPR steering for a Bell-local state, thus confirming EPR steering as a distinct class of nonlocality.
The LHS approach provides a rigorous framework from which to derive a multitude of criteria for the “intermediate” EPR-steering nonlocality—in scenarios *different* to the one originally considered by EPR and Schrödinger. This allows fresh investigations of a form of nonlocality which has a profound historical significance but about which relatively little is known [@opweh-1].
However, an important scenario remains unexplored. This prior work was limited to the bipartite case, as in EPR’s original argument. WJD presented an information-theoretic interpretation for the distinction between entanglement, EPR steering and Bell nonlocality, based on whether an overseer *trusts* each of two parties in a task where they are asked to demonstrate entanglement. That approach motivates our extension to multiple parties.
Further motivation exists, as the $N$-party scenario displays very interesting behaviour. Greenberger, Horne and Zeilinger (GHZ) [@GHZ-1-1] showed that multipartite “GHZ states” exhibit extreme forms of nonlocality, which Mermin, followed by Ardehali, Belinski and Klyshko (MABK) [@mermin], characterised by inequalities whose violation by QM increases exponentially as $2^{(N-1)/2}$ with $N$. Werner and Wolf [@wernerwolfmaxghz-2] later proved the MABK inequalities to be those violated by “the widest margin in quantum theory”, for the two-setting, two-outcome experiments. Surprisingly, this multi-party Bell nonlocality is stable against loss [@braunmann-1]. Cabello et al. [@cabello] showed that above a critical detection efficiency $\eta_{crit}=N/(2N-2)$ there is no LHV model to describe violations of Mermin’s inequality.
In this paper, we introduce a family of locally causal models involving $N$ parties, $T$ of which are trusted, and derive inequalities to test for their failure. Entanglement, EPR steering and Bell nonlocality apply to special cases where all, one or no parties are trusted. These inequalities take the form of MABK but with different thresholds for each level of nonlocality.
We prove that violation of the EPR steering inequalities grows exponentially, as $2^{(N-2)/2}$, for continuous variable (CV) measurements, and as $2^{(N-1)/2}$, for dichotomic measurements. Unlike the Bell violation, however, the dichotomic maximal violation manifests, for all ** $N$, for *both* of two famous choices of measurement settings: EPR and GHZ’s that gives a perfect correlation between outcomes, and Bell’s, that gives statistical correlation only. Furthermore, we show as a consequence that EPR steering is more resistant to loss than Bell nonlocality.
We begin by introducing some notation: we consider $N$ spatially separated parties labelled by $j$ who can choose between a number of experiments. We label the experiments by lower-case letters $x_{j}\in\mathcal{M}_{j}$. The respective outcomes are labelled by upper-case letters $X_{j}\in\mathcal{O}_{x_{j}}$. A sufficient specification of the features of the preparation procedure which are explicitly *known* by the experimenters is labelled by $\kappa$; a sufficient specification of *any* (possibly unknowable and thus “hidden”) variables which may be relevant to the experiments considered is labelled by $\lambda$. Whenever an equation involving those variables appears, it is implicitly assumed that the equation holds for all values of the variables.
Before continuing, we summarise the nonlocality hierarchy for bipartite systems [@hw-1]. The strongest form of nonlocality is *Bell nonlocality*, in which LHV models are falsified [@Bell]. LHV models require that probabilities for joint measurements at sites $A$ (Alice) and $B$ (Bob) can be written in the factorisable form: $$\begin{gathered}
P(X_{A},X_{B}|x_{A},x_{B},\kappa)=\\
\int d\lambda P(\lambda|\kappa)P(X_{A}|x_{A},\lambda,\kappa)P(X_{B}|x_{B},\lambda,\kappa).\label{eq:LHVmodelbipartite}\end{gathered}$$ A *further* assumption is to require that Bob’s “local state” be *quantum*: ie. there must be a quantum state $\rho_{\lambda,\kappa}$ such that for all outcomes $X_{B}$ of all measurements $x_{B}$$$P(X_{B}|x_{B},\lambda,\kappa)=\mathrm{Tr}[E_{X_{B}}\rho_{\lambda,\kappa}]\equiv P_{Q}(X_{B}|x_{B},\lambda,\kappa),$$ where $E_{X_{B}}$ is the POVM element associated with $X_{B}$. With this assumption, we arrive at the asymmetric ** LHS model of WJD [@hw-1]*. EPR Steering* (of Bob’s state by Alice) arises when this model fails [@hw-1; @cavalsteer]. *Entanglement* arises as a failure of the Quantum Separable (QS) model, $P(X_{A},X_{B})=\int d\lambda P(\lambda)P_{Q}(X_{A}|\lambda)P_{Q}(X_{B}|\lambda),$ in which one assumes quantum states for *both* systems.
Now consider the following task [@hw-1]: Charlie wants to demonstrate entanglement between $N$ parties. Initially, he will be satisfied if their correlations cannot be written as a quantum separable (QS) model (leaving $\kappa,x_{A},x_{B}$ henceforth implicit) $$P(X_{1},...,X_{N})=\int d\lambda P(\lambda)\prod_{j=1}^{N}P_{Q}(X_{j}|\lambda).\label{eq:QS_model_multipartite-1-1}$$ Suppose now that Charlie trusts the first $T$ agents (and their apparata), but not the remaining $N-T$ (or their apparata). That is, he does not trust that the measurement outcomes reported by the untrusted group correspond to the quantum observables they report to have measured. In this case, the QS model may be violated even without any entanglement. However, if the observed correlations cannot be reproduced by a model of form $$\begin{gathered}
P(X_{1},...,X_{N})=\\
\int d\lambda P(\lambda)\prod_{j=1}^{T}P_{Q}(X_{j}|\lambda)\prod_{j=T+1}^{N}P(X_{j}|\lambda),\label{eq:LHS_model_multipartite}\end{gathered}$$ with the outcomes of the untrusted parties given by arbitrary (not necessarily quantum) LHV distributions $P(X_{j}|\lambda)$, Charlie will be convinced they share entanglement, since then no locally causal model exists that could be used by an untrusted party to generate the statistics. We denote the multipartite LHS model (\[eq:LHS\_model\_multipartite\]) with $T$ trusted sites and $N-T$ untrusted sites by LHS$(T,N)$. Violation of the LHS($T,N$) model confirms entanglement in the presence of $N-T$ untrusted sites.
Violation of a LHS($N,N$) model is equivalent to a standard entanglement test, while violation of LHS$(0,N)$ model implies Bell nonlocality. Following Ref. [@hw-1], violation of a LHS($1,2$) model is a demonstration of EPR steering. For $T>1$, failure of LHS($T,N$) implies entanglement, but not necessarily EPR steering. The most interesting case is violation of LHS($1,N$). This violation can only occur if EPR steering exists for some bipartition of the $N$ sites: due to the violation of a LHS($1,2$) model between the trusted site and the untrusted sites taken as a group, or to the violation of a LHV model among the non-trusted sites (which in turn implies EPR steering). However, in these cases we cannot interpret the situation as the state in a specific site being steered by the others, so we refer to violation of LHS($1,N$) as a multipartite demonstration of ** EPR steering*.*
We now turn to derive inequalities to demonstrate failure of each member of the family of locally causal models (\[eq:LHS\_model\_multipartite\]). Following [@mermin; @cvbell2-1-1], we construct complex functions $F_{j}^{\pm}=X_{j}\pm iY_{j}$ of measurement outcomes $X_{j},Y_{j}$ at each site $j$. For any LHS($T,N$) model (\[eq:LHS\_model\_multipartite\]), $\langle\prod_{j=1}^{N}F_{j}^{s_{j}}\rangle=\int d\lambda P(\lambda)\prod_{j=1}^{N}\langle F_{j}^{s_{j}}\rangle_{\lambda}$, where $s_{j}\in\{-,+\}$. Here $\langle F_{j}^{\pm}\rangle_{\lambda}=\langle X_{j}\rangle_{\lambda}\pm i\langle Y_{j}\rangle_{\lambda}$ and $\langle X_{j}\rangle_{\lambda}=\sum_{X_{j}}P(X_{j}|\lambda)\, X_{j}$, with $P(X_{j}|\lambda)=P_{Q}(X_{j}|\lambda)$ for the trusted parties, $1\leq j\leq T$. From the variance inequality it then follows that $$|\langle\prod_{j=1}^{N}F_{j}^{s_{j}}\rangle|^{2}\leq\int d\lambda P(\lambda)\prod_{j=1}^{N}|\langle F_{j}^{s_{j}}\rangle_{\lambda}|^{2}.\label{eq:variance_ineq}$$ Since $|\langle F_{j}^{\pm}\rangle_{\lambda}|^{2}=\langle X_{j}\rangle_{\lambda}^{2}+\langle Y_{j}\rangle_{\lambda}^{2}$, it follows from the non-negativity of variances that for any LHV (untrusted) state: $|\langle F_{j}^{\pm}\rangle_{\lambda}|^{2}\leq\langle X_{j}^{2}\rangle_{\lambda}+\langle Y_{j}^{2}\rangle_{\lambda}$. For a local *quantum* (trusted) state, quantum uncertainty relations impose further restrictions. We consider uncertainty relations of the form $\Delta^{2}X_{j}+\Delta^{2}Y_{j}\geq C_{j}$, where $C_{j}$ depends on the operators associated to $x_{j}$ and $y_{j}$. Substituting on we obtain a family of nonlocality criteria: $$|\langle\prod_{j=1}^{N}F_{j}^{s_{j}}\rangle|\leq\left\langle \prod_{j=1}^{T}(X_{j}^{2}+Y_{j}^{2}-C_{j})\prod_{j=T+1}^{N}(X_{j}^{2}+Y_{j}^{2})\right\rangle ^{1/2}.\label{eq:main_ineq}$$
So far no assumption was made about the measurements $x_{j},y_{j}$. We now consider two cases: continuous and dichotomic outcomes. For the continuous case, we assume position-momentum conjugation relations $[\hat{x_{j}},\hat{y_{j}}]=i$ for the trusted sites, which imply the local uncertainty relation $\Delta^{2}X_{j}+\Delta^{2}Y_{j}\geq1$, i.e., $C_{j}=1$. Thus the LHS model (\[eq:LHS\_model\_multipartite\]) implies $$|\langle\prod_{j=1}^{N}F_{j}^{s_{j}}\rangle|\leq\left\langle \prod_{j=1}^{T}(X_{j}^{2}+Y_{j}^{2}-1)\prod_{j=T+1}^{N}(X_{j}^{2}+Y_{j}^{2})\right\rangle ^{1/2}.\label{eq:multipartiteproofcv}$$ With $T=N$, we obtain the entanglement criterion of Hillery and Zubairy [@hillzub-1]; with $T=0$, we obtain the Bell inequality of Cavalcanti *et al.* [@cvbell2-1-1]. We now show these inequalities can be violated by QM. Using quadrature operators $\hat{x}_{j}=(\hat{a}_{j}+\hat{a}_{j}^{\dagger})/\sqrt{2}$, $\hat{y}_{j}=i(\hat{a}_{j}^{\dagger}-\hat{a}_{j})/\sqrt{2}$, where $\hat{a}_{j}^{\dagger},\hat{a}_{j}$ are bosonic creation/annihilation operators ($[\hat{a}_{j},\hat{a}_{k}^{\dagger}]=\delta_{j,k}$), we obtain $F_{j}^{+}=\sqrt{2}\hat{a}_{j}^{\dagger}$ and $F_{j}^{-}=\sqrt{2}\hat{a}_{j}$, and $(\hat{x}_{j}^{2}+\hat{y}_{j}^{2})=2\hat{a}_{j}^{\dagger}\hat{a}_{j}+1=2\hat{n}_{j}+1$, $\hat{n}_{j}$ being the number operator for each site. Symbolising $\hat{a}^{+}=\hat{a}^{\dagger}$ and $\hat{a}^{-}=\hat{a}$, the inequalities (\[eq:multipartiteproofcv\]) will be violated when $$|\langle\hat{a}_{1}^{s_{1}}...\hat{a}_{N}^{s_{N}}\rangle|>\langle\prod_{j=1}^{T}\hat{n}_{j}\prod_{j=T+1}^{N}(\hat{n}_{j}+1/2)\rangle^{1/2}.\label{eq:ataineq}$$ Consider the $N$ sites prepared in a GHZ-type state [@GHZ-1-1] $$|\psi\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes r}|1\rangle^{\otimes N-r}+e^{i\phi}|1\rangle^{\otimes r}|0\rangle^{\otimes N-r}),\label{eq:ghz}$$ where $r\in\{1,...,N\}$; $|n\rangle^{\otimes r}\equiv\bigotimes_{j=1}^{r}|n\rangle_{j}$; $|n\rangle^{\otimes N-r}\equiv\bigotimes_{j=r+1}^{N}|n\rangle_{j}$ and $|n\rangle_{j}$ are the eigenstates of $\hat{n}_{j}$. Taking $\phi=0$, the left-side of is nonzero when $s_{j}=+$ for all $j\leq r$ and $s_{j}=-$ for all $j>r$; or vice-versa. For those parameters, $|\langle\prod_{k=1}^{N}\hat{a}_{k}^{s_{k}}\rangle|=1/2$. For the right-side, note that the ordering of the trusted sites does not need to coincide with the ordering of the sites on Eq. . Inspecting Eq. we see that the trusted sites will annihilate the terms of when their corresponding state is $|0\rangle$. Thus for all $T\geq2$, $r\neq0$, we can choose the ordering on state such that both terms will give zero contribution and inequality will be violated by the same amount in all cases. For $T=1$, $r\neq0$, the right-side is $(3^{r-1}/2^{N})^{1/2}$ and we can violate the inequality for $N\ge3$ . For the optimal case of $r=1$, the ratio of left to right side becomes $2^{N-2/2}$, an exponential increase of EPR-steering with $N$. Bell nonlocality ($T=0$) requires $N>9$, with $r=N/2$ optimal, as shown in [@cvbell2-1-1; @he; @cfrd; @pra-1].
We now examine the dichotomic case using qubits. We choose $\hat{x}_{j}=\sigma_{j}^{\theta}$, $\hat{y}_{j}=\sigma_{j}^{\theta+\pi/2}$ where $\sigma_{j}^{\theta}=\sigma_{j}^{x}\cos\theta+\sigma_{j}^{y}\sin\theta$, and $\sigma_{j}^{x/y}$ are the Pauli spin operators ($\theta$ can be different for each site). Using the local uncertainty relation $\Delta^{2}\sigma_{j}^{x}+\Delta^{2}\sigma_{j}^{y}\geq1$ [@hoftoth-1] ($C_{j}=1$) for the trusted sites and the identity $(\sigma_{j}^{\theta})^{2}=1$, inequality becomes $$|\langle\prod_{j=1}^{N}F_{j}^{s_{j}}\rangle|\leq2^{(N-T)/2}.\label{eq:qubit_ineq}$$ Defining the Hermitian parts of the operator product by $\prod_{j=1}^{N}F_{j}^{s_{j}}=\mathrm{Re}\Pi_{N}+i\mathrm{Im}\Pi_{N}$, inequality implies $$\begin{aligned}
\langle\mathrm{Re}\Pi_{N}\rangle,\langle\mathrm{Im}\Pi_{N}\rangle & \leq & 2^{(N-T)/2},\label{eq:merminsteer}\\
\langle\mathrm{Re}\Pi_{N}\rangle+\langle\mathrm{Im}\Pi_{N}\rangle & \leq & 2^{(N-T+1)/2},\label{eq:merminsteerstat-2}\end{aligned}$$
For $T=N$ these reduce to the separability inequalities of Roy [@royprl]. These inequalities take the form of the MABK inequalities, but for the Bell case ($T=0),$ a stronger bound can be found [@mermin]: for *odd* $N$ only, $$\langle\mathrm{Re}\Pi_{N}\rangle,\mathrm{\langle Im}\Pi_{N}\rangle\leq2^{(N-1)/2},\label{eq:MABKodd}$$ (which is Mermin’s inequality), and for *even* $N$ only, $$\langle\mathrm{Re}\Pi_{N}\rangle+\langle\mathrm{Im}\Pi_{N}\rangle\leq2^{N/2}.\label{eq:mabkeven}$$ (which is the Ardehali-CHSH inequality [@Bell; @mermin]). The reason for the different bounds is that in the Bell case, the set of values of $\langle\prod_{j}F_{j}^{s_{j}}\rangle$ that is defined by all convex combinations of the classical extreme points (i.e., all LHV models) is a square in the complex plane [@wernerwolfmaxghz-2], and thus the edges of the square provide tighter bounds than that given by the maximum modulus of $\langle\prod_{j}F_{j}^{s_{j}}\rangle$. When one or more parties are treated as having local quantum states, however, this set becomes a circle, due to the continuum of pure states allowed by quantum mechanics, and thus the bound given by is tight.
Defining $|0\rangle$ and $|1\rangle$ now as eigenstates of $\sigma^{z}$, the GHZ state ($r=N)$ violates (\[eq:merminsteer\]-\[eq:mabkeven\]) by the maximum amount for QM [@wernerwolfmaxghz-2]. This occurs for Mermin-type inequalities (\[eq:merminsteer\]), (\[eq:MABKodd\]) for $F_{j}=\sigma_{j}^{x}\pm i\sigma_{j}^{y}$ ($j=1,...,N$), which is the case of the EPR-Bohm and GHZ paradoxes [@einstein; @GHZ-1-1] that yield perfect correlations between spatially separated spins. The quantum prediction is:$$\langle\mathrm{Re}\Pi_{N}\rangle,\mathrm{\langle Im}\Pi_{N}\rangle=2^{N-1}.\label{eq:qm1}$$ With $F_{j}=\sigma_{j}^{x}-i\sigma_{j}^{y}$ ($j\neq N$), $F_{N}=\sigma^{-\pi/4}+i\sigma^{\pi/4}$ we get the maximum quantum prediction for the Ardehali-type inequalities (\[eq:merminsteerstat-2\]), (\[eq:mabkeven\]):$$\langle\mathrm{Re}\Pi_{N}\rangle+\langle\mathrm{Im}\Pi_{N}\rangle=2^{N-1/2}.\label{eq:qm2}$$ The ratio of left to right side of (\[eq:merminsteer\]) and (\[eq:merminsteerstat-2\]) is thus $S_{N}=2^{(N+T-2)/2}$, an exponential growth for all $T$.
EPR steering is shown when the inequalities (\[eq:merminsteer\]) and (\[eq:merminsteerstat-2\]) with $T=1$ are violated. Interestingly, while the ratio $S_{N}=2^{(N-1)/2}$ is unchanged from the optimal MABK case, there is the new feature that this ratio is achieved for *both* statistical and perfect correlations, i.e. via *both* inequalities (\[eq:merminsteer\]) and (\[eq:merminsteerstat-2\]). Thus, additional EPR-steering criteria different to the MABK inequalities follow from (\[eq:merminsteerstat-2\]) when $N$ is odd, and from (\[eq:merminsteer\]) when $N$ is even: e.g. EPR steering is confirmed if $|\langle\sigma_{1}^{x}\sigma_{2}^{x}\rangle-\langle\sigma_{1}^{y}\sigma_{2}^{y}\rangle|>\sqrt{2}$. As shown by Roy [@royprl], entanglement criteria also follow from both inequalities when $T=N$: e.g. entanglement is confirmed for the CHSH Bell inequality [@Bell] with a lower theshold: $|\langle\sigma_{1}^{x}\sigma_{2}^{x'}\rangle-\langle\sigma_{1}^{y}\sigma_{2}^{y'}\rangle+\langle\sigma_{1}^{y}\sigma_{2}^{x'}\rangle+\langle\sigma_{1}^{x}\sigma_{2}^{y'}\rangle|>\sqrt{2}$.
Crucial in many nonlocality experiments is the effect of loss [@cabello]. Following [@ystbeamsplit-1], we model loss with a beam-splitter and calculate moments of detected fields, using $a_{det}=\sqrt{\eta}a+\sqrt{1-\eta}a_{vac}$. Here $a_{vac}$ is the operator for a vacuum reservoir mode into which quanta are lost.
To summarise the calculation, we start with the continuous variable case: $[a_{det},a_{det}^{\dagger}]=1$ and inequalities (\[eq:multipartiteproofcv\]-\[eq:ataineq\]) still apply. The left-side of (\[eq:ataineq\]) becomes $\eta_{t}^{T/2}\eta_{u}^{(N-T)/2}/2$ where $\eta_{t}$ ($\eta_{u}$) are efficiencies at trusted (untrusted) sites respectively. Examining the right-side and optimising $r$ of , we can detect violation of LHS($T,N$) for $T\geq2$, with $r\geq2$ and *any* nonzero efficiencies $\eta_{t},\eta_{u}$. To test EPR steering ($T=1$) we need (with $r\geq1$) $\eta_{u}^{N-1}>2^{r-N+1}(\eta_{u}+1/2)^{r-1}$, which reduces to$$\eta_{u}>2^{1/(N-1)}/2$$ for $r=1$. We note an interesting asymmetry: there is sensitivity to loss at the *untrusted* sites only, an effect noticed for the bipartite EPR paradox [@rrmp]. The limiting efficiency of 50% required to demonstrate EPR steering is much more accessible than that required for Bell nonlocality [@cvbell2-1-1] ($\eta_{u}>(1+\sqrt{5})/4\approx81\%$ as $N\rightarrow\infty$).
For the qubit case, violation of inequalities (\[eq:merminsteer\])–(\[eq:mabkeven\]) is the same for EPR steering and Bell nonlocality. However, analysis reveals that in important scenarios the former will be less sensitive to loss. As GHZ states have been prepared for qubits in optical polarisation states [@ghzexp], into the $\pm1$ eigenstates of $s^{z}$: $|0\rangle_{j}\rightarrow|0\rangle_{+j}|1\rangle_{-j}$ and $|1\rangle_{j}\rightarrow|1\rangle_{+j}|0\rangle_{-j}$.$|0\rangle_{+j}|0\rangle_{-j}$ For $T>0$, we use the uncertainty relation $\Delta^{2}s^{x}+\Delta^{2}s^{y}\geq\Delta^{2}n-\Delta^{2}s^{z}+2\langle n\rangle$ [@tothsch] to obtain $|\langle F_{j}^{\pm}\rangle_{\lambda}|^{2}\leq\eta_{t}^{2}$ [@proof].Inequality becomes$${\color{black}|\langle\prod_{j=1}^{N}F_{j}^{s_{j}}\rangle|\leq2^{(N-T)/2}\eta_{t}^{T}.}\label{eq:spin_ineq_loss}$$ With state ($r=N$), we need $\eta_{u}>2^{(2-N-T)/(2(N-T))}$ to detect failure of the LHS($T,N$) model via these inequalities. EPR steering ($T=1$) requires $\eta_{u}>1/\sqrt{2}$ (Fig.1 (b)). We see that for $N=2,\,3$, EPR steering is detectable with lower efficiency than the $\eta_{crit}>N/(2N-2)$ necessary and sufficient for incompatibility of the measurements of (\[eq:merminsteer\]) with a LHV model [@cabello].
In conclusion, we have presented a unified framework to derive criteria for a family of nonlocality models based on differing levels of trust on different parties. Those criteria are sufficient to demonstrate Bell nonlocality, EPR steering and entanglement as special cases. The criteria follow from one general proof, with different bounds for each type of nonocality, thus clarifying the relationship between them. We note that violation of the inequalities presented here (as for MABK inequalities) are sufficient but not necessary conditions for a given quantum state to display the corresponding type of nonlocality. This is evident on examining the criteria for the mixed Werner states [@werner], for which the boundary for entanglement and steering is known [@Peres-1; @hw-1]. A promising avenue is to search for criteria involving more than two settings.
We acknowledge support from the ARC Centre of Excellence program (CE11E0096 and COE348178) and an ARC Postdoctoral Research Fellowship.
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We rearrange [$\Delta^{2}s_{j}^{x}+\Delta^{2}s_{j}^{y}\geq\Delta^{2}n_{j}-\Delta^{2}s_{j}^{z}+2\langle n_{j}\rangle$]{} and substitute [$s^{2}=n(n+2)$]{} to get [$\langle s^{x}\rangle^{2}+\langle s^{y}\rangle^{2}\leq\langle n\rangle^{2}-\langle s^{z}\rangle^{2}\leq\langle n\rangle^{2}=\eta_{t}^{2}$]{}.
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[On Universal $R$-Matrix for Quantized Nontwisted Rank $3$ Affine Lie Algebras]{} .3in [Y.-Z.Zhang]{} and [M.D.Gould]{} .3in [Department of Mathematics, University of Queensland, Brisbane, Qld 4072, Australia]{}
.6in
[**Abstract:**]{}
Explicit formulas of the universal $R$-matrix are given for all quantized nontwisted rank 3 affine Lie algebras $U_q(A_2^{(1)})\,,~U_q(C_2^{(1)})$ and $U_q(G_2^{(1)})$.
Introduction
============
To any Kac-Moody (KM) algebra with a symmetrizable, generalized Cartan matrix in the sense of Kac[@Kac] there corresponds a quantum deformation of its universal enveloping algebra[@Drinfeld][@Jimbo]. Such quantum deformations are found to have a quasitriangular Hopf algebra structure. In particular, there exists a canonical element $R$ in the deformed algebra satisfying the well-known quantum Yang-Baxter relation which plays an important role in CFT’s[@Sierra], quantum integrable models[@Faddeev] [@Baxter] and knot theory[@Witten][@Reshetikhin] [@ZGB]. The canonical element $R$ is called “universal $R$-matrix”.
The explicit form of the $R$-matrix for quantized finite-dimensional simple Lie algebras and Lie superalgebras has been known for some time[@Rosso][@KR][@KT1]. Moreover, the general form of the $R$-matrix for quantum deformations of infinite-dimensional affine Lie algebras was given in general terms[@KT2]. However, the explicit results have only been obtained for the quantized nontwisted rank 2 affine Lie algebra.
The aim of the present Letter is to determine the normalizing coeffients for all quantized nontwisted rank 3 affine Lie algebras. Deriving these coeffients is interesting in several respects (see section 5), but is more relevant to our proof on the unitarity of highest weight modules of quantized affine Lie algebras which we will publish elsewhere.
Preliminaries
=============
We start with the definition of the nontwisted quantum affine Lie algebra $U_q({\cal G}^{(1)})$. Let $A^0=(a_{ij})_{1\leq i,j\leq r}$ be a symmetrizable Cartan matrix. Let ${\cal G}$ stand for the finite-dimensional simple Lie algebra associated with the symmetrical Cartan matrix $A^0_{\rm sym}=(a^{\rm sym}_{ij})=(\alpha_i,\alpha_j),~i,j=1,2,...,r$, where $r$ is the rank of ${\cal G}$. Let $A=(a_{ij})_{0\leq i,j\leq r}$ be a symmetrizable, generalized Cartan matrix in the sense of Kac. Let ${\cal G}^{(1)}$ denote the nontwisted affine Lie algebra associated with the corresponding symmetric Cartan matrix $A_{\rm sym}=(a^{\rm sym}_{ij})=(\alpha_i,\alpha_j), ~i,j=0,1,
..., r$. Then the quantum algebra $U_q({\cal G}^{(1)})$ is defined to be a Hopf algebra with generators: $\{E_i,~F_i,~q^{h_i}~(i=0,1,...,r),~q^d\}$ and relations, $$\begin{aligned}
&&q^h.q^{h'}=q^{h+h'}~~~~(h,~ h'=h_i~ (i=0,1,...,r),~d)\nonumber\\
&&q^hE_iq^{-h}=q^{(h,\alpha_i)} E_i\,,~~q^hF_iq^{-h}=
q^{-(h,\alpha_i)}F_i\nonumber\\
&&[E_i, F_j]=\delta_{ij}\frac{q^{h_i}-q^{-h_i}}{q-q^{-1}}\nonumber\\
&&({\rm ad}_qE_i)^{1-a_{ij}}E_j=0\,,~~~({\rm ad}_{q^{-1}}F_i)^{1-a_{ij}}F_j=0
\,~~~~(i\neq j)\label{relations1}\end{aligned}$$ where $$({\rm ad}_qx_\alpha)x_\beta=[x_\alpha\,,\,x_\beta]_q=x_\alpha x_\beta -
q^{(\alpha\,,\,\beta)}x_\beta x_\alpha\nonumber$$
The algebra $U_q({\cal G}^{(1)})$ is a Hopf algebra with coproduct, counit and antipode similar to the case of $U_q(\cal G)$: explicitly, the coproduct is defined by $$\begin{aligned}
&&\Delta(q^h)=q^h\otimes q^h\,,~~~h=h_i,~d\nonumber\\
&&\Delta(E_i)=q^{-h_i}\otimes E_i+E_i\otimes 1\nonumber\\
&&\Delta(F_i)=1\otimes F_i+F_i\otimes q^{h_i}\,,~~~i=0,1,...,r\end{aligned}$$ Formulae for the counit and antipode may also be given, but are not required below.
Let $\Delta'$ be the opposite coproduct: $\Delta'=T\,\Delta$, $T(x\otimes
y)=y\otimes x\,,~\forall x,y\in U_q({\cal G}^{(1)})$. Then $\Delta$ and $\Delta'$ is related by the universal $R$-matrix $R$ in $U_q({\cal G}^{(1)})\otimes
U_q({\cal G}^{(1)})$ satisfying $$\begin{aligned}
&&\Delta'(x)R=R\Delta(x)\,,~~~~~~x\in U_q({\cal G}^{(1)})\nonumber\\
&&(\Delta\otimes id )R=R^{13}R^{23}\,,~~~~(id\otimes\Delta)R=R^{13}R^{12}\end{aligned}$$
We define an anti-involution $\theta$ on $U_q({\cal G}^{(1)})$ by $$\theta(q^h)=q^{-h}\,,~~\theta(E_i)=F_i\,,~~\theta(F_i)=E_i\,,~~\theta(q)=
q^{-1}$$ which extend uniquely to an algebra anti-involution on all of $U_q({\cal G}^{(1)})$ so that $\theta(ab)=\theta(b)\theta(a)\,,~~\forall a,b\in U_q({\cal G}^{(1)})$. Throughout the paper, we use the notations: $$\begin{aligned}
&&(n)_q=\frac{1-q^n}{1-q}\,,~~[n]_q=\frac{q^n-q^{-n}}{q-q^{-1}}\,,~~
q_\alpha=q^{(\alpha,\alpha)}\nonumber\\
&&{\rm exp}_q(x)=\sum_{n\geq 0}\frac{x^n}{(n)_q!}\,,~~(n)_q!=
(n)_q(n-1)_q\,...\,(1)_q\end{aligned}$$
Rank $2$ Case: $U_q(A_1^{(1)})$
===============================
This section is devoted to a brief review of Khoroshkin and Tolstoy’s construction[@KT2] of the universal $R$-matrix. We start with the rank 2 case. Fix a normal ordering in the positive root system $\Delta_+$ of $A_1^{(1)}$ : $$\alpha,\,\alpha+\delta,\,...,\,\alpha+n\delta,\,...,\,\delta,\,2\delta,\,
...,\,m\delta,\,...\,,\,...\,,\,\beta+l\delta,\,...\,,\beta\label{order1}$$ where $\alpha$ and $\beta$ are simple roots; $\delta=\alpha+\beta$ is the minimal positive imaginary root. Construct Cartan-Weyl generators $E_\gamma\,,~F_\gamma=\theta(E_\gamma)
\,,~~\gamma\in \Delta_+$ of $U_q(A^{(1)})$ as follows: We define $$\begin{aligned}
&&\tilde{E_\delta}=[(\alpha,\alpha)]_q^{-1}[E_\alpha,\,E_\beta]_q\nonumber\\
&&E_{\alpha+n\delta}=(-1)^n\left ({\rm ad}\tilde{E_\delta}\right )^nE_\alpha
\nonumber\\
&&E_{\beta+n\delta}=\left ({\rm ad}\tilde{E_\delta}\right )^nE_\beta\,,...
\nonumber\\
&&\tilde{E}_{n\delta}= [(\alpha,\alpha)]_q^{-1}[E_{\alpha+(n-1)\delta},
\,E_\beta]_q \label{cartan-weyl1}\end{aligned}$$ where $[\tilde{E}_{n\delta},\,\tilde{E}_{m\delta}]=0$ for any $n,\,m >0$. For any $n>0$ there exists a unique element $E_{n\delta}$ [@KT2] satisfying $[E_{n\delta}\,,\,E_{m\delta}]=0$ for any $n,\,m>0$ and the relation $$\tilde{E}_{n\delta}=\sum_{
\begin{array}{c}
k_1p_1+...+k_mp_m=n\\
0<k_1<...<k_m
\end{array}
}\frac{\left ( q^{(\alpha,\alpha)}-q^{-(\alpha,\alpha)}\right )^{\sum_ip_i-1}}
{p_1!\;...\;p_m!}(E_{k_1\delta})^{p_1}...(E_{k_m\delta})^{p_m} \label{ee1}$$ Then the vectors $E_\gamma$ and $F_\gamma=
\theta(E_\gamma)$, $\gamma\in \Delta_+$ are the Cartan-Weyl generators for $U_q(A^{(1)})$.\
One has[@KT2] .1in [**Theorem 3.1:**]{} The universal $R$-matrix for $U_q(A_1^{(1)})
$ may be written as $$\begin{aligned}
R&=&\left ( \Pi_{n\geq 0}\;{\rm exp}_{q_\alpha}((q-q^{-1})(E_{\alpha+n\delta}
\otimes F_{\alpha+n\delta}))\right )\nonumber\\
& &\cdot{\rm exp}\left ( \sum_{n>0}n[n]_{q_\alpha}^{-1}
(q_\alpha-q_\alpha^{-1})(E_{n\delta}\otimes F_{n\delta})\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}\;{\rm exp}_{q_\alpha}((q-q^{-1})
(E_{\beta+n\delta}\otimes
F_{\beta+n\delta}))\right )\cdot
q^{\frac{1}{2}h_\alpha\otimes h_\alpha+c\otimes d+d\otimes c}\label{sl2R}\end{aligned}$$ where $c=h_\alpha+h_\beta$. The order in the product (\[sl2R\]) concides with the chosen normal order (\[order1\]).
For the general case $U_q({\cal G}^{(1)})$, KT proposed the following general construction. Fix some order in the positive root system $\Delta_+$ of the KM Lie algebra ${\cal G}^{(1)}$, which satisfies an additional condition, $$\alpha+n\delta~\leq~k\delta~\leq~(\delta-\beta)+l\delta\label{order2}$$ where $\alpha\,,~\beta\,\in~\Delta^0_+\,,~~\Delta_+^0$ is the positive system of ${\cal G}$; $k\,,\,l\,,\,n\,\geq\,0$ and $\delta$ is a minimal positive imaginary root. Then Cartan-Weyl generators $E_\gamma$ and $F_\gamma=\theta(E_\gamma)\,,~~\gamma
\in \Delta_+$, may be constructed inductively as follows[@KT2]. One starts from simple roots. If, for instance, $\gamma=\alpha+\beta\,,~~\alpha<\gamma<\beta$, is a root and there are no other positive roots $\alpha'$ and $\beta'$ between $\alpha$ and $\beta$ such that $\gamma=\alpha'+\beta'$, then set $$E_\gamma=[E_\alpha\,,\,E_\beta]_q=E_\alpha E_\beta - q^{(\alpha,\beta)}E_\beta
E_\alpha$$ For the root $\delta$, one uses the formula for roots $\alpha+n\delta$ and roots $(\delta-\alpha)+n\delta$, to define $$\begin{aligned}
&&\tilde{E}_\delta^{(i)}=[(\alpha_i,\alpha_i)]_q^{-1}[E_{\alpha_i},\,
E_{\delta-\alpha_i}]_q\,~~~\alpha_i=\alpha,\,\beta\nonumber\\
&&E_{\alpha_i+n\delta}=(-1)^n\left ({\rm ad}\tilde{E}_\delta^{(i)}\right )^n
E_{\alpha_i}\nonumber\\
&&E_{\delta-\alpha_i+n\delta}=\left ({\rm ad}\tilde{E}_\delta^{(i)}\right )^n
E_{\delta-\alpha_i}\,,~~~~ ...\nonumber\\
&&\tilde{E}_{n\delta}^{(i)}= [(\alpha_i,\alpha_i)]_q^{-1}[E_{\alpha_i
+(n-1)\delta},\,E_{\delta-\alpha_i}]_q \label{cartan-weyl3}\end{aligned}$$ The above inductive proceduce may be repeated for other nonsimple roots to obtain new real root vectors $E_{\gamma+n\delta}\,,~~E_{\delta-\gamma+n\delta}$, $~\gamma\in\Delta_+^0$. Finally, the imaginary root vectors $E^{(i)}_{n\delta}$ are defined through $\tilde{E}^{(i)}_{n\delta}$ by the relation (\[ee1\]) with $\alpha$ replaced by $\alpha_i$. Then, the vectors defined above $E^{(i)}_{n\delta}\,,~~
F^{(i)}_{n\delta}=\theta(E^{(i)}_{n\delta})~~(i=1,2)\,,~E_\gamma\,~~
F_\gamma=\theta(E_\gamma)$ are the Cartan-Weyl generators of $U_q({\cal G}^{(1)})$. KT[@KT2] proposed the following: .1in [**Theorem 3.2**]{}: The universal $R$-matrix for $U_q({\cal G}^{(1)})$ may be written in the form, $$\begin{aligned}
R&=&\left (\Pi_{\gamma\in \Delta_+^{\rm re}\,,\,\gamma<\delta}~~{\rm exp}_{q_
\gamma}\left (\frac{q-q^{-1}}{C_\gamma(q)}E_\gamma\otimes F_\gamma\right )
\right )\nonumber\\
& &\cdot {\rm exp}\left (\sum_{n>0}\sum^r_{i,j=1}
C^n_{ij}(q)(q-q^{-1})(E^{(i)}_{n\delta}\otimes F^{(j)}_{n\delta})
\right )\nonumber\\
& &\cdot \left (\Pi_{\gamma\in \Delta_+^{\rm re}\,,\,\gamma>\delta}~~
{\rm exp}_{q_
\gamma}\left (\frac{q-q^{-1}}{C_\gamma(q)}E_\gamma\otimes F_\gamma\right )
\right )\cdot q^{\sum^r_{i,j=1}\,(a^{-1}_{\rm sym})^{ij}h_i\otimes h_j
+c\otimes d+d\otimes c}\label{generalR}\end{aligned}$$ where $c=h_0+h_{\psi}$, $\psi$ is the highest root of ${\cal G}$, $\Delta^{\rm re}_+$ denotes the real root part of $\Delta_+$ and $(C^n_{ij}(q))=(C^n_{ji}(q))\,,~~i,j=1,2,...,r$, is the inverse of the matrix $(B^n_{ij}(q))\,~~i,j=1,2,...,r$ with $$B^n_{ij}(q)=(-1)^{n(1-\delta_{ij})}n^{-1}\frac{q^n_{ij}-q^{-n}_{ij}}
{q_{j}-q^{-1}_{j}}\frac{q-q^{-1}}{q_{i}-q^{-1}_{i}}
\,,~~~~q_{ij}=q^{(\alpha_i,\alpha_j)}\,,~~~q_i\equiv q_{\alpha_i}\label{bij}$$ The $C_\gamma(q)$ is a normalizing constant defined by $$[E_\gamma\,,\,F_\gamma]=\frac{C_\gamma(q)}{q-q^{-1}}\left ( q^{h_\gamma}
-q^{-h_\gamma}\right )$$ The order in the product of $R$-matrix coincides with the chosen order (\[order2\]).
From the point of view of applications, it remains to compute the normalizing coeffients $C_\gamma\,,~\gamma\in\Delta^{\rm re}_+$ explicitly. This is our task in next section.
Rank $3$ Case: $U_q(A_2^{(1)}),~U_q(C_2^{(1)}),~U_q(G_2^{(1)})$
===============================================================
In this section we determine the normalizing constants appearing in (\[generalR\]) for all quantized nontwisted rank 3 affine Lie algebras. The rank 3 nontwisted affine Lie algebras are ${\cal G}^{(1)}=A_2^{(1)}\,,
\, C_2^{(1)}\,,\,G_2^{(1)}$. They correspond to the rank 2 finite-dimensional simple Lie algebras $A_2\,,\,C_2\,,\,G_2$ with symmetrical Cartan matrix $A^0_{\rm sym}=(a^{\rm sym}_{ij})$, $i,j=1,2$ and positive root system $\Delta^0_+$. In what follows we use $A^0_{\rm sym}$ in the form $$A^0_{\rm sym}=(a^{\rm sym}_{ij})=\left (
\begin{array}{cc}
(\alpha,\alpha) & (\alpha,\beta)\\
(\beta,\alpha) & (\beta,\beta)
\end{array}
\right )$$ Explicitly, $$A^0_{\rm sym}=(a^{\rm sym}_{ij})=\left \{
\begin{array}{c}
\left (\begin{array}{cc}
2 & -1\\
-1 & 2
\end{array} \right )\,,~~~~{\rm for}~~ A_2\\
\left (\begin{array}{cc}
2 & -1\\
-1 & 1
\end{array} \right )\,,~~~~{\rm for }~~C_2\\
\left (\begin{array}{cc}
6 & -3\\
-3 & 2
\end{array} \right )\,,~~~~{\rm for}~~G_2
\end{array} \right .$$ The simple roots are $\alpha\,,\,\beta$ and $\delta-\psi$ with $\psi=\alpha+\beta$ for $A_2^{(1)}$, $\psi=\alpha+2\beta$ for $C_2^{(1)}$ and $\psi=2\alpha+3\beta$ for $G_2^{(1)}$.
Now we come to our main concern, i.e. to determine the normalizing constants appearing in the above $R$-matrix (\[generalR\]). First of all, we compute the inverse of (\[bij\]) and obtain $$\begin{aligned}
(C^n_{ij}(q))&=&(C^n_{ji}(q))\nonumber\\
&=&n\,c_{\alpha,\beta}\,\left (
\begin{array}{cc}
\frac{q^n_\alpha-q^{-n}_\alpha}{q-q^{-1}}(q_\beta-q^{-1}_\beta)^2 &
-(-1)^n\frac{q^n_{\alpha\beta}-q^{-n}_{\alpha\beta}}{q-q^{-1}}
(q_\alpha-q^{-1}_\alpha)
(q_\beta-q^{-1}_\beta)\\
-(-1)^n\frac{q^n_{\alpha\beta}-q^{-n}_{\alpha\beta}}{q-q^{-1}}
(q_\alpha-q^{-1}_\alpha)
(q_\beta-q^{-1}_\beta) & \frac{q_\beta^n-q^{-n}_\beta}{q-q^{-1}}
(q_\alpha-q^{-1}_\alpha)^2
\end{array} \right )\end{aligned}$$ where $$c^{-1}_{\alpha\beta}=(q^n_\alpha-q^{-n}_\alpha)(q^n_\beta-q^{-n}_\beta)
-(q^n_{\alpha\beta}-q^{-n}_{\alpha\beta})^2\,,~~~~q_{\alpha\beta}=
q^{(\alpha,\beta)}$$ For other normalizing constants we state following .1in [**Proposition 4.1:**]{} For the quantized nontwisted affine algebra $U_q(A^{(1)}_2)$, we fix the following order in $\Delta_+$ of $A_2^{(1)}$, $$\begin{aligned}
&&\alpha,\,\alpha+\delta,\,...,\,\alpha+m_1\delta,\,...,\,\alpha+\beta,\,
\alpha+\beta+\delta,\,...,\,\alpha+\beta+m_2\delta,\,...,\,\beta,\,
\beta+\delta,\,...,\,\beta+m_3\delta,\,...,\,\delta,\,
2\delta,\,...,\,\nonumber\\
&&k\delta,\,...,\,...\,(\delta-\beta)+l_1\delta,\,...,\,\delta-\beta,\,...,\,
(\delta-\alpha)+l_2\delta,\,...,\,\delta-\alpha,\,...,\,(\delta-\alpha-\beta)
+l_3\delta,\,...,\,\delta-\alpha-\beta\nonumber\\
\label{ordering1}\end{aligned}$$ where $m_i,k,l_i \geq 0\,,~~i=1,2,3$. We set $$\begin{aligned}
&&E_{\alpha+\beta}=[E_\alpha\,,\,E_\beta]_q\,,~~~~~
E_{\delta-\alpha}=[E_\beta\,,\,E_{\delta-\alpha-\beta}]_q\nonumber\\
&&E_{\delta-\beta}=[E_\alpha\,,\,E_{\delta-\alpha-\beta}]_q\end{aligned}$$ and use formula (\[cartan-weyl3\]) for $E_{\gamma+n\delta}$ and $E_{(\delta-\gamma)+n\delta}$, $\gamma\in \Delta_+^0$. Then the root vectors $E_\gamma\,,F_\gamma=\theta(E_\gamma)\,,\gamma\in
\Delta_+^{\rm re}$ satisfy the following relations: $$\begin{aligned}
&&[E_{\alpha+n\delta},F_{\alpha+n\delta}]=([(\alpha+\beta,\beta)]_q)^n\cdot
\frac{q^{h_{\alpha+n\delta}}-q^{-h_{\alpha+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{\beta+n\delta},F_{\beta+n\delta}]=([(\alpha+\beta,\alpha)]_q)^n\cdot
\frac{q^{h_{\beta+n\delta}}-q^{-h_{\beta+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{(\delta-\alpha-\beta)+n\delta},F_{(\delta-\alpha-\beta)+n\delta}]=
(-[(\alpha,\beta)]_q)^n\cdot
\frac{q^{h_{(\delta-\alpha-\beta)+n\delta}}-q^{-h_{(\delta-\alpha-\beta)
+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{\alpha+\beta+n\delta},F_{\alpha+\beta+n\delta}]=
(-[(\alpha,\beta)]_q)^{n+1}\cdot
\frac{q^{h_{\alpha+\beta+n\delta}}-q^{-h_{\alpha+\beta+n\delta}}}{q-q^{-1}}
\nonumber\\
&&[E_{(\delta-\alpha)+n\delta},F_{(\delta-\alpha)+n\delta}]=
([(\alpha+\beta,\beta)]_q)^{n+1}\cdot
\frac{q^{h_{(\delta-\alpha)+n\delta}}-q^{-h_{(\delta-\alpha)+n\delta}}}
{q-q^{-1}}\nonumber\\
&&[E_{(\delta-\beta)+n\delta},F_{(\delta-\beta)+n\delta}]=
([(\alpha+\beta,\alpha)]_q)^{n+1}\cdot
\frac{q^{h_{(\delta-\beta)+n\delta}}-q^{-h_{(\delta-\beta)+n\delta}}}
{q-q^{-1}}\label{a2}\end{aligned}$$ .1in [**Proposition 4.2:**]{} For the quantized nontwisted affine algebra $U_q(C_2^{(1)})$, we fix the order in $\Delta_+$ of $C_2^{(1)}$, according to $$\begin{aligned}
&&\alpha,\,\alpha+\delta,\,...,\,\alpha+m_1\delta,\,...,\,\alpha+\beta,\,
\alpha+\beta+\delta,\,...,\,\alpha+\beta+m_2\delta,\,...,\,\alpha+2\beta,\,
\alpha+2\beta+\delta,\,...,\,\nonumber\\
&&\alpha+2\beta+m_3\delta,\,...,\,
\beta,\,\beta+\delta,\,...,\,
\beta+m_4\delta,\,...,\,\delta,\,2\delta,\,...,\,
k\delta,\,...,\,...\,(\delta-\beta)+l_1\delta,\,...,\,\delta-\beta,\,...,\,
\nonumber\\
&&(\delta-\alpha)+l_2\delta,\,...,\,\delta-\alpha,\,...,\,(\delta-\alpha-\beta)
+l_3\delta,\,...,\,\delta-\alpha-\beta,\,...,\,(\delta-\alpha-2\beta)+l_4
\delta,\,...,\,\delta-\alpha-2\beta\nonumber\\
\label{ordering2}\end{aligned}$$ where $m_i,k,l_i\geq 0\,,~~i=1,2,3,4$. We set $$\begin{aligned}
&&E_{\alpha+\beta}=[E_\alpha\,,\,E_\beta]_q\,,~~~~~
E_{\alpha+2\beta}=[E_{\alpha+\beta}, E_\beta]_q\nonumber\\
&&E_{\delta-\alpha}=[E_\beta, E_{\delta-\alpha-\beta}]_q\,,~~~~~
E_{\delta-\beta}=[E_\alpha, E_{\delta-\alpha-\beta}]_q\nonumber\\
&&E_{\delta-\alpha-\beta}=[E_\beta, E_{\delta-\alpha-2\beta}]_q\end{aligned}$$ and use formula (\[cartan-weyl3\]) for $E_{\gamma+n\delta}$ and $E_{(\delta-\gamma)+n\delta}$, $\gamma\in \Delta_+^0$. Then the root vectors $E_\gamma\,,F_\gamma=\theta(E_\gamma)\,,\gamma\in
\Delta_+^{\rm re}$ satisfy the following relations $$\begin{aligned}
&&[E_{\alpha+n\delta},F_{\alpha+n\delta}]=([(\alpha+2\beta,\beta)]_q)^{2n}\cdot
\frac{q^{h_{\alpha+n\delta}}-q^{-h_{\alpha+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{\beta+n\delta},F_{\beta+n\delta}]=([(\alpha+2\beta,\beta)]_q)^n\cdot
([(\alpha+\beta,\alpha)]_q)^n\cdot
\frac{q^{h_{\beta+n\delta}}-q^{-h_{\beta+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{(\delta-\alpha-2\beta)+n\delta},F_{(\delta-\alpha-2\beta)+n\delta}]=
([(\alpha,\beta)]_q)^{2n}
\cdot\frac{q^{h_{(\delta-\alpha-2\beta)+n\delta}}-
q^{-h_{(\delta-\alpha-2\beta)+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{\alpha+\beta+n\delta},F_{\alpha+\beta+n\delta}]=
(-[(\alpha,\beta)]_q)^{n+1}\cdot
\frac{q^{h_{\alpha+\beta+n\delta}}-q^{-h_{\alpha+\beta+n\delta}}}{q-q^{-1}}
\nonumber\\
&&[E_{\alpha+2\beta+n\delta},F_{\alpha+2\beta+n\delta}]=
([(\alpha,\beta)]_q)^{2(n+1)}\cdot
\frac{q^{h_{\alpha+2\beta+n\delta}}-q^{-h_{\alpha+2\beta+n\delta}}}{q-q^{-1}}
\nonumber\\
&&[E_{(\delta-\alpha-\beta)+n\delta},F_{(\delta-\alpha-\beta)+n\delta}]=
([(\alpha+2\beta,\beta)]_q)^{n+1}\cdot
\frac{q^{h_{(\delta-\alpha-\beta)+n\delta}}-q^{-h_{(\delta-\alpha-\beta)
+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{(\delta-\alpha)+n\delta},F_{(\delta-\alpha)+n\delta}]=
([(\alpha+2\beta,\beta)]_q)^{2(n+1)}\cdot
\frac{q^{h_{(\delta-\alpha)+n\delta}}-q^{-h_{(\delta-\alpha)+n\delta}}}
{q-q^{-1}}\nonumber\\
&&[E_{(\delta-\beta)+n\delta},F_{(\delta-\beta)+n\delta}]=
([(\alpha+2\beta,\beta)]_q)^{n+1}
\cdot ([(\alpha+\beta,\alpha)]_q)^{n+1}
\cdot\frac{q^{h_{(\delta-\beta)+n\delta}}-q^{-h_{(\delta-\beta)+n\delta}}}
{q-q^{-1}}\nonumber\\
\label{c2}\end{aligned}$$ .1in [**Proposition 4.3:**]{} For the quantized nontwisted affine algebra $U_q(G_2^{(1)})$, we fix the order in $\Delta_+$ of $G_2^{(1)}$ as follows: $$\begin{aligned}
&&\alpha,\,\alpha+\delta,\,...,\,\alpha+m_1\delta,\,...,\,\alpha+\beta,\,
\alpha+\beta+\delta,\,...,\,\alpha+\beta+m_2\delta,\,...,\,2\alpha+3\beta,\,
2\alpha+3\beta+\delta,\,...,\,\nonumber\\
&&2\alpha+3\beta+m_3\delta,\,...,\,\alpha+2\beta,\,
\alpha+2\beta+\delta,\,...,\,\alpha+2\beta+m_4\delta,\,...,\,
\alpha+3\beta,\,\alpha+3\beta+\delta,\,...,\,\nonumber\\
&&\alpha+3\beta+m_5\delta,\,...,\,\beta,\,\beta+\delta,\,...,\,
\beta+m_6\delta,\,...,\,\delta,\,2\delta,\,...,\,
k\delta,\,...,\,...\,(\delta-\beta)+l_1\delta,\,...,\,\delta-\beta,\,...,\,
\nonumber\\
&&(\delta-\alpha)+l_2\delta,\,...,\,\delta-\alpha,\,...,\,
(\delta-\alpha-\beta)+l_3\delta,\,...,\,\delta-\alpha-\beta,\,...,\,
(\delta-\alpha-2\beta)+l_4\delta,\,...,\,\nonumber\\
&&\delta-\alpha-2\beta,\,...,\,(\delta-\alpha-3\beta)+l_5
\delta,\,...,\,\delta-\alpha-3\beta,\,...,\,(\delta-2\alpha-3\beta)+l_6
\delta,\,...,\,\delta-2\alpha-3\beta\nonumber\\
\label{ordering3}\end{aligned}$$ where $m_i,k,l_i\geq 0\,,~~i=1,2,...,6$. We set $$\begin{aligned}
&&E_{\alpha+\beta}=[E_\alpha,E_\beta]_q\,,~~~~~E_{\alpha+2\beta}=[E_{\alpha+
\beta},E_\beta]_q\nonumber\\
&&E_{\alpha+3\beta}=[E_{\alpha+2\beta},E_\beta]_q\,,~~~~~E_{2\alpha+3\beta}=
[E_{\alpha+\beta},E_{\alpha+2\beta}]_q\nonumber\\
&&E_{\delta-\alpha-3\beta}=[E_\alpha, E_{\delta-2\alpha-3\beta}]_q\,,~~~~~
E_{\delta-\alpha-2\beta}=[E_\beta, E_{\delta-\alpha-3\beta}]_q\nonumber\\
&&E_{\delta-\alpha-\beta}=[E_\beta, E_{\delta-\alpha-2\beta}]_q\,,~~~~~
E_{\delta-\alpha}=[E_\beta, E_{\delta-\alpha-\beta}]_q\nonumber\\
&&E_{\delta-\beta}=[E_\alpha, E_{\delta-\alpha-\beta}]_q\end{aligned}$$ and use formula (\[cartan-weyl3\]) for $E_{\gamma+n\delta}$ and $E_{(\delta-\gamma)+n\delta}$, $\gamma\in \Delta_+^0$. Then the root vectors $E_\gamma\,,F_\gamma=\theta(E_\gamma)\,,\gamma\in
\Delta_+^{\rm re}$ satisfy the following relations $$\begin{aligned}
&&[E_{\alpha+n\delta},
F_{\alpha+n\delta}]=a^n\cdot([(\alpha+3\beta,\beta)]_q)^n
\cdot\frac{q^{h_{\alpha+n\delta}}-q^{-h_{\alpha+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{\beta+n\delta}, F_{\beta+n\delta}]=a^n\cdot (
[(\alpha+\beta,\alpha)]_q)^n\cdot
\frac{q^{h_{\beta+n\delta}}-q^{-h_{\beta+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{(\delta-2\alpha-3\beta)+n\delta},
F_{(\delta-2\alpha-3\beta)+n\delta}]=b^n
\cdot([(\alpha,\beta)]_q)^{2n}
\cdot\frac{q^{h_{(\delta-2\alpha-3\beta)+n\delta}}-q^{-h_{(\delta-2\alpha
-3\beta)+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{\alpha+\beta+n\delta}, F_{\alpha+\beta+n\delta}]=(-[(\alpha,\beta)
]_q)^{n+1}\cdot\frac{q^{h_{\alpha+\beta+n\delta}}-q^{-h_{\alpha+\beta
+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{\alpha+2\beta+n\delta}, F_{\alpha+2\beta+n\delta}]=b^{n+1}\cdot
\frac{q^{h_{\alpha+2\beta+n\delta}}-q^{-h_{\alpha+2\beta
+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{\alpha+3\beta+n\delta}, F_{\alpha+3\beta+n\delta}]=b^{n+1}\cdot
(-[(\alpha,\beta)]_q)^{n+1}\cdot
\frac{q^{h_{\alpha+3\beta+n\delta}}-q^{-h_{\alpha+3\beta
+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{2\alpha+3\beta+n\delta}, F_{2\alpha+3\beta+n\delta}]=b^{n+1}\cdot
([(\alpha,\beta)])^{2(n+1)}\cdot
\frac{q^{h_{2\alpha+3\beta+n\delta}}-q^{-h_{2\alpha+3\beta
+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{(\delta-\alpha-3\beta)+n\delta}, F_{(\delta-\alpha-3\beta)+n\delta}]=
([(2\alpha+3\beta,\alpha)]_q)^{n+1}\cdot
\frac{q^{h_{(\delta-\alpha-3\beta)+n\delta}}-q^{-h_{(\delta-\alpha-3\beta)
+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{(\delta-\alpha-2\beta)+n\delta}, F_{(\delta-\alpha-2\beta)+n\delta}]=
([(2\alpha+3\beta,\alpha)]_q[(\alpha+3\beta,\beta)]_q )^{n+1}\cdot
\frac{q^{h_{(\delta-\alpha-2\beta)+n\delta}}-q^{-h_{(\delta-\alpha-2\beta)
+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{(\delta-\alpha-\beta)+n\delta},
F_{(\delta-\alpha-\beta)+n\delta}]=a^{n+1}
\cdot \frac{q^{h_{(\delta-\alpha-\beta)+n\delta}}-q^{-h_{(\delta-\alpha-\beta)
+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{(\delta-\alpha)+n\delta}, F_{(\delta-\alpha)+n\delta}]=a^{n+1}\cdot (
[(\alpha+3\beta,\beta)]_q)^{n+1}\cdot
\frac{q^{h_{(\delta-\alpha)+n\delta}}-q^{-h_{(\delta-\alpha)
+n\delta}}}{q-q^{-1}}\nonumber\\
&&[E_{(\delta-\beta)+n\delta}, F_{(\delta-\beta)+n\delta}]=a^{n+1}\cdot
([(\alpha+\beta,\alpha)]_q)^{n+1}\cdot
\frac{q^{h_{(\delta-\beta)+n\delta}}-q^{-h_{(\delta-\beta)
+n\delta}}}{q-q^{-1}}\label{g2}\end{aligned}$$ where $$\begin{aligned}
&&a=[(2\alpha+3\beta,\alpha)]_q\cdot [(\alpha+3\beta,\beta)]_q\cdot
([\alpha+3\beta,\beta)]_q+[(\alpha+2\beta,\beta)]_q)\nonumber\\
&&b=[(\alpha,\beta)]_q\cdot
([(\alpha,\beta)]_q+[\alpha+\beta,\beta)]_q)\end{aligned}$$ .1in [**Proof:**]{} All these propositions 4.1–4.3 can be obtained by direct computations and induction on $n$. $\Box$
Now that all explicit expressions for the constants $C_\gamma$ appearing the $R$-matrix have been obtained, we deduce, from theorem 3.2, .1in [**Theorem 4.1:**]{} For $U_q(A_2^{(1)})$, the universal $R$-matrix takes the explicit form $$\begin{aligned}
R&=&\left (\Pi_{n\geq 0}~{\rm exp}_{q_\alpha}
\left (\frac{q-q^{-1}}{C_{\alpha+n\delta}(q)}E_{\alpha+n\delta}
\otimes F_{\alpha+n\delta}\right )\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}~{\rm exp}_{q_{\alpha+\beta}}
\left (\frac{q-q^{-1}}{C_{\alpha+\beta+n\delta}(q)}E_{\alpha+\beta+n\delta}
\otimes F_{\alpha+\beta+n\delta}\right )\right )\nonumber\\
& &\cdot \left (\Pi_{n\geq 0}~{\rm exp}_{q_\beta}
\left (\frac{q-q^{-1}}{C_{\beta+n\delta}(q)}E_{\beta+n\delta}
\otimes F_{\beta+n\delta}\right )\right )\nonumber\\
& &\cdot {\rm exp}\left (\sum_{n>0}\sum^2_{i,j=1}
C^n_{ij}(q)(q-q^{-1})(E^{(i)}_{n\delta}\otimes F^{(j)}_{n\delta})
\right )\nonumber\\
& &\cdot \left (\Pi_{n\geq 0}~{\rm exp}_{q_{\beta}}
\left (\frac{q-q^{-1}}{C_{(\delta-\beta)+n\delta}(q)}E_{(\delta-\beta)+n\delta}
\otimes F_{(\delta-\beta)+n\delta}\right )\right )\nonumber\\
& &\cdot \left (\Pi_{n\geq 0}~{\rm exp}_{q_\alpha}
\left (\frac{q-q^{-1}}{C_{(\delta-\alpha)+n\delta}(q)}E_{(\delta-\alpha)
+n\delta}
\otimes F_{(\delta-\alpha)+n\delta}\right )\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}~{\rm exp}_{q_{\alpha+\beta}}
\left (\frac{q-q^{-1}}{C_{(\delta-\alpha-\beta)+n\delta}(q)}
E_{(\delta-\alpha-\beta)+n\delta}
\otimes F_{(\delta-\alpha-\beta)+n\delta}\right )\right )\nonumber\\
& &\cdot q^{\sum^2_{i,j=1}\,(a^{-1}_{\rm sym})^{ij}h_i\otimes h_j+c\otimes d+
d\otimes c}\label{aR}\end{aligned}$$ where $C_{\gamma+n\delta}\,~C_{(\delta-\gamma)+n\delta}\,,~\gamma\in\Delta^0
_+$ can be read off from (\[a2\]) and the order in the product of (\[aR\]) is defined by (\[ordering1\]). .1in [**Theorem 4.2:**]{} For $U_q(C_2^{(1)})$, the universal $R$-matrix takes the explicit form $$\begin{aligned}
R&=&\left (\Pi_{n\geq 0}~{\rm exp}_{q_\alpha}
\left (\frac{q-q^{-1}}{C_{\alpha+n\delta}(q)}E_{\alpha+n\delta}
\otimes F_{\alpha+n\delta}\right )\right )\nonumber\\
& &\cdot \left (\Pi_{n\geq 0}~{\rm exp}_{q_{\alpha+\beta}}
\left (\frac{q-q^{-1}}{C_{\alpha+\beta+n\delta}(q)}E_{\alpha+\beta+n\delta}
\otimes F_{\alpha+\beta+n\delta}\right )\right )\nonumber\\
& &\cdot \left (\Pi_{n\geq 0}~{\rm exp}_{q_{\alpha+2\beta}}
\left (\frac{q-q^{-1}}{C_{\alpha+2\beta+n\delta}(q)}E_{\alpha+2\beta+n\delta}
\otimes F_{\alpha+2\beta+n\delta}\right )\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}~{\rm exp}_{q_\beta}
\left (\frac{q-q^{-1}}{C_{\beta+n\delta}(q)}E_{\beta+n\delta}
\otimes F_{\beta+n\delta}\right )\right )\nonumber\\
& &\cdot {\rm exp}\left (\sum_{n>0}\sum^2_{i,j=1}
C^n_{ij}(q)(q-q^{-1})(E^{(i)}_{n\delta}\otimes F^{(j)}_{n\delta})
\right )\nonumber\\
& &\cdot \left (\Pi_{n\geq 0}~{\rm exp}_{q_{\beta}}
\left (\frac{q-q^{-1}}{C_{(\delta-\beta)+n\delta}(q)}E_{(\delta-\beta)+n\delta}
\otimes F_{(\delta-\beta)+n\delta}\right )\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}~{\rm exp}_{q_\alpha}
\left (\frac{q-q^{-1}}{C_{(\delta-\alpha)+n\delta}(q)}E_{(\delta-\alpha)
+n\delta}
\otimes F_{(\delta-\alpha)+n\delta}\right )\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}~{\rm exp}_{q_{\alpha+\beta}}
\left (\frac{q-q^{-1}}{C_{(\delta-\alpha-\beta)+n\delta}(q)}
E_{(\delta-\alpha-\beta)+n\delta}
\otimes F_{(\delta-\alpha-\beta)+n\delta}\right )\right )\nonumber\\
& &\cdot \left (\Pi_{n\geq 0}~{\rm exp}_{q_{\alpha+2\beta}}
\left (\frac{q-q^{-1}}{C_{(\delta-\alpha-2\beta)+n\delta}(q)}
E_{(\delta-\alpha-2\beta)+n\delta}
\otimes F_{(\delta-\alpha-2\beta)+n\delta}\right )\right )\nonumber\\
& &\cdot q^{\sum^2_{i,j=1}\,(a^{-1}_{\rm sym})^{ij}h_i\otimes h_j+c\otimes d+
d\otimes c}\label{cR}\end{aligned}$$ where $C_{\gamma+n\delta}\,~C_{(\delta-\gamma)+n\delta}\,,~\gamma\in\Delta^0
_+$ can be read off from (\[c2\]) and the order in the product of (\[cR\]) is defined by (\[ordering2\]). .1in [**Theorem 4.3:**]{} For $U_q(G_2^{(1)})$, the universal $R$-matrix takes the explicit form $$\begin{aligned}
R&=&\left (\Pi_{n\geq 0}~{\rm exp}_{q_\alpha}
\left (\frac{q-q^{-1}}{C_{\alpha+n\delta}(q)}E_{\alpha+n\delta}
\otimes F_{\alpha+n\delta}\right )\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}~{\rm exp}_{q_{\alpha+\beta}}
\left (\frac{q-q^{-1}}{C_{\alpha+\beta+n\delta}(q)}E_{\alpha+\beta+n\delta}
\otimes F_{\alpha+\beta+n\delta}\right )\right )\nonumber\\
& &\cdot \left (\Pi_{n\geq 0}~{\rm exp}_{q_{2\alpha+3\beta}}
\left (\frac{q-q^{-1}}{C_{2\alpha+3\beta+n\delta}(q)}E_{2\alpha+3\beta+n\delta}
\otimes F_{2\alpha+3\beta+n\delta}\right )\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}~{\rm exp}_{q_{\alpha+2\beta}}
\left (\frac{q-q^{-1}}{C_{\alpha+2\beta+n\delta}(q)}E_{\alpha+2\beta+n\delta}
\otimes F_{\alpha+2\beta+n\delta}\right )\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}~{\rm exp}_{q_{\alpha+3\beta}}
\left (\frac{q-q^{-1}}{C_{\alpha+3\beta+n\delta}(q)}E_{\alpha+3\beta+n\delta}
\otimes F_{\alpha+3\beta+n\delta}\right )\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}~{\rm exp}_{q_\beta}
\left (\frac{q-q^{-1}}{C_{\beta+n\delta}(q)}E_{\beta+n\delta}
\otimes F_{\beta+n\delta}\right )\right )\nonumber\\
& &\cdot {\rm exp}\left (\sum_{n>0}\sum^2_{i,j=1}
C^n_{ij}(q)(q-q^{-1})(E^{(i)}_{n\delta}\otimes F^{(j)}_{n\delta})
\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}~{\rm exp}_{q_{\beta}}
\left (\frac{q-q^{-1}}{C_{(\delta-\beta)+n\delta}(q)}E_{(\delta-\beta)+n\delta}
\otimes F_{(\delta-\beta)+n\delta}\right )\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}~{\rm exp}_{q_\alpha}
\left (\frac{q-q^{-1}}{C_{(\delta-\alpha)+n\delta}(q)}E_{(\delta-\alpha)
+n\delta}
\otimes F_{(\delta-\alpha)+n\delta}\right )\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}~{\rm exp}_{q_{\alpha+\beta}}
\left (\frac{q-q^{-1}}{C_{(\delta-\alpha-\beta)+n\delta}(q)}
E_{(\delta-\alpha-\beta)+n\delta}
\otimes F_{(\delta-\alpha-\beta)+n\delta}\right )\right )\nonumber\\
& &\cdot \left (\Pi_{n\geq 0}~{\rm exp}_{q_{\alpha+2\beta}}
\left (\frac{q-q^{-1}}{C_{(\delta-\alpha-2\beta)+n\delta}(q)}
E_{(\delta-\alpha-2\beta)+n\delta}
\otimes F_{(\delta-\alpha-2\beta)+n\delta}\right )\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}~{\rm exp}_{q_{\alpha+3\beta}}
\left (\frac{q-q^{-1}}{C_{(\delta-\alpha-3\beta)+n\delta}(q)}
E_{(\delta-\alpha-3\beta)+n\delta}
\otimes F_{(\delta-\alpha-3\beta)+n\delta}\right )\right )\nonumber\\
& &\cdot\left (\Pi_{n\geq 0}~{\rm exp}_{q_{2\alpha+3\beta}}
\left (\frac{q-q^{-1}}{C_{(\delta-2\alpha-3\beta)+n\delta}(q)}
E_{(\delta-2\alpha-3\beta)+n\delta}
\otimes F_{(\delta-2\alpha-3\beta)+n\delta}\right )\right )\nonumber\\
& &\cdot q^{\sum^2_{i,j=1}\,(a^{-1}_{\rm sym})^{ij}h_i\otimes h_j+c\otimes d+
d\otimes c}\label{gR}\end{aligned}$$ where $C_{\gamma+n\delta}\,~C_{(\delta-\gamma)+n\delta}\,,~\gamma\in\Delta^0
_+$ can be read off from (\[g2\]) and the order in the product of (\[gR\]) is defined by (\[ordering3\]). .1in [**Remark:**]{} $C_\gamma(q)$ have the following general property $$C_\gamma(q)=C_\gamma(q^{-1})~>~0\,~~~~{\rm for}~~q~>~0\,,~~ q~\neq~1
\label{positive}$$ .1in which follows directly from their definitions. The positivity (\[positive\]) in fact implies useful information on unitarity of representations, which we investigate elsewhere [@ZG].
Concluding Remarks
==================
In this Letter we have obtained the exlicit formulas of the universal $R$-matrix for all quantized nontwisted rank 3 affine Lie algebras $U_q(A_2^{(1)})\,,~~U_q(C_2^{(1)})$ and $U_q(G_2^{(1)})$ by deriving all normalizing coeffients appearing the $R$-matrix. These explicit expressions are relevant with obtaining spectral parameter dependent solutions to the Yang-Baxter equation for the corresponding quantum finite-dimensional simple Lie algebras. The point is that on finite-dimensional loop representations, the infinite products in the expression of the universal $R$-matrix truncate and thus are well defined. Since loop representations (say, evaluation representations) automatically contain a spectral parameter, the universal $R$-matrix, acting on loop representations, automatically realizes the so-called “Yang-Baxterization” procedure [@Jimbo2][@ZGB2]. This is proved for the rank 2 case $U_q(A_1^{(1)})$ in KT’s paper[@KT2]. Therefore, one may expect that our explicit expressions for the rank 3 case, acting on loop representations, will truncate to the spectral parameter dependent solution to quantum Yang-Baxter equation for $U_q(A_2)\,,~~U_q(C_2)$ obtained in [@Jimbo2][@ZGB2] and for $U_q(G_2)$ in [@Kuniba]. .3in
[**Acknowledgements:**]{}
Y.Z.Z. would like to thank Anthony John Bracken for contineous encouragement and suggestions, to thank Loriano Bonora for communication of preprint [@KT2] and to thank M.Scheunert for many patient explanations on quantum groups during July and August of last year. The financial support from Australian Research Council is gratefully acknowledged.
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|
---
author:
- |
Marco Cè\
Scuola Normale Superiore, Piazza della Cavalieri 7, I-56126 Pisa, Italy &\
INFN, sezione di Pisa, Largo B. Pontecorvo 3, I-56127 Pisa, Italy\
E-mail:
- |
\
John von Neumann Institute for Computing (NIC),\
DESY, Platanenallee 6, D-15738 Zeuthen, Germany &\
Insitut für Physik, Humboldt Universität zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany\
E-mail:
- |
Leonardo Giusti\
Università di Milano Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy &\
INFN, sezione di Milano Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy\
E-mail:
- |
Stefan Schaefer\
John von Neumann Institute for Computing (NIC),\
DESY, Platanenallee 6, D-15738 Zeuthen, Germany\
E-mail:
title: 'The large $N$ limit of the topological susceptibility of Yang-Mills gauge theory'
---
Introduction
============
One of the main successes of the large $\Nc$ limit of Yang-Mills theories is the explanation of the large mass of the $\eta'$ meson. The solution is given through the Witten-Veneziano formula [@Witten:1979vv; @Veneziano:1979ec], which relates the mass of the $\eta'$ meson to the topological susceptibility $\chitop$ in the pure Yang-Mills theory
$$\lim_{N \to \infty} \frac{m_{\eta'}^2 F_\pi^2}{2 N_\mathrm{f}} = \lim_{N \to \infty} \chi_{_\mathrm{YM}}
\qquad \text{with} \qquad \chitop=\int d^4x\, \langle q(x)\, q(0) \rangle_{_\mathrm{YM}} \,,
\label{eq:wv}$$
where $F_{\pi}$ is the pion decay constant, $\Nf$ the number of massless flavours and $q=\frac{1}{32 \pi^2} \epsilon_{\mu\nu\rho\sigma} \, \text{Tr} \, F_{\mu\nu}F_{\rho\sigma}$ is the topological charge density. The quantity on the right can only be computed directly on the lattice, provided that one employs a correct definition of the topological charge density $q$.
Our main result is the large $\Nc$ and continuum limit extrapolation of $\chitop$. We use the theoretically clean definition of $\chitop$ through the Yang-Mills gradient flow [@Luscher:2010iy] and open boundary conditions [@Luscher:2011kk] in order to avoid the freezing of the topology. In this contribution we expand on the results presented in Ref. [@Ce:2016awn] by discussing all the systematics involved in the computation of $\chitop$ for each gauge group, and those coming from the continuum and large $\Nc$ extrapolations.
Observables {#sec:obs}
===========
In the continuum, the composite fields we are interested in are the energy density $e^t$ and the topological charge density $q^t$, defined as
$$e^t=\frac{1}{2} \, \text{Tr} \, G_{\mu\nu} G_{\mu\nu} \, , \qquad q^t =\frac{1}{32 \pi^2} \epsilon_{\mu\nu\rho\sigma} \, \text{Tr} \, G_{\mu\nu} G_{\rho\sigma} \, ,
\label{eq:bobs}$$
where $G_{\mu\nu}$ is built in terms of the gauge fields $B_{\mu}$ evaluated at positive gradient flow time $t$ [@Luscher:2010iy].
Using the gradient flow, correlators built out of the fields $e^t$ and $q^t$ are finite and have a trivial renormalization. In particular, the quantity $\chitop^t$ as defined in Eq. has a finite and unambiguous continuum limit, which is independent of $t$, and obeys the correct chiral ward identities to be inserted in the Witten-Veneziano relation [@Ce:2015qha].
In order to compare the theories at different $\Nc$, we need to define a common scale to be used to express our results. In this sense, the reference scale $t_0$ introduced in Ref. [@Luscher:2010iy] for is a good choice, as it can be computed up to very high accuracy with a moderate cost. For general $\Nc$, we want this quantity to be constant at leading order in $1/\Nc$, so we generalize its definition to be
$$t^2 \left\langle e^t \right\rangle \big|_{t=t_0} = 0.1125 \left( \Nc^2 -1 \right)/\Nc \, ,
\label{eq:t0def}$$
such that it coincides with the value of $0.3$ for .
The scale $t_0$ will be used to express all our results in dimensionless units, while we use the value of $\sqrt{t_0} = 0.166$ fm only as a reference, for the clarity of the presentation, to quote values for the lattice spacing and lattice dimensions. From now on all the observables are computed at flow time $t=t_0$ unless stated otherwise.
Lattice details {#sec:latdet}
---------------
We consider Yang-Mills gauge theory on the lattice with the standard Wilson plaquette action and open boundary conditions in the time direction [@Luscher:2011kk]. For each gauge group ($\Nc=4, \, 5, \, 6$), we simulate at three different lattice spacings in a range between $0.096$ fm and $0.065$ fm and a size of the spatial dimension of $L \approx 1.5$ fm. The details of the ensembles are given in Table 1 of Ref. [@Ce:2016awn].
Because of the use of open boundary conditions, the vacuum expectation value of the observables is extracted in a plateau region sufficiently far away from the boundaries. This region is parametrized by the distance to the boundary $d$, so that the sum in the time direction is performed from $x_0 = d$ to $x_0 = T-a -d$. Considering this, the estimator for $\left\langle e^t \right\rangle$ in the lattice is given by
$$\left\langle e ^t \right\rangle = \frac{a^4}{(T-2d)\,L^3} \sum_{x_0=d}^{T-a-d} \left\langle \bar{e}^t(x_0) \right\rangle \qquad \text{with} \qquad \bar{e}^t(x_0) = \sum_{\vec{x}} e^t(\vec{x},x_0) \;,
\label{eq:defelat}$$
where $e^t(\vec{x},x_0)$ is computed through the standard clover definition of the field strength tensor.
Concerning the topological susceptibility, we define its estimator in a similar way as in Ref. [@Bruno:2014ova]
$$\begin{aligned}
\label{eq:defchilat}
\chitop^{t}(r) &= \bar{C}^t(0)+ 2 \sum_{\Delta=a}^r \bar{C}^t(\Delta) \qquad \text{with} \\
\bar{C}^t(\Delta) = \frac{a^4}{(T-2d - \Delta)L^3} & \sum_{x_0=d}^{T-a-d-\Delta} \left\langle \bar{q}^t(x_0) \bar{q}^t(x_0 + \Delta) \right\rangle \, ,
\qquad \bar{q}^t(x_0) = \sum_{\vec{x}} q^t(\vec{x},x_0) \, .
\nonumber
\end{aligned}$$
In this case, the definition of $\chitop^t$ includes an extra parameter, $r$. As we explain in the next section, this parameter can be chosen as to minimize the statistical uncertainties, while keeping the systematic effects under control.
Systematic effects from the definition of the observables {#sec:syseff1}
=========================================================
Open boundaries
---------------
Open boundaries are instrumental to achieve the finer lattice spacings in this work. Although we did not perform a dedicated comparison between open and periodic boundary conditions, the scaling of autocorrelations found for the larger $N$ is compatible with a polynomial scaling law (our evidence even suggests $\tau_{\mathrm{int}} \propto a^{-2}$); in comparison with the exponential growth observed in Ref. [@DelDebbio:2002xa]. The details of our update algorithm are given in Ref. [@Ce:2016awn].
In order to fix the parameter $d$ in Eqs. and , we fit the symmetrized data to an ansatz of the form $f(x_{0}) = A + B e^{-m x_0}$. The criterion to define the plateau region is to require that $|f(d) - A| < \sigma/4$, where $\sigma$ is the average statistical error for $x_0>d$. This guarantees that the systematic effects are negligible compared to the statistical uncertainty. Following this prescription, a good choice for $\bar{e}^t$ and $\bar{C}^t$ is $d=9.5 \sqrt{t_0}$, and $d=7.5 \sqrt{t_0}$, respectively. An example of how this fit works is shown Fig \[fig:OBCeffects\] (left).
![*Left*: $x_0$ dependence of $t_0^2 \left\langle \bar{e}^t(x_0) \right\rangle$ for an ensemble at $\beta = 11.14$. The fit to a one excited state contribution agrees very well with the data. The red vertical line denotes the value of $d=9.5 \sqrt{t_0}$, which defines the plateau region for this observable. *Right*: $\Delta$ dependence of the $\left\langle \bar{q}^t(x_0) \bar{q}^t(x_0 + \Delta) \right\rangle$ correlator. The red (open) symbols show the results when using a standard algorithm and statistics comparable to the ones used for our large $\Nc$ simulations, while in black (filled), we show the precise data obtained using a multilevel approach and approximately $10$ times more statistics. After the value of $\Delta=7.0 \sqrt{t_0}$ (red vertical line), the contribution of the tail is negligible compared to the statistical uncertainty.[]{data-label="fig:OBCeffects"}](Fig1.pdf){width="\textwidth"}
Large distance behaviour of the topological charge correlator
-------------------------------------------------------------
The definition of $\chitop^t$ in Eq. has an extra parameter $r$. For a given statistical accuracy, the existence of an appropriate $r$ is guaranteed from the exponential fall-off of $\bar{C}^t(\Delta)$. In practice however, this behaviour is hidden by the statistical fluctuations of the data, and one has to deal with a severe signal to noise problem. This is particularly relevant in the pure gauge theory, where the large mass of the pseudoscalar glueball produces an extremely fast decay in the signal.
One way to deal with the signal to noise problem is to use multilevel techniques, which have the potential to dramatically improve on the scaling of errors of the standard Monte-Carlo algorithm used in lattice QCD simulations. We use the algorithm described in Ref. [@Vera:2016xpp] to obtain high precision data for an ensemble at $\beta=6.11 \, (a = 0.078 \, \text{fm})$ on a lattice of $L \approx 1.6 \, \text{fm}$. Assuming that the relative contribution of the tail in the sum of the $\bar{C}^t(\Delta)$ correlator does not depend strongly on $\Nc$, the estimation of the tail obtained from the high precision data can be used to truncate the sum in the rest of ensembles.
Figure \[fig:OBCeffects\] (right) shows a comparison between the correlator computed using the multilevel algorithm with a total of $N_0 \times N_1 = 784 \times 280 = 201600$ measurements and the standard algorithm with $N_0 = 15600$ measurements. Clearly, the reduction in errors obtained from the multilevel algorithm is larger than the one expected simply from an increase in statistics.
We use the high precision data to estimate $\bar{D}^t (r) = \sum_{\Delta > r} \bar{C}^t(\Delta)$, and then compare it to $\bar{C}^t(\Delta)$ for each of our ensembles. Basically, at large distances, the contribution of the tail in the correlator is much smaller than the statistical variation, and therefore, summing it up to arbitrarily large values of $r$ increases only the statistical fluctuation, without an improvement in the signal. To find the right value of $r$ at which the systematics from the truncation can be neglected, we impose the condition $\alpha \bar{D}^t (r) < \sigma/4$, where $\sigma$ is the statistical error of $\bar{C}^t(\Delta)$ at $\Delta = r$, and $\alpha$ is a normalization factor to account for possible $\Nc$ dependences in the observable. With this criterion, the choice of $r = 7.0 \sqrt{t_0}$ guarantees that the systematic effects coming from neglecting the tail of the correlator are negligible within our statistics.
Finite volume checks
--------------------
One final source of systematic uncertainty comes from the finite volume used in lattice simulations. All our ensembles have a physical size $L \approx 1.5 \, \fm$, which are slightly larger than the ensembles used in Ref. [@Ce:2015qha]. The statistics in Ref. [@Ce:2015qha] are one order of magnitude larger than ours, and no finite size effects are observed. In order to validate this for the larger $\Nc$, we simulated lattices with $L=1.1 \, \fm$ and $2.3 \, \fm$ for both and . An additional lattice at $L= 2.0 \, \fm$ was also generated in the case of . The results are shown in Fig. \[fig:FinVol\] (left) and show that finite size effects are below the statistical fluctuations.
Large $N$ and continuum limit fits {#sec:syseff2}
==================================
The final part of the analysis is the large $\Nc$ and continuum limit extrapolations. The data used for this purpose is shown Fig. \[fig:allFits\] (left), together with the final extrapolation. In order to assess the systematics from the extrapolations, several fits were performed and a summary is shown in Fig. \[fig:allFits\] (right). The various fit strategies are described in the following.
For the final result all the points are fitted to a global function which accounts for the leading order in the Symanzik and large $\Nc$ expansions
$$t_0^2 \chitop^t(1/\Nc,a) = t_0^2 \chitop^t(0,0) + c_1 \frac{1}{N^2} + c_2 \frac{a^2}{t_0} \, .
\label{eq:GFfit}$$
Given that the scaling violations are of the same order of the statistical errors, a conservative choice is to use only the two finest points for each lattice. In this way, the assumption on the region of validity of the leading order Symanzik expansion is constrained, thus systematics are reduced at the expense of an increase in the statistical uncertainty. We use this approach and furthermore restrict the use of the data only to fit the coefficient $c_2$ in Eq. . Again, not using to fit $c_1$ reduces the systematics from the large $\Nc$ extrapolation. Using this fit strategy (NGF2), we obtain a result for $t^2_0 \chitop(0,0) = 7.03(13) \cdot 10^{-4}$. If one extra point in is used (NGF3), the result $t^2_0 \chitop(0,0)= 7.13(10) \cdot 10^{-4}$ is obtained, which is compatible with the one from NGF2.
Among the rest of fits attempted, the simplest one is to perform a continuum limit fit group by group and later apply the large $\Nc$ extrapolation (LF3). Additionally, one can use Eq. and fit it to all the points without restrictions (GF3), or in a similar fashion, as for NGF3, use the three points from , but only the two finest from the rest of gauge groups (GF2). The former produces a result of $t^2_0 \chitop(0,0) = 7.06(7) \cdot 10^{-4}$, while the latter gives a value of $t^2_0 \chitop(0,0) = 7.09(7) \cdot 10^{-4}$. Both are compatible with the results quoted previously, but notice that the errors are half as small, so the choice made on NGF2 is a more conservative one, accounting for possible systematic effects.
In addition, an extra term of the form $a^2/N^2$ can be added to Eq. . However, our data suggest that both the $1/\Nc$ and the $\mathrm{O}(a^2)$ corrections are small; a fact which is further supported by the $\Nc$ independence of the ratio $\chitop^t/\chitop^{t_0}$ as a function of $a^2/t_0$. This quantity can be captured up to very high accuracy as shown in Fig. \[fig:FinVol\] (right). In spite of this, a fit including the sub-leading $a^2/N^2$ term (GFF3) was also considered in our analysis.
As can be seen in Fig. \[fig:allFits\] (right), the different fit strategies are all compatible, and the fluctuations in the final result cannot be directly associated with a systematic effect. In fact, systematic effects cannot be discerned from the data, so the more conservative choice in NGF2 is the one we choose for our final result.
![*Left:* Results for all the ensembles used for the large $\Nc$ and continuum extrapolations. The data is from Ref. [@Ce:2015qha], while the rest is taken from Ref. [@Ce:2016awn]. The fit corresponds to NGF2. *Right:* Summary of several fits employed. For each fit we report the value of $\chi^2/\mathrm{dof}$ on the upper axis. The band shows the result from the fit NGF2, which we report as the central value for $t_0^2 \chitop(0,0)$ and is compatible with the rest of fits we have tested. []{data-label="fig:allFits"}](Fig3.pdf){width="\textwidth"}
Conclusions {#sec:conclusion}
===========
In this work we have presented the computation of the large $\Nc$ limit of the topological susceptibility $\chitop$ using a theoretically sound definition on the lattice through the Yang-Mills gradient flow. Our final result $t_0^2\chitop = 7.03(13) \cdot 10^{-4}$ has a $2 \%$ error and represents a new verification of the Witten-Veneziano formula that gives mass to the $\eta'$ meson. We have presented a detailed discussion of the systematic effects involved in this calculation and at the level of accuracy of our results, we observe no significant finite $\Nc$ or finite $a$ corrections.
Acknowledgements {#acknowledgements .unnumbered}
================
Simulations were performed at Fermi and Galileo at CINECA (YMlargeN Iscra B project and CINECA-INFN agreement), the ZIB computer center with the resources granted by the North-German Supercomputing Alliance (HLRN), on PAX at DESY (Zeuthen) and on Wilson at Milano-Bicocca. We are grateful to those institutions for computer resources granted. M.G.V. acknowledges the support from the Research Training Group GRK1504/2 “Mass, Spectrum, Symmetry” founded by the German Research Foundation (DFG).
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|
---
abstract: 'In this paper I describe a new optimal Krylov subspace solver for shifted unitary matrices called the Shifted Unitary Orthogonal Method (SUOM). This algorithm is used as a benchmark against any improvement like the two-grid algorithm. I use the latter to show that the overlap operator can be inverted by successive inversions of the truncated overlap operator. This strategy results in large gains compared to SUOM.'
address: |
Department of Physics, University of Tirana\
Bulevardi Zog I, Tirana-Albania\
[*borici@fshn.edu.al*]{}
author:
- 'A. Boriçi'
title: 'The two-grid algorithm confronts a shifted unitary orthogonal method'
---
It is well-known that overlap fermions [@Ne98] lead to much more expensive computations than standard fermions, i.e. Wilson or Kogut-Sussking fermions. This is obvious since for every application of the overlap operator an extra linear system solving is needed. For the time being, it seems that to get chiral symmetry at finite lattice spacing one should wait for a Petaflops computer being built.
However, algorithmic research is far from exhausted. In this paper I give an example that this is the case if one uses the two-grid algorithm [@Borici_MG]. Before I do this, I introduce briefly an optimal Krylov subspace solver for shifted unitary matrices.
SUOM: A NEW OPTIMAL KRYLOV SOLVER
=================================
Consider the task of solving the linear system: $$\label{lin_sys}
Dx = b, ~~~D = c_1 {\text{\fontshape{n}\selectfont 1\kern-.56exl}}+ c_2 V$$ where $V = \gamma_5$sign$(H_W)$ is a unitary matrix, the identity matrix, $H_W$ the Hermitian Wilson operator, $c_1 = (1 + m_q)/2, c_2 = (1 - m_q)/2$ and $m_q$ the bare fermion mass. The overlap operator $D$ is non-Hermitian. For such operators GMRES (Generalised Minimal Residual) and FOM (Full Orthogonalisation Method) are known to be the fastest. It is shown that when the norm-minimising process of GMRES is converging rapidly, the residual norms in the corresponding Galerkin process of FOM exhibit similar behaviour [@CullumGreenbaum96]. But they are based on long recurrences and thus require to store a large number of vectors of the size of matrix columns. However, exploiting the fact that the overlap operator is a shifted unitary matrix one can construct a GMRES type algorithm with short recurrences [@JagelsReichel94].
Similarly, a short recurrences algorithm can be obtained from FOM. The method is based on an observation of Rutishauser [@Rutishauser66] that for upper Hessenberg unitary matrices one can write $H = L U^{-1}$, where $L$ and $U$ are lower and upper bidiagonal matrices. Applying this decomposition for the Arnoldi iteration: $$V Q_k= Q_k H_k + h_{k+1,k} q_{k+1} e_k^T$$ one obtains an algorithm which constructs Arnoldi vectors $Q_k$ by short recurrences [@Borici_SUOM]: $$V Q_k U_k = Q_k L_k + l_{k+1,k} q_{k+1} e_k^T.$$ Projecting the linear system (\[lin\_sys\]) onto the Krylov subspace one gets: $$(c_1 {\text{\fontshape{n}\selectfont 1\kern-.56exl}}_k + c_2 L_k U_k^{-1}) y_k = e_1$$ which can be equivalently written as: $$(c_1 U_k + c_2 L_k) z_k = e_1, ~~~y_k = U_k z_k.$$ Note that the matrix on the left hand side is tridiagonal. It can be shown that one can solve this system and therefore the original system using short recurrences [@Borici_SUOM]. The resulting algorithm is called the Shifted Unitary Orthogonal Method (SUOM) and is given below:
$\rho = ||b||_2; ~q_0 = b/\rho; ~w_0 = q_0$ $l_{00} = q_0^H V q_0$ $\tilde{q} = Vq_0 - l_{00} q_0$ $l_{10} = ||\tilde{q}||_2; ~q_1 = \tilde{q} / l_{10}$ $\tilde{l}_{00} = c_1 + c_2 l_{00}$ $\alpha_0 = \rho/\tilde{l}_{00}; ~x_0 = \alpha_0 w_0;
~r_0 = b - \alpha_0 D w_0$ $u_{k-1k} = - q_{k-1}^H V q_k / q_{k-1}^H V q_{k-1}$ $l_{kk} = q_k^H V q_k + u_{k-1k} q_k^H V q_{k-1}$ $\tilde{q} = (V - l_{kk}) q_k + u_{k-1k} V q_{k-1}$ $l_{k+1k} = ||\tilde{q}||_2$ $q_{k+1} = \tilde{q} / l_{k+1k}$ $\tilde{l}_{kk} = c_1 + c_2 l_{kk}
- c_1 c_2 l_{kk-1} u_{k-1k} / \tilde{l}_{k-1k-1}$ $\alpha_k = - c_2 l_{kk-1} / \tilde{l}_{kk} \alpha_{k-1}$ $w_k = q_k + u_{k-1k} q_{k-1}
- c_1 u_{k-1k} / \tilde{l}_{k-1k-1} w_{k-1}$ $x_k = x_{k-1} + \alpha_k w_k$ $r_k = r_{k-1} - \alpha_k D w_k$ Stop if $||r_k||_2 <$ tol $\rho$
Note that in an actual implementation one can store $Vq_k$ and $Dw_k$ as separate vectors, which can be used in the subsequent iteration to compute $Dw_{k+1}$. Therefore only one multiplication by $V$ is needed at each step.
THE TWO-GRID ALGORITHM
======================
A straightforward application of multigrid algorithms is hopeless in the presence of non-smooth gauge fields. However, the situation is different for the 5-dimensional formulation of chiral fermions where there are no gauge connections along the fifth dimension. Here, I will limit my discussion in the easiest case which consists of two grids: the “fine” grid, which is the continuum along the fifth coordinate and a coarse grid, which is the lattice discretisation of the “fine” grid.
I define chiral fermions on the coarse grid using truncated overlap fermions [@Borici_TOV]. The corresponding 5-dimensional matrix ${\mathcal M}$ in blocked form is given by: $$\hspace{-0.2cm}
\begin{small}
\begin{pmatrix}
D_W-{\text{\fontshape{n}\selectfont 1\kern-.56exl}}& (D_W+{\text{\fontshape{n}\selectfont 1\kern-.56exl}})P_+ & -m_q(D_W+{\text{\fontshape{n}\selectfont 1\kern-.56exl}})P_- \\
(D_W+{\text{\fontshape{n}\selectfont 1\kern-.56exl}})P_- & \ddots & \\
& \ddots & \ddots \\
-m_q(D_W+{\text{\fontshape{n}\selectfont 1\kern-.56exl}})P_+ & & D_W-{\text{\fontshape{n}\selectfont 1\kern-.56exl}}\\
\end{pmatrix}
\end{small}$$ where $P_{\pm} = ({\text{\fontshape{n}\selectfont 1\kern-.56exl}}_4 + \gamma_5)/2$. Let ${\mathcal M}_1$ be the above matrix but with bare quark mass $m_q = 1$ and $P$ the permutation matrix: $$\begin{small}
\hspace{1cm}
\begin{pmatrix} P_+ & P_- & & \\
& P_+ & \ddots & \\
& & \ddots & P_- \\
P_- & & & P_+ \\
\end{pmatrix}
\end{small}$$ It can be shown that the following result hold [@Borici_MG; @Borici_link]:
\[proposition\] Let $P^T {\mathcal M}_1^{-1} {\mathcal M} P \chi = \eta$ be the linear system defined on the 5-dimensional lattice with $\chi = (y,\chi^{(2)},\ldots,\chi^{(N_5)})^T$ and $\eta = (r,o,\ldots,o)^T$. Then $y$ is the solution of the linear system $D^{(N_5)} y = r$, where $D^{(N_5)} \rightarrow D$ as $N_5 \rightarrow \infty$.
This result lends itself to a special two-grid algorithm [@Borici_MG; @Borici_link]. Indeed, $x_5 = a_5$ is the (fifth Euclidean) coordinate of interest since it contains the information about the 4-dimensional physics.
$x_1 \in {\mathbb C}^N; ~r_1 = b - D x_1;
~\text{tol}, \text{tol}_0 \in {\mathbb R}_+$ Let $\eta_i = (r_i,o,\ldots,o)^T \in {\mathbb C}^{NN_5}$ Let $\chi_{i+1} =
(y_{i+1},\chi_{i+1}^{(2)},\ldots,\chi_{i+1}^{(N_5)})^T \in {\mathbb C}^{NN_5}$ Solve ${\mathcal M} P\chi_{i+1} = {\mathcal M}_1 P\eta_i$ until $||{\mathcal M}_1 P\eta_i - {\mathcal M} P\chi_{i+1}||_2
< \text{tol}_0 ~||{\mathcal M}_1 P\eta_i||_2$ $x_{i+1} = x_i + y_{i+1}$ $r_{i+1} = b - D x_{i+1}$ Stop if $||r_{i+1}||_2 <$ tol $||b||_2$
One way of exploiting this is to use “decimation” over the fifth coordinate in order to get the 5d-vector $\eta$. Using proposition \[proposition\] one can evaluate directly the first 4d-component of $\eta$ by $r = b - D x$, $x$ being an approximate solution. The rest can be padded with zero 4d-vectors. The second step is to solve the problem on the coarse grid. Finally, one can extract the 4d-solution $y$ on this grid and correct the “fine” grid solution by $x \leftarrow x + y$. In the second cycle one has to repeat the same decimation method, since the “fine” 5d-operator is not available. Hence, the whole scheme is a restarted two-grid algorithm, which is given here as Algorithm \[two\_grid\_algor\].
COMPARISON OF METHODS
=====================
In Fig. 1 is shown the convergence of variuos algorithms as a function of Wilson matrix-vector multiplication number on a fixed gauge background on a $8^316$ lattice at $\beta = 5.7$. The convergence is measured using the norm of the residual error. For the overlap matrix-vector multiplication is used the double pass Lanczos algorithm (without small eigenspace projection of $H_W$) as described in [@Borici_over]. Together with the algorithms described in the previous sections Fig. 1 shows the performance of Conjugate Residuals (CR), Conjugate Gradients on Normal Equations (CGNE) and CG-CHI. The latter is the CGNE which solves simultaneously the decoupled chiral systems appearing in the matrix $D^HD$. One can observe a gain over CGNE which may be explained due to the reduced number of eigenvalues at each chiral sector. However, this gain is no more than $10\%$. On the other hand SUOM and CR preform rather similar with SUOM being slightly faster in this scale. The gain over CGNE is about a factor two. The Two-Grid algorithm performs the best with a gain of at least a factor 6 over SUOM and more than an order of magnitude over CGNE. This situation repeats itself for a different gauge configuration which is not shown here for the lack of space. However, if the projection of small eigenvalues is used the gain over SUOM/CR should be smaller since the Two-Grid algorithm is much less intensive in the application of the overlap operator. It is exactly the purpose of this comparison to make clear this feature of the Two-Grid algorithm. Finally, it is (not) surprising that SUOM and CR perform similarly: CR can be shown to be an efficient method for normal matrices. Since it is easier to implement CR is more appealing than other Krylov solvers.
The results of this work suggest that the full multigrid algorithm along the fifth dimension should be the next method to be explored.
[**Aknowledgements.**]{} The author would like to thank Andreas Kronfeld and LOC of Lattice 2004 for the kind support and Soros Foundation Albania for granting the travel to Fermilab.
[9]{}
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|
---
abstract: 'The dynamics of close relationships is important for understanding the migration patterns of individual life-courses. The bottom-up approach to this subject by social scientists has been limited by sample size, while the more recent top-down approach using large-scale datasets suffers from a lack of detail about the human individuals. We incorporate the geographic and demographic information of millions of mobile phone users with their communication patterns to study the dynamics of close relationships and its effect in their life-course migration. We demonstrate how the close age- and sex-biased dyadic relationships are correlated with the geographic proximity of the pair of individuals, e.g., young couples tend to live further from each other than old couples. In addition, we find that emotionally closer pairs are living geographically closer to each other. These findings imply that the life-course framework is crucial for understanding the complex dynamics of close relationships and their effect on the migration patterns of human individuals.'
author:
- 'Hang-Hyun Jo'
- Jari Saramäki
- 'Robin I. M. Dunbar'
- Kimmo Kaski
title: Spatial patterns of close relationships across the lifespan
---
Introduction
============
Human societies have successfully been described in the framework of social networks based on dyadic relationships [@Borgatti2009]. In recent years, a number of social networks have been characterised in terms of small-world properties [@Watts1998], broad distributions of the number of neighbours [@Barabasi1999], assortative mixing [@Newman2002] or homophily [@McPherson2001], and community structure [@Fortunato2010]. This is partially due to recent access to large-scale highly-resolved digital datasets on human dynamics and social interaction [@Lazer2009]. Mobile phone datasets in particular have provided a unique opportunity to study the structure and dynamics of human relationships [@Onnela2007; @Onnela2007b; @Eagle2009; @Eagle2010; @miritello2013time; @miritello2013limited]. Although the lack of detail about individuals often undermines the importance of large-scale studies based on anonymised datasets, limited geographic and demographic information of mobile phone users has successfully been used, e.g., in studies of age and sex biases in social relationships [@Palchykov2012; @Kovanen2013].
It is important to stress that humans are embedded not only in social networks but also in geographic space [@Lambiotte2008; @Barthelemy2011]. People move and migrate for a number of reasons. For example, people leave their parental home for education or employment, to get married, to rear a family, or they can move due to divorce and separation [@Kley2011]. In these life-course events, close relationships play a crucial role in shaping migrational patterns. Human mobility patterns have recently been studied by using the datasets of time-resolved location of mobile phone users [@Gonzalez2008; @Krings2009; @Simini2012; @Palchykov2014]. These datasets are limited to periods of a few years at most, so far allowing only cross-sectional analysis. In contrast, the longitudinal approach adopted by social scientists has been used to investigate long-term migration patterns over the human lifespan [@Speare1970; @Courgeau1990], but suffers from the fact that sample size is invariably limited.
Large-scale mobile phone datasets can be used to understand the role of close relationships in the life-course migration, by exploiting geographic and demographic information of mobile phone users. Frequency of contact between a pair of individuals has been established as a reliable index of emotional closeness in relationships, and the frequency of contact by telephone and other digital media like email and text message is known to correlate significantly with the frequency of face-to-face contact [@Roberts2009; @Roberts2011; @Saramaki2014]. Thus one can assume that most of important relationships of individuals are captured by mobile phone communication records, and that the level of emotional closeness in a relationship is reflected in the strength of communication. We also assume that the life-course migration patterns are reflected in the age- and sex-dependent geographic correlations of users. In other words, even though our mobile phone dataset of users of all age groups is necessarily cross-sectional, we can use them to gain insights into longitudinal, life-course migration patterns.
Results
=======
We analyse a large-scale mobile phone call dataset from a single mobile service provider in a European country [@Onnela2007; @Onnela2007b]. This dataset spans the first 7 months of 2007 and contains around 1.9 billion calls between 33 million individual users. Among them, around 5.1 million users subscribed to the provider, meaning that their demographic information is partially available. In order to study the closest relationships of egos (individuals), we rank their alters (friends, family members, acquaintances) in terms of the number of calls to/from the ego, irrespective of whether such alters have the relevant demographic information. Initially we focus on the top-ranked and 2nd-ranked alters. We consider only those ego-alter pairs for which information on age, sex, and geographic location is available for both ego and alter. There are 1.1 million pairs of egos and top-ranked alters and 0.7 million pairs of egos and 2nd-ranked alters.
For each ego-alter pair, we determine the geographic difference index $h_{ij}$ that has the value of $0$ ($1$) if the alter $j$ lives in the same (different) municipality as the ego $i$ (see Methods for determination of the municipality). Let us first set the baseline by calculating the average geographic difference indices. For all pairs of egos and top-ranked alters, $\langle h\rangle \approx 0.58$. The average indices for female egos and for male egos are $0.59$ and $0.58$, respectively, i.e., no sex differences are observed. Here we have considered egos whose number of calls is at least $15$, corresponding to one call per two weeks on average. We like to note that our conclusion is robust against to the variation of this criterion.
Geographic locations of highly-ranked alters and sex differences
----------------------------------------------------------------
The variation in geographic correlations across the lifespan can be revealed by using more detailed information on the egos and alters. We split the ego-alter pairs into four groups based on sex (M for male, F for female): F:F, M:M, F:M, and M:F. We also consider the ages of both egos and alters, and divide ego-alter pairs into two categories where the age difference between egos and alters is $\leq 10$ years and $>10$ years. This division is motivated by the observation that the age distribution of top-ranked alters is bimodal, with one peak approximately around the ego’s own age and another $\approx 25$ years apart, roughly corresponding to one generation [@Palchykov2012]. The average geographic difference indices $\langle h \rangle$, i.e., the fractions of alters living in a different municipality to the ego, are shown in Fig. \[fig:allBFZip\] as a function of the ego’s age for the different sex and age groups.
For top-ranked alters of the opposite sex (F:M, M:F) with age difference $\leq 10$ years, the geographic difference indices for female and male egos are mostly identical within error bars (Fig. \[fig:allBFZip\](a)). The fraction of alters in a different municipality increases up to $\approx 0.7$ for egos who are around 20 years old, then it gradually decreases to $\approx 0.45$ by the mid-40’s, and then remains at approximatively the same level thereafter. If one assumes that this behaviour is caused by romantic ego-alter relationships, a possible interpretation is that young couples live in the same neighbourhood until going to work or college necessitates a larger distance; eventually, individuals settle down together with either the same or another partner at an older age. The indices for same-sex ego-alter pairs with age difference $\leq 10$ years behave differently and show sex dependence. They increase more slowly, reach the maximum in the late 20’s (M:M) or around 30 years (F:F), then decrease and fluctuate (M:M) or slightly decrease and increase again (F:F). After the peak, the M:M curve remains lower than the F:F curve at all ages, indicating that the top-ranked male alters of males live on average geographically closer.
![Geographic difference indices of top-ranked and 2nd-ranked alters who live in a different municipality to ego (top and bottom, respectively). The left panels are for ego-alter pairs with age difference less than 10 years, while the age difference is larger than 10 years in the right panels. We use the notational convention that, for example, “M:F” denotes male ego and female alter. Error bars show the confidence interval with significance level $\alpha=0.05$.[]{data-label="fig:allBFZip"}](Fig1_Jo.pdf){width="\columnwidth"}
![Geographic difference indices of top-ranked alters who live in a different municipality to ego for several demographic groups of egos of ages 25, 40, and 60 (top to bottom), and of females (left) and males (right). The fraction for each age, $a$, has been averaged over $[a-2,a+2]$ for accentuating trends. Error bars show the confidence interval with significance level $\alpha=0.05$. In each panel, lower curves with empty symbols represent age distributions of top-ranked alters, which are magnified $2.5$ times for clearer visualization. Vertical black lines at the ego age are added for guiding the eye.[]{data-label="fig:1BF"}](Fig2_Jo.pdf){width="\columnwidth"}
For top-ranked alters with age difference $>10$ years, the sexes of egos and alters appear to play no role (Fig. \[fig:allBFZip\](b)). For all cases, when the egos are young, $\approx 65\%$ of top-ranked alters live in the same municipality. The fraction of alters in different municipality peaks in the 30’s, shows a local minimum at around mid-40’s, and then increases again. It would be reasonable to expect that this is because children live with their parents until leaving home in their 20’s; this is reflected in the local minimum at around mid-40’s as well. For older egos, children have left home and live elsewhere, increasing the geographic difference index with ego’s age.
![Geospatial layout of egocentric networks for four egos. The red and blue filled circles denote female and male individual users, respectively, and the size of circle is proportional to the age of the user. The width of an edge corresponds to the number of calls between users. Some overlapping circles have been moved a small amount for the sake of clarity.[]{data-label="fig:egoNets"}](Fig3_Jo.pdf){width="\columnwidth"}
The geographic difference indices between egos and their 2nd-ranked alters behave roughly similarly to the top-ranked alters (Fig. \[fig:allBFZip\](c–d)). The main difference is that the indices peak somewhat later; furthermore, for age difference $\leq 10$ years, the F:M and M:F curves no longer overlap, but the 2nd-ranked alters of females live slightly more often in the same municipality with the egos. There are several possible reasons for this behaviour and no definite conclusions can be drawn from demographic data alone. For example, this behaviour might arise from the partners of female egos being ranked 2nd more often than the partners of male egos. In particular women switch their focal interest from their partners to their children as these become adult; however this would only affect the geographic difference indices for female egos old enough to have grown-up children. For younger women, it may relate to the need to have a close female friend in whim to confide; women are more likely to have a same-sex intimate friend than men are. For ego-alter pairs with age difference $>10$ years, ego and alter sexes do not matter, except for female egos in their 30’s–40’s whose 2nd-ranked alters are slightly more often located in the same municipality, compared to corresponding male egos. This effect might be because of mother-child relationships, considering the age difference.
Correlations with alter age
---------------------------
These geographic correlations can be more clearly understood by considering age correlations between highly-ranked alters and their geographic difference indices. We show the age distributions of top-ranked alters and their geographic difference indices for several demographic groups of egos in Fig. \[fig:1BF\]. We use the notational convention that, for example, “F25” denotes the group of 25-year-old female egos. For the F25 and M25 groups (18675 and 21018 pairs, respectively), the curves for geographic difference indices are overall high and relatively flat, showing little correlation between demographic and geographic information. For 40-year-old females (F40, 8550 pairs) (Fig. \[fig:1BF\](c)), the male top-ranked alters are on average slightly older than the egos, and tend to live close to egos; it is reasonable to expect these to be their partners. In contrast, female top-ranked alters of similar age live more often in another municipality. The age distribution of top-ranked alters shows a small peak at around 20 years of age, more pronounced for female egos. As these alters commonly live in the same municipality, this peak can likely be attributed to the children of egos. For the M40 (10933 pairs), the overall pattern is fairly similar. For the F60 and M60 groups (4360 and 5115 pairs, respectively), the data display a lot of variation because of smaller samples. However, top-ranked alters of the same age tend to be more often found nearby. Additionally, there are clear peaks in the distribution of alter ages at around 30 years old; these can again be attributed to parent-child relationships. This finer scale analysis is comparable to the observed geographic correlations in Fig. \[fig:allBFZip\](a–b).
Geography and alter ranks
-------------------------
We next extend our analysis from the highly-ranked alters to whole egocentric networks. Here the alters of an ego are ranked in a descending order according to the number of calls to/from the ego; this ranking typically reflects the emotional closeness to alters [@Saramaki2014]. Figure \[fig:egoNets\] displays the geospatially embedded egocentric networks for four example egos; note the larger geographic spread for the younger egos. The geographic difference indices for groups of egos of specified age and sex and their ranked alters are shown in Fig. \[fig:actLocMuni\]. Overall, the geographic difference indices increase with increasing rank, implying that alters who are emotionally close also tend to be geographically close. This is in line with the results of Ref. [@Lambiotte2008], where it was shown that in a mobile communication network, on average, the probability that a tie belongs to a triangle decreases with the geographic distance of the two individuals. Since it is known that tie strength correlates positively with the density of triangles around the tie [@Onnela2007], this can be seen as indicative of strong ties being more frequently found between individuals who live close to one another. The results of Fig. \[fig:actLocMuni\] confirm this for egocentric networks.
Some sex differences can be observed in Fig. \[fig:actLocMuni\]. Among 25-year-olds, women are more likely to call alters who live in another municipality than are men (Fig. \[fig:actLocMuni\](a)), but this pattern is reversed among 40-year-olds (Fig. \[fig:actLocMuni\](b)). This distinction is lost among 60-year-olds (Fig. \[fig:actLocMuni\](c)). In addition, it may be noted that 25-year-olds of both sexes are more likely to call and get called by alters who live further away than either 40-year-olds or 60-year-olds. This presumably reflects the fact that once married, egos are more likely to focus their attention on people who live closer, irrespective of whether these are other family members or friends.
![Geographic difference indices of alters who live in a different municipality to ego for several demographic groups of egos: F25 and M25 (a), F40 and M40 (b), and F60 and M60 (c). The rank of an alter is determined in terms of the number of calls to/from the ego. Error bars show the confidence interval with significance level $\alpha=0.05$.[]{data-label="fig:actLocMuni"}](Fig4_Jo.pdf){width="\columnwidth"}
We have repeated all the above analyses by calculating the geographic difference indices based on provinces instead of municipalities, and with distance thresholds of 10 km and 50 km. The overall patterns are not affected by the choice of geographic difference index (see Supplementary Figs. \[fig:allDiffLoc\_error\], \[fig:groupAll\_error\], and \[fig:actAll\_error\]).
Discussion
==========
We have analysed a large-scale mobile phone dataset to study the geographic correlations of emotionally close human relationships in the life-course framework. For this, we have made a number of assumptions: (i) Important relationships of mobile phone users are captured by call records, and the emotional closeness of a relationship is reflected in the strength of communication. (ii) The highly-ranked alters of the same age group as egos and of opposite sex than egos are likely to be egos’ partners or spouses, while highly-ranked alters whose age is one generation apart from egos’ age are likely to be egos’ children or parents, irrespective of sex. (iii) The life-course migration patterns are reflected in the age- and sex-dependent geographic correlations of users. Based on these assumptions, we find that young couples tend to live further apart from each other than old couples. Female top-ranked alters of the same age and sex group as egos tend to live further apart than male top-ranked alters of the same age and sex group as egos. In all cases, the geographic correlation of close relationships with age difference larger than 10 years shows similar patterns independent of both sexes of egos and their highly-ranked alters. These findings are consistent with the finer scale analysis considering age correlations between highly-ranked alters and their geographic difference indices.
In addition, we find that emotionally closer friends tend to live closer to the ego. This is an important finding because it speaks against the common notion that people only take the trouble to call those who live further away from them (i.e., those individuals they cannot easily see face-to-face). In other words, the phone, particularly mobile phone, is not an alternative way of contacting distant alters, but rather a supplementary mechanism for contacting those alters that one also sees face-to-face.
Methods
=======
Data preparation and filtering
------------------------------
We use the mobile phone call dataset from a single mobile service provider in a European country. The dataset spans the first 7 months of 2007 and contains around 1.9 billion calls between 33 million individual users. Among them, around 5.1 million users subscribed to the operator, called company users. For company users, we have demographic information like sex and age, as well as geographic information like zip code and the most common location. The zip code is based on billing information, and the most common location is available in terms of latitude and longitude, i.e., the coordinate of GSM tower where the user spent the most time. The most common location is not necessarily the same as the location by zip code. Since the most common location is more informative than the zip code for studying behavioural patterns, we determine a zip code for the municipality nearest to the most common location of each user, which replaces the zip code based on the billing information whenever possible. By using the latitude and longitude of each municipality, the geographic distance between municipalities can be also calculated. Finally, the number of users whose information on sex, age, and zip code is available in the dataset is 3.2 million, i.e., 1.4 million females and 1.8 million males.
We find that for most egos, the age distribution of their alters shows a peculiarly sharp peak at the ego’s own age. Such peaks are most likely anomalies due to multiple phones being registered to a single individual but used by his or her family members. In order to filter these anomalies, we assume that the number of alters of the same age as egos is equal to the bigger of numbers of alters who are one year younger or older than egos.
Error estimation
----------------
Finite sample sizes lead to errors in geographic difference indices. For each point of geographic difference index, the number of ego-alter pairs and the number of such pairs satisfying the condition are given as $n$ and $m$, respectively. Here the condition can be such that the alter lives in a different municipality to ego. Since the condition can be either satisfied or not, meaning a binary variable, we introduce a fraction $x$ of pairs satisfying the condition. The conditional probability of finding $m$ such pairs among $n$ pairs is $p(x|n,m)=x^m(1-x)^{n-m}/B_1(m+1,n-m+1)$, where $B_z(a,b)=\int_0^z x^{a-1}(1-x)^{b-1}dx$ denotes an incomplete beta function. For a given significance level $\alpha$, one may estimate the confidence interval $(x_{\rm min},x_{\rm max})$. Here the value of $x_{\rm min}$ is determined such that the cumulative probability for $x\in (0,x_{\rm min})$ is equal to $\alpha/2$, i.e., $$\frac{B_{x_{\rm min}}(m+1,n-m+1)}{B_1(m+1,n-m+1)}=\frac{\alpha}{2}.$$ From this equation, $x_{\rm min}$ is obtained as the inverse incomplete beta function. $x_{\rm max}$ is similarly obtained from $$\frac{B_{x_{\rm max}}(m+1,n-m+1)}{B_1(m+1,n-m+1)}=1-\frac{\alpha}{2}.$$ Then, $(x_{\rm min},x_{\rm max})$ defines the confidence interval. These calculations are repeated for all points of geographic difference index.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank A.-L. Barabási for the data used in this research. Financial supports by the Aalto University postdoctoral programme (H.-H.J.), by the Academy of Finland, project no. 260427 (J.S.), and by an ERC Advanced grant (R.I.M.D.) are gratefully acknowledged.
[27]{} natexlab\#1[\#1]{}url \#1[`#1`]{}urlprefix
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---
abstract: 'Gamma-ray bursts (GRBs), which have been observed up to redshifts $z \approx 9.5$ can be good probes of the early universe and have the potential of testing cosmological models. The analysis by Dainotti of GRB Swift afterglow lightcurves with known redshifts and definite X-ray plateau shows an anti-correlation between the time when the plateau ends (the plateau end time) and the calculated luminosity at that time (or approximately an anti-correlation between plateau duration and luminosity). We present here an update of this correlation with a larger data sample of 101 GRBs with good lightcurves. Since some of this correlation could result from the redshift dependences of these intrinsic parameters, namely their cosmological evolution we use the Efron-Petrosian method to reveal the intrinsic nature of this correlation. We find that a substantial part of the correlation is intrinsic and describe how we recover it and how this can be used to constrain physical models of the plateau emission, whose origin is still unknown. The present result could help clarifing the debated issue about the nature of the plateau emission.'
author:
- 'Maria Giovanna Dainotti, Vahe’ Petrosian , Jack Singal , Michal Ostrowski'
title: 'Determination of the intrinsic Luminosity Time Correlation in the X-ray Afterglows of GRBs'
---
Introduction
============
GRBs are the farthest sources, seen up to redshift $z=9.46$ [@Cucchiara2011], and if emitting isotropically they are also the most powerful, (with $E_{iso} \leq 10^{54}$ erg s$^{-1}$), objects in the Universe. In spite of the great diversity of their prompt emission lightcurves and their broad range spanning over 7 orders of magnitude of $E_{iso}$, some common features have been identified from investigation of their afterglow light curves. A crucial breakthrough in this field has been the observation of GRBs by the *Swift* satellite which provides a rapid follow-up of the afterglows in several wavelengths revealing a more complex behavior of the X-ray lightcurves than a broken power law generally observed before [@OB06; @Sak07]. The [*Swift*]{} afterglow lightcurves manifest several segments. The second segment, when it is flat, is called the plateau emission. A significant step forward in determining common features in the afterglow lightcurves was made by fitting them with an analytical expression [@W07], called hereafter W07.
This provides the opportunity to look for universal features that could provide a redshift independent measure of the distance, as in studies of correlations between GRB isotropic energy and peak photon energy of the $\nu F_{\nu}$ spectrum, $ E_{iso}$-$E_{peak}$, [@Lloyd1999; @amati09], the beamed total energy $E_{\gamma}$-$E_{peak}$ [@G04; @Ghirlanda06], $L$-$V$ luminosity-Variability, [@N00; @FRR00], $L$-$E_{peak}$ [@Yonekotu04] and possibly others [@S03]. Impacts of detector thresholds on cosmological standard candles have also been considered [@Shahmoradi09; @Petrosian1998; @Petrosian1999; @Petrosian2002; @Cabrera2007]. Unfortunately, because of large dispersion [@Butler09; @Yu09] and absence of good calibration none of these correlations allow the use of GRBs as ‘standard candles’ as has been done e.g. with type Ia Supernovae.
Dainotti et al. (2008, 2010), using the W07 phenomenological law for the lightcurves of long GRBs, discovered a formal anti-correlation between the X-ray luminosity at the end of the plateau $L_X$ and the rest frame plateau end- time, $T^{*}_a=T^{obs}_a/(1+z)$, (hereafter LT), described as :
$$\log L_X = \log a + b \log T^*_{a},
\label{feq}$$
where $T^{*}_a$ is in seconds and $L_X$ is in erg/s. The normalization and the slope parameters $a$ and $b$ are constants obtained by the D’Agostini fitting method [@Dago05]. Dainotti et al. 2011a attempted to use the LT correlation as possible redshift estimator, but the paucity of the data and the scatter prevents from a definite conclusion at least for a sample of 62 GRBs. In addition, a further step to better understand the role of the plateau emission has been made with the discovery of new significant correlations between $L_X$, and the mean luminosities of the prompt emission, $<L_{\gamma,prompt}>$ [@Dainotti2011b].
The LT anticorrelation is also a useful test for theoretical models such as the accretion models, [@Cannizzo09; @Cannizzo11], the magnetar models [@Dall'Osso; @Bernardini2011; @Bernardini2012; @Rowlinson2010; @Rowlinson2013], the prior emission model [@Yamazaki09], the unified GRB and AGN model [@Nemmen2012] and the fireshell model [@Izzo2012]. Furthermore, it has been recovered within also other observational correlations [@Ghisellini2008; @Sultana2012; @Qi2012]. Finally, it has been applied as a cosmological tool [@Cardone09; @Cardone2010; @Postnikov2013]. Here, we study an updated sample of 101 GRBs and we investigate whether the LT correlation is intrinsic or induced by cosmological evolution of $L_X$ and $T^{*}_a$, and/or observational biases due to the instrumental threshold. This step is necessary to cast light on the nature of the plateau emission, to provide further constraints on the theoretical models, and possibly to assess the use of the LT correlation as a model discriminator. In section \[Data\] we describe the data and the results from correlation test carried using the data. In section \[intrinsic correlations\] we use the EP method to determine the correlation between $L_X$ and $T^*_{a}$. In section \[density and luminosity\] the cumulative density and luminosity are defined and derived. This is followed by a discussion section.
Lightcurve Data and raw correlations {#Data}
====================================
We have analyzed the sample of all GRB X-ray afterglows with known redshifts detected by [*Swift*]{} from January 2005 up to May 2011, for which the light curves include early X-ray data and therefore can be fitted by the W07 model. Willingale proposed a functional form for $f(t)$ :
$$f(t) = \left \{
\begin{array}{ll}
\displaystyle{F_i \exp{\left ( \alpha_i \left( 1 - \frac{t}{T_i} \right) \right )} \exp{\left (
- \frac{t_i}{t} \right )}} & {\rm for} \ \ t < T_i \\
~ & ~ \\
\displaystyle{F_i \left ( \frac{t}{T_i} \right )^{-\alpha_i}
\exp{\left ( - \frac{t_i}{t} \right )}} & {\rm for} \ \ t \ge T_i \\
\end{array}
\right .
\label{eq: fc}$$
for both the prompt (the index ‘i=*p*’) $\gamma$-ray and initial X -ray decay and for the afterglow ( “i=*a*") modeled so that the complete lightcurve $f_{tot}(t) = f_p(t) + f_a(t)$ contains two sets of four parameters $(T_{i},F_{i},\alpha_i,t_i)$. The transition from the exponential to the power law occurs at the point $(T_{i},F_{i}e^{-t_i/T_i})$ where the two functional sections have the same value and gradient. The parameter $\alpha_{i}$ is the temporal power law decay index and the time $t_{i}$ is the initial rise timescale.
In previous papers W07, Dainotti et al. (2008,2010) fitted the [*Swift*]{} Burst Alert Telescope (BAT)+ X-Ray Telescope (XRT) lightcurves of GRBs were fitted to Eq. (\[eq: fc\]) assuming that the rise time of the afterglow started at the time of the beginning of the decay phase of the prompt emission, $T_p$, namely $t_a=T_p$. In this paper, we search for an independent measure of the parameters of the afterglow, thus we leave $t_a$ to be a free parameter. In the majority of the cases we have $t_a \geq 0$. We use the redshifts available in the literature [@X09], in Greiner web page http://www.mpe.mpg.de/ jcg/grbgen.html, and in the Circulars Notice arxive (GCN). (We exclude GRBs with uncertain redshift measurement.) The complete set of GRBs with definite known redshift till May 2011 is $> 120$, but not all GRBs show a well defined plateau emission. [**The fitting procedure fails either when it gives unreasonable values, or when the determination of confidence interval in 1 $\sigma$ doesn’t fulfill the Avni 1976 prescriptions, for more details see http://heasarc.nasa.gov/xanadu/xspec/manual/XspecSpectralFitting.html. The latter prescriptions require for a proper evaluation of the error bars the computation in the 1 $\sigma$ confidence interval for every parameter varying the parameter value until the $\chi^2$ increases by $3.5$ above the minimum (or the best-fit) value, because we are in a tree-parameter space. These rules define the amount that the $\chi^2$ is allowed to increase, which depends on the confidence level one requires, and on the number of parameters whose confidence space is being calculated.**]{}
The source rest-frame luminosity in the [*Swift*]{} XRT bandpass, $(E_{min}, E_{max})=(0.3,10)$ keV at time $T_a$, is computed from the Equation:
$$L_X (E_{min},E_{max},T_a)= 4 \pi D_L^2(z) \, F_X (E_{min},E_{max},T_a) \times \textit{K},
\label{eq: lx}$$
where $D_L(z)$ is the GRB luminosity distance [^1], $F_X$ is the measured X-ray energy flux and $\textit{K}= (1+z)^{-1 +\beta_{a}}$ is the so called *K*-correction for X-ray power law index $\beta_{a}$ [@Evans2009; @Dainotti2010].
**The error bars on the normalization parameter and the slope quoted of the $L_X$ and $T^*_{a}$ are computed with the method of D’Agostini (2005), which is a suitable method where the errors on both variables are comparable (which is the case here) and it is not possible to decide which one is the independent variable to be used in the $\chi^2$ fitting analysis. Moreover, the relation $L_X = a T_{a}^b$ may be affected by an intrinsic scatter $\sigma_{int}$ of unknown nature that has to be taken into account. Thus, to determine the parameters $(a, b,\sigma_{int})$, we follow the [@Dago05] Bayesian approach and maximize the likelihood function ${\cal{L}}(a, b, \sigma_{int}) = \exp{[-L(a, b, \sigma_{int})]}$ where:**
$$\begin{aligned}
L(a, b, \sigma_{int}) & = &
\frac{1}{2} \sum{\ln{(\sigma_{int}^2 + \sigma_{y_i}^2 + b^2
\sigma_{x_i}^2)}} \nonumber \\
~ & + & \frac{1}{2} \sum{\frac{(y_i - a - b x_i)^2}{\sigma_{int}^2 + \sigma_{y_i}^2 + b^2
\sigma_{x_i}^2},}
\label{eq: deflike}\end{aligned}$$
$(x_i, y_i) = (\log{L_X}, \log{T_a})$ and the sum is over the ${\cal{N}}$ objects in the sample. Note that, actually, this maximization is performed in the two parameter space $(b, \sigma_{int})$ since $a$ may be estimated analytically as:
$$a = \left [ \sum{\frac{y_i - b x_i}{\sigma_{int}^2 + \sigma_{y_i}^2 + b^2
\sigma_{x_i}^2}} \right ] \left [\sum{\frac{1}{\sigma_{int}^2 + \sigma_{y_i}^2 + b^2
\sigma_{x_i}^2}} \right ]^{-1}
\label{eq: calca}$$
so that we will not consider it anymore as a fit parameter.
[^2] Initially, we had a sample of 116 GRBs with firm redshift including 11 IC, and with the evaluation of the observables $T_a$, $F_a$, $\alpha_a$ but not for all of them we were able to fulfill the Avni prescriptions mentioned above. Among the 116 GRBs, 104 had the proper evaluation of the error measurements, but 3 of them had an error energy parameter $\sigma_{E} \equiv \sqrt{\sigma_{L^*_{X}}^2 + \sigma_{T^*_a}^2} > 1$, (for definition about this parameter and its use, see Dainotti et al. 2011b) therefore we discarded, because such values of the errors have no physical meaning. We thus have a sample of 101 GRBs. To ensure that the inclusion of the IC does not introduce biases in the evaluation of the power slope for the $L_X$-$T^*_a$ correlation for long GRBs, we checked the slopes of the sample with and without the 8 IC bursts. The two power slopes are compatible within 1 $\sigma$. We pointed out that in a previous paper [@Dainotti2010] we did not introduce the IC bursts because these represented more than $14 \%$ of the sample, while in the current sample they represents only $8 \%$. For the whole sample without the IC we found the power law slope $b=-1.27 \pm _{-0.26}^{+0.18}$, while for the whole sample $b=-1.32 \pm _{-0.17}^{+0.18}$.
The Spearman correlation coefficient for the larger sample ($\rho=-0.74$) is higher than $\rho=-0.68$ obtained for a subsample of 66 long duration GRBs analyzed in Dainotti et al. 2010. The probability of the correlation (of the 101 long GRBs) occurring by chance within an uncorrelated sample is $P \approx 10^{-18}$ [@Bevington].
Figure \[fig1\], left panel, shows the $L_X$-$T^*_a$ distribution of 101 GRBs with $0.08 \leq z \leq 9.4$ and includes afterglows of 93 long and 8 short bursts with extended emission [@nb2010], called the Intermediate class (IC), see Table 1 [^3]
$GRB$ $z$ $Fx$ $dFx$ $beta_a$ $dbeta_a$ $log Ta*$ $dlogTa$ $log Lx$ $dlog Lx$ $class$
--------- -------- ---------- ---------- ---------- ----------- ----------- ---------- ---------- ----------- ---------
50315 1.949 1.16e-11 1.56e-12 1.47 1.23 3.97 0.09 47.49 0.56 long
50318 1.44 1.0e-8 1.41e-9 0.93 0.18 1.62 0.59 50.09 0.62 long
50401 2.9 5.41e-11 1.41e-11 0.87 0.23 3.19 0.04 48.58 0.12 long
050416A 0.6535 2.82e-11 3.82e-12 1.16 0.32 2.86 0.09 46.70 0.11 long
050505 4.27 4.93e-12 3.84e-12 1.09 0.04 3.67 0.09 48.02 0.34 long
050525A 0.606 2.92e-9 6.81e-10 1.04 0.15 2.29 0.10 48.84 0.11 long
050603 2.82 1.10e-12 6.64e-13 0.91 0.10 4.25 0.25 46.82 0.27 IC
50730 3.97 2.58e-11 1.55e-12 0.54 0.05 3.46 0.01 48.58 0.04 long
50802 1.71 2.20e-11 1.49e-12 0.82 0.08 3.39 0.02 47.63 0.04 long
050820A 2.612 6.28e-11 5.12e-12 0.91 0.10 3.40 0.03 48.53 0.06 long
50824 0.83 5.37e-13 1.10e-13 0.95 0.14 4.91 0.15 45.24 0.10 long
050904 6.29 5.79e-12 6.16e-13 0.61 0.02 3.15 0.40 48.09 0.46 long
050922C 2.198 8.54e-12 2.29e-12 0.92 0.24 3.38 0.09 47.48 0.16 long
051016B 0.9364 3.22e-12 5.60e-13 0.83 0.15 3.83 0.11 46.14 0.09 long
051109A 2.35 2.51e-11 7.74e-12 0.93 0.02 3.40 0.11 48.01 0.13 long
051221A 0.5465 7.91e-13 1.06e-13 0.95 0.18 4.47 0.07 44.96 0.08 IC
60108 2.03 1.69e-12 2.99e-13 1.00 0.24 3.80 0.09 47.17 0.14 long
60115 3.53 3.51e-12 6.62e-13 0.96 0.21 3.06 0.11 47.59 0.14 long
60124 2.297 4.02e-11 3.25e-12 0.97 0.14 3.75 0.03 48.20 0.07 long
60206 4.05 5.69e-11 1.61e-11 1.29 0.59 3.12 0.08 48.95 0.35 long
060210 3.91 4.84e-12 2.65e-12 1.05 0.04 3.77 0.22 47.90 0.24 long
60218 0.0331 1.32e-12 5.34e-13 3.51 0.45 5.29 0.13 42.52 0.18 long
060223A 4.41 1.14e-11 5.98e-12 1.02 0.12 1.99 0.22 48.37 0.24 long
60418 1.49 1.47e-10 2.17e-11 1.04 0.22 2.68 0.07 48.30 0.11 long
060502A 1.51 5.79e-12 5.86e-13 1.04 0.11 3.94 0.08 46.91 0.06 IC
060510B 4.9 3.51e-13 3.96e-14 1.57 0.12 3.78 0.48 47.39 0.5 long
60512 2.1 1.60e-12 5.69e-13 1.08 0.28 3.31 0.21 46.75 0.20 long
60522 5.11 1.88e-12 5.80e-13 1.14 0.28 3.17 0.14 47.70 0.21 long
60526 3.21 4.21e-12 7.22e-13 0.95 0.11 3.27 0.10 47.57 0.09 long
60604 2.68 2.31e-12 2.92e-13 1.08 0.10 3.87 0.06 47.13 0.07 long
60605 3.8 6.48e-12 1.03e-13 1.03 0.11 3.32 0.05 47.94 0.09 long
060607A 3.082 4.17e-12 5.29e-13 0.57 0.06 3.77 0.02 47.53 0.06 long
60614 0.125 1.54e-12 2.05e-13 0.88 0.05 5.01 0.04 43.79 0.06 IC
60707 3.43 3.74e-12 1.40e-13 1.34 0.18 3.81 0.16 47.59 0.19 long
60714 2.71 1.71e-11 1.52e-12 0.90 0.15 3.07 0.05 48.01 0.09 long
60729 0.54 7.97e-12 2.58e-13 1.03 0.04 4.88 0.02 46.95 0.02 long
60814 0.84 2.75e-11 2.92e-12 1.10 0.11 3.71 0.04 46.96 0.06 long
However, as mentioned above, because both $L_X$ and $T^*_a$ depend on redshift ($L_X$ increasing and $T^*_a$ decresing with $z$) and the sample covers a broad redshift range all or part of the anticorrelation might be induced by these dependencies. It is therefore important to determine the extent of this effect and determine the true or intrinsic correlation. In addition any cosmological evolution in $L_X$ and/or $T^*_a$ will affect the degree of the observed anti-correlation.
**Fig.\[fig1\], central panel, shows the colour coded fitted lines. The distribution of the subsamples presents different power law slopes when we divide the whole sample into 5 redshift bins (see Dainotti et al. 2011 for a comparison with a smaller sample) thus having 20 GRBs in each subsample. The objects in different bins exhibit some separation into different regions of the $L_X$-$T^*_a$ plane. The results are shown in fig \[fig1\] (central) with the fitted lines. In the right panel of Fig. \[fig1\] we show the power slope of the redshift bins with the mean values of the redshif bins.**
![[**Left Panel**]{} $L_X$ vs $T^*_a$ distribution for the sample of 101 GRB afterglows with the fitted correlation shown by the dashed line. The red points are the IC bursts. [**Central Panel**]{}: The same distribution divided in 5 equipopulated redshift bins shown by different colours: black for $z < 0.89$, magenta for $0.89 \leq z \leq 1.68$, blue for $1.68 < z \leq 2.45$, green $2.45 < z \leq 3.45$, red for $ z \geq 1.76$. Solid lines shows the fitted correlations. [**Right panel**]{} The variation of the power law slope (and its error range) vith the mean value of the redshift bins.[]{data-label="fig1"}](fig1A.eps "fig:"){width="0.33\hsize"}![[**Left Panel**]{} $L_X$ vs $T^*_a$ distribution for the sample of 101 GRB afterglows with the fitted correlation shown by the dashed line. The red points are the IC bursts. [**Central Panel**]{}: The same distribution divided in 5 equipopulated redshift bins shown by different colours: black for $z < 0.89$, magenta for $0.89 \leq z \leq 1.68$, blue for $1.68 < z \leq 2.45$, green $2.45 < z \leq 3.45$, red for $ z \geq 1.76$. Solid lines shows the fitted correlations. [**Right panel**]{} The variation of the power law slope (and its error range) vith the mean value of the redshift bins.[]{data-label="fig1"}](fig1B.eps "fig:"){width="0.33\hsize"} ![[**Left Panel**]{} $L_X$ vs $T^*_a$ distribution for the sample of 101 GRB afterglows with the fitted correlation shown by the dashed line. The red points are the IC bursts. [**Central Panel**]{}: The same distribution divided in 5 equipopulated redshift bins shown by different colours: black for $z < 0.89$, magenta for $0.89 \leq z \leq 1.68$, blue for $1.68 < z \leq 2.45$, green $2.45 < z \leq 3.45$, red for $ z \geq 1.76$. Solid lines shows the fitted correlations. [**Right panel**]{} The variation of the power law slope (and its error range) vith the mean value of the redshift bins.[]{data-label="fig1"}](fig1C.eps "fig:"){width="0.33\hsize"}
![[**Left Panel**]{}: The bivariate distribution of $L_X$ and redshift with two different flux limits. The instrumental XRT flux limit, $1.0 \times 10^{-14}$ erg cm$^{-2}$ (dashed green line) is too low to be representative of the flux limit, $1.5 \times 10^{-12}$ erg cm$^{-2}$ (solid red line) better represents the limit of the sample. [**Right panel**]{}: The bivariate distribution of the rest frame time $T^*_a$ and the redshift. The chosen limiting value of the observed end-time of the plateau in the sample, $T_{a,lim}= 242$ s. The red line is the limiting rest frame time, $T_{a,{\rm lim}}/(1+z)$.[]{data-label="fig2"}](fig2A.eps "fig:"){width="0.5\hsize"} ![[**Left Panel**]{}: The bivariate distribution of $L_X$ and redshift with two different flux limits. The instrumental XRT flux limit, $1.0 \times 10^{-14}$ erg cm$^{-2}$ (dashed green line) is too low to be representative of the flux limit, $1.5 \times 10^{-12}$ erg cm$^{-2}$ (solid red line) better represents the limit of the sample. [**Right panel**]{}: The bivariate distribution of the rest frame time $T^*_a$ and the redshift. The chosen limiting value of the observed end-time of the plateau in the sample, $T_{a,lim}= 242$ s. The red line is the limiting rest frame time, $T_{a,{\rm lim}}/(1+z)$.[]{data-label="fig2"}](fig2B.eps "fig:"){width="0.5\hsize"}
![[**Left Panel**]{}: Test statistic $\tau$ vs. $\alpha$, the slope of the LT correlation. [**Right Panel**]{}: Test statistic $\tau$ vs. $\alpha^{*}=1/\alpha$, the power slope of the reciprocol of the LT correlation defined by Eq. \[eq:alphareciprocol\]. The vertical lines show the best value $\tau=0$, and the 1$\sigma$ range for $| \tau | \leq 1$ for $\alpha$ and $\alpha^{*}$. Note that we expect $\alpha=1/\alpha^{*}$ which is the case within 1$\sigma$ showing the consistency of our results. The $\tau$ values for the earlier sample of 53 GRBs are also shown by (green) dotted lines. This is also consistent with the result from the current sample of 101 GRBs.[]{data-label="Fig3"}](fig3A.eps "fig:"){width="0.51\hsize"} ![[**Left Panel**]{}: Test statistic $\tau$ vs. $\alpha$, the slope of the LT correlation. [**Right Panel**]{}: Test statistic $\tau$ vs. $\alpha^{*}=1/\alpha$, the power slope of the reciprocol of the LT correlation defined by Eq. \[eq:alphareciprocol\]. The vertical lines show the best value $\tau=0$, and the 1$\sigma$ range for $| \tau | \leq 1$ for $\alpha$ and $\alpha^{*}$. Note that we expect $\alpha=1/\alpha^{*}$ which is the case within 1$\sigma$ showing the consistency of our results. The $\tau$ values for the earlier sample of 53 GRBs are also shown by (green) dotted lines. This is also consistent with the result from the current sample of 101 GRBs.[]{data-label="Fig3"}](fig3B.eps "fig:"){width="0.51\hsize"}
As evident for each bin, we again found an anticorrelation similar to the whole sample, but the mean values of the slopes are larger (smaller in absolute values) indicating flatter relations, except for the first redshift bin, than for the whole sample. As shown in the right panel of Fig. \[fig1\] (left) there is some indication that the slope steepens for higher redshifts.
This is the first indication that some of the anticorrelation may be induced by the above mentioned effects [^4]. However, in all cases these differences are all less then $3 \sigma$. We expect the correlation slope be closer to the one of the subsamples than the whole sample, because each subsample has a smaller redshift range $\delta_z$, which decreases the effect of the redshift dependence and/or redshift evolution. In addition, this test disfavors a strong redshift evolution in the correlation. In the next section we give a more quantitative analysis of these results with the Efron & Petrosian (EP) method [@Efron1992] which is able to determine the intrinsic correlation among variables in a truncated bivariate distribution.
Determination of intrinsic correlations {#intrinsic correlations}
=======================================
The first important step for determining the distribution of true correlations among the variables is quantification of the biases introduced by the observational and sample selection effects. In the case under study the selection effect or bias that distorts the statistical correlations are the flux limit and the temporal resolution of the instrument. To account for these effects we apply the Efron & Petrosian technique, already successfully applied for GRBs [@Petrosian2009; @Lloyd2000; @Kocevski2006]. Other methodologies to treat selection biases have also been investigated [@Collazzi2008].
[**The EP method reveals the intrinsic correlation because the method is specifically designed to overcome the biases resulting from incomplete data. Moreover, it identifies and removes also the redshift evolution present in both variables, time and luminosity.**]{}
The EP method uses a modified version of the Kendall $\tau$ statistic to test the independence of variables in a truncated data. Instead of calculating the ranks $R_{i}$ of each data points among all observed objects, which is normally done for an untruncated data, the rank of each data point is determined among its “associated sets" which include all objects that could have been observed given the observational limits. A full discussion of the method is provided in the literature [@Singal2011] and references cited therein.
**Here we give a brief summary of the algebra involved in the EP method. This method uses the Kendall rank test to determine the best-fit values of parameters describing the correlation functions using the test statistic**
$$\tau = {{\sum_{i}{(\mathcal{R}_i-\mathcal{E}_i)}} \over {\sqrt{\sum_i{\mathcal{V}_i}}}}
\label{tauen}$$
to determine the independence of two variables in a data set, say ($x_i,y_i$) for $i=1, \dots, n$. Here $R_i$ is the rank of variable $y$ of the data point $i$ in a set associated with it. For a untruncated data (i.e. data truncated parallel to the axes) the [*associated set*]{} of point $i$ includes all of the data with $x_j < x_i$. If the data is truncated one must form the [*associated set*]{} consisting only of those points of lower $x$ value that would have been observed if they were at the $x$ value of point $i$ given the truncation, see definition below.
If ($x_i,y_i$) were independent then the rank $\mathcal{R}_i$ should be distributed continuously between 0 and 1 with the expectation value $\mathcal{E}_i=(1/2)(i+1)$ and variance $\mathcal{V}_i=(1/12)(i^{2}+1)$. Independence is rejected at the $n \, \sigma$ level if $\vert \, \tau \, \vert > n$. Here the mean and variance are calculated separately for each associated set and summed accordingly to produce a single value for $\tau$. This parameter represents the degree of correlation for the entire sample with proper accounting for the data truncation.
With this statistic, we find the parametrization that best describes the luminosity and time evolution. This means that we have to determine the limiting flux, $F_{lim}$, which gives the minimum observed luminosity for a given redshift, $L_x= 4 \pi D_L^2(z) \, F_X K$ as shown in Fig. \[fig2\]. The nominal limiting sensitivity of XRT, $F_{lim}=10^{-14}$ [erg cm]{}$^{-2}$ ${\rm s}^{-1}$, is too low to describe the truncation of our sample, dashed line. This is because there is a limit in the plateau end times, $T^{*}_{a,{\rm lim}}= 242/(1+z)$ s, right panel of Fig. \[fig2\]. Therefore, as pointed out by Cannizzo et al. 2011 this restriction increases the flux threshold to $10^{-12}$ erg cm$^{-2}$. Therefore, taking into account the above minimum plateau end time we have investigated several limiting fluxes to determine a good representative value while keeping an adequate size of the sample. We have chosen the limiting flux $F_{lim} = 1.5 \times $10$^{-12}$ erg cm$^{-2}$, shown by the red solid line, which allows 90 GRBs in the sample.
Cosmological evolutions
-----------------------
The first step required for this kind of investigation is the determination of whether the variables $L_X$ and $T^*_a$, are correlated with redshift or are statistically independent of it. For example, the correlation between $L_X$ and the redshift, $z$, is what we call luminosity evolution, and independence of these variables would imply absence of such evolution. The EP method prescribed how to remove the correlation by defining new and independent variables.
Following the approach used for quasars and blazars [@Singal2011; @Singal2012b; @Singal2012a], we determine the correlation functions, g(z) and f(z) when determining the evolution of $L_X$ and $T^{*}_a$ so that de-evolved variables $L'_{X} \equiv L_X/g(z)$ and $T'_a \equiv T^*_a/f(z)$ are not correlated with z. The evolutionary function are parametrized by simple correlation functions
$$g(z)=(1+z)^{k_{Lx}}, f(z)=(1+z)^{k_{T^{*}a}}
\label{lxev}$$
so that $L'_{X}=L_X/g(z)$ refer to the local ($z=0$) luminosities. This is an arbitrary choice. One can chose any other fiducial redshift by defining $g(z)=[(1+z)/(1+z_{fid})]^{k_Lx}$. We have also tried this approach obtaining compatible results with the presented ones. The associated set for the source $i$ to obtain the luminosity evolution is : $$\label{eq:Ji1}
J_{i} \equiv \{j:z_{j} < z_{max}(L_i) \} \vee \{j: L_j > L_i \},$$
where $z_{\rm max}(L_i)$ is the maximum redshift at which object $i$ with $L_j$ could be placed and still be included in the survey. The objects of all the sample are indicated with $i$, while the objects in the associated sets are denoted with $j$. With the the simbol $\vee$ we intend the union of the sets.
Analogously, to obtain the plateau end time evolution factor the associated set for a given object $i$ are :
$$\label{eq:Ji2}
J_{i} \equiv \{j: z_j > z_{min,i} \} \vee \{j: T_j > T_i \},$$
where $z_{\rm min}(T_{a_i})$ is the minimum redshift at which object $i$ could be placed and still be included in the survey given its plateau duration and the limiting time of the observation.
With the specialized version of Kendell’s $\tau$ statistic, the values of $k_{L_x}$ and $k_{T^{*}a}$ for which $\tau_{L_x} = 0$ and $\tau_{T^{*}a} = 0$ are the ones that best fit the luminosity and plateau end time evolution respectively, with the 1$\sigma$ range of uncertainty given by $| \tau_x | \leq 1$. Plots of $\tau_{L_x}$ and $\tau_{T^{*}a}$ versus $k_{L_x}$ and $\tau_{T^{*}a}$ are shown in Fig. \[Fig4\]. With $k_{L_x}$ and $k_{T^{*}a}$ we are able to determine the de-evolved observables $T{'}_a$ and $L{'}_X$.
![ [**Left:**]{} Test statistic $\tau$ vs. $k_{L_x}$, the luminosity evolution defined by Eq. \[eq:Ji1\]. Right panel Test statistic $\tau$ vs. $k_{T^{*}_a}$, the time evolution defined by Eq. \[eq:Ji2\]. The red line represents the full sample of 101 GRBs, while the green line represents the small sample of 47 GRBs in common with the previous sample of 77 GRBs.[]{data-label="Fig4"}](fig4A.eps "fig:"){width="50.00000%" height="48.00000%"} ![ [**Left:**]{} Test statistic $\tau$ vs. $k_{L_x}$, the luminosity evolution defined by Eq. \[eq:Ji1\]. Right panel Test statistic $\tau$ vs. $k_{T^{*}_a}$, the time evolution defined by Eq. \[eq:Ji2\]. The red line represents the full sample of 101 GRBs, while the green line represents the small sample of 47 GRBs in common with the previous sample of 77 GRBs.[]{data-label="Fig4"}](fig4B.eps "fig:"){width="50.00000%" height="50.00000%"}
We evident there is no discernable luminosity evolution, $k_{L_x}=-0.05_{-0.55}^{+0.35}$, but there is a significant evolution in $T^{*}_a$, $k_{T^{*}a}=-0.85_{-0.30}^{+0.30}$.
Intrinsic LT correlation
------------------------
This is the first time the EP method has been applied in a parameter space for a bivariate correlation which involves a luminosity and a time, while previously the EP applications have been done in a luminosity-luminosity space [@Singal2012a]. Thefore, we stress that this means different trend in the data truncation as we have shown in Fig \[fig2\]. In the \[$L_X-T^{*}_a$\] variable space we apply the EP method to define the associated sets as:
$$\label{eq:Ji}
J_{i} \equiv \{j: L^{'}_{\rm min}(z_j) < L^{'}(z_i)\} \vee \{ j: L^{'}_j > L^{'}_i \} \vee \{ j: T^{'}_{a_{min}}(z_j) < T^{'}{_a(z_i)} \},$$
where $L^{'}_{\rm min}(z_j)$ and $T^{'}_{a_{min}}(z_j)$ are respectively the de-evolved minimum luminosity and de-evolved plateau time at redshift $z_j$ that object $j$ could have and still be included in the survey given the flux limits, its redshift, and the limiting time of the observation : $L_{\rm min}(z_j)=4 \pi D_L^2(z_j) \, F_{lim} \textit{K}$ and $T^{*}_{a_{min}}(z_j)=T_{a,{\rm lim}}/(1+z_j)$. Using the Kendall $\tau$ rank test we determine whether $L_X$, $T^{*}_a$ are independent or not. The test shows some dependence, so we apply a coordinate transformation by defining a new luminosity $L^{'}_X$ as
$$\label{eq:Ldeevolved}
\log_{10} L^{'}_X = \log Lx + \alpha \log T^{*}_a.
\label{alphae}$$
We then vary $\alpha$ and determine the value of $\tau$ (the correlation between $L^{'}_X$ and $T^{'}_a$) as a function of $\alpha$. The value of $\alpha$ which gives $\tau_{\alpha}=0$ determines the correlation between $L_X$ and $T^{*}_a$ with the 1$\sigma$ range of uncertainty given by $|\tau_{\alpha} | \leq 1$.
Fig. \[Fig3\] shows variation of $\tau_{alpha}$ with $\alpha$. As can be seen independence is achieved ($\tau=0$) for $\alpha=1.07$ with a 1 $\sigma$ range of $\alpha=(-1.21,-0.98)$, therefore $\alpha=-1.07_{-0.14}^{+0.09}$. This means that $L_X$ and $T^{*}_a$ are correlated with the intrinsic slope of $1.07$ and that the significance of this correlation is at 12 $\sigma$ level. The $\alpha$ value is flatter than the one obtained from the raw data of the whole sample (parameter b) and it is compatible with the average value of the slopes of the subsamples shown in Table 1.
[**With the EP method we are able both to overcome the problem of selection effects and to determine the intrinsic value of the slope, because we removed the induced correlation by observables due to the time evolution and luminosity evolution dividing the respective time and luminosity for the respective evolution functions as it is explained above. Any differences between the correlation obtained from our methods and the present one in the raw data is assumed to arise from selection effects and partly to time evolution. The evolution seen in Figure 1 (central panel) is due to observational biases and partly to time evolution. In fact, we have determined that there is no cosmological evolution of $L_X$, and that the evolution of $T^{*}_a$ becomes significant at high redshift only.**]{}
We also present results for a sample of 53 GRBs which are in common between a previous sample of 77 GRBs [@Dainotti2010] and the present one (see green line of Fig. \[Fig3\] and Fig. \[Fig4\]). For the sample of 53 GRBs we have adopted as a limiting flux $1.8 \times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$ adopting the same criterion as for the larger sample. In this case we have 47 GRBs remaining above the adopted flux limit, again resulting in retain 90 $\%$ of the sample. We note that there is compatibility within 1$\sigma$ range among the power law slopes ($\alpha$) of the two samples. [**The two samples are fitted with a different fitting procedure, one procedure leaves $t_a$ free to vary (101 sample) and the other fixes $t_a=T_p$, where $T_p$ is the beginning of the decay phase of the prompt emission, therefore we here stress that we still find compatible results proving that the LT correlation intrinsic slope is independent from the particular adopted procedure.**]{}
In addition, for a consistency check we have used an inverted transformation:
$$\label{eq:alphareciprocol}
\log_{10}(T^{'}_a) = \log T^{*}_a + \alpha^{*} \log L_X,$$
and followed the same procedure to determine $\tau$ as a function of $\alpha^{*}$. We expect $\alpha^{*}=1/\alpha$. The result is shown in the right panel of Fig. \[Fig.7\]. Values of $\alpha^{*}=-0.71^{-0.10}_{+0.12}$ are compatible within 2 $\sigma$ with $1/\alpha=-0.93 \pm 0.09$ obtained from the previous transformation. This is a further demonstration that the method is well build and the results are robust. However, for an exact compatibility we would expect $\alpha^{*}=1/\alpha$ (See also the Appendix).
The cumulative local luminosity and density functions {#density and luminosity}
=====================================================
Since we found no luminosity evolution the cumulative distribution of $\Phi(> L)=\int^{\infty}_{L} \Psi (L^{'}) dL^{'}$ according to our method [@Petrosian1992] is given as:
$$\Phi_i\!(L_{i}) = \prod_{k}{(1 + {1 \over n\!(k)})}
\label{phieq}$$
Here $n(k)$ is the number of objects in the associated sets of object k, namely these with $L>L_k$ and $z<z_{max}$.
The density rate evolution $\rho\!(z)^{'}$ and the LF, (with ${'}$ we indicate the differential simbol), which gives the number of objects per unit comoving volume $V$ per unit source luminosity can be computed with the EP method. The method gives the cumulative functions $\sigma(<z)^{'}=\int_0^z{\rho}(z')\,[dV(z')/dz']\,dz'$ and $\phi(>L')=\int_{L'}^\infty \psi(L'')\,dL''$. The differential functions $\rho^{'}$ and $\psi$ are obtained by differentiation.
One can define the cumulative density function as follow:
$$\sigma\!(z_j) = \prod_{i}{(1 + {1 \over m\!(i)})}$$
where $i$ runs over all objects with a redshift lower than or equal to $z$, and $m(i)$ is the number of objects with a redshift lower than the redshift of object $i$ [*which are in object j’s associated set*]{}. In this case, the associated set is again those objects with X-ray luminosity that would be seen if they were at object $i$’s redshift. The use of only the associated set for each object removes the biases introduced by the data truncation. We show the distribution in the right panel of Fig. \[Fig.5\] of the cumulative density distribution corrected (red points), which is contrasted with the raw distribution (black points).
As evident the correction for the cumulative density starts to apply for $z=1$, namely we have a higher density of GRBs than the one we observe for $z>1$. In Fig. \[Fig.5\] (left panel) the corrected cumulative luminosity function agrees with the raw observed luminosity distribution until $L_{X}=10^{48}$ erg/s while for higher values of the luminosity the two distributions separate.
![ [**Left panel**]{} : cumulative redshift distribution, N(>z) of the raw data (black points) and the cumulative distribution, $\sigma(z)$ corrected by the EP method (red points) and described in Section \[density and luminosity\].[**Right panel**]{}: the cumulative local luminosity distribution of raw data (black point) and corrected by the EP method (red points).[]{data-label="Fig.5"}](fig5A.eps "fig:"){width="0.50\hsize"} ![ [**Left panel**]{} : cumulative redshift distribution, N(>z) of the raw data (black points) and the cumulative distribution, $\sigma(z)$ corrected by the EP method (red points) and described in Section \[density and luminosity\].[**Right panel**]{}: the cumulative local luminosity distribution of raw data (black point) and corrected by the EP method (red points).[]{data-label="Fig.5"}](fig5B.eps "fig:"){width="0.50\hsize"}
To obtain the differential distribution $\psi(L)$ and $\rho^{'}(z)$ we fitted the cumulative luminosity function with a polinomial of order $7$, while for the cumulative density we divide the distribution in two parts, one with $z \leq 1$ as we can see from Fig. \[Fig.6\], blue line and the other part for $z \geq 1$, green line. The two fitting lines are both a polinomial of order $5$.
![[**Left Panel** ]{}: cumulative intrinsic luminosity function determined by the EP method along with a fitted function as discussed in Section \[density and luminosity\]. [**Right panel** ]{}: the cumulative intrinsic density distribution with two fitted functions lines, as discussed in Section \[density and luminosity\]. The blue line till $z=1$, while the green line for $z>1$.[]{data-label="Fig.6"}](fig6A.eps "fig:"){width="0.50\hsize"} ![[**Left Panel** ]{}: cumulative intrinsic luminosity function determined by the EP method along with a fitted function as discussed in Section \[density and luminosity\]. [**Right panel** ]{}: the cumulative intrinsic density distribution with two fitted functions lines, as discussed in Section \[density and luminosity\]. The blue line till $z=1$, while the green line for $z>1$.[]{data-label="Fig.6"}](fig6B.eps "fig:"){width="0.50\hsize"}
Discussion
==========
The obtained $\tau_{\alpha}$ vs $\alpha$ plot (Right panel of Figure \[Fig3\]), clearly demonstrates the existence at the 12 $\sigma$ level of a significant LT correlation, characterized by the value found for the power law slope relating the luminosity and the plateau end times.
Therefore, the presented analysis, with the intrinsic value of the power law slope of the LT correlation, provides new constraints for physical models of GRB explosion mechanisms. With this new determination of the correlation power law slope we discuss the consequences of these findings for GRB physical models. The LT relation is predicted by several theoretical models [@Cannizzo09; @Cannizzo11; @Dall'Osso; @Yamazaki09; @Obrien2012; @Bernardini2011] and in other observational ones [@Ghisellini2009; @Qi2012], proposed for the physical GRB evolution in the time $T_a$. Recently, Oates et al. 2012 pointed out the existence of an anticorrelation between the luminosity at $200$ s and the decay slope of the optical lightcurve. This correlation is related to the LT one considered here. Racusin et al. (in prep.) recover the Oates et al. correlation in the X-ray band. Therefore, it is now even more challenging to understand the meaning of the $L_X$ and $T^{*}_a$ correlation, which becomes the principal X-ray afterglow correlation from which further correlations in other wavelengths can be derived. From a theoretical point of view the Cannizzo model predicts a correlation slope ($3/2$) which is in agreement with our intrinsic correlation power law only within $3\sigma$, while the model of Yamazaki predicts a less steep decay which is in agreement in 1$\sigma$ with the presented results. The LT correlation is also recovered for short GRBs [@Obrien2012] within the magnetar scenario. Any physical interpretation of the LT correlation should be based on the intrinsic power slope and not that obtained from the raw observed quantities. In fact, assuming the observed power law as a key feature to discriminate among physical models could lead to misleading results based either on data truncation or on redshift evolution. We conclude that determining the intrinsic correlations among, and distributions of, the observables is a necessary step before any possible and plausible usage of the LT correlation as a theoretical model discriminator, distance estimator, and as useful cosmological tool. Therefore, this paper opens a new perspective not only on the interpretation of the LT correlation but also of the other existing GRB correlations and prepares for a new possible future approach for the usage of GRBs in cosmology.
Appendix
========
![ [**Upper Panel**]{}: The distribution of the redshift (left), of the time $T^{*}_a$ (middle) and of the spectral index (right) for the sample of 101 GRBs used in the analysis. [**Lower panel**]{}: $L_X$-$T^*_a$ distribution for a set of 101 simulated GRBs discussed in Appendix (left); Test statistic $\tau$ vs. the LT correlation power slope parameter $\alpha$ (center), and the reciprocol of it, $\alpha^{*}$, (right) for the simulated set. The 1$\sigma$ range of best fit values is where $| \tau | \leq 1$ shown by the vertical dotted lines.[]{data-label="Fig.7"}](fig7A.eps "fig:"){width="0.33\hsize"} ![ [**Upper Panel**]{}: The distribution of the redshift (left), of the time $T^{*}_a$ (middle) and of the spectral index (right) for the sample of 101 GRBs used in the analysis. [**Lower panel**]{}: $L_X$-$T^*_a$ distribution for a set of 101 simulated GRBs discussed in Appendix (left); Test statistic $\tau$ vs. the LT correlation power slope parameter $\alpha$ (center), and the reciprocol of it, $\alpha^{*}$, (right) for the simulated set. The 1$\sigma$ range of best fit values is where $| \tau | \leq 1$ shown by the vertical dotted lines.[]{data-label="Fig.7"}](fig7B.eps "fig:"){width="0.32\hsize"} ![ [**Upper Panel**]{}: The distribution of the redshift (left), of the time $T^{*}_a$ (middle) and of the spectral index (right) for the sample of 101 GRBs used in the analysis. [**Lower panel**]{}: $L_X$-$T^*_a$ distribution for a set of 101 simulated GRBs discussed in Appendix (left); Test statistic $\tau$ vs. the LT correlation power slope parameter $\alpha$ (center), and the reciprocol of it, $\alpha^{*}$, (right) for the simulated set. The 1$\sigma$ range of best fit values is where $| \tau | \leq 1$ shown by the vertical dotted lines.[]{data-label="Fig.7"}](fig7C.eps "fig:"){width="0.33\hsize"} ![ [**Upper Panel**]{}: The distribution of the redshift (left), of the time $T^{*}_a$ (middle) and of the spectral index (right) for the sample of 101 GRBs used in the analysis. [**Lower panel**]{}: $L_X$-$T^*_a$ distribution for a set of 101 simulated GRBs discussed in Appendix (left); Test statistic $\tau$ vs. the LT correlation power slope parameter $\alpha$ (center), and the reciprocol of it, $\alpha^{*}$, (right) for the simulated set. The 1$\sigma$ range of best fit values is where $| \tau | \leq 1$ shown by the vertical dotted lines.[]{data-label="Fig.7"}](fig7D.eps "fig:"){width="0.33\hsize"} ![ [**Upper Panel**]{}: The distribution of the redshift (left), of the time $T^{*}_a$ (middle) and of the spectral index (right) for the sample of 101 GRBs used in the analysis. [**Lower panel**]{}: $L_X$-$T^*_a$ distribution for a set of 101 simulated GRBs discussed in Appendix (left); Test statistic $\tau$ vs. the LT correlation power slope parameter $\alpha$ (center), and the reciprocol of it, $\alpha^{*}$, (right) for the simulated set. The 1$\sigma$ range of best fit values is where $| \tau | \leq 1$ shown by the vertical dotted lines.[]{data-label="Fig.7"}](fig7E.eps "fig:"){width="0.32\hsize"} ![ [**Upper Panel**]{}: The distribution of the redshift (left), of the time $T^{*}_a$ (middle) and of the spectral index (right) for the sample of 101 GRBs used in the analysis. [**Lower panel**]{}: $L_X$-$T^*_a$ distribution for a set of 101 simulated GRBs discussed in Appendix (left); Test statistic $\tau$ vs. the LT correlation power slope parameter $\alpha$ (center), and the reciprocol of it, $\alpha^{*}$, (right) for the simulated set. The 1$\sigma$ range of best fit values is where $| \tau | \leq 1$ shown by the vertical dotted lines.[]{data-label="Fig.7"}](fig7F.eps "fig:"){width="0.33\hsize"}
For a further test of the robustness of the main conclusion of this work we have applied the analysis methods discussed here to a simulated observational data set with a known intrinsic LT correlation. As is clear from the top panel of \[Fig.7\] the distributions of three observables in the real data, the time, $T^{*}_a$, the spectral index, $\beta_a$,and the redshift, $z$, can be approximated with normal distributions with mean values which are $<\beta_a>=1.05$, $<T^{*}_a>=3.5$ and $<z>=2.09$. Therefore, we have created a Monte Carlo population with these distributions. The luminosities are determined by applying an LT correlation with $\log L_X \approx -1.9 \log T^{*}_a$, $-1.9$ in this case being the imposed $\alpha$ slope of LT correlation (see left lower panel Fig. \[Fig.7\]). We then compute the simulated fluxes from the simulated $\beta$, $z$ and $L_X$. We have imposed the same limiting flux used for the real observational data to form an ‘observed’ simulated data set on which we then apply the analysis method discussed in this work. Application of the method successfully recover the known intrinsic power law slope of the LT correlation and its inverse as is shown in lower central and right panels of Fig. \[Fig.7\].
Acknowledgments
===============
This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. M.D. is grateful to Richard Willingale and Paul Obrien for comments on the paper and Qin Rong Chen for fruitful discussions. M.D is also grateful to the Polish MNiSW through the grant N N203 579840, the Fulbright Scholarship and the Ludovisi- Blanceflor Foundation. M. O. is grateful to the Polish National Science Centre through the grant DEC-2012/04/A/ST9/00083.
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[^1]: We assume a $\Lambda$CDM flat cosmological model with $\Omega_M = 0.291$ and $H_0 = 71 {\rm Km s}^{-1} {\rm Mpc}^{-1}$
[^2]: We pointed out here that since this method takes into account the hidden errors thus gives greater error estimates than the ones obtained with the Marquardt Levemberg algorithm (Marquardt 1963).
[^3]: for a complete table of the fitting parameters see http://www.oa.uj.edu.pl/M.Dainotti
[^4]: Note also as a result the intercept or normalization parameters a for the individual bins are smaller than the sample as a whole
|
---
abstract: |
In stochastic analysis, a standard method to study a path is to work with its signature. This is a sequence of tensors of different order that encode information of the path in a compact form. When the path varies, such signatures parametrize an algebraic variety in the tensor space. The study of these signature varieties builds a bridge between algebraic geometry and stochastics, and allows a fruitful exchange of techniques, ideas, conjectures and solutions.
In this paper we study the signature varieties of two very different classes of paths. The class of rough paths is a natural extension of the class of piecewise smooth paths. It plays a central role in stochastics, and its signature variety is toric. The class of axis-parallel paths has a peculiar combinatoric flavour, and we prove that it is toric in many cases.
author:
- 'Laura Colmenarejo[^1], Francesco Galuppi[^2], Mateusz Michałek[^3] [^4]'
bibliography:
- 'biblio.bib'
title: Toric geometry of path signature varieties
---
Introduction {#sec:intro}
============
A *path* is a continuous map $X\colon [0,1]\to{\mathbb{R}}^d$. This very simple mathematical object can be used to interpret a wide range of situations. From a physical transformation to a meteorological model, from a medical experiment to the stock market, everything that involves parameters changing with time can be described by a path. This makes paths priceless tools in many branches of mathematics, as well as in a number of applied sciences. The downside is that, being a continuous object, explicit computations on a path are not easy to handle. A typical way to overcome this problem is to find invariants that are simpler to understand and can provide us enough information. For paths, this problem was faced in [@Chen]. Assume that $X$ is piecewise smooth, and fix $k\in{\mathbb{N}}$. Let $X_i$ be the composition of $X$ with the projection to the $i$-th coordinate. Chen defined the *$k$-th signature* of $X$ to be the order $k$ tensor $\sigma^{(k)}(X)$ whose $(i_1\ldots i_k)$-th entry is $$\int_{0}^{1}\int_{0}^{t_k}\dots\int_{0}^{t_3}\int_{0}^{t_2}\dot{X}_{i_1}(t_1)\cdot\ldots\cdot\dot{X}_{i_k}(t_k)dt_1\dots dt_k.$$ By convention, $\sigma^{(0)}(X)=1$. The sequence $$\sigma(X):=(\sigma^{(k)}(X)\mid k\in{\mathbb{N}})$$ is called the *signature* of $X$. Sometimes it is also convenient to consider a truncated signature $\sigma^{\le m}(X):=(\sigma^{(k)}(X)\mid k\in\{0,\dots,m\})$. In this context, it is natural to ask how much these tensors tell us about $X$. In [@chenuniqueness Theorem 4.1] Chen proved that, up to a mild equivalence relation, the signature allows to uniquely recover a piecewise smooth path. Signatures are appreciated not only in stochastics, but also in topological data analysis, financial mathematics or machine learning among others (see [@GyurkoLyonsKontkowskiField; @ChevyrevKormilitzin; @ChevyrevNandaOberhauser; @ChevyrevOberhauser]). Chen’s iterated integrals have deep connections with de Rham homotopy theory, as shown in [@Hain2002]. In this paper we are interested in the geometric side, illustrated in Section \[subsec:geometric side\]. We study the so-called *signature varieties* – roughly speaking the geometric locus of all possible signatures. In Section \[sec:rough veronese\] we consider the variety ${\mathcal{R}}_{d,k,m}$ of signatures of rough paths. Our results often rely on toric geometry, which allows us to answer several questions about their equations and singularities:
- Answering a question raised in [@AFS18] and [@G18], we show that the ideal of ${\mathcal{R}}_{d,k,m}$ may be generated in arbitrary high degree - Proposition \[pro:rough veronese is not gen in bounded degree\]. However, we prove that quadratic polynomials define a (possibly reducible) variety, of which ${\mathcal{R}}_{d,k,m}$ is an irreducible component - Proposition \[prop:quad\].
- We characterize the cases in which ${\mathcal{R}}_{d,k,m}$ is an embedding of the weighted projective space. We find conditions that make it (projectively) normal and examples when it is not - Subsection \[sub:GeomRVV\].
- We study the asymptotic behavior of the degree of ${\mathcal{R}}_{d,k,m}$ and we give explicit formulas for some special cases - Subsection \[sub:degRVV\].
In Section \[sec:axis parallel\] we study the geometry of the signature variety ${\mathcal{A}}_{\nu,k}$ of axis parallel paths:
- We provide an easy, combinatorial parametrization of the variety - Lemma \[lem:CombDescription\].
- We apply the above description to prove that ${\mathcal{A}}_{\nu,k}$ is toric in several cases - Section \[sub: axis toricness\]. We study the dimension of ${\mathcal{A}}_{\nu,k}$ exhibiting defective cases - Section \[sub:axisDim\].
Finally, we use our knowledge of axis paths to prove a general formula for the determinant of the signature matrix of any path, Theorem \[thm:det is a square\] and Corollary \[cor:shuffle determinant is a shuffle square\].
In order to present those results we need to recall the algebraic background.
The tensor algebra
------------------
The $k$-th signature of a path $X$ belongs to $({\mathbb{R}}^d)^{\otimes k}$. We now introduce an ambient space for the whole signature $\sigma(X)$.
The *tensor algebra* over ${\mathbb{R}}^d$ is the graded ${\mathbb{R}}$-vector space $$T(({\mathbb{R}}^d)):= {\mathbb{R}}\times{\mathbb{R}}^d\times({\mathbb{R}}^d)^{\otimes 2}\times\ldots$$ of formal power series in the non-commuting variables $x_1,\dots,x_d$. It is an ${\mathbb{R}}$-algebra with respect to the tensor product . The algebraic dual of $T(({\mathbb{R}}^d))$ is the graded free ${\mathbb{R}}$-algebra $$T({\mathbb{R}}^d)={\mathbb{R}}\langle x_1,\dots,x_d\rangle$$ of polynomials in the non-commuting variables $x_1,\dots,x_d$.
These spaces and their rich algebraic structures are well studied. In this section we recall the features we need, and we refer to [@Reutenauer] for a detailed treatment. Next result, proven in [@C57 Section 2], gives a first taste of the correlation between the tensor algebra and signatures of paths. Recall that the *concatenation* of two paths $X$ and $Y$ is the path $X\sqcup Y\colon [0,1] \longrightarrow \mathbb{R}^d$ given by $$(X\sqcup Y)(t)=\begin{cases} X(2t) & \mbox{ if } t\in \left[ 0,\frac{1}{2}\right], \\
X(1)+Y(2t-1) & \mbox{ if } t\in \left[\frac{1}{2},1\right].
\end{cases}$$
\[lem:Chen’s identity\] If $X, Y\colon [0,1]\to{\mathbb{R}}^d$ are piecewise smooth paths, then $\sigma(X\sqcup Y)=\sigma(X)\otimes\sigma(Y)$ as formal power series in $T(({\mathbb{R}}^d))$.
We now introduce a useful notation.
For $T\in T(({\mathbb{R}}^d))$, denote by $T_{i_1\dots i_k}$ the $(i_1\dots i_k)$-th entry of the order $k$ element of $T$. For $y\in{\mathbb{R}}$, we set $T_y(({\mathbb{R}}^d)):=\{T\in T(({\mathbb{R}}^d))\mid T_0=y\}$ to be the space of tensor sequence with constant entry 0. Moreover, we will identify a degree $k$ monomial $x_{i_1}\cdot\ldots\cdot x_{i_k}$ with the word $w=i_1\ldots i_k$ in the alphabet $\{1,\dots,d\}$. The number $k$ is called the *length* of $w$ and it is denoted by $\ell(w)$. The degree 0 monomial corresponds to the empty word $e$. In this way, the tensor product of the two monomials corresponding to the words $v$ and $w$ is the word obtained by writing $v$ followed by $w$, and it is called the *concatenation product*. The natural duality pairing $$\langle-,-\rangle\colon T(({\mathbb{R}}^d))\times T({\mathbb{R}}^d)\to{\mathbb{R}}$$ is given by $\langle T, i_1\ldots i_k\rangle=T_{i_1\dots i_k}$, and extended by linearity.
Besides the concatenation of words, there is another product on $T({\mathbb{R}}^d)$. It has a strong combinatoric flavour, but it will also allow us to define very interesting algebraic objects.
We define the *shuffle product* of two words recursively by $$\begin{aligned}
\begin{array}{l}
e \shuffle w = w \shuffle e = w, \text{ and }\\
(w_1\otimes a) \shuffle (w_2\otimes b) = \left(w_1\shuffle (w_2\otimes b)\right)\otimes a + \left((w_1\otimes a )\shuffle w_2\right)\otimes b.
\end{array}\end{aligned}$$ Less formally, $v\shuffle w$ is the sum of all order-preserving interleavings of $v$ and $w$.
For instance, $1\shuffle 23=123+213+231$. Despite its apparently complicated definition, the shuffle product enjoys good properties. For instance, $(T({\mathbb{R}}^d),\shuffle, e)$ is a commutative algebra. Moreover next Lemma, proven in [@Reutenauer Proof of Corollary 3.5], shows that the shuffle product behaves nicely with respect to the signatures.
\[lem:shuffle identity\] If $X $ is a piecewise smooth path, then $$\langle \sigma(X),v\rangle\cdot\langle \sigma(X),w\rangle=\langle \sigma(X),v\shuffle w\rangle$$ for all words $v,w\in T({\mathbb{R}}^d)$.
It follows that signatures do not fill the tensor space $T(({\mathbb{R}}^d))$, but rather they live in a subset. In order to make this observation precise, we introduce an important subspace of the tensor algebra.
Consider the Lie bracketing $[T,S]=TS-ST$ on $T(({\mathbb{R}}^d))$. We define ${\mathop{\rm Lie}\nolimits}({\mathbb{R}}^d)\subset T_0(({\mathbb{R}}^d))$ to be the free Lie algebra generated by $x_1,\dots,x_d$, that is, the smallest vector subspace of $T(({\mathbb{R}}^d))$ that contains $x_1,\dots,x_d$ and is closed with respect to the bracketing.
The Lie group associated to ${\mathop{\rm Lie}\nolimits}({\mathbb{R}}^d)$ is an important object. One way to define it is as the image of ${\mathop{\rm Lie}\nolimits}({\mathbb{R}}^d)$ under the exponential map.
Define $\exp:T_0(({\mathbb{R}}^d))\to T_1(({\mathbb{R}}^d))$ by the formal power series $$\exp(T):=\sum_{n=0}^{\infty}\frac{T^{\otimes n}}{n!}.$$ We denote ${\mathcal{G}}({\mathbb{R}}^d):=\exp({\mathop{\rm Lie}\nolimits}({\mathbb{R}}^d))$.
By construction, $({\mathcal{G}}({\mathbb{R}}^d),\otimes, e)$ is a group. However, there is another way to characterize it in terms of shuffle product. By [@Reutenauer Theorem 3.2], $${\mathcal{G}}({\mathbb{R}}^d)=\{T\in T_1(({\mathbb{R}}^d))\mid \langle T,v\rangle\cdot\langle T,w\rangle=\langle T,v\shuffle w\rangle
\mbox{ for all words } v,w\in T({\mathbb{R}}^d)\}.$$ Now we see the first clear connection with the signatures. By the shuffle identity, ${\mathcal{G}}({\mathbb{R}}^d)$ contains the signatures of all piecewise smooth paths.
For our purposes, we need to point out that every definition we recalled has a truncated version. Namely, one can fix $m\in{\mathbb{N}}$ and consider $$T^m({\mathbb{R}}^d):=\bigoplus_{k=0}^m({\mathbb{R}}^d)^{\otimes k},$$ where tensors of order greater than $m$ are set to zero. Inside $T^m({\mathbb{R}}^d)$ there are ${\mathcal{G}}^m({\mathbb{R}}^d)$ and ${\mathop{\rm Lie}\nolimits}^m({\mathbb{R}}^d)$. The restriction of the map $\exp$ is defined in the same way, and we will write $\exp^{(m)}$ to denote the map $T_0^m(({\mathbb{R}}^d))\to T_1^m(({\mathbb{R}}^d))$.
Another feature of the group ${\mathcal{G}}({\mathbb{R}}^d)$ is that, for every $m\in{\mathbb{N}}$, ${\mathcal{G}}^m({\mathbb{R}}^d)$ not only contains, but actually coincides with the set of all truncated signatures of smooth paths (see [@AFS18 Theorem 4.4]) and it is a Lie group (see [@FrizVictoir2010 Theorem 7.30]). Moreover, it is defined by finitely many polynomials - namely, the shuffle relations with words of length at most $m$ - hence it is an algebraic variety. It is irreducible by [@AFS18 Theorem 4.10]. On the other hand, ${\mathop{\rm Lie}\nolimits}^m({\mathbb{R}}^d)$ is a finite dimensional vector space. In order to provide it with a basis, it is time to introduce another powerful combinatoric concept.
A non-empty word $w$ is *Lyndon* if, whenever we write $w=pq$ as the concatenation of two nonempty words, we have $w<q$ in the lexicographic order. In this case there is a unique pair $(p,q)$ of nonempty words such that $w=pq$ and $q$ is minimal with respect to lexicographic order. The *bracketing* of $w$ is $[p,q]=pq-qp$.
As shown in [@Reutenauer Corolllary 4.14] and [@AFS18 Proposition 4.7], the bracketings of all Lyndon words of length at most $m$ form a basis for ${\mathop{\rm Lie}\nolimits}^m({\mathbb{R}}^d)$. Therefore, in order to compute its dimension - and thus the dimension of the variety ${\mathcal{G}}^m({\mathbb{R}}^d)$ - it is enough to count Lyndon words. First, recall that the Möbius function $\mu:{\mathbb{N}}\to{\mathbb{N}}$ sends a natural number $t$ to $$\mu(t)=\begin{cases}
0 & \mbox{ if $t$ is divisible by the square of a prime},\\
1 & \mbox{ if $t$ is the product of an even number of distinct primes},\\
-1 & \mbox{ if $t$ is the product of an odd number of distinct primes}.\\
\end{cases}$$ Then the number of Lyndon words of length $l$ in the alphabet $\{ 1,\dots,d\}$ is $\sum_{t\mid l}\frac{\mu(t)}{l}d^{\frac{l}{t}},$ and therefore $$\dim{\mathop{\rm Lie}\nolimits}^m({\mathbb{R}}^d)=\sum_{l=1}^m \sum_{t\mid l}\frac{\mu(t)}{l}d^{\frac{l}{t}}.$$
Signatures from a geometric viewpoint {#subsec:geometric side}
-------------------------------------
Chen’s theorem assumes the knowledge of the $k$-th signature for every $k\in{\mathbb{N}}$. What happens when we know only one of them? In [@AFS18], the authors consider the problem from an algebraic geometry perspective. If we fix a certain class of paths and the order $k$ of the tensors, then the $k$-th signature $\sigma^{(k)}$ is an algebraic map into $({\mathbb{R}}^d)^{\otimes k}$. The closure of the image of $\sigma^{(k)}$ is called the *$k$-signature variety*. We get a first example by considering the class of all smooth paths in ${\mathbb{R}}^d$. If $X$ is smooth, then the $k$-th signature of $X$ is just the $k$-th entry of $\sigma(X)\in{\mathcal{G}}({\mathbb{R}}^d)$. This leads to the following definition.
The *universal variety* ${\mathcal{U}}_{d,k}\subset({\mathbb{R}}^d)^{\otimes k}$ is the projection of ${\mathcal{G}}({\mathbb{R}}^d)$ onto the $k$-th factor. It is the closure of the set of all $k$-th signatures of smooth paths.
Since every ${\mathcal{G}}^m({\mathbb{R}}^d)$ is an irreducible variety, ${\mathcal{U}}_{d,k}$ is irreducible as well. By [@AFS18 Theorem 6.1], its dimension coincides with $\dim{\mathcal{G}}^k({\mathbb{R}}^d)$. Equivalently, we can compute it as the dimension of the vector space ${\mathop{\rm Lie}\nolimits}^k({\mathbb{R}}^d)$, that is the number of Lyndon words of length at most $k$ in the alphabet $\{1,\dots,d\}$. This geometric approach provides a way to translate questions about paths into questions about the signature variety. For instance, the crucial problem of reconstructing uniquely the path from its $k$-signature translates into the injectivity of $\sigma^{(k)}$. This leads to study the fibers of $\sigma^{(k)}$ and therefore the dimension of the signature variety. Provided that a preimage can be reconstructed, such computation consists in solving a system of polynomial equations, and we can get an idea of how difficult that is by looking at the degree of the variety or at the degrees of the generators of its ideal. Taking a step further, the singular locus can point out that some signatures are different from the others, so we have a meaningful way to distinguish special paths in our class.
There are many interesting questions when it comes to study a signature variety, and we do not hope to give full answers to all of them. Path recovery from the third signature is the main goal of [@PSS18]. Further interplay between the tensor algebra and the signatures is explored in [@ColmenarejoPreiss]. Our contribution to this topic is a detailed study of the rough paths signature variety, presented in Section \[sec:rough veronese\], as well as the analysis of the axis parallel signature variety, which we describe in Section \[sec:axis parallel\]. Both varieties present several geometric subtleties. When dealing with them, we could take advantage of a surprisingly rich combinatorial structure, that allows us to integrate a projective geometric approach with more computational techniques.
The rough Veronese variety {#sec:rough veronese}
==========================
Researchers in stochastic analysis usually work with paths that are not piecewise smooth, and therefore do not have a signature in the sense of Chen’s definition. One of the most important examples is the class of rough paths. The interest in rough paths is quickly growing. Applications include the study of controlled ODEs and stochastic PDEs (see [@LyonsCaruanaLevy2004]) as well as sound compression (see [@LyonsSidorova2005]), not to mention mathematical physics (see [@FrizGassiatLyons2015]). The main references on rough paths are [@FrizVictoir2010] and [@FrizHairer2014]. In this section we recall what the signature of a rough path is, and study the corresponding signature variety. The signature variety of all rough paths is the whole universal variety. In the same way as polynomial paths are an interesting subclass of smooth paths, in [@AFS18 Section 5.4] the authors consider a nice subclass of rough paths, parametrized by ${\mathop{\rm Lie}\nolimits}^m({\mathbb{R}}^d)$. Their signature variety ${\mathcal{R}}_{d,k,m}$ exhibits similarities with the classical Veronese variety and it was therefore named the *rough Veronese variety* - see Definition \[def:rough veronese\]. Another reason to study ${\mathcal{R}}_{d,k,m}$ is that we can deduce many properties of the universal varieties from it. Indeed, ${\mathcal{U}}_{d,k}={\mathcal{R}}_{d,k,m}$ for every $m\ge k$ and moreover, ${\mathcal{U}}_{d,k}={\mathcal{R}}_{d,k,k}$ is a cone over ${\mathcal{R}}_{d,k,k-1}$ (see [@G18 Proposition 24]).
Preliminaries
-------------
One purpose of the definition of a rough path is to generalize the concept of smooth path. We consider then a smooth path $X$, and without loss of generality we assume $X(0)=0$. Fix $t\in [0,1]$. In the definition of $k$-th signature we can replace the integral on $[0,1]$ with an integral on $[0,t]$. This is the same as restricting $X$ to $[0,t]$, hence we will denote such integral by $\sigma^{(k)}(X_{|[0,t]})$. For every $k$, we notice that $\sigma^{(k)}(X_{|[0,t]})$, as a function of $t$, is a path $[0,1]\to({\mathbb{R}}^d)^{\otimes k}$. If we look at the full signature $\sigma(X_{|[0,t]})$, we get a path $[0,1]\to{\mathcal{G}}({\mathbb{R}}^d)$. Notice that the signature of $X$ is the endpoint of such path. By [@G18 Lemma 9], this ${\mathcal{G}}({\mathbb{R}}^d)$-valued path satisfies the Hölder-like inequality $$\label{eq:holder}
\left| \sigma^{(k)}(X_{|[s,t]})\right| \apprle |t-s|^k,$$ where $f(x)\apprle g(x)$ means that there is a constant $c$ such that $f(x)\le c\cdot g(x)$ for every $x$. Summing up, a smooth path $X\colon [0,1]\to{\mathbb{R}}^d$ induces a path $\sigma(X_{|[0,\cdot]})\colon [0,1]\to{\mathcal{G}}({\mathbb{R}}^d)$ satisfying inequality (\[eq:holder\]). If we want a rough path to be a generalization of a smooth path, we can use this property, also allowing different exponents. Let $p_k:T(({\mathbb{R}}^d))\to({\mathbb{R}}^d)^{\otimes k}$ be the projection.
A *rough path* of order $m$ is a path ${\bf X}\colon [0,1]\to{\mathcal{G}}^m({\mathbb{R}}^d)$ such that $|p_k({\bf X}(s)^{-1}\otimes {\bf X}(t))| \apprle |t-s|^\frac{k}{m}$ for every $k\in\{1,\dots,m\}$ and every $s,t\in [0,1]$. The inverse is taken in the group ${\mathcal{G}}^m({\mathbb{R}}^d)$.
As we anticipated, we will focus on a special subclass of rough paths of order $m$, indexed by elements $L\in{\mathop{\rm Lie}\nolimits}^m({\mathbb{R}}^d)$.
For $L\in{\mathop{\rm Lie}\nolimits}^m({\mathbb{R}}^d)$, consider the path ${\bf X}_L\colon [0,1]\to{\mathcal{G}}^m({\mathbb{R}}^d)$ sending $t$ to $\exp^{(m)}(tL)$. By [@FrizVictoir2010 Exercise 9.17], this is indeed an order $m$ rough path, and we define its *signature* to be its endpoint $\sigma({\bf X}_L):=\exp(L)\in{\mathcal{G}}({\mathbb{R}}^d)$.
We want to parametrize the variety containing all $\sigma^{(k)}({\bf X}_L)$ when $L$ ranges in ${\mathop{\rm Lie}\nolimits}^m({\mathbb{R}}^d)$, so we are interested in the image of $p_k\circ\exp:{\mathop{\rm Lie}\nolimits}^m({\mathbb{R}}^d)\to ({\mathbb{R}}^d)^{\otimes k}$. While $\exp$ is not an algebraic map, $p_k\circ\exp$ is. In general, the image of an algebraic map is only a semialgebraic subset of the real affine space $({\mathbb{R}}^d)^{\otimes k}$. However, it is simpler to work with complex projective varieties, so we will follow a common approach in applied algebraic geometry and consider the Zariski closure of the image, take the complexification and pass to the projectivization.
\[def:rough veronese\] The *rough Veronese variety* ${\mathcal{R}}_{d,k,m}$ is the closure of the image of the composition $$f_{d,k,m}\colon {\mathop{\rm Lie}\nolimits}^m({\mathbb{R}}^d)\xrightarrow{\exp}{\mathcal{G}}({\mathbb{R}}^d)\xrightarrow{p_k}({\mathbb{R}}^{d})^{\otimes k} \rightarrow ({\mathbb{R}}^{d})^{\otimes k}\otimes{\mathbb{C}}=({\mathbb{C}}^{d})^{\otimes k}\dashrightarrow{\mathbb{P}}^{d^k-1}
.$$
In general it is not easy to work out all the invariants of a given variety. Luckily, we will shortly see that the rough Veronese variety is toric. In other words it is the closure of the image of a monomial map. There are a lot of tools and techniques that make toric varieties accessible and easy to work with. Classical references on toric varieties are [@CLS11; @BerndBook]. First, we fix some notation.
\[not:SR\] For every $d\ge 2$, let $W_{d,m}$ be the set of Lyndon words of length at most $m$ in the alphabet with $d$ letters. Let $S_{d,m}:={\mathbb{C}}[x_w\mid w\in W_{d,m}]$ be a polynomial ring with as many variables as Lyndon words. On this algebra we define a grading by setting the weight of $x_w$ to be the length of $w$.
Let $A$ be the set of all monomials in ${\mathbb{C}}[x_w\mid w\in W_{d,m}]$ of weighted degree $k$. Define another polynomial ring $R_{d,k}:={\mathbb{C}}[y_\alpha\mid \alpha\in A]$ with as many variables as elements of $A$, and the usual polynomial grading.
Our starting point will be the following result, proven in [@G18 Proposition 18]. Not only we know that ${\mathcal{R}}_{d,k,m}$ is toric, but we also know what are the monomials parametrizing it.
\[thm:toricR\] Up to projectivization, ${\mathcal{R}}_{d,k,m}$ is isomorphic to the closure of the image of the map $\operatorname{Spec}S_{d,m}\rightarrow\operatorname{Spec}R_{d,k}$ given by all monomials of weighted degree $k$. That is, the kernel of the map $R_{d,k}\rightarrow S_{d,m}$ is the homogeneous prime ideal defining ${\mathcal{R}}_{d,k,m}$.
More precisely, it is the image of a weighted projective space by the map given by all sections of $\mathcal{O}(k)$, i.e. all monomials of (weighted) degree $k$. Such a map does not have to be an embedding. In fact, $\mathcal{O}(k)$ does not need to be a Cartier divisor.
The classical $k$-Veronese variety is the image of the map given by all degree $k$ monomials in the usual grading. Theorem \[thm:toricR\] shows that ${\mathcal{R}}_{d,k,m}$ can be seen as a weighted version of the Veronese variety, thereby justifying its name.
Theorem \[thm:toricR\] allows us to investigate ${\mathcal{R}}_{d,k,m}$ with the tools of toric geometry. In some of our combinatoric arguments, we will find it convenient to use the following notation.
For any integers $a,b$ we denote by $(a^b)$ a sequence or multiset of $b$ elements equal to $a$. For example $(1^3,2^2)=(1,1,1,2,2)$.
We start by describing the ideal of this variety, i.e. the polynomials that vanish on it.
Polynomials defining the rough Veronese variety
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The classical Veronese variety is defined by quadrics, so it is natural to ask whether ${\mathcal{R}}_{d,k,m}$ enjoys the same property. Despite holding in many examples (see [@AFS18 Section 4.3] and the computations in [@G18]), the conjecture that ${\mathcal{R}}_{d,k,m}$ is always defined by quadrics is false. In [@G18 Proposition 26] we see that the ideal ${\mathcal{R}}_{2,20,14}$ has a cubic generator. One could still hope to bound the degree of the generators. Surprisingly no such bound exists, as we will shortly prove. First we need the following lemma from linear algebra.
\[lem: magic square\] For every $n\ge 3$, there exist $k_n\in{\mathbb{N}}$ and an $n\times n$ matrix $M_n$ with positive integer entries satisfying the following properties:
1. the sum of the entries of each row and each column of $M_n$ equals $k_n$,
2. the only way to decompose the multiset of entries of $M_n$ into $n$-tuples summing to $k_n$ is by rows or by columns,
3. each column, as a multiset, is distinct from each row.
We refer to the matrix $M_n$ as a *rigid $k_n$-square matrix*.
We proceed by induction on $n$. If $n=3$, we take $k_3=12$ and $$M_3=\left( \begin{matrix}
4 & 5 & 3\\6 & 5 & 1\\
2 & 2 & 8
\end{matrix}\right) .$$ Assume the statement holds for $n$ and let us prove it for $n+1$. By induction hypothesis, there exist $k_n\in{\mathbb{N}}$ and an $n\times n$ matrix $M_n$ with the required properties. Let $\lambda>n^2+1$ be a natural number and define $k_{n+1}=\lambda k_n+n+1$. Moreover, define the $(n+1)\times (n+1)$ matrices $$A=\left(
\begin{tabular}{cccc|c}
&&& & 0\\
&$\lambda M_n$ && & 0\\
&& && $\vdots$\\
&&& & 0\\\hline\bigstrut
0& $\dots$& 0& 0 & $\lambda k_n$
\end{tabular}\right) \mbox{ and }
B=\left(
\begin{tabular}{cccc|c}
0&0&\dots& 0& $n+1$\\
$\vdots$&$\vdots$& & $\vdots$& $\vdots$\\
0&0& $\dots$& 0& $n+1$\\
1&1& $\dots$& 1& 1\\\hline\bigstrut
$n$& $n$& $\dots$& $n$ & $-n^2+n+1$
\end{tabular}\right).$$ Then we take $M_{n+1}:= A+B$. Since $M_n$ has positive entries, the same holds for $M_{n+1}$. Moreover, the sum of the entries of each row and each column of $M_{n+1}$ equals $k_{n+1}$. Fix a partition of the entries of $M_{n+1}$ into $n+1$ multisets $\{r_{i,1},\dots,r_{i, n+1}\}$, for $1\leq i\leq n+1$, each summing up to $k_{n+1}$. Without loss of generality, we assume that $r_{1,1}=M_{n+1,n+1}=\lambda k_n-n^2+n+1$. None of the other $r_{1,j}$’s can be larger than or equal to $\lambda$, otherwise their sum would exceed $M_{n+1,n+1}+\lambda>\lambda k_n+n+1=k_{n+1}$, which is a contradiction. Therefore the only possible choices for the other $r_{1,j}$’s are either from the $(n+1)$-st row or the $(n+1)$-st column of $M_{n+1}$. If $r_{1,j_0}=1$ for some $j_0$, then $r_{1,j}=n+1$ for $j\in\{2,\dots,\hat{j_0},\dots,n\}$, hence $\{r_{1,j}\}_j$ must be the last column of $M_{n+1}$. Otherwise, we have $r_{1,j}\geq n$ for all $2\leq j\leq n+1$, and as $\sum_{j=1}^{n+1} r_{1,j}=k_{n+1}$ we must have $r_{1,j}=n$ for $2\leq j\leq n+1$. In this case, $\{r_{1,j}\}_j$ must be the last row of $M_{n+1}$. We now consider the other multisets $r_{i,j}$, for $i>1$. By the previous argument, they induce a partition of either the first $n$ columns or rows of $M_{n+1}$. We first consider the induced partition on $A $, i.e. the partition of $\lambda M_n$. We claim that, restricted to that matrix, each part must sum up to $\lambda k_n$. Otherwise, one part would need to have a smaller sum, and by divisibility by $\lambda$, it would be at most $\lambda k_n-\lambda$. However, none of the $n+1$ entries in the first $n$ columns or first $n$ rows in $B$ sums up to $\lambda+n+1$. This would contradict the fact that the multiset sums up to $k_{n+1}$.
We notice that at this point we are not allowed to use the inductive assumption on $M_n$. Indeed, although we know that the induced partition of $\lambda M_n$ has correct sum, we do not know that each induced multiset has exactly $n$ elements. However, we know that the multisets induced on $B $ must sum up exactly to $n+1$. This implies that each of the first $n$ elements of the last column (if $\{r_{1,j}\}_j$ was the last row) or last row (if $\{r_{1,j}\}_j$ was the last column) must belong to a different multiset $\{r_{i,j}\}_j$. Hence, there is precisely one such element in each such multiset. It follows that in each $\{r_{i,j}\}_j$ there are precisely $n$ elements from the upper left $n\times n$ submatrix of $M_{n+1}$. Hence, by induction, the induced partition of $\lambda M_n$ is the same as a partition by columns or by rows. Suppose it is a partition by columns. It is only compatible with a partition of $M_{n+1}$ by extending it by $n$. Further, looking at $B$, each partition must contain exactly one $n$ and one $1$. Thus, the considered partition is the same as the one given by columns. The argument for a row partition is similar.
Thanks to Lemma \[lem: magic square\], we are now able to produce instances of ${\mathcal{R}}_{d,k,m}$ generated in arbitrarily high degree.
\[pro:rough veronese is not gen in bounded degree\] For every $n\ge 3$ and every $d\geq 2$, there exist $k,m\in{\mathbb{N}}$ such that $k\ge m$ and the ideal of ${\mathcal{R}}_{d,k,m}$ is not generated in degree $n$.
By Lemma \[lem: magic square\], there exists $k\in{\mathbb{N}}$ and a rigid $k$-square matrix $M$ of size $(n+1)\times(n+1)$. Let $m$ be the largest entry of $M$. By Theorem \[thm:toricR\], the ideal $I$ of ${\mathcal{R}}_{d,k,m}$ is the kernel of the ring homomorphism $${\varphi}:{\mathbb{C}}[y_\alpha\mid \alpha\in A]\to{\mathbb{C}}[x_w\mid w\in W_{d,m}]$$ sending $y_\alpha$ to the weighted monomial $\alpha$.
For every $i\in\{1,\dots,m\}$, fix once and for all a variable of weight $i$. This means that we fix a length $i$ Lyndon word $w$ and we associate to $i$ the variable $x_w$. For every $j\in\{1,\dots,n+1\}$, consider the monomial $\alpha_j$ defined by the product of $n+1$ variables $x_w$ whose weights are the entries of the $j$-th row of $M$. By construction, this monomial has weighted degree $k$ and therefore $\alpha_1,\dots,\alpha_{n+1}\in A$. In a similar way, we can consider the monomials $\beta_1,\dots,\beta_{n+1}$ defined by the columns of $M$. We define $$f:=y_{\alpha_1}\cdot\ldots\cdot y_{\alpha_{n+1}}-y_{\beta_1}\cdot\ldots\cdot y_{\beta_{n+1}}.$$ This is a degree $n+1$ element of ${\mathbb{C}}[y_\alpha\mid \alpha\in A]$. Moreover, since the degrees appearing in the rows of $M$ are the same as the degrees appearing in the columns, $f\in\ker({\varphi})=I$. Let us show that $f$ is not generated by degree $n$ elements. Let $q_1,\dots,q_r$ be the polynomials spanning the degree $n$ part of $I$. Since ${\mathcal{R}}_{d,k,m}$ is toric, we can assume these are binomials [@CLS11 Proposition 1.1.9]. Hence, $q_i=g_i-h_i$, where each $g_i$ and each $h_i$ is a degree $n$ monomial. Suppose by contradiction that $f$ can be generated by $q_1,\dots,q_r$. Then there exist variables $l_1,\dots,l_r$ such that $f$ is the sum $$f=l_1(g_1-h_1)+\dots+l_r(g_r-h_r).$$ Without loss of generality, assume that $y_{\alpha_1}\cdot\ldots\cdot y_{\alpha_{n+1}}=l_1g_1$. Since $l_1g_1-l_1h_1\in I$, we know that $\varphi(l_1h_1)=\varphi(l_1g_1)$. As a multiset, the variables in $\varphi(l_1h_1)$ are the entries of the matrix $M$. Furthermore, the variables of $l_1h_1$ provide a partition into $n+1$ sets of cardinality $n+1$, each with sum $k$. Since $h_1\neq g_1$, by assumption on $M$, this partition must correspond to the column partition of $M$. Hence, $l_1(h_1-g_1)=f$. This is a contradiction, because no variable divides $f$ by construction.
In applications, sometimes one has a signature coming from experiments and wants to know whether it belongs to a path of a certain class. In other words, we have a point in ${\mathcal{U}}_{d,k}$ and we want to understand if it belongs to a given signature subvariety. From this viewpoint Proposition \[pro:rough veronese is not gen in bounded degree\] may seem disappointing. In order to check whether a given point belongs to ${\mathcal{R}}_{d,k,m}$, we might have to evaluate polynomials of very high degree. The good news is that in most cases it is enough to only evaluate quadrics. More precisely, ${\mathcal{R}}_{d,k,m}$ is always defined by quadrics outside of a coordinate linear subspace of large codimension. In order to prove that, we recall a classical lemma from toric geometry.
\[lem:toric generator\] Let $s\in{\mathbb{N}}$ and let $I$ be the homogeneous toric ideal associated to a set of lattice points $A\subset{\mathbb{Z}}^n$. Let $g$ be a binomial which we write as $p_1+\ldots+p_t-(p_1'+\ldots+p_t')$, where $p_i,p_j'\in A$. Then $g\in I(s)$ if and only if starting from the multiset $\{p_i\}$ we can reach the multiset $\{p_j'\}$ in finitely many steps of the following form. In each step we replace $p_{i_1},\ldots,p_{i_s}$ with some other $r_{i_1},\ldots,r_{i_s}\in A$ such that $p_{i_1}+\ldots+p_{i_s}=r_{i_1}+\ldots+r_{i_s}$.
A toric ideal is known to be binomial. Thus, $f\in I(s)$ if and only if: $$f=\sum_{i=1}^k m_ib_i,$$ where $m_i$ are monomials and $b_i\in I$ are binomials of degree at most $s$. The proof is by induction on $k$. The monomial of $f$ corresponding to $p_1+\dots+p_t$ must appear on the right hand side, say in $m_1b_1$. Then $f-m_1b_1$ is obtained by applying one step of the procedure described in the lemma. We conclude by induction.
\[prop:quad\] Fix $R=R_{d,k}$ as in Notation \[not:SR\]. Let $I\subset R$ be the ideal of ${\mathcal{R}}_{d,k,m}$ and let $I(2)\subset I$ be the vector subspace of degree two forms. Let $V\subset{\mathbb{P}}^{d^k-1}$ be the variety defined by $I(2)$. Then ${\mathcal{R}}_{d,k,m}$ is an irreducible component of $V$.
Since $I(2)\subset I$, it is clear that ${\mathcal{R}}_{d,k,m}\subset V$. We want to find a polynomial $f\in R\setminus I$ such that $(I(2):f^\infty)=I$. This implies that $I$ and $I(2)$ coincide in the localization $R_f$ and therefore $V$ and ${\mathcal{R}}_{d,k,m}$ coincide in the affine open subset $(f\neq 0)$. In particular, ${\mathcal{R}}_{d,k,m}$ is an irreducible component of $V$.
Let $\mu=\dim{\mathop{\rm Lie}\nolimits}^m({\mathbb{R}}^d)$ and let $A\subset{\mathbb{N}}^\mu$ be the set of lattice points associated to the monomials defining the toric variety ${\mathcal{R}}_{d,k,m}$. By Theorem \[thm:toricR\], points of $A$ are of the form $$(w_{1,1},\ldots,w_{1,a_1},w_{2,1},\ldots,w_{2,a_2},\ldots,w_{m,1},\ldots,w_{m,a_m})\text{ where }\displaystyle{\sum_{i=1}^m\sum_{j=1}^{a_i} iw_{i,j}=k}.$$ By [@CLS11 Proposition 1.1.9], there exists a set of binomial generators of $I$. Each of such binomials corresponds to an integral relation $p_1+\ldots+p_t=p_1'+\ldots+p_t'$, where all $p_i$ and $p_i'$ are in $A$ (the sum of points corresponds to the product of variables). Our task is to find $f\notin I$ such that $f^ng\in I(2)$ for every generator $g=p_1+\ldots+p_t-p_1'-\ldots-p_t'$ of $I$ for some $n$. We will define $f$ to be a variable. In toric words, $f$ corresponds to a lattice point $p\in A$ . By Lemma \[lem:toric generator\], we want to find $p\in A$ such that $p+\ldots+p+p_1+\ldots+p_t$ can be turned in $p+\ldots+p+p_1'+\ldots+p_t'$ by repeatedly replacing *a pair* of summands with another pair having the same sum. Let $$f=p:=(k,0,\dots, 0)$$ and $p_1=(w_{1,1},\ldots,w_{1,a_1},w_{2,1},\ldots,w_{2,a_2},\ldots,w_{m,1},\ldots,w_{m,a_m})$. Assuming $w_{2,1}>0$, we can replace $p+p_1$ by $$(k-2,0,\dots,0,1,0,\dots,0)+(w_{1,1}+2,w_{1,2},\ldots,w_{1,a_1},w_{2,1}-1,w_{2,2}\dots,w_{2,a_2} ,\ldots,w_{m,a_m}),$$ i.e. we replace two on the first coordinate with one on the coordinate corresponding to $w_{2,1}$. By iterating this process, we add more copies of $p$ so that we can replace $p+\ldots+p+p_1$ by points for which *at most one coordinate* $w_{i,j}$ is nonzero for $(i,j)\neq(1,1)$, and that coordinate is equal to one. We do the same to $p_2,\dots,p_t$, so that we can replace $p+\ldots+p+p_1+\ldots+p_t$ by a sum of points for which at most one coordinate $w_{i,j}$ is nonzero for $(i,j)\neq(1,1)$, and that coordinate is equal to one. We apply the same process to $p_1'+\ldots+p_t'$. Since now both sides are broken into this kind of simple pieces, and since the condition of having weighted sum $k$ is always preserved, the only possibility is that the summands are pairwise the same.
In the proof of Proposition \[prop:quad\] we could have chosen the point $p$ to be any point with coordinates $w_{i,j}=0$ for $i>1$. This proves that $I(2)$ and $I$ define the same scheme (in particular, the quadrics define the correct set) outside of the locus where *all* these coordinates vanish. Therefore the components of the variety defined $I(2)$ (embedded or not) must be supported on a coordinate subspace of large codimension.
Normality of the rough Veronese Variety {#sub:GeomRVV}
---------------------------------------
A classical approach to study geometry of a projective toric variety is to look at the associated lattice polytope [@CLS11; @BerndBook; @Fulton]. In our case the central object is presented in the following definition.
\[def:lattice polytope\]For $k\in{\mathbb{N}}$ and $w=(w_1,\dots,w_r)\in{\mathbb{N}}^r$, we define the lattice polytope $P(w,k)$ as the convex hull of $$\left\lbrace (t_1,\dots,t_r)\in{\mathbb{N}}^r\ \left|\ \sum_{i=1}^{r}w_it_i=k\right. \right\rbrace .$$
In this notation, the polytope associated to ${\mathcal{R}}_{d,k,m}$ is $P((1^{a_1},2^{a_2},\dots,m^{a_m}),k)$, where $a_i$ is the number of Lyndon words of length $i$ in the alphabet $\{1,\dots,d\}$. Many authors, like [@CLS11; @Fulton], require a toric variety to be *normal*. This is equivalent to the fact that the toric variety may be represented by a fan. Hence, the first important task is to decide when ${\mathcal{R}}_{d,k,m}$ is normal. One advantage is that normality can be checked on the polytope.
\[lem:easy facts about normality\] A polytope $P$ is normal if and only if $kP$ is normal for every $k\in{\mathbb{N}}$. Moreover, if $X_P$ is the associated toric variety, then
1. $X_P$ is normal if and only if $P$ is very ample,
2. $X_P$ is projectively normal (i.e. the affine cone over it is normal) if and only if $P$ is normal.
This results are proven in [@CLS11 Chapter 2], together with many other features of normal polytopes.
Lemma \[lem:easy facts about normality\] will help us to determine when ${\mathcal{R}}_{d,k,m}$ is normal. We start with the following result, that simplifies the problem.
\[lem:reduce to single variables\]$P((w_1^{a_1},w_2^{a_2},\dots,w_m^{a_m}),k)$ is normal if and only if $P((w_1,w_2,\dots,w_m),k)$ is normal. In particular, the normality of ${\mathcal{R}}_{d,k,m}$ does not depend on $d$.
By induction, it is enough to check what happens when we add or discard one of the entries. In order to simplify notation, set $$P:=P((w_1,\dots,w_{i-1},w_i,w_i',w_{i+1},\dots,w_m),k),$$ where $w_i=w_i'$ and we do not assume that the $w_j$’s are distinct. Observe that one of the facets of $P$ is precisely the polytope $$Q:=P((w_1,\dots, w_i,w_{i+1},\dots,w_m),k).$$ Since every face of a normal polytope is normal, the first implication follows.
Assume now that $Q$ is normal. Consider the linear surjection $\sim:P\to Q$, defined by the sum of the two entries corresponding to $w_i$ and $w_i'$. If $t=(t_1,\ldots,t_{i-1},t_i,t_i',t_{i+1},\dots,t_m)\in P$, then $\tilde{t}=(t_1,\ldots,t_{i-1},t_i+t_i',t_{i+1},\dots,t_m)\in Q$. Note that $\sim$ is also surjective on lattice points, that is $\sim:P\cap{\mathbb{N}}^r\to Q\cap{\mathbb{N}}^{r-1}$ is surjective. Let $p$ be a lattice point in some multiple $sP$ of $P$. We can write $p=\lambda_1p_1+\ldots+\lambda_sp_s$, where $p_1,\dots,p_s\in P$ are lattice points and $\lambda_1+\ldots+\lambda_s=s$. Since $\tilde{p}\in sQ$, $Q$ is normal by hypothesis and $\sim$ is surjective on lattice points, there are $x_1,\ldots,x_s\in P\cap{\mathbb{N}}^r$ such that $\tilde{p}=\lambda_1\tilde{p}_1+\ldots+\lambda_s\tilde{p}_s=\tilde{x}_1+\ldots+\tilde{x}_s$. Let $x_i\in P$ be the preimage of $\tilde{x}_i$ having 0 in the entry corresponding to $w_i'$, and let $x=x_1+\ldots+x_s$. Since $\tilde{p}=\tilde{x}$, $p$ and $x$ coincide on every coordinate except the ones corresponding to $w_i$ and $w_i'$. Moreover, these entries have the same sum, and such sum is an integer by construction of $x$. Then it is enough to increase the zero entry of some suitable $x_i$, keeping such sum untouched, to obtain from $x_1,\dots, x_s$ another set of $s$ lattice points of $P$ whose sum is $p$. Therefore $P$ is normal.
In general, the rough Veronese variety does not need to be normal. As we show using the code available online [@CGM19], $P ((1, 2, \ldots, 9), 18)$ is not very ample, so ${\mathcal{R}}_{d,18,9}$ is not normal for any $d\geq 2$.
In algebraic geometry, given a map $f=(f_1,\dots,f_s)$ defined by homogeneous polynomials of the same degree, it is natural to study the induced *rational map* $\tilde f$ between projective spaces. The map $\tilde f$ may be not defined everywhere - the locus where all of the polynomials $f_i$ vanish is called the *base locus* or *indeterminacy locus* of $\tilde f$. In our setting, the monomials defining ${\mathcal{R}}_{d,k,m}$ are of degree $k$, however in variables that are of different degrees. This implies that instead of considering the classical projective space, we need the *weighted projective space*. Theorem \[thm:toricR\] tells us that ${\mathcal{R}}_{d,k,m}$ is the closure of the image of a rational map $${\mathbb{P}}(1^{a_1},\dots,m^{a_m})\dashrightarrow{\mathbb{P}}^{d^k-1}.$$ The codomain is the usual projective space and the domain is the weighted projective space with $\sum_i a_i$ variables of weights as given in the brackets. The map is given by all monomials of degree $k$.
Before we proceed further, let us recall that maps from an algebraic variety $X$ to a projective space are studied through line bundles (or equivalently Cartier divisors) on $X$ [@Lazarsfeld]. The theory of line bundles on weighted projective spaces is very well understood. The Picard group equals ${\mathbb{Z}}$ where $\bf{1}$ is an ample generator. The global sections of this generator may be identified with all monomials of degree $l$ equal to the least common multiple of all the degrees appearing in the weighted projective space. Thus, it is customary to denote it by $\mathcal{O}(l)$. In particular, the elements of the Picard group will be denoted by $\mathcal{O}(s)$, for $s$ divisible by $l$. Although the class group is abstractly also equal to ${\mathbb{Z}}$, it is larger. The inclusion of the Picard group in the class group ${\mathbb{Z}}\hookrightarrow{\mathbb{Z}}$ is given by multiplication by $l$. In particular, the elements of the class group will be denoted by $\mathcal{O}(s)$, for arbitrary $s$.
Let ${\mathbb{P}}(w_1,\dots,w_s)$ be any weighted projective space and let $l={\mathop{\rm lcm}\nolimits}(w_1,\dots,w_s)$. The map from ${\mathbb{P}}(w_1,\dots,w_s)$ to a projective space given by all monomials of degree $k$ does not have a base locus if and only if $l|k$.
If $w_1\nmid l$, then $[1:0:\dots:0]$ is a base point. If all $w_i|l$ then the monomials $x_i^{\frac{l}{w_i}}$ do not have a base locus.
As there is some confusion in the literature, we stress that even if $l|k$ the map from the previous lemma does not have to be an embedding. In other words $\mathcal{O}(l)$, although always ample, may be not very ample.
The polytope $P((1,6,10,15),30)$ is not very ample. Equivalently, the map ${\mathbb{P}}(1,6,10,15)\to {\mathbb{P}}^{17}$ given by all monomials of weighted degree 30 is not an embedding. This can be checked with the software *Normaliz* [@Normaliz], or by studying the polytope.
This raises the following question: when is ${\mathcal{R}}_{d,k,m}$ an embedding of the weighted projective space ${\mathbb{P}}(1^{a_1},\dots, m^{a_m})$? We will prove that this happens if and only if $k$ is a multiple of all natural numbers between 1 and $m$. Equivalently, $\mathcal{O}(k)$ is a very ample line bundle if and only if ${\mathop{\rm lcm}\nolimits}(2,\dots,m)\mid k$. We will need the following technical lemma.
\[lem:there are enough primes\] If $m\ge 7$ and $m\neq 10$, then there exist two distinct prime numbers strictly larger than $\frac{m}{2}$ and at most equal to $m$ such that their sum is not a power of two.
If $m\le 56$, we can easily find the required primes. $$\begin{array}{|c|c|}
\hline
32\le m\le 56 & 29\mbox{ and }31\\
\hline 20 \le m \le 31& 17\mbox{ and }19\\
\hline 14 \le m \le 19& 11\mbox{ and }13\\
\hline 11\le m \le 13& 7\mbox{ and }11\\
\hline m\in\{7,8,9\}& 5\mbox{ and }7\\\hline
\end{array}$$ Assume then $m\ge 57$. For $x\in{\mathbb{N}}$, let $\pi(x)$ be the number of prime numbers smaller or equal than $x$. In [@RS62], the authors show that $$\frac{x}{\log x} < \pi(x) < 1.3 \frac{x}{\log x}$$ for every $x\geq 17$. Now, $$\pi(m)-\pi\left(\frac{m}{2}\right)\ge \frac{m}{\log m} - 1.3\frac{m}{2 \log \left(\frac{m}{2}\right)},$$ and the latter is at least 3 for $m\geq 57$, so there are at least three prime numbers $a<b<c$ between $\frac{m}{2}$ and $m$. If $b+c$ is not a power of 2, we are done. Otherwise we have $b+c=2^n$. Then $$2^{n-1}=\frac{b+c}{2}\le m<a+b<b+c=2^n,$$ so $a+b$ cannot be a power of 2.
\[thm: normality of rough Veronese\] If $i\mid k$ for every $i\in\{1,\dots,m\}$, then ${\mathcal{R}}_{d,k,m}$ is projectively normal.
For $m\le 6$, normality can be explicitly checked with the software *Polymake* [@polymake], so we assume $m\ge 7$. By Lemmas \[lem:easy facts about normality\] and \[lem:reduce to single variables\], we may assume that $k={\mathop{\rm lcm}\nolimits}(1,\dots,m)$ and study the map $${\mathbb{P}}(1,2,\ldots,m)\to{\mathbb{P}}^N$$ defined by all monomials of weighted degree $k$. Let $P:=P((1,2,\ldots,m),k)$ be the associated polytope. For $s\in{\mathbb{N}}$, consider the dilation $sP$ of $P$. A lattice point of $sP$ is of the form $(a_1,\dots,a_m)\in{\mathbb{N}}^m$ with $a_1+2a_2+\ldots+ma_m=sk$. We can write it as a multiset $$M=\{1^{a_1},\dots,m^{a_m}\}.$$ By induction on $s$, we only need to prove that there exists a submultiset $S$ of $M$ whose entries sum up to $k$. Let us modify $M$ to $M'$ in the following way. If there are $i,j\in M$ such that $i,j\le \frac{m}{2}$, then discard $i$ and $j$ and add $i+j$. Notice that $i+j\le m$. If $S'$ is a submultiset of $M'$ whose entries sum up to $k$, then either $S$ does not contain $i+j$, and therefore $S=S'$ is a submultiset of $M$ as well, or $S'$ contains $i+j$, and we define $S$ by replacing back $i+j$ with $i,j$. By iterating this argument, we can assume that there is at most one $i\in\{1,\dots,m\}$ such that $i\le\frac{m}{2}$.
The resulting multiset $M'$ is of the form $\left\{i_1^{b_{i_1}},\dots, i_t^{b_{i_t}} \right\}$. Let $k'={\mathop{\rm lcm}\nolimits}(i_1,\dots,i_t)$ and consider the map $${\mathbb{P}}(i_1,i_2,\ldots,i_t)\to{\mathbb{P}}^T$$ given by all the monomials of weighted degree $k'$. Its image is associated to the polytope $P':=P((i_1,i_2,\ldots,i_t),k')$. By construction, $k'$ divides $k$ and $\dim P'\le t-1 \le \frac{m}{2}$. Observe that any point of $P'$ has entries summing up to $k'$. Since the entries of $M'$ sum up to $sk$, $M'$ is a point in $rP'$ for $r = \frac{sk}{k'}$. In fact, $M'\in s\left(\frac{k}{k'}P'\right)$ and so, it is enough to prove that $\frac{k}{k'}P'$ is normal. This will imply that $M'$ is a sum of lattice points of $\frac{k}{k'}P'$, and therefore it admits the required submultiset $S'$.
If there exists a prime number $\frac{m}{2}\le p \le m$ such that $p\nmid k'$, then $p\mid \frac{k}{k'}$ and so $\frac{k}{k'} \ge p \ge \frac{m}{2} \ge \dim P'$. In this case, $\frac{k}{k'}P'$ is normal by [@CLS11 Theorem 2.2.12]. This guarantees the existence of the desired $S'$. Otherwise, assume $m\neq 10$. By Lemma \[lem:there are enough primes\], there exist two prime numbers $p_1, p_2$ between $\frac{m}{2}$ and $m$ such that $p_1+p_2$ is not a power of $2$. Up to relabelling, we can assume that $b_{p_1}\le b_{p_2}$.
Once more, we have to modify $M'$ by making some replacements. For $b_{p_1}$ times, discard an entry $p_1$ and an entry $p_2$ and add an entry $p_1+p_2$. Denote by $M''$ the resulting multiset and by $P''$ the associated polytope, that satisfies $\dim P'' \le \frac{m}{2}$. As before, it is enough to find a submultiset $S''$ of $M''$ such that the entries of $S''$ sum up to $k$. By construction, the entries of $M''$ still sum up to $sk$ and their least common multiple $k''$ divides $k$. Indeed, the sum $p_1+p_2$ is even and not a power of two, thus all powers of prime numbers that divide it are smaller or equal to $m$. Notice that neither $p_1$, nor any of its multiples appear in $M''$. Therefore, $p_1 \mid k$ and $p_1\nmid k''$, which implies that $p_1\mid \frac{k}{k''}$ and so $\frac{k}{k''} \ge p_1 \ge \frac{m}{2} \ge \dim P''$. By [@CLS11 Theorem 2.2.12], $\frac{k}{k''}P''$ is normal. This guarantees the existence of the desired $S''$.
The last case $m=10$ can be explicitly checked with the software Normaliz.
The map ${\mathbb{P}}(1^{a_1},\dots,m^{a_m})\to{\mathbb{P}}^N$ given by all monomials of weighted degree $k$ is an embedding if and only if $i\mid k$ for every $i\in\{1,\dots,m\}$. If so, ${\mathcal{R}}_{d,k,m}\simeq{\mathbb{P}}(1^{a_1},\dots,m^{a_m})$ is embedded as a projectively normal variety.
By Theroem \[thm: normality of rough Veronese\], the polytope associated to ${\mathcal{R}}_{d,k,m}$ is normal and thus very ample. This implies that the map is an embedding and the image is projective normal.
Dimension and Degree of the rough Veronese variety {#sub:degRVV}
--------------------------------------------------
In this section we provide formulas for the dimension and degree of ${\mathcal{R}}_{d,k,m}$. We work under the assumption that $k\geq m$, as otherwise some variables do not appear at all in the parametrization and hence we can easily reduce to this case. Recall that the *normalized volume* of a polytope $P\subset{\mathbb{R}}^n$, denoted by $\operatorname{vol}P$, is $n!$ times its Lebesgue measure.
\[pro: dim and deg of rough veronese\] Let $d,k,m\in{\mathbb{N}}$ with $d\ge 2$ and $k\ge m$. Let $W_{d,m}$ be the set of Lyndon words of length at most $m$ in the alphabet with $d$ letters.
1. The dimension of ${\mathcal{R}}_{d,k,m}$ is $\# (W_{d,m})-1$.
2. Let $l={\mathop{\rm lcm}\nolimits}(1,2,\dots,m)$ and let $\Delta$ be the convex hull of the integral points $$\left(\frac{l}{1},0,\dots,0\right),\left( 0,\frac{l}{1},0,\dots,0\right),\dots,\left(0,\dots,0,\frac{l}{m}\right),$$ where the number of occurrences of $\frac{l}{i}$ is the number of Lyndon words of length $i$. Then $$\label{equat:degree of rough Veronese}
\deg{\mathcal{R}}_{d,k,m}\le\operatorname{vol}\left( \frac{k}{l}\Delta\right)$$ and $$\lim_{k\rightarrow\infty}\frac{\deg {\mathcal{R}}_{d,k,m}}{ \operatorname{vol}\left(\frac{k}{l}\Delta\right) }=1.$$ Moreover, equality holds in (\[equat:degree of rough Veronese\]) if and only if $l|k$.
Let $a_i$ be the number of Lyndon words of length $i$. We know that ${\mathcal{R}}_{d,k,m}$ is the toric variety associated to the polytope $$P:=P((1^{a_1},\dots, m^{a_m}),k).$$
1. The dimension of ${\mathcal{R}}_{d,k,m}$ was already computed in [@AFS18 Remark 6.5] by proving that the fibers of the map $f_{d,k,m}$ have dimension 0. However, here we show a different proof based on toric techniques. Note that $\dim{\mathcal{R}}_{d,k,m}\le\dim{\mathop{\rm Lie}\nolimits}^m({\mathbb{R}}^d)-1=\# (W_{d,m})-1$, as this is the dimension of the parameterizing weighted projective space. Thus, in order to prove the statement we only have to show that $P$ has maximal possible dimension. In fact, we prove that the lattice points of $P$ generate the lattice $$L=\left\{(x_{1,1},\dots,x_{1,a_1},x_{2,1},\dots,x_{m,a_m})\in{\mathbb{Z}}^{\#W_{d,m}}\mid k\mbox{ divides }\sum_{i=1}^m\sum_{j=1}^{a_i}ix_{i,j}\right\}.$$ If $i\in\{1,\ldots,m\}$, $j\in\{1,\ldots,a_i\}$ and $(i,j)\neq (1,1)$, then there exists a lattice point $p\in P$ with $x_{i,j}=1$ and $x_{i',j'}=0$, unless $(i',j')=(i,j)$ or $(i',j')=(1,1)$. Take any $z\in L$. By adding and subtracting $p$ of the above type, we can reduce to the situation when $z$ has only the first coordinate nonzero. In such a case, the claim is obvious.
2. We note that $P$ is the convex hull of lattice points in $\frac{k}{l}\Delta$. Thus, $$\deg{\mathcal{R}}_{d,k,m}=\operatorname{vol}P\leq \operatorname{vol}\frac{k}{l}\Delta.$$ Now, equality holds if and only if $P= \frac{k}{l}\Delta$. However, the latter is a lattice polytope if and only if $l|k$. Finally, note that $P((1^{a_1},\dots, m^{a_m}),k)$ is injected into $P((1^{a_1},\dots, m^{a_m}),k+1)$ simply by adding $(1,0,\dots,0)$. Hence $$\operatorname{vol}\left\lfloor\frac{k}{l}\right\rfloor\Delta\leq \operatorname{vol}P=\deg {\mathcal{R}}_{d,k,m}\leq \operatorname{vol}\left\lceil\frac{k}{l}\right\rceil\Delta,$$ which finishes the proof.
Many software-aided computations on the dimension, the degree and even the generators of the ideal of ${\mathcal{R}}_{d,k,m}$ are presented in [@G18]. We now want to use Proposition \[pro: dim and deg of rough veronese\] to deal with this numbers by using toric geometry.
Fix $d=m=2$. Since $W_{2,2}=\{1,2,12\}$, $\dim{\mathcal{R}}_{2,k,2}=2$. We now compute the degree for small values of $k$. Since ${\mathcal{R}}_{2,k,2}$ is a surface, we will deal with 2-dimensional polytopes. Here, $l={\mathop{\rm lcm}\nolimits}(1,2)=2$ and we denote $P:=P((1^2,2),k)$.
1. As in Notation \[not:SR\], $R_{2,2}={\mathbb{C}}[x^2,xy,y^2,a]$ and $P=\Delta$ is the convex hull of the points $(2,0,0),(0,2,0),(0,0,1)$. It is a triangle $$\begin{tikzpicture}
\filldraw[fill=black!15!white, draw=black] (0,0) -- (2,0) -- (1,1) -- cycle;
\draw (1,0) -- (1,1);
\fill[black] (0,0) circle (0.06cm);
\fill[black] (1,0) circle (0.06cm);
\fill[black] (2,0) circle (0.06cm);
\fill[black] (1,1) circle (0.06cm);
\end{tikzpicture}$$ of normalized area 2, hence $\deg {\mathcal{R}}_{2,2,2}=2$. As [@G18 Example 14] shows, ${\mathcal{R}}_{2,2,2}$ is the cone over a smooth conic in ${\mathbb{P}}^3$.
2. Since $R_{2,3}={\mathbb{C}}[x^3,x^2y,xy^2,y^3,xa,ya]$, $\frac{3}{2}\Delta={\mathop{\rm Conv}\nolimits}\left\lbrace (3,0,0),(0,3,0),(0,0,\frac{3}{2})\right\rbrace $, while $P={\mathop{\rm Conv}\nolimits}\{(3,0,0),(0,3,0),(1,0,1),(0,1,1)\}$. $$\begin{tikzpicture}
\draw (0,0) -- (3,0) -- (1.5,1.5) -- cycle;
\filldraw[fill=black!15!white, draw=black] (0,0) -- (3,0) -- (2,1) -- (1,1) -- cycle;
\draw (1,1) -- (1,0) -- (2,1) -- (2,0);
\fill[black] (0,0) circle (0.06cm);
\fill[black] (1,0) circle (0.06cm);
\fill[black] (2,0) circle (0.06cm);
\fill[black] (3,0) circle (0.06cm);
\fill[black] (1,1) circle (0.06cm);
\fill[black] (2,1) circle (0.06cm);
\end{tikzpicture}$$ Therefore, $\deg{\mathcal{R}}_{2,3,2}=\operatorname{vol}P=4<\frac{9}{2}=\operatorname{vol}\frac{3}{2}\Delta$.
3. In this case, $R_{2,4}={\mathbb{C}}[x^4,x^3y,x^2y^2,xy^3,y^4,x^2a,xya,y^2a,a^2]$, and so, $P=2\Delta={\mathop{\rm Conv}\nolimits}\{(4,0,0),(0,4,0),(0,0,2)\}$ and $\deg {\mathcal{R}}_{2,4,2}=\operatorname{vol}P=8$. $$\begin{tikzpicture}
\filldraw[fill=black!15!white, draw=black] (0,0) -- (4,0) -- (2,2) -- cycle;
\draw (3,1) -- (1,1) -- (1,0) -- (2,1) -- (2,0) -- (3,1) -- (3,0);
\draw (2,1) -- (2,2);
\fill[black] (0,0) circle (0.06cm);
\fill[black] (1,0) circle (0.06cm);
\fill[black] (2,0) circle (0.06cm);
\fill[black] (3,0) circle (0.06cm);
\fill[black] (4,0) circle (0.06cm);
\fill[black] (1,1) circle (0.06cm);
\fill[black] (2,1) circle (0.06cm);
\fill[black] (3,1) circle (0.06cm);
\fill[black] (2,2) circle (0.06cm);
\end{tikzpicture}$$
4. As in the case $k=3$, $P\subsetneq\frac{5}{2}\Delta$. More precisely, $\frac{5}{2}\Delta={\mathop{\rm Conv}\nolimits}\left\lbrace (5,0,0),(0,5,0),(0,0,\frac{5}{2})\right\rbrace $, and $P={\mathop{\rm Conv}\nolimits}\{(5,0,0),(0,5,0),(1,0,2),(0,1,2)\}$. $$\begin{tikzpicture}
\draw (0,0) -- (5,0) -- (2.5,2.5) -- cycle;
\filldraw[fill=black!15!white, draw=black] (0,0) -- (5,0) -- (3,2) -- (2,2) -- cycle;
\draw (1,1) -- (1,0) -- (2,1) -- (2,0) -- (3,1) -- (3,0) -- (4,1) -- (4,0);
\draw (1,1) -- (4,1);
\draw (2,2) -- (2,1) -- (3,2) -- (3,1);
\fill[black] (0,0) circle (0.06cm);
\fill[black] (1,0) circle (0.06cm);
\fill[black] (2,0) circle (0.06cm);
\fill[black] (3,0) circle (0.06cm);
\fill[black] (4,0) circle (0.06cm);
\fill[black] (5,0) circle (0.06cm);
\fill[black] (1,1) circle (0.06cm);
\fill[black] (2,1) circle (0.06cm);
\fill[black] (3,1) circle (0.06cm);
\fill[black] (4,1) circle (0.06cm);
\fill[black] (2,2) circle (0.06cm);
\fill[black] (3,2) circle (0.06cm);
\end{tikzpicture}$$ Hence, $\deg{\mathcal{R}}_{2,5,2}=\operatorname{vol}P=12<\frac{25}{2}=\operatorname{vol}\frac{5}{2}\Delta$.
In this way, it is straightforward to see that, up to a linear isometry, the simplex $\frac{k}{2}\Delta$ is the triangle with vertices $(k,0,0),(0,k,0)$ and $\left( 0,0,\frac{k}{2}\right)$. If $k$ is even, then they are lattice points, $P=\frac{k}{2}\Delta$, and $$\deg{\mathcal{R}}_{2,k,2}=\operatorname{vol}P=\operatorname{vol}\frac{k}{2}\Delta=\frac{k^2}{2}.$$ If $k$ is odd, then $P$ is the trapezium with vertices $(k,0,0),(0,k,0),\left( 1,0,\frac{k-1}{2}\right)$ and $\left(0 ,1,\frac{k-1}{2}\right)$, implying that $$\deg{\mathcal{R}}_{2,k,2}=\operatorname{vol}P=\frac{k^2-1}{2}<\frac{k^2}{2}=\operatorname{vol}\frac{k}{2}\Delta.$$
Axis-parallel paths {#sec:axis parallel}
===================
Besides ${\mathcal{R}}_{d,k,m}$, the universal variety contains other interesting subvarieties. One of them is the signature variety ${\mathcal{L}}_{d,k,m}$ of piecewise linear paths. Up to translation, a piecewise linear path $X$ with $m$ steps is the concatenation of $m$ linear paths, each represented by a vector $v_i$. This decomposition is unique, provided that $v_{i+1}$ is not a multiple of $v_i$ for any $i$. In this section, we study the subfamily of *axis-parallel paths*.
Let $\{e_1,\dots, e_d\}$ be the standard basis for $\mathbb{R}^d$. A piecewise linear path $X=v_1\sqcup\ldots\sqcup v_m$ is an axis-parallel path (or simply an axis path) if there are $a_1,\dots,a_m\in{\mathbb{R}}$ such that $v_i=a_ie_{\nu_i}$ for every $i$, where $\nu_i\in \{1,\dots,d\}$.
In other words, each step is a multiple of a basis vector. Therefore, an axis-parallel path is characterized by two sequences. One of them is the sequence $\nu=(\nu_1,\dots,\nu_m)$, called *shape* of $X$, that stores in each $\nu_i\in \{1,2,\dots,d\}$ the direction of the $i$-th step. The other sequence, $a =(a_1,\dots,a_m)\in{\mathbb{R}}^m$, stores the length of each step and we call it the *sequence of lengths*. Notice that $\ell(\nu)=\ell(a)=m$. When we study an axis-parallel path $X\colon [0,1]\to{\mathbb{R}}^d$, we may assume that the image is not contained in any hyperplane, that is, it is nondegenerate. This means that $\{\nu_1,\dots,\nu_m\}=\{1,\dots, d\}$. The $k$-th signature of an axis-parallel path $X$ can be computed combinatorially in a very nice way. Recall that a *partition* of a set $S$ is a collection of subsets (called blocks) such that their union is $S$. Now each sequence $\nu$ induces a partition $\pi_\nu=\{\pi_1|\pi_2|\dots|\pi_d\}$ of the set $\{1,\dots, m\}$, defined by $(\pi_\nu)_i=\{j\in\{1,\dots,m\}\mid \nu_j=i\}$. For instance, if $\nu=(1,2,1,3,3,1)$ then $\pi_\nu = \{1,3,6|2|4,5\}$. We will write $\pi$ instead of $\pi_\nu$ when there are no ambiguities. We can now introduce the main character of this section.
Fix $\nu=(\nu_1,\dots,\nu_m)$ and $k\in{\mathbb{N}}$. Let $d=\max(\nu_1,\ldots,\nu_m)$. For an axis parallel path $X=a_1e_{\nu_1}\sqcup\ldots\sqcup a_me_{\nu_m}$, let $g_{\nu,k}(X):=\sigma^{(k)}(X)$ be its $k$-th signature. As we did in Definition \[def:rough veronese\], we pass to the complex projective space and we define the *axis paths variety* $\mathcal{A}_{\nu,k}$ to be closure of the image of the composition $${\mathbb{R}}^m\xrightarrow{g_{\nu,k}}({\mathbb{R}}^d)^{\otimes k}\rightarrow({\mathbb{R}}^d)^{\otimes k}\otimes{\mathbb{C}}=({\mathbb{C}}^d)^{\otimes k}\dashrightarrow{\mathbb{P}}^{d^k-1}.$$ Instead of $g_{\nu,k}$, we can also consider $$G_{\nu,k}:{\mathbb{R}}^m\to{\mathbb{R}}^d\times ({\mathbb{R}}^d)^{\otimes 2}\times\ldots\times ({\mathbb{R}}^d)^{\otimes k}$$ by sending $a\mapsto (g_{\nu,1}(a),\dots,g_{\nu,k}(a))$. In this case we denote by ${\mathcal{A}}_{\nu,\le k}\subset {\mathbb{C}}^d\times({\mathbb{C}}^d)^{\otimes 2}\times\ldots\times ({\mathbb{C}}^d)^{\otimes k}$ the closure of the complexification of the image of $G_{\nu,k}$.
There is a nice way to write down the polynomials defining the map $g_{\nu,k}$. The following result is a consequence of [@AFS18 Corollary 5.3].
\[lem:CombDescription\] Let $X$ be the axis-parallel path of shape $\nu$ and sequence of lengths $a$. Then the $(i_1\dots i_k)$-th entry of the $k$-th signature is $$\begin{aligned}
\sigma(X)_{i_1\dots i_k} = \sum_{(j_1,\dots,j_k)} \frac{1}{s_1!s_2!\cdots s_m!} a_{j_1}a_{j_2}\cdots a_{j_k},\end{aligned}$$ where we sum over all the non-decreasing sequences $(j_1,j_2,\dots,j_k)$ such that $j_l\in \pi_{i_l}$ for $l\in\{1,\dots, k\}$, and $s_l$ counts the number of times that $l$ appears in $(j_1,j_2,\dots,j_k)$.
Lemma \[lem:CombDescription\] allows us to find the $k$-th signature of an axis-parallel path and to explicitly write the map $g_{\nu,k}$. We implemented a Macaulay2 code to compute the ideal of ${\mathcal{A}}_{\nu,k}$. This code can be found in [@CGM19].
Let $\sigma$ be the 4-th signature of an axis path of shape $\nu= (1,2,1,3,2,3,1,4)$ and sequence of lengths $a=(a_1,\dots, a_8)$. Then $\pi_\nu = \{1,3,7|2,5|4,6|8\}$. By Lemma \[lem:CombDescription\], $$\begin{aligned}
\sigma_{1234} &=& a_1a_2a_4a_8 + a_1a_2a_6a_8 + a_1a_5a_6a_8 +
a_3a_5a_6a_8, \\
\sigma_{2314} &=& a_2a_4a_7a_8 + a_2a_6a_7a_8 + a_5a_6a_7a_8, \\
\sigma_{4123} &=& 0, \\
\sigma_{1124} &=& \frac{1}{2}a_1^2a_2a_8 + \frac{1}{2}a_1^2a_5a_8 + a_1a_3a_5a_8 + \frac{1}{2}a_3^2a_5a_8.\end{aligned}$$
\[rmk:fill the universal\] As any signature variety, the axis paths variety is contained in the universal variety. Namely, ${\mathcal{A}}_{\nu,k}\subset{\mathcal{U}}_{d,k}$ for every $d\geq \max(\nu_1,\ldots,\nu_m)$. It is known that ${\mathcal{L}}_{d,k,m}$ coincides with ${\mathcal{U}}_{d,k}$ for $m$ large enough, because every piecewise smooth path can be approximated by using piecewise linear paths. Since every piecewise linear path can be approximated by an axis path, we deduce that for every $d$ and $k$, there exists $\nu$ such that $\max(\nu_i)=d$ and ${\mathcal{A}}_{\nu,k}={\mathcal{U}}_{d,k}$.
It is worth to write down what Chen’s identity means for the axis paths signature variety.
\[cor:Chen for axis paths\] Consider two sequences $\nu_1$ and $\nu_2$. If $X_1$ and $X_2$ are axis-parallel paths of shape $\nu_1$ and $\nu_2$ respectively, then $$g_{\nu_1\nu_2,k}(X_1\sqcup X_2)=\sum_{a+b= k}g_{\nu_1,a}(X_1)\otimes g_{\nu_2,b}(X_2).$$ In particular, ${\mathcal{A}}_{\nu_1\nu_2,\le k}$ is a projection of the Segre product ${\mathcal{A}}_{\nu_1,\le k}\times{\mathcal{A}}_{\nu_2,\le k}$.
In Section \[sec:rough veronese\] we saw how important it is for a variety to be toric, so it is natural to ask whether ${\mathcal{A}}_{\nu,k}$ enjoys such property.
Toricness of ${\mathcal{A}}_{\nu,k}$ {#sub: axis toricness}
------------------------------------
It is not difficult to find instances when the axis paths variety is toric. As an example we can take $\mathcal{A}_{\nu,1} = \mathbb{R}^{d}$, or consider $\nu=(1,2,\dots,d)$ and obtain the Veronese variety ${\mathcal{V}}_{d,k}={\mathcal{A}}_{\nu,k}$. A less trivial example is given by Remark \[rmk:fill the universal\]. Thanks to Lemma \[lem:CombDescription\], we are able to perform efficient computations and check that many other occurrences of ${\mathcal{A}}_{\nu,k}$ are toric, and even find their polytopes.
In [@CGM19] we compute the ideal of ${\mathcal{A}}_{(1,2,1),3}$ and we check that it is toric. Further computations on the ideal allow us to determine its polytope $P$. Indeed, ${\mathcal{A}}_{(1,2,1),3}$ is a degree 6 surface in ${\mathbb{P}}^6$, so $P$ is a two-dimensional lattice polytope of normalized area 6 that contains exactly 7 lattice points. By checking the Hilbert and Erhart polynomials and the Betti table, we see that there are only two possible polytopes.
By looking at the intersections of the tangent spaces at singular points with the variety, we see that the correct one is the one on the right.
In [@G18] the second author presents a change of coordinates, based on the exponential description of the tensor signature, for which the universal variety $\mathcal{U}_{d,k}$ is the image of a monomial map, and therefore it is toric. However, such change of coordinates fails to make every ${\mathcal{A}}_{\nu,k}$ toric as well. For instance, it does not turn ${\mathcal{A}}_{(1,2,1),3}$ into a variety defined by binomials. One could still hope to find another change of coordinates that makes both ideals of $\mathcal{A}_{\nu,k}$ and $\mathcal{U}_{d,k}$ binomial. Unfortunately, this is not true.
No change of coordinates in ${\mathbb{P}}^7$ can make the ideals of both $\mathcal{A}_{(1,2,1),3}$ and $\mathcal{U}_{3,3}$ toric. The polytopes $P$ and $Q$ of $\mathcal{U}_{3,3} $ and $\mathcal{A}_{(1,2,1), 3}$ are
There are two cases how $\mathcal{A}_{(1,2,1),3}$ could be a toric subvariety of $\mathcal{U}_{3,3}$. Either it is a toric divisor or the inclusion is a toric morphism. A careful technical analysis shows that neither is possible.
Despite previous Example, the axis paths variety turns out to be toric in many cases.
\[pro:partial results on toricness\] Let $d=\max(\nu_1,\ldots,\nu_m)$. Then
1. ${\mathcal{A}}_{\nu, 3}$ is toric for every $d\le 2$.
2. ${\mathcal{A}}_{\nu, 2}$ is toric for every $d \leq 3$.
3. $\mathcal{A}_{(1,2,\dots, d,j), 2}$, with $1\leq j< d$, is toric.
It is not difficult to check all possible cases for the first two items. The computation can be found in [@CGM19]. For the third item, we have $m=d+1$. Let us apply the change of coordinates $$\begin{cases} \tilde{a}_j = a_j + a_{d+1}\\ \tilde{a}_i = a_i \text{ for } i\neq j \end{cases}$$ on the domain ${\mathbb{R}}^{d+1}$. By Lemma \[lem:CombDescription\], we know that the entries of $\sigma^{(2)}(X)$ are $$\sigma^{(2)}(X)_{il}=\begin{cases}
a_ia_l=\tilde{a}_i\tilde{a}_l & \mbox{for $i,l\neq j$}\\
a_i(a_j+a_{d+1})=\tilde{a}_i\tilde{a}_j& \mbox{for $i<l=j$}\\
(a_j+a_{d+1})^2=(\tilde{a}_j)^2& \mbox{for $i=l=j$}\\
a_{i}a_{d+1}=\tilde{a}_i\tilde{a}_{d+1}& \mbox{for $i>l=j$}\\
0&\mbox{for $i=j<l$}\\
a_ja_i&\mbox{for $i=j>l$}.
\end{cases}$$ Finally, if we perform the change of coordinates $\sigma^{(2)}(X)_{jd}\mapsto \sigma^{(2)}(X)_{jd}+\sigma^{(2)}(X)_{j,d+1}$, the map becomes monomial.
There are further cases in which we can prove that the axis paths variety is toric.
\[lem:adding extra letter\] If the ideal of ${\mathcal{A}}_{\nu,\leq k}$ is binomial after a linear change of coordinates, then the same holds for the ideal of ${\mathcal{A}}_{\nu (d+1),k}$.
Let $X$ be an axis path of shape $\nu$ and sequence of length $(a_1,\dots,a_m)$. Since $d+1$ does not appear in $\nu$, in the set partition $\pi_{\nu (d+1)}$ the $(d+1)$-st block only contains the entry $a_{m+1}$. Therefore, by Lemma \[lem:CombDescription\], $$\begin{aligned}
\sigma(X)_{i_1\dots i_{\tilde{k}} (d+1)^{k-\tilde{k}}} =
\begin{cases}
\frac{1}{(k-\tilde{k})!}(a_{m+1})^{k-\tilde{k}}\sigma(X)_{i_1\dots i_{\tilde{k}}} & \mbox{if $d+1$ does not appear in } (i_1,\dots, i_{\tilde{k}})\\
0 & \mbox{otherwise.}
\end{cases}
\end{aligned}$$
Hence, ${\mathcal{A}}_{\nu (d+1),k}$, up to diagonal change of coordinates, is equal to the projectivization of ${\mathcal{A}}_{\nu,\leq k}$.
We finish this section with the following natural question.
\[quest:toricness\] Is every variety ${\mathcal{A}}_{\nu,k}$ toric?
Our results provide a positive answer in all relatively small cases. However, since we found no general technique, we expect that the answer may be negative. The main obstacle to provide a nontoric example is that a first potential candidate is already too large to be dealt with, either with ad hoc geometric arguments or general computational methods [@katthan2017polynomial].
Dimension of ${\mathcal{A}}_{\nu,k}$ {#sub:axisDim}
------------------------------------
As we stressed in the introduction, a fundamental problem when we deal with signatures is to determine whether it is possible to recover the path from the signature. The best case scenario is injectivity, when we can uniquely reconstruct the path. In general this is too much to hope, so we focus on a weaker property, and we want to know when $g_{\nu,k}$ is generically finite. Therefore we are interested in the dimension of the fibers, or equivalently in $\dim {\mathcal{A}}_{\nu,k}$. Since ${\mathcal{A}}_{\nu,k}\subset{\mathcal{U}}_{d,k}$ is the image of ${\mathbb{R}}^{\ell(\nu)}$ under a polynomial map, we have $$\dim\mathcal{A}_{\nu,k}\le\min\{\ell(\nu),\dim({\mathcal{U}}_{d,k})\}.$$ In general, the inequality can be strict. For instance, the dimension of ${\mathcal{A}}_{(1,2,1,2,3),3}$ is 4, while $\ell(\nu)=5$ and $\dim({\mathcal{U}}_{3,3})=7$. In order to better study the dimension of our axis-parallel signature variety, we introduce the following definition.
For $d,k\in{\mathbb{N}}$, we define the set $$N_{d,k}:=\{\nu\mid \max(\nu)\le d\mbox{ and $G_{\nu,k}$ is not generically finite}\}.$$ We say that ${\mathcal{A}}_{\nu,k}$ is *defective* if $\nu\in N_{d,k}$ for $d\ge \max(\nu)$. Otherwise, we say that ${\mathcal{A}}_{\nu,k}$ has the expected dimension.
It may seem that we made a choice in using $G_{\nu,k}$ instead of $g_{\nu,k}$ in the previous definition. However, the $k$-th signature of a generic piecewise smooth path $X$ defines all the previous signatures up to finitely many choices (see [@AFS18 Section 6]). Therefore, $$N_{d,k}=\{\nu\mid \max(\nu)\le d\mbox{ and $g_{\nu,k}$ is not generically finite}\}.$$ For the same reason, $N_{d,k+1}\subset N_{d,k}$ for every $k$. Each $N_{d,k}$ is a language, i.e. a set of words. It would be very interesting to completely characterise it - cf. Conjecture \[con:nondef\]. First, we prove that $N_{d,k}$ is absorbing with respect to concatenation.
\[lem:concatenation remains defective\] Let $\nu_1,\nu_2$ be two shapes, let $\nu_1\nu_2$ be their concatenation and let $d=\max(\nu_1\nu_2)$. If $\nu_1\in N_{d,k}$, then both $\nu_1\nu_2$ and $\nu_2\nu_1$ belong to $N_{d,k}$.
Take a general point of ${\mathcal{A}}_{\nu_1\nu_2,\le k}$. It is of the form $\sigma^{\le k}(X)$ for some general axis-parallel path $X$. Since $X$ has shape $\nu_1\nu_2$, we can write it as a concatenation $X_1X_2$, for general paths $X_1$ of shape $\nu_1$ and $X_2$ of shape $\nu_2$. By hypothesis, $\nu_1\in N_{d,k}$, and so there exist infinitely many paths $Y$ of shape $\nu_1$ such that $g_{\nu_1,a}(Y)=g_{\nu_1,a}(X_1)$ for every $a\le k$. By Proposition \[cor:Chen for axis paths\], $$g_{\nu_1\nu_2,k}(YX_2)=
\sum_{a+b= k}g_{\nu_1,a}(Y)\otimes g_{\nu_2,b}(X_2)=
\sum_{a+b= k}g_{\nu_1,a}(X_1)\
\otimes g_{\nu_2,b}(X_2)=
g_{\nu_1\nu_2,k}(X),$$ hence the fiber containing $X$ is not finite and therefore $\nu_1\nu_2\in N_{d,k}$. In the same way it is possible to prove that $\nu_2\nu_1\in N_{d,k}$.
On the other hand, if ${\mathcal{A}}_{\nu,k}$ is not defective, then we can add a new letter at any point in $\nu$ and the resulting variety is still of expected dimension.
\[lem:adding a new letter does not affect defectiveness\] Let $\nu\notin N_{d,k}$ and let $d=\max(\nu)$. If we write $\nu=\nu_1\nu_2$ and we take a new letter $l>d$, then $\nu_1\cdot l\cdot\nu_2\notin N_{l,k}$.
Let us pick a general element in the affine cone over ${\mathcal{A}}_{\nu_1\cdot l\cdot\nu_2,k}$. First we know that, up to finite number of choices, we can identify the parameter $a$ associated to $l$, as $\sigma_{(l^k)}=\frac{a^k}{k!}$. We may also identify the other parameters as the signatures indexed by numbers from one to $d$ are the same for $\nu$ and $\nu_1\cdot l\cdot\nu_2$.
We noticed several times that the universal variety can be obtained as ${\mathcal{A}}_{\nu,k}$ for sufficiently long $\nu$. Now we want to prove that this can be done in an efficient way, namely by using a sequence with as many entries as $\dim{\mathcal{U}}_{d,k}$.
\[lem:filling path\] For every $d$ and $k$, there is a shape $\nu$ such that $\max(\nu)=d$, ${\mathcal{A}}_{\nu,k}={\mathcal{U}}_{d,k}$ and $\ell(\nu)=\dim{\mathcal{U}}_{d,k}$.
By Remark \[rmk:fill the universal\], there is a $\nu'$ satisfying the first two properties. We build $\nu$ as a subsequence of $\nu'$ by deleting those entries that do not increase the dimension of the variety in the following way. If $\nu'$ satisfies the last requirement too, we are done. Suppose then that $\nu'=(\nu_1,\dots,\nu_s)$ for $s>\dim {\mathcal{U}}_{d,k}$. By construction, there is an index $i\in\{1,\dots, s-1\}$ such that ${\mathcal{A}}_{(\nu_1,\dots,\nu_{i}),k}={\mathcal{A}}_{(\nu_1,\dots,\nu_{i+1}),k}$. If $i=s-1$, take $\nu':=(\nu_1,\dots, \nu_{s-1})$ and proceed by induction on the length $\nu'$. Otherwise, let $\nu:=(\nu_1,\dots,\widehat{\nu_{i+1}},\dots,\nu_s)$, where the entry with the hat is omitted. Consider the sequences $$\begin{aligned}
\alpha :=(\nu_1,\dots,\nu_{i}),\nonumber\quad
\alpha':=(\nu_1,\dots,\nu_{i+1}),\nonumber\quad
\beta :=(\nu_{i+2},\dots,\nu_s),\nonumber
\end{aligned}$$ so that $\nu'=\alpha'\beta$ and $\nu=\alpha\beta$. By Proposition \[cor:Chen for axis paths\], the map $$\sigma^{\le k}\colon{\mathbb{R}}^{\ell(\nu')}\to{\mathcal{A}}_{\nu',\le k}$$ factors as $${\mathbb{R}}^{\ell(\nu')}\to{\mathcal{A}}_{\alpha',\le k}\times{\mathcal{A}}_{\beta,\le k}\xrightarrow{\text{Segre}} {\mathcal{A}}_{\nu',\le k}.$$ We also have another map $${\mathbb{R}}^{\ell(\nu)}\to{\mathcal{A}}_{\alpha,\le k}\times{\mathcal{A}}_{\beta,\le k}\xrightarrow{\text{Segre}} {\mathcal{A}}_{\nu,\le k}.$$ Since ${\mathcal{A}}_{\alpha,\le k} = {\mathcal{A}}_{\alpha',\le k}$, we conclude that ${\mathcal{A}}_{\nu,\le k}={\mathcal{A}}_{\nu',\le k}$. We continue the process until there is no such index $i$. The resulting subsequence of $\nu'$ satisfies the requirements of the statement.
As an easy consequence, we notice that if $d=2$ there is only one possible shape $(1,2,1,2,\dots)$. Since it is unique, it satisfies the properties of Lemma \[lem:filling path\] and therefore ${\mathcal{A}}_{\nu,k}$ has always the expected dimension.
Now, we look back to our first defective example. The reason why ${\mathcal{A}}_{(1,2,1,2,3),3}$ is defective is that there is a subsequence $(1,2,1,2)$ such that ${\mathcal{A}}_{(1,2,1,2),3}={\mathcal{U}}_{2,3}$ but $\dim({\mathcal{U}}_{2,3})=3$. This means that we are filling a 3-dimensional universal variety with a sequence of length four. We conjecture that this behavior is the only obstruction to having the expected dimension.
\[con:nondef\] Let $\nu=(\nu_1,\dots,\nu_m)$ be a sequence with $d=\max(\nu)$. Then ${\mathcal{A}}_{\nu,k}$ is defective if and only if there is a subsequence $\nu^\prime=(\nu_i,\dots, \nu_{i+r})$ of $\nu$ with $d^\prime=\max(\nu^\prime)$, such that ${\mathcal{A}}_{\nu^\prime,k}={\mathcal{U}}_{d^\prime,k}$ but $r+1>\dim({\mathcal{U}}_{d^\prime,k})$.
Determinant of axis-parallel signatures
---------------------------------------
Let us start with an observation about the entries of the $k$-th signature for a special family of axis-parallel paths.
\[lem:lone entries are factors of slices\] Let $k\in{\mathbb{N}}$ and let $X$ be an axis-parallel path with shape $\nu$ and sequence of length $(a_1,\dots,a_m)$. If an entry $\nu_i$ of $\nu$ appears only once in the $j$-th block of $\pi$, then $a_i$ divides all the entries of each $j$-th slice of $\sigma^{(k)}(X)$. If moreover $k=2$, then $a_i^2\mid\det(\sigma^{(2)}(X))$.
If we look at one of the $j$-th slices, we are fixing one of the indices of $\sigma^{(k)}(X)$ to be $j$. By Lemma \[lem:CombDescription\], every monomial of the sum is a multiple of $a_i$. When $k=2$, we are dealing with a square matrix in which both the $j$-th row and the $j$-th column of $\sigma^{(2)}(X)$ are multiples of $a_i$. In order to conclude that $a_i^2\mid\det(\sigma^{(2)}(X))$, it is enough to check that $a_i^2$ divides the diagonal entry $\sigma^{(2)}(X)_{jj}$, and this follows from Lemma \[lem:CombDescription\].
This is just a special case of a much more general phenomenon, corroborated by many experiments with Sage [@sage] and the code included in [@CGM19]. The determinant of the second signature of an axis-parallel path is always the square of a polynomial in $a_1,\dots,a_m$. This subsection is devoted to explaining this result and its consequences. In order to correctly state it, we need some definitions.
\[def:good shape\] For any shape $\nu=(\nu_1,\dots,\nu_m)$ with $d=\max \nu$, we say that a subsequence $\mu=(\nu_{i_1},\dots,\nu_{i_d})$ is a *good subshape* if $i_1<\dots <i_d$ and $\{\nu_{i_1},\dots,\nu_{i_d}\}=\{1,\dots, d\}$. Moreover, we define the *sign* of a good subshape, $\operatorname{sgn}\mu$, as the sign of the permutation $(\nu_{i_1},\dots,\nu_{i_d})\in S_d$.
Without loss of generality we assumed that $\{1,\dots,d\}=\{\nu_1,\dots,\nu_m\}$, so good subshapes always exist.
\[def:Pdet\] Let $X$ be an axis path of shape $\nu$ and let $a_i$ be the parameter associated to $\nu_i$. We define the degree $d$ homogeneus polynomial $P(a)\in \CC[a_1,\dots,a_m]$ by $$P(a):=\sum_{\mu=(\nu_{i_1},\dots,\nu_{i_d})}(\operatorname{sgn}\mu) \prod_{j=1}^d a_{i_j},$$ where the sum is taken over all good subshapes $\mu$ of $\nu$. Moreover, we denote by $\det_2 \nu\in \CC[a_1,\dots,a_m]$ the determinant $\det\left(\sigma^{(2)}(X)\right)$. By Lemma \[lem:CombDescription\], it is homogeneous of degree $2d$.
\[thm:det is a square\] With the notation from Definition \[def:Pdet\], we have $$2^d\det\left(\sigma^{(2)}(X)\right)=P(a)^2.$$
The proof we found is quite technical and we present it in Appendix \[app:proof\]. However, Theorem \[thm:det is a square\] has several interesting consequences. Since we can write ${\mathcal{U}}_{d,2}$ as an axis paths variety, we now know that the determinant of the signature matrix of any path is a square. In particular, if $X$ is any path, then $\det\sigma^{(2)}(X)\ge 0$, hence the real part of ${\mathcal{U}}_{d,2}$ lies in the semialgebraic set $\{M\in{\mathbb{R}}^{d\times d}\mid \det M\ge 0\}$. In other words, real points of ${\mathcal{U}}_{d,2}$ with negative determinant are not signature matrices of paths, but rather they come from taking the Zariski closure. It is even more interesting to think in terms of the shuffle identity.
\[example:determinant 2x2 is a shuffle square\] Let $X\colon [0,1]\to{\mathbb{R}}^2$ be any path. By Lemma \[lem:shuffle identity\], $$\begin{aligned}
\det(\sigma^{(2)}(X))&=\det\left( \begin{matrix}
\langle \sigma(X), 11\rangle &\langle \sigma(X), 12\rangle\\
\langle \sigma(X), 21\rangle &\langle \sigma(X), 22\rangle
\end{matrix}\right)\\
& = \langle \sigma(X), 11\rangle\cdot\langle \sigma(X), 22\rangle-\langle \sigma(X), 12\rangle \cdot\langle \sigma(X), 21\rangle\\
& = \langle \sigma(X), 11\shuffle 22-12\shuffle 21\rangle\\
& = \frac{1}{4}\langle \sigma(X), (12-21)^{\shuffle 2}\rangle.\end{aligned}$$
In this case it was not too difficult to realize that $4(11\shuffle 22-12\shuffle 21)=(12-21)^{\shuffle 2}$ and therefore that $\det(\sigma^{(2)}(X))$ is a square. Note that $12-21$ is twice the Lévy area defined by the path. This is a general behavior.
Let $S_d$ be the symmetric group in $d$ elements. Define $${\mathop{\rm inv}\nolimits}_d=\sum_{\rho\in S_d}\operatorname{sgn}(\rho)\rho(1)\dots\rho(d)\in T({\mathbb{R}}^d).$$ It is the sum of $d!$ words of length $d$, so it is a degree $d$ element of the shuffle algebra.
On one hand, this has a geometric meaning: as illustrated in [@DR18 Section 3.2], ${\mathop{\rm inv}\nolimits}_d$ has an interpretation in terms of the *signed volume* of the convex hull of the path. On the other hand, there is a relation with signatures.
\[lem: invariant and good underlinings\] Let $a=(a_1,\dots,a_m)\in{\mathbb{R}}^m$ and let $X\colon [0,1]\to{\mathbb{R}}^d$ be the axis path corresponding to the sequence of lengths $a$ and shape $\nu$. Then $P(a)=\langle\sigma(X),{\mathop{\rm inv}\nolimits}_d\rangle$.
By Lemma \[lem:CombDescription\], if $i_1,\dots, i_d$ are pairwise disjoint then $\langle\sigma(X),i_1\dots i_d\rangle$ equals the sum of products $\prod_{l=1}^d a_{j_l}$ where $\nu_{j_l}=i_l$ and $j_1<\dots<j_d$. In particular, it is a subsum in the definition of $P(a)$ corresponding to the good subshapes associated to the permutation $(i_1,\dots,i_d)$. Summing up over all possible permutations $(i_1,\dots,i_d)$ with signs we obtain the statement of the lemma.
It follows that Theorem \[thm:det is a square\] has an important consequence on the shuffle algebra.
\[cor:shuffle determinant is a shuffle square\] Consider the matrix $$A=\left( \begin{matrix}
11 & 12 &\dots &1d\\21 & 22 &\dots &2d\\
\vdots & \vdots &\ddots &\vdots\\
d1 & d2 &\dots &dd\\
\end{matrix}\right) $$with coefficients in the shuffle algebra $(T({\mathbb{R}}^d),\shuffle,e)$. Let $\det_\shuffle(A)$ be its determinant, where the product is the shuffle. Then $2^d\det_\shuffle(A)=({\mathop{\rm inv}\nolimits}_d)^{\shuffle 2}$.
It is enough to prove that $\langle T,({\mathop{\rm inv}\nolimits}_d)^{\shuffle 2}\rangle=2^d \langle T,\det_{\shuffle}(A)\rangle$ for every $T\in T(({\mathbb{R}}^d))$. Since the linear span of ${\mathcal{G}}({\mathbb{R}}^d)$ is the whole $T(({\mathbb{R}}^d))$, by linearity we just have to show that such equality holds when $T\in{\mathcal{G}}({\mathbb{R}}^d)$. Let us pick then $T\in{\mathcal{G}}({\mathbb{R}}^d)$. Then there is an axis path $X$, corresponding to the sequence of lengths $a\in{\mathbb{R}}^m$, such that $\langle T,({\mathop{\rm inv}\nolimits}_d)^{\shuffle 2}\rangle=\langle \sigma(X),({\mathop{\rm inv}\nolimits}_d)^{\shuffle 2}\rangle$ and $\langle T,\det_{\shuffle}(A)\rangle=\langle\sigma(X),\det_{\shuffle}(A)\rangle$. By Theorem \[thm:det is a square\] and Lemma \[lem: invariant and good underlinings\], $$2^d\langle \sigma(X),\left.\det\right._{\shuffle}(A)\rangle=2^d\det(\sigma^{(2)}(X))=P(a)^2=\langle \sigma(X),({\mathop{\rm inv}\nolimits}_d)^{\shuffle 2}\rangle.\qedhere$$
Even though the techniques we use are based on the combinatorics of ${\mathcal{A}}_{\nu,k}$, Corollary \[cor:shuffle determinant is a shuffle square\] has nothing to do with axis paths, nor with signatures at all. It would be very interesting to find a generalization for $k>2$. In our opinion, this can give new insight on the properties of the shuffle algebra.
\[sec:appendix\]
Proof of Theorem \[thm:det is a square\] {#app:proof}
========================================
For this proof we fix an axis path $X$ with shape $\nu\in{\mathbb{R}}^m$ and sequence of lengths $a\in{\mathbb{R}}^m$. The first step will be to reduce the problem to the case in which every entry of $\nu$ appears at most twice. Let us start with an observation.
\[rmk: we can relabel\] The symmetric group $S_d$ acts on ${\mathbb{R}}^d$ by permuting the basis elements. In the same way, it acts on the sequence $\nu$, on the path $X$ and thus on the polynomials $P$ and $\det_2\nu$. The determinant is invariant , while the sign of $P$ changes according to the sign of the permutation. However, $P^2$ is invariant. This means that we are allowed to relabel the entries of $\nu$. In other words, we can always pick $\sigma\in S_d$ and replace $(\nu_1,\dots,\nu_m)$ with $(\sigma(\nu_1),\dots,\sigma(\nu_m))$.
In order to prove that two polynomials are equal, it is enough to prove that they have the same coefficient on each monomial.
For a polynomial $Q$ and a monomial $M$ let $Q_{|M}$ be the coefficient of $Q$ corresponding to $M$.
Since both $P^2$ and $\det_2\nu$ are homogeneous of degree $2d$, we only have to take care of degree $2d$ monomials.
\[lem: no triple entries\] Suppose that there are three (not necessarily distinct) indices $i,j,l\in\{1,\dots,m\}$ such that $\nu_i=\nu_j=\nu_l$.
1. Let $M\in{\mathbb{C}}[a_1,\dots,a_m]_{2d}$ be such a monomial that $a_ia_ja_l\mid M$. Then $P^2_{|M}=(\det_2\nu)_{|M}=0$.
2. Let $\nu'$ be the sequence obtained from $\nu$ by removing $\nu_l$ and let $P'$ be the associated polynomial. If $M\in{\mathbb{C}}[a_1,\dots,\hat{a_l},\dots,a_m]_{2d}$ is a monomial, then $P'_{|M}=P_{|M}$ and $\det_{2}\nu_{|M}=\det_{2}\nu'_{|M}$.
Thanks to remark \[rmk: we can relabel\], we may assume $\nu_i=\nu_j=\nu_l=1$. By Lemma \[lem:CombDescription\], the variables $a_i,a_j,a_l$ only appear in the first row and in the first column of the signature matrix. Furthermore, among the monomials of the diagonal entry $\sigma(X)_{11}$ there are degree at most 2 monomials in the variables $a_i,a_j,a_l$, while away from the diagonal all monomials contain only one of these variables, with exponent 1. Therefore $a_ia_ja_l$ does not divide any monomial appearing in $\det_2\nu$. On the other hand, every good subshape $\mu$ of $\nu$ contains the entry 1 exactly once, so each monomial of $P$ contains at most one among $a_i,a_j,a_l$, and with exponent 1. In $P^2$ there can be monomials containing the product of two among $a_i,a_j,a_l$, but not containing all three of them. The second statement follows from the definitions of $P$ and $\det_2\nu$.
Thanks to Lemma \[lem: no triple entries\], from now on we can restrict our attention to sequences $\nu$ with no triple entries. Now we want to take care of the case in which an entry appears only once. We find it useful to focus on a particular monomial.
Assume that $\nu$ has no triple entries. If $\mu=(\nu_{i_1},\dots,\nu_{i_\ell})$ is a subsequence of $\nu$, we set $$e(i_j)=
\begin{cases}
1 & \mbox{if $\nu_{i_j}$ appears twice in } \mu\\
2 & \mbox{if $\nu_{i_j}$ appears onece in } \mu\\
\end{cases}$$ and we define the monomial $$M_{\mu}=\prod_{j=1}^{\ell}a_{i_j}^{e(i_j)}\in{\mathbb{C}}[a_1,\dots,a_m].$$
Since we are assuming that $\nu$ has no triple entries, $\deg(M_\nu)=2d$.
\[lem: it is enough to look at M\] If a monomial $M$ appears in either $P^2$ or $\det_2\nu$ and it is divisible by all variables $a_1,\dots,a_m$, then $M=M_\nu$.
Let $M$ be such a monomial. By Lemma \[lem: no triple entries\], $M\mid M_\nu$. Indeed, if $\nu_i$ appears in $\nu$ at most once, then $a_i^3\nmid M$. If $\nu_i=\nu_j$ appears exactly twice then $a_ia_j\mid M$, but $a_i^2,a_j^2\nmid M$. As the degrees are the same, equality follows.
The next results will explain what happens to the coefficient of $M_\nu$ when $\nu$ has a non-repeated entry.
\[lem:nonrepeated entry P\] Assume $\nu$ contains at least one non-repeated entry. Suppose that the last one occurs in $\nu_{m-l}$. Let $\nu_0$ be the sequence obtained from $\nu$ by discarding $\nu_{m-l}$ and let $P_0$ be the associated polynomial. Then $$(P(a)^2)_{|M_\nu}=(-1)^l(P_0(a)^2)_{|M_{\nu_0}}.$$
By Remark \[rmk: we can relabel\], we can assume that $\nu_{m-l}=d$. Since $M_\nu$ is a monomial of $P^2$, to compute $P^2_{|M_\nu}$ we have to consider the contribution of all pairs $(\alpha,\beta)$ of good subshapes of $\nu$. Let $(\alpha,\beta)$ be a pair contributing to $M_\nu$. The variable of any non-repeated entry - including $d$ - has to appear in both $\alpha$ and $\beta$. On the other hand, the variable of an entry that appears twice appears once in $\alpha$ and once in $\beta$. Therefore there is a bijection between pairs $(\alpha,\beta)$ of good subshapes of $\nu$ contributing to $M_\nu$ and pairs $(\alpha_0,\beta_0)$ of good subshapes of $\nu_0$ contributing to $M_{\nu_0}$, where $\alpha_0$ is the sequence obtained from $\alpha$ by removing $d$, and $\beta_0$ is defined in a similar way from $\beta$. If we set now $a$ (resp. $b$) to be the number of entries of $\alpha$ (resp $\beta$) after the discarded one, then $$\begin{aligned}
(P(a)^2)_{|M_\nu}&=\sum_{(\alpha,\beta)}\operatorname{sgn}(\alpha)\operatorname{sgn}(\beta)=\sum_{(\alpha_0,\beta_0)}(-1)^a\operatorname{sgn}(\alpha_0)(-1)^b\operatorname{sgn}(\beta_0)\\
&=(-1)^l\sum_{(\alpha_0,\beta_0)}\operatorname{sgn}(\alpha_0)\operatorname{sgn}(\beta_0)=(-1)^l(P_0(a)^2)_{|M_{\nu_0}}.\qedhere
\end{aligned}$$
Before we prove the analogous statement for the polynomial $\det_2\nu$, we introduce a combinatorial interpretation of the coefficient ${\det}_2(\nu)_{|M_\nu}$. By Laplace expansion, the contributions to ${\det}_2(\nu)_{|M_\nu}$ come from directed graphs (with possible loops) with vertices corresponding to symbols in $\nu$ and:
- if a symbol $i$ appears twice in $\nu$ then precisely one such vertex is outgoing and one is incoming;
- if a symbol $i$ appears once in $\nu$ then it has degree two and has one incoming and one outgoing edge - this is the only case where a vertex can be a loop;
- all edges are from left to right.
Each such graph contributes $\pm \frac{1}{2^m}$ where $m$ is the number of loops and the sign is the sign of the corresponding permutation.
Let $\nu=(1,2,1)$. We have two possible graphs. One with edges $(1,2)$ and $(2,1)$. It encodes the transposition $(1,2)$ and contributes with $-1$. The other one has an edge $(1,1)$ and a loop over two. It encodes the identity permutation and contributes $\frac{1}{2}$.
\[lem:nonrepeated entry det\] With the same hypothesis as in Lemma \[lem:nonrepeated entry P\], $${\det}_2(\nu)_{|M_\nu}=\frac{(-1)^l}{2}\cdot {\det}_2(\nu_0)_{|M_{\nu_0}}.$$
We look at the possible cases for the position of $d$.
If $d$ is the last symbol then the $d$-th column of the second signature matrix has only one nonzero entry, which equals $\frac{1}{2}$ times the square of the variable associated to $d$. Applying Laplace expansion we obtain the formula in this case.
We now assume that $d$ is the last but one entry. The contributing graphs are of two types.
If $d$ is a loop then (by forgetting the loop) we obtain graphs on $\nu_0$. Hence, the contribution of those graphs is $\frac{1}{2}({\det}_2(\nu_0))_{|M_{\nu_0}}$.
If $d$ is not a loop, it must have an incoming edge, say $(a,d)$ and an outgoing edge $(d,b)$, where $b$ must be the last symbol in $\nu$. We may replace this by an edge $(a,b)$. In the cycle presentation of the permutation we removed $d$ from one cycle, i.e. changed the sign of the permutation. As we did not change the number of loops the contribution equals $-({\det}_2(\nu_0))_{|M_{\nu_0}}$. Summing the two contributions we obtain the result in this case.
We may now assume that $d$ is the last letter in $\nu$ that appears once and further there are at least two symbols after it, i.e. $\nu=\dots dab\dots$. Let $\nu'=\dots abd$ be $\nu$ with $d$ moved two places forward. We will prove that $({\det}_2(\nu))_{|M_{\nu}}=({\det}_2(\nu'))_{|M_{\nu'}}$. By induction this will finish the proof of the lemma.
We may identify graphs contributing to $({\det}_2(\nu))_{|M_{\nu}}$ with those contributing to $({\det}_2(\nu'))_{|M_{\nu'}}$ with the exception of graphs for which there is an edge from $d$ either to $a$ or $b$. Further, if there is an edge $(d,a)$ and the considered $b$ is an outgoing vertex, the encoded permutation is $x\rightarrow d \rightarrow a$, $b\rightarrow c$ for some $x$ and $c$. We may associate to it a graph on $\nu'$ with edges $(x,a),(b,d),(d,c)$. This does not change the sign of permutation. We proceed in the same way if there is an edge $(d,b)$ and $a$ is outgoing. The remaining graphs are those where there is an edge from $d$ to $a$ or $b$ and the other vertex is incoming. We will prove that the sum of contributions of such graphs equals zero.
Consider a graph with edges $(d,a),(c,b)$ for some $c$. We associate to it a graph with edges $(c,a),(d,b)$. In the cycle decomposition of the permutation this operation either joins two cycles or decomposes one cycle into two, i.e. changes the sign of the permutation. In particular, the contribution of each pair equals zero.
We also see that we obtain all graphs on $\nu'$, apart from those for which there is an edge from $a$ or $b$ to $d$ and the other vertex is outgoing. Just as above one can show that the contribution of such graphs equals zero.
We can now reduce to the case in which every entry of $\nu$ appears exactly twice.
\[cor: reduce to double entries\] Assume Theorem \[thm:det is a square\] holds for sequences in which every entry appears exactly twice. Then it holds for every sequence.
We compare the coefficients of monomials in both polynomials. If a monomial is different from $M_\nu$, then it does not contain one of the variables. In this case Lemma \[lem: it is enough to look at M\] allows us to consider a subsequence of $\nu$ and conclude by induction on $\ell(\nu)$. Consider then $M_\nu$. By repeatedly applying Lemmas \[lem:nonrepeated entry P\] and \[lem:nonrepeated entry det\], we can discard all non-repeated entries and conclude by hypothesis.
From now on we may assume that each of the letters $1,\dots,d$ appears exactly twice in $\nu$. In particular, $d=2m$. As in the proof of Corollary \[cor: reduce to double entries\], we can assume, by induction on $\ell(\nu)$, that $P^2_{|M}=\det_2\nu_{|M}$ for every monomial $M\neq M_\nu$. In order to prove that the two polynomials coincide, it is enough to find a point $q\in{\mathbb{R}}^{2m}$ such that $M_\nu(q)\neq 0$ and $P^2(q)=\det_2\nu(q)$. Since we suppose that every entry of $\nu$ appears exactly twice, we can define $q\in{\mathbb{R}}^{2m}$ by $$q_i=\begin{cases}
1 & \mbox{if $\nu_i$ appears for the first time in the $i$-th entry of } \nu,\\
-1 & \mbox{if $\nu_i$ appears for the second time in the $i$-th entry of } \nu.
\end{cases}$$ The following is a straightforward consequence of Lemma \[lem:CombDescription\].
The matrix $\sigma^{(2)}(q)$ is skew-symmetric. If $i\neq j$, then $$\sigma^{(2)}(q)_{ij}=\begin{cases}
-1 & \mbox{if the subsequence of $\nu$ with the symbols $i$ and $j$ is } jiji,\\
1 & \mbox{if the subsequence of $\nu$ with the symbols $i$ and $j$ is } ijij,\\
0 & \mbox{otherwise}.
\end{cases}$$
To finish the proof it is enough to show that $P(q)^2=2^d\det_2(\nu)(q)$. As the second signature matrix is skew-symmetric, it is enough to prove that $P(q)=2^{d/2}\operatorname{Pf}_2(\nu)(q)$, where $\operatorname{Pf}$ is the Pfaffian. We note that when $\nu =(1,1,2,2,\dots,d,d)$ the claim is easy as both sides equal zero. The following two lemmas allow to reduce any $\nu$ to this case, finishing the proof.
Let $\nu_i,\nu_{i+1}$ be two consecutive entries in $\nu$. Let $\nu'$ be the sequence obtained from $\nu$ by switching $\nu_i$ and $\nu_{i+1}$, and let $q'\in{\mathbb{R}}^{2m}$ be the corresponding point, defined in the same way we defined $q$. Let $\nu''$ be the sequence obtained from $\nu$ by removing both occurrences of the symbol $\nu_i$ and both occurrences of the symbol $\nu_{i+1}$. In a similar fashion we define $q''\in{\mathbb{R}}^{2m-4}$. By Remark \[rmk: we can relabel\], we may assume without loss of generality that $\nu_i=d-1$ and $\nu_{i+1}=d$.
Let $$e(d)=\begin{cases}
1 & \mbox{if $d$ appears for the first time in the $(i+1)$-st entry of } \nu\\
-1 & \mbox{if $d$ appears for the second time in the $(i+1)$-st entry of } \nu.
\end{cases}$$ In a similar way we define $e(d-1)$. Then
1. $\operatorname{Pf}_2(\nu)(q)=\operatorname{Pf}_2(\nu')(q)+e(d-1)e(d)\operatorname{Pf}_2(\nu'')(q'')$ and
2. $P_\nu(q)=P_{\nu'}(q)+2e(d-1)e(d)P_{\nu''}(q')$.
<!-- -->
1. The change from $\nu$ to $\nu'$ changes the second signature matrix by replacing the lower right $2\times 2$ submatrix either from $\begin{pmatrix}
0 & 0\\
0&0
\end{pmatrix}$ to $\pm
\begin{pmatrix}
0 & 1\\
-1 & 0
\end{pmatrix}$ or the other way round. The formula follows from the standard Laplace expansion for Pfaffians.
2. The good underlinings that involve at most one of the exchanged $d-1$ and $d$ provide the same contribution both to $\nu$ and $\nu'$. It remains to investigate the contribution of good underlinings containing both $d-1$ and $d$. These contribute to $\nu$ and $\nu'$ with opposite signs and are exactly the contributions of $P_{\nu''}(q')$. The sign depends only on the property if the underlined variables we forget are taken with plus or minus.
We conclude by induction on the number of permutations needed to transform $\nu$ to $1122\dots dd$ and the length of $\nu$.
[^1]: Laura Colmenarejo, University of Massachusetts at Amherst (Amherst, US), and Max Planck Institute for Mathematics in the Sciences (Leipzig, Germany)
[^2]: Francesco Galuppi, Max Planck Institute for Mathematics in the Sciences (Leipzig, Germany), `galuppi@mis.mpg.de`
[^3]: Mateusz Michałek, Max Planck Institute for Mathematics in the Sciences (Leipzig, Germany), `michalek@mis.mpg.de`, and Polish Academy of Sciences, Institute of Mathematics (Warsaw, Poland), and Aalto University (Aalto, Finland). MM was supported by the Polish National Science Center Project 2013/08/A/ST1/00804 affiliated at the University of Warsaw.
[^4]: The authors would like to thank Lara Bossinger for the discussion in the early stage of this project, Avinash Kulkarni for the helpful suggestions and Joscha Diehl for his feedback.
|
---
abstract: 'Recently, the gamma-ray telescopes [*AGILE*]{} and [*Fermi*]{} observed several middle-aged supernova remnants (SNRs) interacting with molecular clouds. A plausible emission mechanism of the gamma rays is the decay of neutral pions produced by cosmic ray (CR) nuclei (hadronic processes). However, observations do not rule out contributions from bremsstrahlung emission due to CR electrons. TeV gamma-ray telescopes also observed many SNRs and discovered many unidentified sources. It is still unclear whether the TeV gamma-ray emission is produced via leptonic processes or hadronic processes. In this Letter, we propose that annihilation emission of secondary positrons produced by CR nuclei is a diagnostic tool of the hadronic processes. We investigate MeV emissions from secondary positrons and electrons produced by CR protons in molecular clouds. The annihilation emission of the secondary positrons from SNRs can be robustly estimated from the observed gamma-ray flux. The expected flux of the annihilation line from SNRs observed by [*AGILE*]{} and [*Fermi*]{} is sufficient for the future Advanced Compton Telescope to detect. Moreover, synchrotron emission from secondary positrons and electrons and bremsstrahlung emission from CR protons can be also observed by the future X-ray telescope NuSTAR and ASTRO-H.'
author:
- |
Yutaka Ohira$^{1}$[^1], Kazunori Kohri$^{1}$ and Norita Kawanaka$^{2}$\
$^{1}$Theory Center, Institute of Particle and Nuclear Studies, KEK, 1-1 Oho, Tsukuba 305-0801, Japan\
$^{2}$Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
date: 'Accepted 2011 December 28. Received 2011 December 27; in original form 2011 November 21'
title: 'Positron annihilation as a cosmic-ray probe'
---
\[firstpage\]
acceleration of particles – cosmic rays – supernova remnants.
Introduction
============
The origin of cosmic rays (CRs) is a longstanding problem in astrophysics. Supernova remnants (SNRs) are thought to be the origin of Galactic CRs. The most popular SNR acceleration mechanism is diffusive shock acceleration [e.g. @blandford87]. In fact, [*Fermi*]{} and [*AGILE*]{} show that middle-aged SNRs ($\sim10^4~{\rm yrs~old}$) interacting with molecular clouds emit GeV gamma rays with broken power law spectra [e.g. @abdo09; @abdo10; @tavani10; @giuliani11] (Hereafter, these references are referred to as OBS). Considering a stellar wind before the supernova explosion, the molecular cloud has been swept away by the stellar wind. The inner radius of the molecular cloud becomes typically about a few tens of parsecs. Then, the SNR collides with the molecular cloud at typically $10^4~{\rm yrs}$ later and the broken power law spectra of GeV gamma rays can be interpreted as the inelastic collision between molecular clouds and CR nuclei that have escaped from the SNR (hadronic processes) [@ohiraetal11a]. However, observations do not rule out contributions from bremsstrahlung emission due to CR electrons (leptonic processes). SNRs have been also observed by TeV gamma-ray telescopes, and it is still unclear whether the TeV gamma-ray emission is produced via inverse Compton scattering of CR electrons (leptonic processes) or hadronic processes. There are many TeV-unID sources in our galaxy, and their emission mechanisms are also still unclear. Old or middle-aged SNRs are the candidates of the TeV-unID sources [@yamazaki06; @ohiraetal11b]. Thus, it is crucial to identify the emission mechanism of gamma rays. Positrons and neutrinos are produced in hadronic processes. On the other hand, they are not produced in leptonic processes.[^2] Therefore, direct or indirect detections of these particles will enable us to identify the emission mechanism in gamma-ray sources.
Recently, the CR positron excess has been provided by PAMELA [@adriani08]. Although the origin of this excess has been investigated in many theoretical studies [@kashiyama11; @kawanaka10; @ioka10 and references therein], it still remains unclear. SNRs have been discussed as the candidates of the positron sources [@fujita09a; @blasi09; @shaviv09; @biermann09; @mertsch09]. Hence, it is important to observe the positron production at SNRs.
Gamma-ray telescopes showed that the photon flux above $100\,{\rm MeV}$ from SNRs with an age of about $10^{4}\,{\rm yr}$ is $F_{>100\,{\rm MeV}}=\int_{100\,{\rm MeV}}^{\infty} {\rm d}E \left({\rm d}F/{\rm d}E\right)=10^{-7}-10^{-6}\,{\rm photon\,s^{-1}\,cm^{-2}}$ (OBS). If the gamma rays have a hadronic origin, about the same numbers of positrons are produced. When those positrons lose their energy sufficiently, about $80$ percent of them would make positroniums with ambient electrons, and one fourth of them would decay into two photons with the energy of $511\,{\rm keV}$ [e.g. @prantzos10]. This means that we can expect the $511\,{\rm keV}$ photon flux of the order of $10^{-8}-10^{-7}\,{\rm photon\,s^{-1}\,cm^{-2}}$. This flux is sufficiently high for five-years observations of the future Advanced Compton Telescope (ACT) to detect.
In this Letter, considering the interaction between CR protons and molecular clouds, we estimate annihilation emission of secondary positrons produced by the CR protons. We partly refer to analyses of previous works concerning the $511\,{\rm keV}$ line from the Galactic center [@agaronyan81; @beacom06; @sizun06] and the Galactic plane [@stecker67; @stecker69], and we apply their treatments to middle-aged SNRs interacting with molecular clouds. Recent review of positron annihilation can be found in @prantzos10.
We first provide some timescales for CR protons, secondary positrons and electrons inside a molecular cloud (Section \[sec:2\]). We then solve energy spectra of the secondary positrons and electrons (Section \[sec:3\]) and calculate the annihilation line flux as well as the continuum spectrum (Section \[sec:4\]). Section \[sec:5\] is devoted to the discussion.
Relevant timescales {#sec:2}
===================
![Timescales of interaction for CRs in a molecular cloud with the number density $n=300\,{\rm cm^{-3}}$, the size $R_{c} = 30\,{\rm pc}$ and the magnetic field $B=30\,{\rm \mu G}$. The blue and green lines show the cooling times of secondary positrons (and electrons) and CR protons, respectively. The red line shows the annihilation time of positrons. The black solid line shows the diffusion time with $\chi = 0.01$.[]{data-label="fig1"}](f1.eps){width="80mm"}
In this section, we briefly summarize relevant timescales of CRs in a molecular cloud. Physical quantities of molecular clouds associated with gamma-ray sources have not been understood in detail. Observations suggest that the gas number density is about $n=10^2-10^3\,{\rm cm^{-3}}$, the size is of the order of $10\,{\rm pc}$ (OBS), and the magnetic field in molecular clouds is about $B=1-100\,{\rm \mu G}$ [@crutcher10]. In this Letter, for the molecular cloud we adopt values of the gas number density $n=300\,{\rm cm^{-3}}$, the size $R_{c} = 30\,{\rm pc}$ and the magnetic field $B=30\,{\rm \mu G}$. We consider ionization and inelastic collisions with nuclei as cooling processes of CR protons. On the other hand, we include ionization, bremsstrahlung, synchrotron emission, and inverse Compton scattering with the cosmic microwave background (CMB) as cooling processes of secondary positrons and electrons. For SNRs observed by [*Fermi*]{} and [*AGILE*]{}, SNRs interact with molecular clouds and the expansion of the SNRs is strongly decelerated. Therefore, we here do not consider the adiabatic cooling.
We here define $\tau_{\rm cool}= E/\dot{E}$ as a cooling time, where $E$ and $\dot{E}$ are the kinetic energy and the energy loss rate, respectively. We adopt an expression of $\dot{E}$ for protons given in @mannheim94, and that for positrons and electrons given in @strong98. The timescale of direct annihilation of positrons and electrons is represented by $$\tau_{\rm a,\pm}(E)=\frac{1}{n_{\rm \mp}\sigma_{\rm a}(E)v(E)}~~,$$ where $n_{\rm \pm}, \sigma_{\rm a}$ and $v$ are the number density of positrons or electrons, the cross section of the annihilation [@dirac30] and the velocity of secondary electrons or positrons, respectively. The electron density, $n_{-}$, is dominated by thermal electrons and the positron density, $n_{+}$, is very small, so that we can neglect the annihilation of secondary electrons.
The escape time through diffusion is written by $$\tau_{\rm d}(E)=\frac{R_{\rm c}^2}{4D(E)}~~,$$ where $D(E)$ is the diffusion coefficient. It is notable that so far the diffusion coefficient around an SNR has not been understood well. Thus we assume the diffusion coefficient in molecular clouds to be $$D(E)=10^{28}~\chi \frac{v(E)}{c}\left( \frac{E}{10\,{\rm GeV}}\right)^{0.5} {\rm cm^2\,s^{-1}}~~,$$ where $c$ is the velocity of light. Here $\chi=1$ might be reasonable as the Galactic mean value [@berezinskii90], but $\chi=0.01$ should be preferred in surroundings of SNRs [@fujita09b; @fujita10; @torres10; @li10], so that we use $\chi=0.01$ in this Letter.
In Fig. \[fig1\] we show those relevant timescales. The blue line shows the cooling time of secondary electrons and positrons where the synchrotron cooing dominates above $100\,{\rm GeV}$, the bremsstrahlung cooling dominates from $1\,{\rm GeV}$ to $100\,{\rm GeV}$, and the ionization loss dominates below $1\,{\rm GeV}$. The green line shows the cooling time of CR protons where the pion production cooing dominate above $1\,{\rm GeV}$ and the ionization loss dominates below 1 GeV. CR protons (about $1\,{\rm GeV}$) producing most positrons do not cool and escape as long as the SNR age is smaller than the cooling time of $1\,{\rm GeV}$ protons. Cooling times of ionization, bremsstrahlung emission and inelastic collision are inversely proportional to the gas density. Therefore, spectral evolutions of all CRs below $1\,{\rm GeV}$ depends on only the density. The black line shows the escape time due to diffusion with $\chi=0.01$. The escape of secondary electrons and positrons is negligible compared with the energy loss.
The typical energy of positrons produced by the CR protons is about $E_{\rm t}\sim 100\,{\rm MeV}$ [@murphy87]. Then its cooling time is about $3\times10^4\,(n/300\,{\rm cm}^{-3})^{-1}\,{\rm yr}$. Therefore, SNRs with an age longer than its cooling time can produce low energy positrons. Once positrons have cooled to $100\,{\rm eV}$, they form positroniums through charge exchange. There are four possible spin configurations of the positroniums. One of them has the total spin $0$ (singlet) and the three others have the total spin $1$ (triplet). The singlet state produces $511\,{\rm keV}$ line photons, and the triplet state produces continuum photons below $511\,{\rm keV}$. Because both the singlet and the triplet state are produced at the same rate, one fourth of positroniums can decay into two $511\,{\rm keV}$ line photons.
It is remarkable that the annihilation time is approximately $10$ times longer than the cooling time of positrons at around $10-100\,{\rm MeV}$. Then while the positron is being cooled, approximately $10$–$20$ percent of positrons directly annihilate with electrons. They produce continuum photons above $511\,{\rm keV}$.
Energy spectra of secondary\
positrons and electrons {#sec:3}
============================
![Energy spectra of CR protons, secondary positrons and electrons for $s=2.8$ and $nt=2.8\times10^{14}\,{\rm cm^{-3}\,s}$. The green, red and blue lines show CR protons, secondary positrons and electrons spectra, respectively.[]{data-label="fig2"}](f2.eps){width="80mm"}
In this section, to calculate the photon spectrum from secondary positrons and electrons produced by CR protons, we calculate energy spectra of positrons and electrons, $f_{\pm}(t,E)$. An evolution of CR spectra is described by $$\frac{\partial f}{\partial t}+\frac{\partial}{\partial E}\left(\dot{E}(E)f\right)+\frac{f}{\tau_{\rm a}(E)}+\frac{f}{\tau_{\rm d}(E)}=Q(t,E)~~,
\label{eq:df}$$ where $Q(t,E)$ is the source term of CRs, the third and the fourth terms of the left hand side are sink terms due to annihilation and diffusion, respectively. As shown in Fig. \[fig1\], the diffusion escape for secondary electrons and positrons is negligible, so that we neglect the diffusion escape. Then the solution for secondary positrons and electrons, $f_{\pm}$, can be expressed by $$\begin{aligned}
f_{\pm}(t,E)&=&\frac{1}{|\dot{E}(E)|}\int_E^{E_0(t,E)}{\rm d} \epsilon ~Q_{\pm}(t',\epsilon) \nonumber \\
&&~~~~~~~~~~~~\times \exp\left(-\int_E^{\epsilon} \frac{{\rm d}\epsilon'}{\tau_{\rm a, \pm}|\dot{E}|}\right) ~~,
\label{eq:f}\end{aligned}$$ where $t'$ is time for cooling from $E_0$ to $\epsilon$, and defined by $$t'=\int_{\epsilon}^{E_0} \frac{{\rm d}\epsilon'}{|\dot{E}(\epsilon')|}=t-\int_{E}^{\epsilon} \frac{{\rm d}\epsilon'}{|\dot{E}(\epsilon')|}~~,$$ and $E_0(t,E)$ is the initial energy of positrons or electrons before they cool to $E$ and defined by $$\int_{E}^{E_0} \frac{{\rm d}\epsilon}{|\dot{E}(\epsilon)|}=t~~.$$ We can neglect the annihilation loss for secondary electrons because the timescale is much longer than the SNR age.
Using the code provided by @kamae06 [@karlsson08], we calculate secondary positrons and electrons source spectra, $Q_{\pm}$, and the gamma-ray spectrum produced by decaying unstable hadrons such as pions and kaons. In addition, for the high-energy electron source $Q_{-}$, we consider knock-on electrons produced by Coulomb collisions with CR protons [@abraham66]. The knock-on electrons contribute somewhat to the continuum emission as bremsstrahlung emission.
To obtain the source term of secondary positrons and electrons $Q_{\pm}(t,E)$, we need to calculate the spectral evolution of CR protons. CR protons are injected from SNRs to molecular clouds in an energy-dependent way [@ohiraetal10]. Moreover, the diffusion escape from molecular clouds should be considered for CR protons above $10\,{\rm GeV}$ (see Fig. \[fig1\]). However, we do not need to calculate the precise spectrum of CR protons above $1\,{\rm GeV}$ because thanks to [*AGILE*]{} and [*Fermi*]{}, we have already known spectra of CR protons above $1\,{\rm GeV}$. That is, we do not need to calculate the spectral evolution due to the diffusion escape for CR protons. On the other hand, we have to calculate the evolution of the spectrum of CR protons below $1\,{\rm GeV}$ because gamma-ray spectra do not tell us the spectrum of CR protons below $1\,{\rm GeV}$. Furthermore, the injection of CR protons is thought to stop when SNR shock collides with molecular clouds [@ohiraetal11a], that is, the injection of CR protons stopped of the order of $10^4\,{\rm yrs}$ ago. Therefore, we assume that CR protons are injected as a delta function in time and an effective steep-spectrum, $Q_{\rm CR}$, instead of neglecting the diffusion escape $$Q_{\rm CR}(t,E)= q_{\rm CR}(E)\delta(t)~~,$$ where $q_{\rm CR}(E)$ is $$q_{\rm CR}(E)\propto \left(E+m_{\rm p}c^2\right)\left\{E\left(E+2m_{\rm p}c^2\right)\right\}^{-\frac{1+s}{2}}~~,$$ where $m_{\rm p}$ is the proton mass. This expression gives an injection term of $p^{-s}$, where $p$ is the momentum of the CR proton. Observations show that spectra of CR protons have broken power-law forms and the spectral index above the break is $s=2.7-2.9$ except for W44 (OBS). In this Letter, we adopt the single power law with $s=2.8$ (This assumption does not affect the flux of the annihilation line significantly). Then, the solution to equation (\[eq:df\]) for CR protons, $f_{\rm CR}$, can be expressed by $$f_{\rm CR}(t,E)= \frac{\dot{E}(E_0(t,E))}{\dot{E}(E)}q_{\rm CR}(E_0(t,E))~~.
\label{eq:fcr}$$ As mentioned in previous section, the cooling of CRs below $1\,{\rm GeV}$ depends on only the density. Hence, the solution (equation (\[eq:fcr\])) depends on only $nt$.
To calculate source terms of secondary electrons and positrons, $Q_{\pm}(t',E)$, in equation (\[eq:f\]), we approximately use the present spectrum of CR protons, $f_{\rm CR}(t,E)$, by changing $Q_{\pm}(t',E)$ into $Q_{\pm}(t,E)$ in equation (\[eq:f\]). This is because CR protons (about $1\,{\rm GeV}$) producing most positrons do not cool and escape for SNRs observed by [*Fermi*]{} and [*AGILE*]{}. The solution for the secondary positrons and electrons (equation (\[eq:f\])) depends on only $nt$ below $100\,{\rm GeV}$. Hereafter, we use $nt$ as the parameter to describe the system (for example, $nt=2.8\times10^{14}\,{\rm cm^{-3}\,s}$ for $n=300\,{\rm cm}^{-3}$ and $t=3\times 10^4\,{\rm yr}$). After the cooling time of CR protons above $1\,{\rm GeV}$ ($nt>1.9\times10^{15}\,{\rm cm^{-3}\,s}$), all CRs have already cooled and we do not expect any emission.
In Fig. \[fig2\] we show the energy spectra of the CR protons, secondary positrons and electrons given in equations (\[eq:f\]) and (\[eq:fcr\]), where $s=2.8, nt=2.8\times10^{14}\,{\rm cm^{-3}\,s}$. For CR protons (green line), the spectrum below $100\,{\rm MeV}$ is modified from the initial spectrum $q_{\rm CR}(E)$ by ionization. The electron spectrum (blue lines) is dominated by knock-on electrons below $10\,{\rm MeV}$.
A steady-state solution is obtained by changing $E_0$ to infinity in equation (\[eq:f\]). The steady-state spectrum of positrons below the typical energy of the source term $E_{\rm t}$ is approximately obtained by using the following approximation. $$Q_+(E)= q_+\delta(E-E_{\rm t})~~,$$ where $q_+$ is the total production number of the secondary positrons per unit time. Then, the steady-state spectrum below $E_{\rm t}$ is given by $$f_{+}(E)=\frac{q_{+}}{|\dot{E}(E)|}\exp\left(-\int_E^{E_{\rm t}} \frac{{\rm d}\epsilon'}{\tau_{\rm a, +}|\dot{E}|}\right)~~.
\label{eq:fss}$$
Radiation spectrum {#sec:4}
==================
![Photon spectrum from an SNR with $F_{>100\,{\rm MeV}}=10^{-6}\,{\rm photon\,s^{-1}\,cm^{-2}}, nt=2.8\times 10^{14}\,{\rm cm^{-3}\,s}$ and $s=2.8$. The black dashed line shows the total spectrum. The red, blue and green lines show the annihilation spectrum, the bremsstrahlung spectrum from secondary positrons and electrons, and the bremsstrahlung spectrum from CR protons, respectively.[]{data-label="fig3"}](f3.eps){width="80mm"}
![The line shows $F_{511\,{\rm keV}}/F_{>100\,{\rm MeV}}$ as a function of $nt$ for $s=2.8$.[]{data-label="fig4"}](f4.eps){width="80mm"}
In this section, we calculate the radiation spectrum from secondary positrons and electrons at $nt=2.8\times10^{14}\,{\rm cm^{-3}\,s}$. In this case, primary CR electrons are negligible as long as the ratio of primary CR electrons to CR protons is $K_{\rm ep} < 0.01$ (see Fig. \[fig2\]). We calculate synchrotron emission, inverse Compton emission with CMB and bremsstrahlung emission from secondary positrons and electrons [@strong00], bremsstrahlung emission from CR protons [@schuster03], and annihilation emission from secondary positrons [e.g. @sizun06]. Energy spectra of secondary positrons and electrons, obtained from equation (\[eq:f\]), are shown in Fig. \[fig2\].
When positrons cool to $\sim 10^2\,{\rm eV}$, they form positroniums through charge exchange. The production rate of the positroniums, $Q_{\rm Ps}$, is obtained by $$Q_{\rm Ps}(t)=|\dot{E}(100~{\rm eV})|f_{+}(t,100~{\rm eV})~~.
\label{eq:qps}$$ One fourth of positroniums decay into two $511\,{\rm keV}$ photons. Thus the photon flux of the $511\,{\rm keV}$ line, $F_{511\,{\rm keV}}$, is given by $$F_{511{\rm keV}}(t)=\frac{Q_{\rm Ps}(t)}{8\pi d^2}~~.
\label{eq:f511}$$ where $d$ is the source distance. The line width of the annihilation line of positroniums emitted from molecular clouds is $6.4\,{\rm keV}$ [@guessoum05]. We here use a Gaussian profile with the $6.4\,{\rm keV}$ width as the line structure.
Fig. \[fig3\] shows the photon spectrum normalized so as to make $F_{>100\,{\rm MeV}}=10^{-6}\,{\rm photon\,s^{-1}cm^{-2}}$, where $nt=2.8\times 10^{14}\,{\rm cm^{-3}\,s}$ and $s=2.8$. We again note that this condition is typical for middle-aged SNRs observed by gamma-ray telescopes. Annihilation emission (red line) dominates at around the $511\,{\rm keV}$ range. It is notable that synchrotron emission and inverse Compton emission with CMB induced by secondary positrons and electrons do not contribute in the energy range of Fig. \[fig3\]. Moreover, synchrotron emission by secondary positrons and electrons depends on the magnetic field, the maximum energy of CR protons and the spectral index of CR protons ($s$). Bremsstrahlung emission of CR protons (green line) also depends on $s$. Note that bremsstrahlung emission of CR protons can be observed by the future X-ray telescope, NuSTAR [@harrison05] and ASTRO-H [@takahashi10].
For the estimation of the annihilation line, we can neglect the escape loss as long as $\tau_{\rm cool}<\tau_{\rm d}$ for $E<100\,{\rm MeV}$. Moreover, most positrons are cooled by the ionization loss as long as $B<1\,{\rm mG}\,(n/300\,{\rm cm^3})^{1/2}$, so that the magnetic field ($B$), the size of molecular clouds ($R_{\rm c}$) and the normalization of the diffusion coefficient ($\chi$) are not important. Hence, the ambiguity of the annihilation line is only $nt$. Fig. \[fig4\] shows the flux ratio, $F_{511\,{\rm keV}}/F_{>100\,{\rm MeV}}$, as a function of $nt$, where $s=2.8$. The ratio does not depend on the number of CRs and the distance. The flux ratio, $F_{511\,{\rm keV}}/F_{>100\,{\rm MeV}}$, becomes constant after $nt=2.8\times 10^{14}\,{\rm cm^{-3}\,s}$ because almost all positrons have cooled to $100~{\rm eV}$ by that time. From Fig. \[fig4\] we expect $F_{511\,{\rm keV}} \sim 10^{-7}\,
{\rm photon\,cm^{-2}\,s^{-1}}$ from SNRs with $nt=10^{14}-10^{15}\,
{\rm cm^{-3}\,s}$ and with $F_{>100\,{\rm MeV}}\sim10^6\,
{\rm photon\,cm^{-2}\,s^{-1}}$. The expected flux of the annihilation line, $F_{511\,{\rm keV}}$, can be sufficient for five-years observations of ACT to detect [@boggs06]. Therefore we can expect to detect the annihilation line of secondary positrons produced by the CR protons in SNRs observed by [*AGILE*]{} and [*Fermi*]{}.
Discussion and Summary {#sec:5}
======================
In this Letter, we have investigated the MeV emission spectrum due to CR protons from SNRs interacting with molecular clouds. We found that for typical middle-aged SNRs observed by [*AGILE*]{} and [*Fermi*]{}, secondary positrons can cool to an energy sufficient to make positroniums. We calculated annihilation emission of the positrons and other emissions. The expected flux of the annihilation line is sufficient for the future gamma-ray telescope, such as ACT, to detect. Therefore, we propose that annihilation emission from secondary positrons is a important tool as a CR probe. Moreover, synchrotron emission from secondary positrons and electrons, and bremsstrahlung emission from CR protons can be also observed by the future X-ray telescope, NuSTAR [@harrison05] and ASTRO-H [@takahashi10].
All particles with energies smaller than $1\,{\rm GeV}$ lose their energy due to ionization, as shown in Fig. \[fig1\]. Not only the $511\,{\rm keV}$ line but also the ionization rate is also a probe of CR nuclei [@goto08; @indriolo10]. @becker11 proposed that ${\rm H}_2^{+}$ and ${\rm H}_3^{+}$ lime emissions should be observed from molecular clouds if there are many CRs. Low energy CR protons and knock-on electrons might be measured by using atomic lines [@gabriel79; @tatischeff03], so that atomic lines also become a probe of CR nuclei. Quantitative estimations will be addressed in future work. Moreover, CR nuclei can produce nuclear excitation lines by inelastic collisions and productions of unstable nuclei [e.g. @nath94; @tatischeff03; @summa11]. Therefore, CR compositions around SNRs can be also investigated by the nuclear excitation lines. Recent observations and theoretical studies for CR compositions are also remarkable [@ahn10; @ohira11]
The spectra of secondary positrons and electrons produced by CR protons are different from a single power law because of their cooling and injection spectra (see Fig. \[fig2\]). The spectra of secondary positrons and electrons below $100\,{\rm MeV}$ would become harder than the CR proton spectrum around $1\,{\rm GeV}$. The number of secondary positrons and electrons can be larger than that of primary CR electrons when the SNR age is larger than about 10 percent of the cooling time of CR protons due to the inelastic collision. Even when the number of the primary CR electrons is larger than that of secondary positrons and electrons, the spectrum is affected by the cooling. These cooling effects of positrons and electrons might be a reason why radio spectra are different from expected from gamma-ray spectra observed by [*Fermi*]{} [@uchiyama10]. However, the radio emission from SNRs may be originated from inside the SNR, but observed gamma rays may be originated from outside the SNR. That is, a different component may produce the radio emission. We may have to build more detailed models to compare the theory and observed date in future. It is an interesting future work to compare between observations and the theory for individual SNRs.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank K. Ioka for useful comments. This work is supported in part by grant-in-aid from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, No. 21684014 (Y. O.), No. 22740131 (N. K.). K.K. was partly supported by the Center for the Promotion of Integrated Sciences (CPIS) of Sokendai, and Grant-in-Aid for Scientific Research on Priority Areas No. 18071001, Scientific Research (A) No.22244030 and Innovative Areas No. 21111006.
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\[lastpage\]
[^1]: E-mail: ohira@post.kek.jp
[^2]: Strictly speaking, leptonic processes can produce positrons via $e^{-}+\gamma\rightarrow e^-+e^-+e^+$ but this reaction rate is suppressed by the fine structure constant compared with the gamma-ray production rate of inverse Compton scattering.
|
---
abstract: |
Agile approaches tend to focus solely on scoping and simplicity rather than on problem solving and discovery. This hampers the development of innovative solutions. Additionally, little has been said about how to capture and represent the real user needs. To fill this gap, some authors argue in favor of the application of “Creative thinking” for requirements elicitation within agile software development. This synergy between creativeness and agility has arisen as a new means of bringing innovation and flexibility to increasingly demanding software.
The aim of the present study is therefore to employ a systematic review to investigate the state-of-the-art of those approaches that leverage creativity in requirements elicitation within Agile Software Development, as well as the benefits, limitations and strength of evidence of these approaches.
The review was carried out by following the guidelines proposed by Dr. Kitchenham. The search strategy identified 1451 studies, 17 of which were eventually classified as primary studies. The selected studies contained 13 different and unique proposals. These approaches provide evidence that enhanced creativity in requirements elicitation can be successfully implemented in real software projects. We specifically observed that projects related to user interface development, such as those for mobile or web applications, are good candidates for the use of these approaches. We have also found that agile methodologies such as Scrum, Extreme Programming or methodologies based on rapid modelling are preferred when introducing creativity into requirements elicitation. Despite this being a new research field, there is a mixture of techniques, tools and processes that have already been and are currently being successfully tested in industry. Finally, we have found that, although creativity is an important ingredient with which to bring about innovation, it is not always sufficient to generate new requirements because this needs to be followed by user engagement and a specific context in which proper conditions, such as flexibility, time or resources, have to be met.
author:
- Ainhoa Aldave
- 'Juan M. Vara'
- David Granada
- Esperanza Marcos
bibliography:
- 'JSS\_SLR.bib'
date: 'Received: date / Accepted: date'
title: 'Leveraging creativity in requirements elicitation within agile software development: a systematic literature review'
---
Introduction {#sec:introduction}
============
Background studies {#sec:researchContext}
==================
Systematic literature review process {#sec:SLRProcess}
====================================
Data extraction {#sec:dataExtraction}
===============
Data synthesis and results {#sec:dataSynthesisResults}
==========================
Conclusions and further work {#sec:conclusions}
============================
This research has been partially funded by the Regional Government of Madrid under the FORTE-CM (S2018/TCS-4314) project and the MADRID (TIN2017-88557-R) project, financed by the Spanish Ministry of Economy and Business.
**Appendix I - List of studies selected for the review (Primary Studies)**
==========================================================================
- Hollis, B., & Maiden, N. (2013). Extending agile processes with creativity techniques. IEEE software, 30(5), 78-84
- O’Driscoll, K. (2016). The agile data modelling & design thinking approach to information system requirements analysis. Journal of Decision Systems, 25(sup1), 632-638.
- Gamble, M. T. (2016). Can metamodels link development to design intent?. In Proceedings of the 1st International Workshop on Bringing Architectural Design Thinking into Developers’ Daily Activities, 14-17
- Newman, P., Ferrario, M. A., Simm, W., Forshaw, S., Friday, A., & Whittle, J. (2015). The role of design thinking and physical prototyping in social software engineering. 37th International Conference on Software Engineering, 2, 487-496
- Patton, J. (2002). Hitting the target: adding interaction design to agile software development. In OOPSLA 2002 Practitioners Reports, 1-ff
- Lucena, P., Braz, A., Chicoria, A., & Tizzei, L. (2016). IBM Design Thinking Software Development Framework. In Brazilian Workshop on Agile Methods, 98-109
- Lombriser, P., Dalpiaz, F., Lucassen, G., & Brinkkemper, S. (2016). Gamified requirements engineering: model and experimentation. In International Working Conference on Requirements Engineering,171-187
- Mahmud, I., & Veneziano, V. (2011). Mind-mapping: An effective technique to facilitate requirements engineering in agile software development. In Computer and Information Technology (ICCIT), 2011, 157-162
- Maiden, N. (2011). What Time Is It, Eccles?. IEEE software, 28(4), 84-85
- Wanderley, F., da Silveira, D. S., Araujo, J., & Lencastre, M. (2012). Generating feature model from creative requirements using model driven design. In Proceedings of the 16th International Software Product Line Conference, 2, 18-25
- Wanderley, F., & Araujo, J. (2013). Generating goal-oriented models from creative requirements using model driven engineering. In Model-Driven Requirements Engineering (MoDRE), 2013 International Workshop on,1-9
- Wanderley, F., Silveira, D., Araujo, J., Moreira, A., & Guerra, E. (2014). Experimental evaluation of conceptual modelling through mind maps and model driven engineering. In ICCSA 2014, 200-214
- Boness, K., & Harrison, R. (2007). Goal sketching: Towards agile requirements engineering. In Software Engineering Advances, ICSEA 2007, 71-71
- Boness, K., & Harrison, R. (2008). Goal sketching with activity diagrams. In Software Engineering Advances, 2008. ICSEA’08. The Third International Conference on, 277-283
- Boness, K., Harrison, R., & Liu, K. (2008). Goal sketching: An agile approach to clarifying requirements. International Journal on Advances in Software, IARIA, 1(1)
- Hastreiter, I., Krause, S., Schneidermeier, T., & Wolff, C. (2014). Developing UX for Collaborative Mobile Prototyping. In International Conference of Design, User Experience, and Usability, 104-114
- Sulmon, N., Derboven, J., Zaman, B., & Montero, M. (2012). Mapping Participatory Design Methods to the Cognitive Process of Creativity to Facilitate Requirements Engineering. Information Systems Research and Exploring Social Artefacts: Approaches and Methodologies, 221-241
|
---
abstract: 'Ultracompact dark matter (DM) minihalos at masses at and below $10^{-12}$ $M_\odot$ arise in axion DM models where the Peccei-Quinn (PQ) symmetry is broken after inflation. The minihalos arise from density perturbations that are generated from the non-trivial axion self interactions during and shortly after the collapse of the axion-string and domain-wall network. We perform high-resolution simulations of this scenario starting at the epoch before the PQ phase transition and ending at matter-radiation equality. We characterize the spectrum of primordial perturbations that are generated and comment on implications for efforts to detect axion DM. We also measure the DM density at different simulated masses and argue that the correct DM density is obtained for $m_a = 25.2 \pm 11.0 \, \, \mu\mathrm{eV}$.'
author:
- Malte Buschmann
- 'Joshua W. Foster'
- 'Benjamin R. Safdi'
bibliography:
- 'Bibliography.bib'
title: 'Early-Universe Simulations of the Cosmological Axion'
---
The quantum chromodynamics (QCD) axion is a well-motivated dark-matter (DM) candidate capable of producing the present-day abundance of DM while also resolving the strong *CP* problem of the neutron electric dipole moment [@Peccei:1977ur; @Peccei:1977hh; @Weinberg:1977ma; @Wilczek:1977pj; @Preskill:1982cy; @Abbott:1982af; @Dine:1982ah]. The axion is an ultralight pseudo-scalar particle whose mass primarily arises from the operator $a G \tilde G/f_a$, with $a$ the axion field, $G$ the QCD field strength, $\tilde G$ its dual, and $f_a$ the axion decay constant. Below the QCD confinement scale, this operator generates a potential for the axion; when the axion minimizes this potential it dynamically removes the neutron electric dipole moment, thus solving the strong *CP* problem. In the process the axion acquires a mass $m_a \sim \Lambda_{\rm QCD}^2 / f_a$, with $\Lambda_{\rm QCD}$ the QCD confinement scale. The standard ultraviolet completion of the axion low-energy effective field theory is that the axion is a pseudo-Goldstone boson of a symmetry, called the Peccei-Quinn (PQ) symmetry, which is broken at the scale $f_a$ [@Kim:1979if; @Shifman:1979if; @Dine:1981rt; @Zhitnitsky:1980tq; @Srednicki:1985xd].
The cosmology of the axion depends crucially on the ordering of PQ symmetry breaking and inflation. If the PQ symmetry is broken before or during inflation, then inflation produces homogeneous initial conditions for axion field and generically the cosmology is relatively straightforward [@Marsh:2015xka]. In this work we focus on the more complex scenario where the PQ symmetry is broken after reheating. Immediately after PQ symmetry breaking, the initial axion field is uncorrelated on scales larger than the horizon, with neighboring Hubble patches coming into causal contact in the subsequent evolution of the Universe. This leads to complicated dynamical phenomena, such as global axion strings, domain walls, and non-linear field configurations called oscillons (also referred to as axitons) [@Hogan:1988mp; @Kolb:1993zz; @Kolb:1993hw; @Kolb:1994fi; @Zurek:2006sy; @Enander:2017ogx; @Vaquero:2018tib].
We perform numerical simulations to evolve the axion field from the epoch directly before PQ symmetry breaking to directly after the QCD phase transition. Once the field has entered the linear regime after the QCD phase transition, we analytically evolve the free-field axion to matter-radiation equality. The central motivations for this work are to (i) quantify the spectrum of small-scale ultracompact minihalos that emerges through the non-trivial axion self-interactions and initial conditions, and (ii) to determine the $m_a$ that leads to the correct DM density in this scenario.
The post-inflation PQ symmetry breaking cosmological scenario has been the subject of considerable numerical and analytic studies. It has been conjectured that this cosmology gives rise to ultra-dense compact DM minihalos with characteristic masses $\sim$$10^{-13}$-$10^{-11}$ $M_\odot$, though we show that the typical masses are actually smaller than this, and initial DM overdensities of order unity [@Kolb:1993hw; @Kolb:1993zz; @Kolb:1994fi; @Tinyakov:2015cgg; @Davidson:2016uok; @Fairbairn:2017sil; @Vaquero:2018tib]. In this work we compute the minihalo mass function precisely, combining state-of-the-art numerical simulations with a self-consistent cosmological picture. Understanding this mass function is important as it affects the ways that we look for axions in this cosmological scenario. For example, it has been claimed that microlensing by minihalos and pulsar timing surveys [@Dror:2019twh] may constrain the post-inflation PQ symmetry breaking axion scenario [@Fairbairn:2017sil], but these analyses rely crucially on the form of the mass function at high overdensities and masses. The axion minihalos may also impact indirect efforts to detect axion DM through radio signatures [@Pshirkov:2007st; @Huang:2018lxq; @Hook:2018iia; @Safdi:2018oeu; @Bai:2017feq; @safdi2019xxx].
A precise knowledge of the $m_a$ that gives the observed DM density is of critical importance for axion direct detection experiments [@Shokair:2014rna; @Du:2018uak; @Brubaker:2016ktl; @Kenany:2016tta; @Brubaker:2017rna; @TheMADMAXWorkingGroup:2016hpc; @Kahn:2016aff; @Ouellet:2018beu; @Ouellet:2019tlz; @Chaudhuri:2014dla; @Silva-Feaver:2016qhh]. We find , which is within range of [*e.g.*]{} the HAYSTAC program [@Brubaker:2017rna]. Our axion mass estimate is similar to that found in recent simulations [@Klaer:2017ond] but disagrees substantially with earlier semi-analytic estimates [@Davis:1986xc; @Davis:1989nj; @Battye:1994au; @Wantz:2009it; @Hiramatsu:2010yu; @Kawasaki:2014sqa; @Ballesteros:2016xej]. The minihalo mass function is also important for interpreting the results of the laboratory experiments. If a large fraction of the energy density of DM is in compact minihalos, it is possible that the expected DM density at Earth is quite low or highly time dependent, which means that direct detection experiments would need to be more sensitive than previously thought or use an alternate observing strategy.
The original simulations that tried to estimate the minihalo mass function were performed in [@Kolb:1993hw] on a grid of size 100$^3$. Ref. [@Kolb:1993hw] found oscillons (soliton-like oscillatory solutions) that contribute to the high-overdensity tail of the mass function. Note that oscillons are analogous to the breather solutions found in the Sine-Gordon equation (see [*e.g.*]{} [@Visinelli:2017ooc]). Recently [@Vaquero:2018tib] performed updated simulations on a grid of size 8192$^3$. Our results expand on and differ from those presented in [@Vaquero:2018tib] in many ways, such as through our initial state that begins before the PQ phase transition, measurement of the overall DM density, evolution to matter-radiation equality, and accounting of non-Gaussianities.
{width="32.50000%"} {width="32.50000%"} {width="32.50000%"}
\[fig:QCDevo\]
[**Simulation setup:**]{} We begin our simulations with a complex scalar PQ field $\Phi$, with Lagrangian [$$\label{eom} \begin{split}
\mathcal{L}_{PQ} = &\frac{1}{2} | \partial \Phi | ^2 - \frac{\lambda}{4} \left(|\Phi|^2 - f_a^2\right)^2 - \frac{\lambda T^2}{6} |\Phi|^2 \\
&- m_a(T)^2 f_a^2 [1 - \cos \mathrm{Arg}(\Phi)],
\end{split}$$]{} with $T$ the temperature, $\lambda$ the PQ quartic coupling strength, and $m_a(T)$ the temperature-dependent axion mass generated by QCD [@Hiramatsu:2012gg]. We fix $\lambda = 1$ for definiteness, though this does not affect our final results. The parametrization of the temperature-dependent mass is adopted from the leading-order term in the fit in [@Wantz:2009it]. Explicitly, the axion mass is parametrized by $$m_a(T)^2 = \mathrm{min}\bigg[\frac{\alpha_a \Lambda^4}{f_a^2 (T / \Lambda)^n}, \, m_a^2\bigg],$$ for , and , though in the Supplementary Material (SM) we consider alternate parameterizations. The growth of the mass is truncated when it reaches its zero-temperature value, which occurs at independent of the axion decay constant. The zero-temperature mass is given by eV [@diCortona:2015ldu].
For the PQ-epoch simulations we begin well before the breaking of the PQ symmetry at a time when the PQ field is described by a thermal spectrum. The simulation is performed by evolving the equations of motion on a uniformly spaced grid of side-length $L_{PQ}= 8000$ in units of $1/(a_1 H_1)$, with $a_1$ ($H_1$) the scale factor (Hubble parameter) at the temperature when $H_1 = f_a$, at a resolution of $1024^3$ grid-sites. We use a standard leap-frog algorithm in the kick-drift-kick form with an adaptive time-step size and with the numerical Laplacian calculated by the seven-point stencil. It is convenient to use the rescaled conformal time $\tilde \eta = \eta / \eta_1$, where $\eta_1$ is the conformal time at which point $H(\eta_1) \equiv H_1 = f_a$. The simulation begins at $\tilde \eta_i = 0.0001$ and proceeds with initial time-step $\Delta \tilde \eta_i = 0.004$ until $\tilde\eta = 250$, after which a variable time-step calculated by $\Delta\tilde\eta_i(250 / \tilde \eta)$ is used to maintain temporal resolution of the oscillating PQ fields. Convergence was tested by re-running small time intervals of the simulation at smaller time steps. The PQ fields evolve from their initial thermal configuration until the PQ phase transition occurs at $\tilde \eta \approx 280$, after which the radial mode $|\Phi / f_a |$ acquires its vacuum expectation value (VEV). We simulate until $\tilde \eta_f = 800$ in order to proceed to a time at which fluctuations around the radial mode VEV have become highly damped.
Note that the difference in $\tilde \eta$ between $\tilde \eta = 1$ and the PQ phase transition is proportional to $\sqrt{m_{\rm pl} / f_a}$, with $m_{\rm pl}$ the Planck mass. The actual choice of $f_a$ here does not play an important role since we evolve the axion-string network into the scaling regime. In the left panel of Fig. \[fig:QCDevo\] we show the final state of our simulation at the completion of the PQ simulation. The string network is seen in yellow, with the blue colors indicating regions of higher than average axion density. The length of the simulation box at this point is around $8000 /(a(\tilde \eta_f) H(\tilde \eta_f))$, and we indeed find that there is around one string per Hubble patch as would be expected in the scaling regime.
We use the final state of the PQ-epoch simulation as the initial state in our QCD-epoch simulation. To do so we assume that the axion-string network remains in the scaling regime between the two phase transitions (see, [*e.g.*]{}, [@Hiramatsu:2010yu]). Recently [@Gorghetto:2018myk] found evidence for a logarithmic deviation to the scaling solution and we confirm this behavior in the SM. However, we perform tests to show that this deviation to scaling likely has a minimal impact on both the minihalo mass function and on the DM density, though we still assign a systematic uncertainty to our DM density estimate from the scaling violation.
Anticipating requiring greater spatial resolution for late-times in our QCD simulation, we increased the resolution of our simulation to $2048^3$ grid-sites with a nearest-neighbor interpolation algorithm. We re-interpreted the physical dimensions of our box from side-length $L_{\rm PQ} = 8000$ in PQ spatial units to $L_{\rm QCD} = 4$ in units of $1/(a_1 H_1)$. These units are defined such that $H_1 \equiv H( \eta_1^{\rm QCD}) = m_a(\eta_1^{\rm QCD})$ at conformal time $\eta_1^{\rm QCD}$. Further, we use the dimensionless parameter $\hat \eta = \eta / \eta_1^{\rm QCD}$. While our PQ simulation ended at $\tilde \eta_f = 800$ in PQ units, the start time in the QCD phase transition is taken to be $\hat \eta_i = 0.4$ in the QCD units. Modes enter the horizon as their co-moving wavenumber becomes comparable to the co-moving horizon scale, which scales linearly with $\eta$. Therefore, by maintaining the ratio $L_{\rm PQ} /\tilde \eta_f = L_{\rm QCD} / \hat \eta_i$, we preserve the status of our modes with respect to horizon re-entry.
We then evolve the equations of motion with our initial step size now chosen to be $\Delta \hat \eta_i = 0.001$. As before, we adaptively refine our time step size, using time-step $\Delta \hat \eta_i (1.8 / \hat \eta)^{3.34}$ after $\hat \eta = 1.8$, to maintain resolution of the oscillating axion field. We simulate until $\hat \eta_f = 7.0$, periodically checking if all topological defects have collapsed. When this occurs, we switch to axion-only equations of motion for computational efficiency, since past this point the radial mode does not play an important role.
The conformal time $\hat \eta_c$ at which the mass growth was cut off corresponds to the physical value of the axion decay constant since it relates the temperature $T_1$ at which the axion begins to oscillate and the cutoff temperature $T_c \approx 100$ MeV at which the axion reaches its zero-temperature mass. We performed simulations at five values of $\hat \eta_c$ uniformly spaced between $2.8$ and $3.6$. These values are chosen to access different values of $f_a$ while still preserving a hierarchy between $\hat \eta_c$ and our simulation end time in order to provide sufficient time for the field to relax. At each of the five values of $\hat \eta_c$, we performed simulations at five different values of the parameter $\tilde \lambda$, defined by $\tilde \lambda \equiv \lambda f_a^2 / m_a(\hat \eta_1)^2$. This parameter can be interpreted as the squared mass of the radial PQ mode relative to the axion mass, at conformal time $\hat \eta_1$. In order for excitations of the radial mode to be well-resolved in our simulation, we require that the resolution of our simulation $\Delta \bar x$, with $\bar x = a_1 H_1 x$ and $x$ the spatial coordinate, be such that $1 / (\hat \eta \tilde \lambda^{1/2} \Delta \bar x) > 1$, making simulations for realistic axion parameters $\tilde \lambda$ impossible. We break the relation between $\tilde \lambda$ and $f_a$ and consider $\tilde\lambda=[1024,1448,3072,3584,5504]$ in order to study the impact of this parameter.
We illustrate three important phases of the QCD-epoch simulation in Fig. \[fig:QCDevo\]. The left-most panel is the initial state discussed previously in the context of the PQ-epoch simulation final-state. When $m_a(\hat \eta) = 3 H(\hat \eta)$ at $\hat \eta \approx 1.22$, strings grow longer and become less numerous, with domain walls forming on surfaces bounded by the strings. This is illustrated in the middle panel, with red colors indicating domain walls. As the temperature continues to decrease with increasing $\hat \eta$, strings and domain walls tighten and decrease in size until they collapse. By $\hat \eta \gtrsim 2.0$, the network collapses in its entirety. Shortly thereafter, we observe the formation of oscillons [@Kolb:1993hw; @Amin:2010jq; @Vaquero:2018tib]. We note that the oscillon field configuration is relativistic, so that near the origin of the oscillons the oscillation wavelength is $\sim$$m_a(\hat \eta)^{-1}$, which is rapidly shrinking with increasing time. After the zero-temperature mass is reached, oscillons stop shrinking and slowly dissipate at varying rates until the full field enters the linear regime. White regions in the right-most panel of Fig. \[fig:QCDevo\] denote regions of high axion energy density, which are mostly inhabited by oscillons.
At the end of the simulation, the field has relaxed into the linear regime ([*e.g.*]{}, axion self-interactions are unimportant), but the field remains mildly relativistic because axion radiation is produced during the string-network collapse and during the oscillon collapse. It is therefore important to continue evolving the axion field until a time nearer to matter-radiation equality to allow the field to become non-relativistic everywhere and also to allow the compact but high-momentum overdensities to spread out. We perform this evolution analytically by exactly solving the linear axion equations of motion in Fourier space. We end this evolution shortly before matter-radiation equality ($T \sim$ keV), at which time proper velocities have frozen out but local radiation domination is preserved at all locations in our simulation box so that gravitational effects remain negligible.
[**Analysis and results.—**]{} We provide [Supplementary Data](https://zenodo.org/record/2653964#.XPQfs9NKjOR) [@malte_buschmann_2019_2653964] containing the final state from our most realistic QCD-epoch simulation, after having performed the evolution to near matter-radiation equality. Note that the axion field after the QCD phase transition is highly non-Gaussian and phase-correlated at small scales and cannot accurately be reconstructed from the power spectrum. In fact considering that we find large overdensities $\delta$ ($\delta \sim 10$), with $\delta = (\rho - \bar \rho) / \rho$ and $\bar \rho$ ($\rho$) the average (local) DM density, the field could not possibly be Gaussian at these scales, considering that Gaussian random fields have symmetric over and under-densities but under-densities with $\delta < -1$ would have negative DM density.
We may try to estimate the present-day mass function by performing a clustering analysis on the final states. In particular, we expect that the large overdensities will detach from the cosmic expansion, due to reaching locally matter-radiation equality before the rest of the Universe, and collapse onto themselves under gravity. Thus by clustering the 3-D spatial energy density distribution from the simulation slightly before matter-radiation equality and quantifying the distribution of masses and overdensities that we find, we can make predictions for the spectrum of minihalo masses and concentrations today.
From the final-state we construct an overdensity field $\delta(x)$, and we identify overdensities as closed regions of positive $\delta$. Under this definition 50% of the total mass is in overdensities. In practice, we identify these regions by first finding all positive local maxima, then recursively identifying all neighboring grid sites that are larger than 20% of the corresponding local maxima. This threshold is necessary to have a clear boundary between different overdensities, though the final mass function is not strongly dependent on the specific choice of 20%. Additionally, we discard overdensities that consist of less than 80 grid sites to avoid discretization issues in the final result. Note that we discard only about 0.8% of the total mass that would otherwise be assigned to an overdensity due to the 80 grid-site limit and 20% threshold. We assign to each overdensity a mass $M$ and a single mean concentration parameter $\delta$.
An illustration of our clustering procedure is shown in Fig. \[fig:cluster\].
{width="48.00000%"}
\[fig:cluster\]
In that figure we show a 2-dimensional slice through the overdensity field for our most realistic simulation with $\hat\eta_c = 3.6$ and $\tilde \lambda = 5504$. Note that in the left panel we show the field at $\hat \eta = 7$ at the end of the QCD simulation while in the right panel we show the same slice slightly before matter-radiation equality, denoted by $\hat \eta_{\rm MR} = 10^6$ and corresponding to $T \sim$ keV. While a large overdensity left over from oscillon decay, along with corresponding rings of relativistic axion radiation, is visible in the left panel, that structure largely disperses in the subsequent evolution to $\hat \eta_{\rm MR}$. Two-dimensional slices through the boundaries of the clustered regions are shown in red in Fig. \[fig:cluster\].
![Differential mass distribution for axion minihalos for our most realistic simulation, as described in Fig. \[fig:cluster\], computed by clustering the overdensity field at $\hat \eta_{\rm MR}$. The shaded “unresolved" region denotes the parameter space that is beyond our resolution limit. Small statistical uncertainties are displayed as grey error bands. ](plots/Mdelta.pdf){width="48.00000%"}
\[fit:QCDaxionspectra\]
We characterize the minihalo mass function through the distribution $d^2f / d(\log M) / d\delta$, where $f$ represents the fraction of mass in overdensities of mean overdensity $\delta$ and mass $M$ with respect to the total mass in minihalos. We compute the mass function for all of the 25 simulations at varying $\tilde \lambda$ and $\hat \eta_c$. To perform the extrapolation to the physical $f_a$ ($\hat \eta_c$), we use the following procedure. First, we normalize the total DM density found in the simulation at $\hat \eta_{\rm MR}$ to the value that would give the observed DM density today. Then we perform the clustering algorithm to determine $d^2f / d(\log M) / d\delta$. We rescale all of the masses by $\left[ (a_1 H_1)^{\text{sim}} / (a_1 H_1)^\text{target} \right]^3$, where $(a_1 H_1)^{\text{sim}}$ is the simulated horizon size at $\hat \eta = 1$ and $(a_1 H_1)^\text{target} $ is the horizon size at the target $f_a$. The shift accounts for the fact that the characteristic scale of the overdensities is expected to be set by the horizon volume when the axion field begins to oscillate (see, [*e.g.*]{}, [@Fairbairn:2017sil; @Vaquero:2018tib] and the SM). The effect of this shift is to move all of the masses to lower values, since the target $m_a$ is larger than those we simulate. The resulting mass function for our most realistic simulation is shown in Fig. \[fit:QCDaxionspectra\]. As we show in the SM, after applying the mass shift the mass functions appear to give relatively consistent results between the different $\hat \eta_c$, though the agreement is not perfect at high $M$. As a result, we cannot exclude the possibility that simulating to the target $\hat \eta_c$ would give different results, especially at high masses, compared to our extrapolations. On the other hand, the effect of $\tilde \lambda$ appears to be minimal, since this parameter only affects the decay of the string network.
We may also compare our determinations of the total DM density produced during the QCD phase transition to previous analyses (see [*e.g.*]{} [@Davis:1986xc; @Davis:1989nj; @Battye:1994au; @Wantz:2009it; @Hiramatsu:2010yu; @Kawasaki:2014sqa; @Ballesteros:2016xej; @Klaer:2017ond]). Our results are summarized in Fig. \[fig:DM-density\], where we show the DM density today that we find for our top four $\hat \eta_c$, converted to $f_a$, for our most physical $\tilde \lambda$. The uncertainties in our $\rho_a$ measurements are determined from the variance between the different $\tilde \lambda$ simulations, and while some small dependence on $\tilde \lambda$ is expected, we find that this dependence is subdominant to statistical noise and no trend is detectable in our data. We also include a conservative $10\%$ systematic uncertainty that accounts for our unphysical fixing of the effective number of degrees of freedom $g_*$ throughout our simulation, a 15% systematic uncertainty from violations to scaling between the PQ and QCD phase transitions, and the uncertainty on the measured value of $\Omega_a$ in our Universe [@Aghanim:2018eyx] (see the SM for details).
![The DM density $\Omega_a$ as a function of the axion decay constant $f_a$, with statistical uncertainties (black) and correlated systematic uncertainties (red) indicated, for our top four simulations. We compare our results to those in [@Klaer:2017ond] (Klaer and Moore), which agree relatively well with our own, and [@Kawasaki:2014sqa] (Kawasaki et al.), which predicts significantly higher $\Omega_a$ relative to what we find. ](plots/Energy_Density.pdf){width="48.00000%"}
\[fig:DM-density\]
In Fig. \[fig:DM-density\] we compare our results to the best-fit simulation result from [@Klaer:2017ond], which like us numerically evolved the axion-string system through the QCD phase transition, albeit with a different formalism, and also the semi-analytic calculations from [@Kawasaki:2014sqa]. Our results are in reasonable agreement with those in [@Klaer:2017ond] and significantly disagree with those in [@Kawasaki:2014sqa]. Note that we self-consistently account for all production mechanisms for axion DM in our simulation, including string decay in the few decades before the QCD phase-transition. It is the late-time axion production, right before the QCD phase transition, which is most important since it is the least redshifted [@Kawasaki:2014sqa]. The source of the discrepancy could be due in part to the fact that by artificially separating the production mechanisms, [@Kawasaki:2014sqa] over-counted the DM density produced (see [@Klaer:2017ond]). Additionally, the highly non-linear axion dynamics at the QCD epoch likely violate the number-conserving assumptions made by [@Kawasaki:2014sqa].
We may estimate the $f_a$ that gives the correct DM density by fitting our results to a power-law $\Omega_a \sim f_a^\alpha$. We find the best-fit index $\alpha = 1.24 \pm 0.04$, only including statistical uncertainties, which is marginally compatible with the analytic calculations in [@Kawasaki:2014sqa; @Klaer:2017ond] that predict . Fixing $\alpha$ to the theoretical value, we find $\Omega_a = (0.102 \pm 0.02) \times (f_a/ 10^{11} \mathrm{GeV})^{1.187}$, now incorporating the correlated systematic uncertainties, which leads to the prediction that the correct DM density is achieved for $f_a = (2.27 \pm 0.33) \times 10^{11} \, \, \mathrm{GeV}$ ($m_a = 25.2 \pm 3.6 \, \, \mu\mathrm{eV}$) in agreement with [@Klaer:2017ond]. Note that if we fit for $\alpha$ instead of fixing $\alpha$ to the theoretical value we find $m_a = 17.4 \pm 4.5 \, \mu\mathrm{eV}$; the difference between the two $m_a$ estimates could be due to a systematic difference between the theoretically predicted $\alpha$ and the actual dependence of $\Omega_a$ on $f_a$. In light of this we use the difference between the two $m_a$ estimates as an estimate of the systematic uncertainty from the extrapolation to $f_a$ below those simulated. We additionally include a $\sim$27% uncertainty on $m_a$ from uncertainties in the mass growth of the axion (see the SM for details), leading to the prediction $m_a = 25.2 \pm 11.0$ $\mu$eV.
[**Discussion.—**]{} We performed high-resolution simulations of axion DM in the cosmological scenario where the PQ symmetry is broken after inflation, starting from the epoch before the PQ phase transition and evolving the field until matter-radiation equality. After matter-radiation equality one should still evolve the axion field gravitationally down to lower redshifts, which we plan to do in future work. Our mass function is an estimate of the resulting mass function one would find after simulating the gravitational collapse. It is possible that the true halos will be slightly larger in mass due to [*e.g.*]{} accretion of surrounding DM.
We may try to estimate the halo sizes based upon when we expect the halos to collapse gravitationally. Under the assumption, for example, that the final density profile is a constant-density sphere of radius $R$ (which is likely not a good approximation but still is useful to get a sense of the halo sizes), then the halo density today was argued to be approximately $\rho \approx 140 \rho_{\rm eq} \delta^3 (\delta + 1)$, where $\rho_{\rm eq}$ is the DM density at matter-radiation equality [@Kolb:1994fi]. This implies, for example, that a $M = 10^{-14}$ $M_\odot$ subhalo with an initial average overdensity $\delta = 3$ will have a characteristic size of $\sim$$1 \times 10^6$ km. The implications for direct and indirect axion detection efforts ([*e.g.*]{}, non-trivial time dependence) are likely substantial and will be the subject of future work. One immediate implication, however, is that the axion minihalos are likely out of reach for microlensing and pulsar timing surveys [@Dror:2019twh], given the small minihalo masses.
[*We thank Jiji Fan for collaboration on early stages of this project, and we thank Asimina Arvanitaki, Masha Baryakhtar, Gus Evrard, Andrew Long, Nicholas Rodd, Jesse Thaler, Ken Van Tilburg, and Kathryn Zurek for useful comments and discussion. This work was supported in part by the DOE Early Career Grant de-sc0019225. JF received additional support from a Leinweber Graduate Fellowship. This research was supported in part through computational resources and services provided by Advanced Research Computing at the University of Michigan, Ann Arbor. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231.* ]{}
**Supplementary Material for Early-Universe Simulations of the Cosmological Axion**\
Malte Buschmann, Joshua W. Foster, and Benjamin R. Safdi\
[*Leinweber Center for Theoretical Physics, Department of Physics, University of Michigan, Ann Arbor, MI 48109*]{}
This Supplementary Material contains additional results and explanations of our methods that clarify and support the results presented in the main Letter. We begin with a detailed explanation of the equations of motion and initial conditions used in our simulations. Next, we present extended results for the overdensity spectrum and DM density. We then present a modified simulation that allows us to quantify the systematic uncertainty in the DM density determination by assuming a fixed number of relativistic degrees of freedom. Additionally, we quantify the uncertainty on the DM density induced by uncertainties in the mass growth of the axion, and finally we consider the effects of violations to the scaling solution on our final results.
Simulation Equations of Motion {#sec:SupEOM}
==============================
Our phenomenological Lagrangian describing the PQ field is adopted from the construction of [@Hiramatsu:2012gg] and is of the form $$\mathcal{L}_{PQ} = \frac{1}{2} | \partial \Phi | ^2 - \frac{\lambda}{4} \left(|\Phi|^2 - f_a^2\right)^2 - \frac{\lambda T^2}{6} |\Phi|^2
- m_a(T)^2 f_a^2 [1 - \cos \mathrm{Arg}(\Phi)],$$ where $\Phi$ is the complex PQ scalar, $T$ is the temperature, $\lambda$ is the PQ quartic coupling strength, $f_a$ is the PQ-scale identified as the axion decay constant, and $m_a(T)$ is the temperature-dependent axion mass [@Hiramatsu:2012gg]. The parametrization of the temperature-dependent mass is adopted from the leading order term in the fit in [@Wantz:2009it]. Explicitly, the axion mass is parametrized by $$m_a(T)^2 = \mathrm{min}\bigg[\frac{\alpha_a \Lambda^4}{f_a^2 (T / \Lambda)^n}, \, m_a\bigg],$$ for $\alpha = 1.68 \times 10^{-7}$, $\Lambda = 400 \, \mathrm{MeV}$ and $n = 6.68$. The growth of the mass is truncated at $T \approx 100 \, \mathrm{MeV}$. The zero-temperature mass is given by $$m_a^2 = \frac{m_\pi^2 f_\pi^2}{f_a^2} \frac{m_u m_d}{(m_u + m_d)^2},$$ where $m_\pi$ is the pion mass, $f_\pi$ is the pion decay constant, $m_{u/d}$ is the up/down quark mass. Details of the temperature-dependent axion mass, or equivalently, the topological susceptibility, remain uncertain, especially at low temperatures. Note that we do not explore here how our results are affected by uncertainties in the temperature-dependent axion mass, though doing so is a worthwhile direction for future work.
Decomposing the complex scalar as $\Phi = \phi_1 + i \phi_2$, and assuming a radiation-dominated cosmological background, leads to equations of motion in metric coordinates of the form $$\begin{gathered}
\ddot \phi_1+ 3 H \dot \phi_1 - \frac{1}{R^2} \nabla^2 \phi_1 + \frac{1}{3} \lambda \phi _1 \bigg[3 \left(\phi _1^2+\phi _2^2 - f_a^2\right)+T^2\bigg] -\frac{m_a(T)^2 \phi _2^2 }{\left(\phi _1^2+\phi _2^2\right){}^{3/2}} = 0 \\
\ddot \phi_2 + 3 H \dot \phi_2 - \frac{1}{R^2} \nabla^2 \phi_2 + \frac{1}{3} \lambda \phi _2 \bigg[3 \left(\phi _1^2+\phi _2^2 - f_a^2\right)+T^2\bigg] + \frac{m_a(T)^2 \phi _1 \phi _2}{\left(\phi _1^2+\phi _2^2\right)^{3/2}}= 0 \,.
\label{eq: full}\end{gathered}$$ Over temperatures $T \gtrsim 100 \, \mathrm{MeV}$, the number of relativistic degrees of freedom $g_*$ in the Standard Model is expected to vary only mildly. For simplicity, we therefore assume $g_* = 81$, which is a typical value adopted at high temperatures (though later in the SM we explore the systematic uncertainty introduced by this assumption). It is useful to define a dimensionless conformal time $\hat \eta$ such that $$\hat \eta = \frac{R}{R(T = T_1)} = \frac{R}{R_1} = \left(\frac{t}{t_1}\right)^{1/2} \,,
\label{eq:etaref}$$ where $R$ is the scale factor and the time $t_1$ (with $T(t_1) \equiv T_1$) is a reference time that will be defined differently in the PQ and QCD epoch simulations.
The axion-mass term is not included in our PQ-epoch simulations. In our QCD-epoch simulations, on the other hand, the mass term is included and drives the dynamics. In this case, the mass grows until the cutoff temperature $T_c$ at which point the axion mass reaches its zero-temperature value; the corresponding conformal time is given by $\hat \eta_c = R(T = T_c) / R_1$. Rewriting with the dimensionless coordinates, we then find $$\begin{gathered}
\psi_1'' + \frac{2}{\hat \eta} \psi_1' - \bar \nabla^2 \psi_1 + \frac{1}{H_1^2} \left[ \lambda \psi _1 \bigg(\hat \eta^2 f_a^2 \left(\psi _1^2+\psi _2^2 - 1\right)+ \frac{1}{3} T_1^2\bigg) -m_a^2(T_1) \hat \eta^2 \mathrm{min}(\hat \eta, \hat \eta_{c})^{n}\left(\frac{ \psi _2^2 }{\left(\psi _1^2+\psi _2^2\right){}^{3/2}} \right)\right]= 0 \\
\psi_2'' + \frac{2}{\hat \eta}\psi_2' - \bar \nabla^2 \psi_2 + \frac{1}{H_1^2} \left[\lambda \psi _2 \bigg(\hat \eta^2 f_a^2 \left(\psi _1^2+\psi _2^2 - 1\right)+ \frac{1}{3} T_1^2\bigg)+ m_a^2(T_1)\hat \eta^2 \mathrm{min}(\hat \eta, \hat \eta_{c})^{n}\left( \frac{ \psi _1 \psi _2}{\left(\psi _1^2+\psi _2^2\right)^{3/2}} \right)\right]= 0 \,,\end{gathered}$$ where $\phi = f_a \psi$, primes denote derivatives with respect to $\hat \eta$, and the spatial gradient is taken with respect to $\bar x = a_1 H_1 x$.
The PQ Epoch
------------
Simulations in the PQ epoch occur at $T \gg \Lambda_{\rm QCD}$ and so the temperature-dependent axion mass may be neglected. We therefore take our equations of motion to be $$\begin{gathered}
\psi_1'' + \frac{2}{\tilde\eta} \psi_1' - \bar \nabla^2 \psi_1 + \lambda \psi _1\left[\tilde\eta^2 \left(\psi _1^2+\psi _2^2 - 1\right)+ \frac{T_1^2}{3 f_a^2}\right] = 0 \\
\psi_2'' + \frac{2}{\tilde\eta}\psi_2' - \bar \nabla^2 \psi_2 +\lambda \psi _2 \left[\tilde\eta^2 \left(\psi _1^2+\psi _2^2 - 1\right)+ \frac{T_1^2}{3 f_a^2} \right]= 0 \,,\end{gathered}$$ and we fix $\tilde\eta = 1$ to be the time at which $H_1 = f_a$. Note that for our PQ-epoch simulations we refer to $\hat \eta$, defined in , as $\tilde \eta$ in order to avoid confusion with the dimensionless conformal time $\hat \eta$ used in the QCD-epoch simulations. The ratio $(T_1 / f_a)^2$ is determined by $$\left(\frac{T_1}{f_a}\right)^2 \approx 8.4 \times 10^5 \left(\frac{10^{12} \, \mathrm{GeV}}{f_a}\right) \,.$$ In principle, it would seem that axions of different decay constants would require different simulations in the PQ epoch. However, this ratio is degenerate with our choice of physical box size and dynamical range in $\tilde\eta$ in a particular simulation, allowing us to perform only one PQ simulation and interpret its output as the initial state of the axion field for several different values of $f_a$. The key assumption behind this, however, is that at late times after the PQ phase transition the field enters the scaling regime so that we may reinterpret the output of the PQ simulation in the appropriately rescaled box as the initial state of the QCD simulation at much lower temperatures. Note that the value of $\lambda$ is a free parameter, which we naturally choose to be $\lambda = 1$ though it has little effect.
### Initial Conditions for a PQ Scalar {#sec:PQinitial}
We generate initial conditions for our PQ scalar by taking it to be described by a thermal distribution characterized by the temperature $T$ at the initial early time. As can be read off from the Lagrangian, each of the two fields has an effective mass of the form $$m_{\rm eff}^2 = \lambda \left( \frac{T^2}{3} - f_a^2 \right) \,.$$ Correlation functions of the initially-free massive scalar fields are given by $$\begin{aligned}
\langle \phi_i(x) \phi_j(y) \rangle &= \delta_{ij} \int \frac{dk}{2 \pi}\frac{n_k}{\omega_k} e^{i k \cdot(x - y)} \\
\langle \dot \phi_i(x) \dot \phi_j(y) \rangle &= \delta_{ij} \int \frac{dk}{2 \pi}n_k \omega_k e^{i k \cdot(x - y)} \\
\langle \dot \phi_i(x) \phi_j(y) \rangle &= 0 \,,\end{aligned}$$ where overdots denote differentiation with respect to time, and we have defined $$n_k = \frac{1}{e^{\omega_k / T}-1}, \qquad \omega_k = \sqrt{k^2 + m_{\rm eff}^2}.$$ In momentum space, these correlation functions take the form $$\begin{aligned}
\langle \phi_i (k) \phi_j(k') \rangle &= \frac{2 \pi n_k}{\omega_k}\delta(k+k') \delta_{ij}\\
\langle \dot \phi_i (k) \dot \phi_j(k') \rangle &= 2 \pi n_k\omega_k\delta(k+k') \delta_{ij}\\
\langle \dot \phi_i(k) \phi_j(k') \rangle &= 0 \,.\end{aligned}$$ Our simulations occur on a discrete lattice of finite size, so the correlation functions above lead to initial conditions set by a realization of a Gaussian random field specified in Fourier space by $$\begin{aligned}
\qquad \langle \phi_i(k) \rangle &= 0, \qquad \langle | \phi_i(k) |^2 \rangle = \frac{n_k}{\omega_k}L, \\
\langle \dot \phi_i(k) \rangle &= 0, \qquad\langle | \dot \phi_i(k) |^2 \rangle = n_k \omega_k L \,.\end{aligned}$$ Note that we include the 50 lowest $k$-modes in each of the three directions when constructing the initial conditions, and we have verified that including more modes does not affect our results.
Early Times in the QCD Epoch
----------------------------
During the QCD epoch, $T \sim \Lambda_{\rm QCD}$, and so the axion mass is non-negligible. Here, we define $\hat\eta = 1$ to be the time at which $H_1 = m_a(T_1)$, with the axion field beginning to oscillate shortly thereafter when $m_a = 3 H$. The equations of motion are then given by $$\begin{gathered}
\psi_1'' + \frac{2}{\hat\eta} \psi_1' - \bar \nabla^2 \psi_1 + \tilde \lambda \hat\eta^2 \psi_1 (\psi_1^2 + \psi_2^2 - 1)- \mathrm{min}(\hat\eta, \hat\eta_{c})^{n}\hat\eta^2 \left( \frac{\psi_2^2}{(\psi_1^2 + \psi_2^2)^{3/2}}\right)= 0 \\
\psi_2'' + \frac{2}{\hat\eta} \psi_2' - \bar \nabla^2 \psi_2 + \tilde \lambda \hat\eta^2 \psi_2 (\psi_1^2 + \psi_2^2 - 1)+ \mathrm{min}(\hat\eta, \hat\eta_{c})^{n}\hat\eta^2 \left( \frac{\psi_1\psi_2}{(\psi_1^2 + \psi_2^2)^{3/2}}\right)= 0 \,,\end{gathered}$$ where we have neglected the $T_1$ contribution to the PQ scalar mass as it is small compared to $f_a$. The parameter $\tilde \lambda$ is defined by $$\tilde \lambda = \lambda\left(\frac{ f_a}{m_a(T_1)}\right)^2$$ and can be interpreted as the squared mass of the radial mode $|\Phi / f_a |$. For physical parameters we expect $\tilde \lambda \gg 1$, though in practice we find that the final results are relatively independent of $\tilde \lambda$ for moderately sized values of the parameter, as described in the main text and later in the SM. Indeed, our choices for $\tilde \lambda$ allow us to resolve the radial mode mass by more than a few grid-spacings, satisfying the requirement of [@Gorghetto:2018myk] to accurately study the axion spectrum from string radiation, unlike [@Vaquero:2018tib]. There exist additional criteria on the largeness of $\tilde \lambda$ such that the metastability of topological defects is preserved despite the unphysical smallness of simulated $\tilde \lambda$ in comparison with the rapidly increasing axion mass. At all times prior to expected defect collapse, our choices of $\tilde \lambda$ satisfy the simplest construction of these conditions [@Vaquero:2018tib], with our choice of $\tilde \lambda = 5504$ satisfying the most stringent criteria established in [@Fleury:2015aca]. We note that we are largely unable to differentiate between simulations at any two particular values of $\tilde \lambda$, and that our choice of values appear to have minimal impact, as illustrated further below.
Late Times in the QCD Epoch
---------------------------
The presence of topological defects in the axion field at early times during the QCD epoch requires that we fully simulate both degrees of freedom of the PQ field. Once the topological defects have collapsed, however, we are free to use the axion-only equations of motion. Our axion is defined by $a = f_a \mathrm{arctan2}(\phi_1, \phi_2)$ and has the Lagrangian $$\mathcal{L} = \frac{1}{2} (\partial a)^2 - m_a^2(T) f_a^2 \left[ 1- \cos\left(a \over f_a \right) \right]\,,$$ along with corresponding equations of motion $$\theta'' + \frac{2}{\hat\eta} \theta' - \bar \nabla^2 \theta + \mathrm{min}(\hat\eta, \hat\eta_{c})^{n}\hat\eta^2 \sin\theta = 0 \,.$$ Above, we define $\theta = a / f_a$. Evolving these equations of motion is formally equivalent to freezing out excitations of the radial mode by taking $\tilde \lambda \rightarrow \infty$, which more accurately recovers the true physics of the evolution of the axion field for realistic values of $f_a$. Note that the coordinate $\bar x$ and $\hat\eta$ here are identical to those used in evolving the two degrees of freedom of the complex scalar performed prior to defect collapse.
Analytically Evolving in the Fixed-Mass Small-Field Limit
---------------------------------------------------------
At late times when the axion mass has reached its zero-temperature value and the axion field has redshifted considerably so that $|\theta| \ll 1$, the equations of motion are linear and well-approximated by $$\theta'' + \frac{2}{\hat\eta} \theta' - \nabla^2 \theta + \hat\eta_c^n \hat\eta^2 \theta = 0.$$ We may solve this equation analytically by going to Fourier space and adopting an ansatz for the solution as $$\theta(\hat\eta) = f(\hat\eta) \exp(i \mathbf{k} \cdot \mathbf{x}) \,.$$ This ansatz leads to the equation $$f''(\hat\eta )+\frac{2 f'(\hat\eta)}{\hat\eta }+ f(\hat\eta ) \left(\hat\eta ^2 \hat\eta _c^n+\mathbf{k}^2\right)=0 \,,$$ which has the general solution $$f(\hat\eta) = \frac{\exp(-\frac{i}{2}\hat\eta ^2 \hat\eta _c^{n/2}) }{\hat\eta} \left[ C_1 H_{-\frac{1}{2} \hat\eta _c^{-n/2}
\left(\hat\eta _c^{n/2}+i \mathbf{k}^2\right)}\left(\sqrt[4]{-1} \hat\eta \hat\eta
_c^{n/4}\right)+C_2 \,_1F_1\left(\frac{1}{4} \hat\eta _c^{-n/2} \left(\hat\eta _c^{n/2}+i\mathbf{k}^2\right);\frac{1}{2};i \hat\eta ^2 \hat\eta _c^{n/2}\right) \right] \,,$$ for coefficients $C_1$ and $C_2$ determined by boundary conditions, and where $H_n$ and $\,_1F_1$ are the analytic continuations of the Hermite polynomials and the confluent hypergeometric function of the first kind, respectively. From this analytic solution, we can transfer late-time field configurations from our simulation to arbitrary large $\hat\eta$. Differentiation with respect to $\hat\eta$ may be straightforwardly performed to find $f'(\hat\eta)$ at large $\hat\eta$ as well. The computation of the analytically continued Hermite polynomials and hypergeometric functions was performed with the python package `mpmath`.
We directly compare the differential mass spectrum at $\hat\eta=7$ with the same field analytically evolved to $\hat\eta=\eta_\text{MR}$ in Fig. \[fig:MDelta7\]. While the basic differential shape is the same, the $\hat\eta=7$ results have a much wider distribution in $\delta$. In particular, all overdensities above $\delta>10$ have vanished by the time matter-radiation equality is reached. However, the peak of the distribution is still around $\delta=1$. Evolving the fields down to matter-radiation equality is important because many of the modes are generated with high momentum at the QCD epoch, causing the large overdensities to disperse by the time of matter-radiation equality.
{width=".49\textwidth"} {width=".49\textwidth"}\
{width=".49\textwidth"} {width=".49\textwidth"}
\[fig:MDelta7\]
Studying the (Over)Density Field
================================
Our interest in this work is studying the energy density field $\rho$ and the overdensity field $\delta = (\rho - \bar \rho) / \rho$ realized in the axion field from our simulations. The axion energy density for the axion field $a = f_a \theta$ is computed by the Hamiltonian density $$\mathcal{H} = f_a^2 \left[\frac{1}{2}\dot \theta^2 + \frac{1}{2 R^2}( \nabla \theta)^2 + m_a^2(1-\cos\theta)\right],$$ which can be rewritten in simulation units as $$\mathcal{H} = m_a^2 f_a^2\left[ \frac{\theta'^2 + (\bar\nabla \theta)^2}{2 \hat\eta_c^{6.68} \hat\eta^2} + (1 - \cos \theta) \right] \,,$$ assuming $\hat\eta > \hat\eta_c$. At late times, the Hamiltonian is approximately $$\label{eq:hamiltonian}
\mathcal{H} \approx \frac{m_a^2 f_a^2}{2}\left( \frac{\theta'^2}{\hat\eta_c^{n} \hat\eta^2} + \theta^2 \right) \,,$$ when all modes in the simulation are non-relativistic and the field values are small.
Oscillons
---------
Large overdensities right after the QCD phase transition are caused by oscillons. Oscillons are, in contrast to strings and domain walls, not topological defects but arise due to non-linearities in the equation of motions, forming at locations where the the axion self-interaction dominates the Hubble friction. As a result, the first oscillons form at the location of collapsed strings and domain walls, where the axion remains excited and reaches large field values. However, at later times, oscillons are observed forming throughout the simulation box. The dynamics of the oscillons are highly non-trivial, especially as the axion self-interaction increases in strength with the growing axion mass.
Oscillons decrease in size over time following the oscillation wavelength $\sim$$m_a(T)^{-1}$, as axions in the core are relativistic. Good spatial resolution is therefore needed to resolve them. In order to study their behavior we perform a 2D (two spatial dimensions, one time) simulation using the same simulation setup in the PQ- and QCD-epoch as in 3D. We find that there is no qualitative difference between 2D and 3D simulations regarding oscillons, but going to 2D allows us to increase the spatial resolution to $4096^2$ grid sites and to subsequently increase $\hat\eta_c$.
We illustrate the evolution of an oscillon in Fig. \[fig:oscillon\]. Two scenarios are considered with different truncation points of the mass growth, $\hat\eta_c=4.0$ and $\hat\eta_c=6.0$. Note how the radius of the oscillon decreases as long as $m_a(T)$ is increasing. The circles in \[fig:oscillon\] have radius $m_a(T)^{-1}$, and the oscillon cores are seen to track this scale. Subsequently, if the mass growth is truncated at $\hat\eta_c=4.0$, the radius of the oscillon is constant as well. When the mass growth is cut-off, the density contrast at the core of the oscillon slowly decreases over time and the oscillons dissipate, as can be seen in the two lower right panels in Fig. \[fig:oscillon\].
{width="100.00000%"}
\[fig:oscillon\]
Calculating the Axion Relic Abundance
-------------------------------------
To calculate the axion DM abundance as a function of $m_a$, we first need to understand the relationship between the mass cutoff conformal time $\hat \eta_c$ and the decay constant $f_a$. Here we use the relation $T_1 / \hat\eta_c = T_c$, with $T_c \approx 100$ MeV. This allows us to solve for $f_a$ in terms of $\hat\eta_c$. The energy densities are calculated from the axion field and its derivatives according to after numerically evolving until $\hat\eta = 7$, then analytically evolving until $\hat\eta_{\rm MR} = 10^6$, at which point the contribution of the gradient term to the energy density is negligible. As a side note, our definition of $\hat\eta_{\rm MR}$ actually puts us at slightly earlier times than global matter-radiation equality. This us because matter-radiation equality is, locally, reached earlier for the largest overdensities and because we want to make sure that gravitational interactions can be neglected. In particular, note that the temperature corresponding to $\hat \eta_{\rm MR}$ is given by $T_{\rm MR} = T_c \eta_c / \hat \eta_{\rm MR}$. For our most realistic simulation with $\eta_c = 3.6$ this corresponds to $T_{\rm MR} \approx 0.5$ keV. However, if we reinterpret the final state for a more realistic axion with $m_a \approx 25$ $\mu$eV, which has a higher $\eta_c$, then $T_{\rm MR} \approx 4$ keV. In practice, though, the exact value of $T_{\rm MR}$ is not important because by these temperatures the proper motions in the axion field are frozen out and the field is thus not evolving non trivially. As a consequence our results (both for the DM density and for the spectrum of overdensities) are not sensitive to small (or even relatively large) changes to the exact value of $\hat \eta$ that we evolve to.
Note that we present our results in terms of the DM density fraction today $\Omega_a$, which is defined as the ratio of the average energy density today in DM relative to the observed critical energy density. We compute statistical error bars at each value of $f_a$ from the variance as a function of $\tilde \lambda$ at fixed $\hat\eta_c$. We note that no trend is visible in the data for the dependence of $\Omega_a$ on $\tilde \lambda$, as is shown in Fig. \[fig: DM-lambda\].
{width=".4\textwidth"}
\[fig: DM-lambda\]
The statistical noise is inferred from the spread in $\Omega_a$ values, which are determined from the output at $\eta_{\rm MR}$, between different $\tilde \lambda$. The observed variations are consistent with the expected noise from Poisson counting statistics due to having a finite number of overdensities within the simulation box.
In Fig. \[fig:Relic\] we show our results for $\Omega_a$ as a function of $f_a$, compared to earlier predictions in [@Kawasaki:2014sqa] and [@Klaer:2017ond].
{width=".75\textwidth"}
\[fig:Relic\]
For reference, we also include predictions for the relic abundance based on the field value and the time derivative at $\hat\eta = 7$. Here it is less straightforward to determine the DM axion abundance, relative to taking the results at $\hat \eta_{\rm MR}$, as some of the modes in the simulation are still relativistic. This introduces an additional systematic uncertainty, since the field is not completely red-shifting like radiation at this time. For these reasons it is important to evolve the field until it is completely non-relativistic before measuring the DM density.
Because the ratio of the axion mass density to entropy density is constant after the axions have become non-relativistic and the number of axions is conserved, we can redshift our energy density from our matter-radiation equality $\hat \eta_{\rm MR}$ to today. Then, we compare this energy density to the most up-to-date measurement of the average DM density in the Universe today $\rho_{\rm DM} = 33.5 \pm 0.6$ $M_\odot / {\rm kpc}^{3}$ [@Aghanim:2018eyx]. Note that we have propagated all cosmological uncertainties other than those on $N_{\rm eff}$, which we have fixed to the Standard Model value. These cosmological uncertainties introduce an approximately $3\%$ correlated uncertainty across the results of our simulations. We additionally have an approximately $8\%$ uncertainty due to our assumption of fixed $g_*$, which is examined in greater detail later in this Supplement. These uncertainties are the dominant ones in our results, and we emphasize that they have not been typically considered in determinations of the DM axion mass. From the $\Omega_a$ data, for the various $f_a$ values simulated, we may extrapolate to predict the $f_a$ for an axion which produces the observed DM relic abundance by fitting a simple power law relation of the form $$\Omega_a(f_a) = c_1 \cdot f_a^\alpha \,,
\label{eq:Omega}$$ as discussed in the main body of this work. Note that we expect $\alpha = (6 + n)/(4+n) $, where $n$ is the index of the axion mass growth. We assume this scaling is valid to make our estimate for the $m_a$ that gives the correct DM abundance. The relation between $\alpha$ and $n$ is expected to arise for the following reason. Let us estimate the axion DM density from an axion with a constant initial misalignment angle $\theta_i$. The present-day axion abundance as produced by the misalignment mechanism can be estimated by $$\rho_a(T_0) = \rho_a(T_3) \frac{m_a(T_0)}{m_a(T_3)}\frac{g_*(T_0)T_0^3}{g_*(T_1) T_3^3} \,,$$ where $T_0$ is the present-day temperature, $T_3$ is the temperature at which the axion began to oscillate ($m_a(T_3) = 3 H(T_3)$), and $g_*(T)$ the number of effective degrees of freedom at temperature $T$.
The initial axion abundance $\rho_a(T_1)$ is given $$\rho_a(T_1) = \frac{m_a(T_1)^2 f_a^2 }{2}\theta_i^2 \,,$$ Anharmonicity factors can be included, but have no temperature or $f_a$ dependence. The temperature $T_3$ depends on $f_a$ through the relation $T_3 \propto f_a^{-2/(4+n)}$. Substituting these relations in and keeping only terms which depend on $f_a$, we have $$\rho_a(T_0) \propto f_a^{(6+n) / (4+n)} \frac{g_*(T_0)}{g_*(T_3)} \,.$$ We thus expect the relic abundance to scale with $f_a$ like $\rho_a \propto f_a^{(6 + n) / (4 + n)}$. Note that the DM abundance from string and domain wall production is calculated similarly in [@Kawasaki:2014sqa], and although our results are not consistent with those presented in that work, the abundance calculation they present proceeds similarly, yielding string and domain wall production that scale like $f_a^{(6+n) / (4+n)}$ as well.
On the other hand, we may also calculate the the $m_a$ that gives the correct DM abundance by using our fit value for $\alpha$, as defined in , instead of the theoretical value. Doing so leads to a slightly lower $m_a$ estimate, as described in the main text.
Tests of the Overdensity Field Gaussianity
------------------------------------------
In typical cosmological contexts, overdensity fields are treated under the assumption that they are Gaussian random fields. For a real-space Gaussian field, we may Fourier transform the field and find that the squared magnitude of each mode is independently exponentially distributed with mean set by the power-spectrum and with the phase of each mode independently uniformly distributed on $[0, 2\pi)$ [@doi:10.1137/1.9780898718980]. For reference, in Fig. \[fig:PowerSpectra\] we show our power spectra $\Delta_k^2$ at fixed $\tilde \lambda$ across our various choices for $\hat\eta_c$. Note that we construct the power spectra from the fields that have been evolved until $\hat\eta = \hat \eta_{\rm MR}$. However, as we demonstrate below, the power spectrum fails to accurately describe the overdensity field we realize in our simulations because the field is highly non-Gaussian at small scales. As a result, standard tools for predicting structure formation that rely upon an underlying Gaussian overdensity field, such as the Press-Schechter formalism, cannot be applied to predict the spectrum of structures that form from the overdensities in the axion field, at least on the very smallest scales.
{width=".75\textwidth"}
\[fig:PowerSpectra\]
First, we note that the largest field values taken within the overdensity fields at the state realized by the analytic evolution until $\hat\eta = \hat \eta_{\rm MR}$ are $\mathcal{O}(10)$, whereas the minimum value the overdensity field can take is $-1$ by construction. This is trivially incompatible with the interpretation of the overdensity field as a Gaussian random field, which would have symmetric variance about its mean of $0$. For our overdensity fields to realize $\mathcal{O}(10)$ maxima with $-1$ as a construction-imposed minimum, there must exist considerable phase-correlations between Fourier modes, contrary to the uncorrelated phases of a Gaussian random field.
We also may inspect the distribution of power at each mode in the Fourier transformed overdensity field. If the overdensity field were Gaussian, then the power in each mode would be exponentially distributed with mean set by the value of the mean power spectrum. To test this, we plot the probability distribution $dP/dx$ of $x = |\hat \delta(\mathbf{k})|^2 / \langle|\hat \delta(\mathbf{k})|^2 \rangle_{|\mathbf{k}| = k}$, with $\hat \delta({\bf k})$ the Fourier transformed overdensity field at momentum ${\bf k}$, as measured in the final states of our field at $\hat\eta= \hat \eta_{\rm MR}$. We compare the observed distributions with the expected Gaussian random field assumption of an exponential distribution with unit mean in Fig. \[fig:PowerDist\]. Dramatic deviations from the expected behavior are observed for large $|\mathbf{k}|$. We stress, however, that in addition to these distributions departing from the expected exponential distributions, the real and imaginary components across modes are also highly phase correlated on small scales.
{width="1.\textwidth"}
\[fig:PowerDist\]
Minihalo Mass Spectrum
----------------------
In this subsection we give additional details and results for the minihalo mass and density spectrum. In addition to the technical difficulties associated with a non-Gaussian overdensity field, computational limitations prevent us from performing realistic simulations of $f_a \sim 10^{11} \, \mathrm{GeV}$ axions, which would require us to simulate until $\hat\eta_c \approx 15$. We instead interpret our simulation results at smaller $\hat\eta_c$ in appropriate units to rescale these results to the target $f_a \approx 2\times10^{11} \, \mathrm{GeV}$. We do so with the following methods. The total axion mass contained within some set of grid-sites in our simulation can be computed from the Hamiltonian as $$M_{\rm tot} = a(\hat\eta)^3 \int d^3 x \mathcal{H} \approx a(\hat\eta) \sum (\Delta x)^3 \mathcal{H} = \left(\frac{a(\hat\eta)^3 \Delta \bar x}{a_1 H_1}\right)^3\sum \mathcal{H} = \left(\frac{\hat\eta \Delta \bar x}{H_1}\right)^3 \sum (1 + \delta) \bar \rho \,,
\label{eq:MHmass}$$ where $\bar \rho$ is computed by the average of our Hamiltonian in in the simulation box. We calculate $H_1$ from $T_1$ based on our choice of $f_a$, then rescale $\bar \rho$ to the value of the axion energy density at the time $\hat\eta$ such that the correct relic abundance is realized today. In this manner, we aim to rescale all dimension-full quantities related to $f_a$ to our target $f_a$. In particular, we rescale the DM density $\bar \rho$ to give the correct DM density realized in our Universe, and we also rescale the minihalo masses by the factor $\propto (a_1 H_1)^{-3}$ appearing in to those for the target $f_a$.
{width=".66\textwidth"}
\[fig:M-rescale\]
We illustrate the rescaling procedure in Fig. \[fig:M-rescale\]. In that figure we show the differential mass distribution of minihalos $df / d \log M$ as a function of minihalo mass $M$. These mass distributions have been rescaled such that $\bar \rho$ matches the actual DM density. However, the solid curves do not have the $a_1 H_1$ Hubble volume rescaling included. The dashed curves, on the other hand, apply the Hubble volume rescaling factor but for a target $\hat \eta_c$ of $\hat \eta_c = 3.6$, which is that corresponding to the black curve. Clearly there are still differences between the rescaled dashed curves and the black curve, which tells us that there are dynamical effects that arise from changing $\hat \eta_c$ that are not captured by the simple rescaling. This should not be too surprising considering that [*e.g.*]{} the mass growth affects the oscillon stability, which determines the high-mass part of the distribution. In our work we rescale the mass function to the target $f_a$ as described above, but it is important to keep in mind that this almost certainly results in a systematic uncertainty from the fact that we do not capture the full oscillon dynamical range in doing so. Also note that all of the mass functions abruptly drop off at low halo masses. This is due to our resolution limit on the finite lattice. We also cannot rule out the possibility that the low-mass tail continues down to much smaller masses.
{width="49.00000%"} {width="49.00000%"}\
{width="49.00000%"} {width="49.00000%"}
\[fig:MDProjections\]
We compare the single-differential mass fractions for different values of $\hat\eta_c$ and $\tilde\lambda$ as a function of $\delta$ and $M$ in Fig. [\[fig:MDProjections\]]{}. Note that here we have applied the rescaling factors for the masses to our true target $f_a$, which is that which gives the correct DM density. First of all we note that there is no dependence on $\tilde\lambda$ visible in our parameter range within than statistical scatter. As for the differential distribution as a function of $\delta$, there is also no clear dependence on $\hat\eta_c$ visible. The only place where a clear dependence on $\hat\eta_c$ is visible is in the mass fraction as a function of $M$. Here, the peak values shift to smaller masses upon increasing $\hat\eta_c$, even after including the rescaling factors.
It is useful to have an approximate analytic formula for the differential mass fraction. We find that the differential mass fraction as function of $\delta$ can be accurately described by a Crystal Ball function based on a generalized Gaussian and a power-law high-end tail together with a suppression factor at high-$\delta$: $$\frac{df}{d\delta} = \frac{A}{1+\left(\frac{\delta}{\delta_F}\right)^S}
\begin{cases}
e^{-\left[\ln{\left(\frac{\delta}{\delta_G}\right)/\sqrt{2}\sigma}\right]^d} & \text{for } \ln{\left(\frac{\delta}{\delta_G}\right)}\leq\sigma\alpha\\
B\left[C+\frac{1}{\sigma}\ln{\left(\frac{\delta}{\delta_G}\right)}\right]^{-n} & \text{for }\ln{\left(\frac{\delta}{\delta_G}\right)}>\sigma\alpha
\end{cases} \,.$$ The parameters $B$ and $C$ are given by $$B=e^{-\left(\frac{|\alpha|}{\sqrt{2}}\right)}\left[\left(\frac{\sqrt{2}}{|\alpha|}\right)^d\frac{|\alpha|n}{d}\right]^n,\qquad
C=|\alpha|\left[\left(\frac{\sqrt{2}}{|\alpha|}\right)^d\frac{n}{d}-1\right] \,,$$ and they are chosen such that $df/d\delta$ and its first derivative are continuous. $A$ is not a free parameter as $\int_{0}^\infty d\delta (df/d\delta) =1$ must hold. The fit parameters from our most realistic simulation with $\hat\eta_c=3.6$ and $\tilde\lambda=5504$ are given by
------------------------------------ ------------------------------------ ------------------------------------ ---------------
$\sigma=0.448\pm0.008\qquad\qquad$ $n =115\pm8$ $\delta_G=1.06\pm0.02\qquad\qquad$ $S=4.7\pm1.6$
$ d=1.93\pm0.02$ $\alpha =-0.21\pm0.07\qquad\qquad$ $\delta_F=3.4\pm1.2$ .
------------------------------------ ------------------------------------ ------------------------------------ ---------------
{width="66.00000%"}
\[fig:Kolb\]
This fit allows us to make a precise comparison with previous work by Kolb and Tkachev [@Kolb:1995bu]. We present in Fig. \[fig:Kolb\] the cumulative mass fraction that is in overdensities larger than $\delta_0$, $$F(\delta>\delta_0) = \int_{\delta_0}^{\infty}\frac{df}{d\delta} d\delta.$$ Unsurprisingly, we find considerably less mass in highly concentrated overdensities relative to [@Kolb:1995bu]. Whereas [@Kolb:1995bu] predicts roughly $10\%$ of the mass is in overdensities with $\delta=10$ or more, we find a similar result only when using the simulation output at $\hat\eta=7$. Once evolved to matter-radiation equality, that percentage falls to $\sim$$0.1$%.
Equations of Motion for Varying Relativistic Degrees of Freedom
===============================================================
In this section we investigate the systematic effect on our results from the assumption of fixed $g_*$. In truth, the value of $g_*$ is not fixed at $g_* \approx 81$ but instead varies rather sharply in the temperature range of interest; it varies from as large as roughly $100$ to as little as roughly $10$ around the time of the QCD phase transition. This does not represent a dire shortcoming of our simulation procedure, however, as varying $g_*$ should only nontrivially affect the dynamics of the axion field during times when axion number density is not a conserved quantity. By $\hat\eta \approx 3$ most of the field has become linear, except for the isolated oscillon configurations, which means that the axion number density is mostly conserved at this time and beyond. The variation in $g_*$ before $\hat \eta \approx 3$ for our target $f_a$ is relatively minor. To quantify this impact, however, we perform 2D simulations accommodating the varying $g_*$.
With a change of variable we may rewrite the axion equations of motion, in the two-field formalism during the QCD epoch, as $$\begin{gathered}
\phi_1'' + \left(\frac{R_1 \ddot R}{\dot R^2} + \frac{3 }{\eta} \right)\phi_1' - \frac{R_1^2 \dot R_1^2}{R^2 \dot R^2}\nabla^2 \phi_1 + \frac{\dot R_1^2}{\dot R^2}\left[ \tilde \lambda \phi _1 \left(\phi _1^2+\phi _2^2 - 1\right) -\frac{m_a(T)^2\phi _2^2 }{ H_1^2 \left(\phi _1^2+\phi _2^2\right){}^{3/2}} \right] = 0 \\
\phi_2'' + \left(\frac{R_1 \ddot R}{\dot R^2} + \frac{3}{\eta} \right)\phi_2' - \frac{R_1^2 \dot R_1^2}{R^2 \dot R^2} \nabla^2 \phi_2 +\frac{\dot R_1^2}{\dot R^2}\left[\tilde \lambda \phi _2 \left(\phi _1^2+\phi _2^2 - 1\right) + \frac{m_a(T)^2 \phi _1 \phi _2}{H_1^2\left(\phi _1^2+\phi _2^2\right)^{3/2}}\right]= 0 \,,\end{gathered}$$ where we define $\tilde \lambda = \lambda f_a^2 /H_1^2$ as before. Citing standard references [@Kolb:1990vq], we have $$\begin{gathered}
H \approx 1.660 g_*(T)^{1/2} \frac{T^2}{m_{Pl}} \\
t \approx 0.3012 g_*^{-1/2} \frac{m_{Pl}}{T^2} \\
R \approx 3.699 \times 10^{-10} g_{*}(T)^{-1/3}\frac{\mathrm{MeV}}{T} \,.\end{gathered}$$ Using these relations, we may compute $$\begin{aligned}
\frac{R_1 \ddot R}{\dot R^2} &= \frac{\left(\frac{t_1}{t}\right)^{1/2} \left(\frac{g(t)}{g\left(t_1\right)}\right)^{1/12} \left(13
t^2 \dot g(t)^2-12 t g(t) \left(t \ddot g(t)+\dot g(t)\right)-36 g(t)^2\right)}{\left(t
\dot g(t)-6 g(t)\right)^2} \\
&= -\frac{1}{\hat \eta} \left[ \frac{-13 t^2 \dot g(t)^2+12 t g(t) \left(t \ddot g(t)+\dot g(t)\right)+36 g(t)^2}{\left(t
\dot g(t)-6 g(t)\right)^2}\right] \\
&= - {f_1(\hat \eta) \over \hat \eta} \,.\end{aligned}$$ Above, we have defined $$f_1(\hat \eta) = \frac{-13 t^2 \dot g(t)^2+12 t g(t) \left(t \ddot g(t)+\dot g(t)\right)+36 g(t)^2}{\left(t\dot g(t)-6 g(t)\right)^2} \,,$$ where the right hand side is evaluated at the time $t$ corresponding to the conformal time $\hat \eta$. Similarly, we evaluate $$\frac{R_1^2 \dot R_1^2}{R^2 \dot R^2} = f_2(\hat \eta), \qquad \frac{\dot R_1^2}{\dot R^2} = \hat \eta^2 f_2(\hat \eta) \,,$$ for $$f_2(\hat \eta) = \frac{\left(\frac{g(t)}{g\left(t_1\right)}\right){}^{7/3} \left(t_1
\dot g\left(t_1\right)-6 g\left(t_1\right)\right){}^2}{\left(t \dot g(t)-6
g(t)\right)^2} \,.$$ Finally, we define $f_3(\hat \eta) = m_a(\hat \eta(T))^2 / H_1^2 $. Combining these results, the equations of motion take the form $$\begin{gathered}
\phi_1'' + \left(\frac{3}{\hat \eta} - \frac{f_1(\hat \eta)}{\hat \eta}\right)\phi_1' - f_2(\hat \eta) \nabla^2 \phi_1 +\eta^2 f_2(\hat \eta) \left[ \tilde \lambda \phi _1 \left(\phi _1^2+\phi _2^2 - 1\right) -\frac{f_3(\hat \eta) \phi _2^2 }{\left(\phi _1^2+\phi _2^2\right){}^{3/2}} \right] = 0 \\
\phi_2'' + \left(\frac{3}{\hat \eta} - \frac{f_1(\hat \eta)}{\hat \eta}\right)\phi_2' - f_2(\hat \eta) \nabla^2 \phi_2 +\eta^2 f_2(\hat \eta) \left[\tilde \lambda \phi _2 \left(\phi _1^2+\phi _2^2 - 1\right) + \frac{f_3(\hat \eta) \phi _1 \phi _2}{\left(\phi _1^2+\phi _2^2\right)^{3/2}}\right]= 0.\end{gathered}$$ In the single-field formalism, these results are analogously applied to obtain $$\theta'' + \left(\frac{3}{\hat \eta} - \frac{f_1(\hat \eta)}{\hat \eta}\right)\theta' - f_2(\hat \eta) \bar \nabla^2 \theta +\eta^2 f_2(\hat \eta)f_3(\hat \eta) \sin \theta= 0 \,.$$
In Fig. \[fig:geff\] we show the functions $f_1$, $f_2$, and $f_3$ entering into the equations of motion as functions of $\hat \eta$. Note that for $f_3$, we normalize against $\tilde f_3(\hat \eta)$, which we define to be $f_3$ but with a fixed $g_*$. In the absence of a varying $g_*$, all of the curves appearing in Fig. \[fig:geff\] would be identically one.
{width=".66\textwidth"}
\[fig:geff\]
To test the impact of an evolving $g_*$, we adopt the parametrization of $g_*$ from [@Wantz:2009it] and simulate, in two spatial dimensions, for an axion with $f_a = 4.83 \times 10^{15} \, \mathrm{GeV}$. When we assumed $g_*$ was constant, this axion reached its zero-temperature mass at $\hat \eta = 3.6$, but accounting for the changing value of $g_*$, the axion now reaches its zero-temperature mass at $\hat \eta \approx 5.5$. As before, we conclude our simulation at $\hat \eta = 7$, and we calculate a relic abundance that is $7.7\%$ smaller than it is in the fixed $g_*$ case. We note that this scenario represents something of a worst-case scenario for the impact of $g_*$ on the dynamics because $g_*$ varies significantly during the epoch where axion number density is not conserved for this choice of $f_a$, and so we adopt this as a quantification of the systematic error associated with adopting a fixed $g_*$. For our target $f_a$, $g_*$ varies less, relative to the example illustrated, when the axion is in the non-linear regime and so we expect the effect of varying $g_*$ to be less important in this case.
Testing the Impact of the Mass Parametrization
==============================================
Precise details regarding the temperature dependence of the axion mass remain uncertain. While we have chosen to use the parametrization of [@Wantz:2009it] with index $n = 6.68$ as done in [@Hiramatsu:2010yu; @Kawasaki:2014sqa], an alternative result is provided in [@Borsanyi:2016ksw]. In that work, an index of $n \approx 8.2$ is found at high temperatures, though we do note that an increasingly shallow dependence on $T$ is realized at lower temperatures. Motivated by power-law fits to this numerical result and informed by considerations of the changing number of degrees of freedom, recent works have taken an index of $n = 7.6$ in [@Klaer:2017ond] and $n = 7.3$ in [@Vaquero:2018tib] to study the axion field. In this section, we use the extreme value $n = 8.2$ to estimate the maximal effect that uncertainties in the mass growth may have on the determination of the DM density.
We perform simulations for different $n$ in two spatial dimensions. This is done for computational efficiency, and while we do not expect such a simplification to significantly affect the main conclusions we caution some care should be taken when interpreting these results for this reason. We fix $\tilde \lambda = 5504$ and $f_a \approx 4.8 \times 10^{14} \, \mathrm{GeV}$. This choice of $f_a$ corresponds to $\hat \eta_c = 3.6$ in the $n=6.68$ parametrization. However, the value of $\hat \eta_c$ depends on our choice of $n$ and is $\hat \eta_c \approx 3.1$ for the choice of $n = 8.2$, since the mass grows faster in that case. We re-simulate with this alternative choice of index until $\hat \eta = 7$ and then recompute the present-day axion abundance by analytically transferring the simulation fields to the same late time physical temperature. We find that there is a $\sim$$10\%$ enhancement in the expected relic abundance with $n = 8.2$ versus $n = 6.68$. This is somewhat surprising, considering that the analytic estimate predicts that higher $n$ should result in a lower DM abundance at fixed $f_a$. To understand how this result affects the final determination of the axion mass, we fit the predicted scaling $\Omega_a \sim f_a^{(n+6) / (n+4)}$ for the DM abundance using $n = 8.2$ and find the $m_a$ that gives the correct DM abundance. The result is that with $n = 8.2$ we find that the $m_a$ that gives the correct DM abundance is enhanced by $\sim$27% compared to the $n = 6.68$ case. We account for this 27% uncertainty as an additional systematic uncertainty in our final determination of the axion mass.
Testing Deviations from the String Scaling Regime
=================================================
While our simulation was performed in two stages, it can be understood as a single simulation in which the PQ phase transition and the beginning of the QCD phase transition are separated by approximately an order of magnitude in temperature. By comparison, for a physically motivated hierarchy, we would expect these two epochs in our simulation to be separated by at least 11 orders of magnitude in temperature. As a result, our simulations might be expected to be highly unphysical. However, it has been conjectured that the axion field and associated defect network enters a scaling regime some time after the PQ phase transition (see, [*e.g.*]{}, [@Hiramatsu:2010yu]). If this conjecture is true and our field configuration has entered the scaling regime before the axion begins to oscillate, our simulation should be expected to give a good description of the physics of interest despite the abbreviated hierarchy.
Recent work has found evidence for logarithmic deviations to the number of strings per Hubble patch in the scaling regime [@Gorghetto:2018myk]. In this section we confirm that we also observe such deviations. This implies that we are not fully justified in taking the final state of our PQ-epoch simulation, fast-forwarding through the rest of the radiation dominated epoch to the QCD phase transition, and then restarting our simulation directly before the QCD phase transition. This is because the axion-string network should change logarithmically during the evolution between the phase transitions. Below, we provide evidence for the logarithmic deviation to scaling and then perform simulations to address the impact of this deviation on our determination of the axion mass $m_a$ and the spectrum of DM minihalos.
The average number of strings per Hubble patch is commonly defined by [@Hiramatsu:2010yu; @Fleury:2015aca; @Gorghetto:2018myk] $$\label{eq:LogForm}
\xi(\tilde \eta) = \frac{l(\tilde \eta) t(\tilde \eta)^2}{L(\tilde \eta)^3} \,,$$ where $l(\tilde \eta)$, $L(\tilde \eta)$, and $t(\tilde \eta)$ are the physical total string length in the box, the physical box length, and the physical time, each a function of $\tilde \eta$, respectively. We measure the string length by first identifying grid sites that are next to a string. This is achieved by forming a loop in each of the three dimensions around a test grid site. The grid site is flagged once at least one change larger than $\pi$ in the axion field between consecutive grid sites is found. In a 2D slice this implies the 4 closest grid sites that surround the string core are tagged, such that we use the number of tagged grid sites divided by 4 as a measure for the string length. Note that this is a rough estimate for the string length and more sophisticated methods exist [@Hiramatsu:2010yu].
We compute $\xi(\tilde \eta)$ at 13 points in $\tilde \eta$ in our PQ simulation, with results illustrated in Fig. \[fig:TotalStringLength\]. As in [@Gorghetto:2018myk], we find that $\xi$ depends logarithmically on $\tilde \eta$ after the PQ phase transition. Note that the shaded region denotes $\tilde \eta$ before the PQ phase transition, where it does not make sense to talk about axion strings. We fit the model $$\xi = \alpha \log\left(\frac{T}{T_{PQ}}\right) + \beta$$ to the $\{ \tilde \eta, \xi \}$ data, where $T_{PQ}$ denotes the temperature of the PQ phase transition, and we find $\alpha \approx -2.60$ and $\beta \approx 1.27$. Note that our values for $\xi$ at comparable $\tilde \eta$ are significantly larger than those found in [@Gorghetto:2018myk]. Part of this discrepancy could be due to the way in which we measure string length versus in that work, which may introduce an overall rescaling between our two results. We are in good agreement, however, with [@Gorghetto:2018myk] regarding the distribution of string length in long and short strings. As in that work, we find that approximately $80\%$ of the string length resides in long strings, much larger than a Hubble length, at all times in our simulation as seen in Fig. \[fig:LongStringLength\].
{width=".56\textwidth"}
\[fig:TotalStringLength\]
{width=".56\textwidth"}
\[fig:LongStringLength\]
Since we do observe logarithmic scaling violations, it is important to determine the impact of these corrections on the minihalo mass spectrum and the DM relic abundance. In particular, we find that $\xi$ should be around a factor of 15 higher at the QCD phase transition than it is for the final state of our most realistic PQ-epoch simulation. The string density $\xi$ at the beginning of the QCD simulation depends on the simulation box size as $\xi_{\rm QCD} \propto L_{\rm QCD}^{-2}$, where $L_{\rm QCD}$ is the box size in units of $1/(a_1 H_1)$ when $H = m_a$, for a fixed initial state. Thus by performing new simulations with the same initial conditions and run parameters as our fiducial analysis (namely $\hat \eta_c = 3.6, \tilde \lambda = 5504$, and starting at $\hat \eta_i = 0.4$) but changing $L_{\rm QCD}$ from 4 to $L_{\rm QCD} = 3$ and $L_{\rm QCD}=2$, we may enhance $\xi_{\rm QCD}$ by a factor of $16/9$ and $4$, respectively, compared to our fiducial simulation. While these value still fall short of the physically motivated enhancement $\sim$$15$, such simulations still allow us to see if there is a trend in how $\xi$ affects observables such as the DM density. We do caution that modifying $L_{\rm QCD}$ in this way is somewhat unphysical as it changes the horizon entry status of modes in the simulation box from the end of the PQ simulation as compared to the beginning of the QCD simulation.
{width=".66\textwidth"}
\[fig:StringDensityRelicAbundance\]
{width=".56\textwidth"}
\[fig:ReinterpSims\]
The results of varying $L_{\rm QCD}$ in order to modify $\xi$ are shown in Fig. \[fig:StringDensityRelicAbundance\], where we see no discernible trends in the dependence of the relic abundance on $\xi$. Note that the uncertainties in Fig. \[fig:StringDensityRelicAbundance\] are estimates of the statistical uncertainty. As the box size gets smaller the statistical uncertainty increases. However, we caution that these are estimates only, as we have not run multiple independent simulations for each box size due to computational limitations. It is possible that the true uncertainties at small box sizes are larger than indicated due to the fact that there are a small number of [*e.g.*]{} domain walls that form these cases. Still, to be maximally conservative given the available datasets, we estimate the difference between the $L_{\rm QCD} = 2$ and $L_{\rm QCD} = 3$ values for $\Omega_a$ as a systematic uncertainty induced from the deviation to scaling. However, we cannot be sure that this difference is not a result of statistics or from the way in which we artificially mock up initial conditions with higher $\xi$ values. The systematic uncertainty we assign from these tests is $15\%$ correlated between different $f_a$ points.
We show in Fig. \[fig:ReinterpSims\] the impact on the minihalo mass spectrum. Again, all simulations largely agree within their error bands (estimated from statistical uncertainties), indicating that an increase in $\xi$ has only a marginal effect on the late-time axion field. Note that computational resources limit us to just these three additional simulations, and we leave a more detailed investigation of the dependence of $\Omega_a$ on $\xi$ to future work.
|
---
abstract: 'Deuterium plays a crucial role in cosmology because the primordial D/H abundance, in the context of big bang nucleosynthesis (BBN) theory, yields a precise measure of the cosmic baryon content. Observations of D/H can limit or measure the true primordial abundance because D is thought to be destroyed by stars and thus D/H monotonically decreases after BBN. Recently, however, Mullan & Linsky have pointed out that D arises as a secondary product of neutrons in stellar flares which then capture on protons via $n+p \rightarrow d + \gamma$, and that this could dominate over direct D production in flares. Mullan & Linsky note that if this process is sufficiently vigorous in flaring dwarf stars, it could lead to significant non-BBN D production. We have considered the production of D in stellar flares, both directly and by $n$ capture. We find that for reasonable flare spectra, $n/d \la 10 $ and $(n+d)/\li6 \la 400$, both of which indicate that the $n$-capture channel does not allow for Galactic D production at a level which will reverse the monotonic decline of D. We also calculate the 2.22 MeV $\gamma$-ray line production associated with $n$ capture, and find that existing COMPTEL limits also rule out significant D production in the Galaxy today. Thus, we find flares in particular, and neutron captures in general, are not an important Galactic source of D. On the other hand, we cannot exclude that flare production might contribute to recent FUSE observations of large variations in the local interstellar D/H abundance; we do, however, offer important constraints on this possibility. Finally, since flare stars should inevitably produce [*some*]{} $n$-capture events, a search for diffuse 2.22 MeV $\gamma$-rays by INTEGRAL can further constrain (or measure!) Galactic deuterium production via $n$-capture.'
author:
- 'Tijana Prodanović and Brian D. Fields'
title: 'On Non-Primordial Deuterium Production by Accelerated Particles'
---
Introduction
============
In the past two decades big-bang nucleosynthesis (BBN) has become one of the most important cosmological probes [@osw and refs. therein]. Its success lies in the good agreement of predictions for the abundances of four light elements D,$^3$He,$^4$He and $^7$Li with their observations. A key element is deuterium, which is considered to be the best cosmic “baryometer” among the light elements [@st] because of its strong dependence on the baryon density, or equivalently the baryon-to-photon ratio $\eta$. Furthermore, now that the cosmic microwave background anisotropy measurements of WMAP have independently measured $\eta$ to high precision [@wmap], D takes a new role as a probe of both early universe physics and of astrophysics [@cfo2; @cfo3]. Therefore, it is crucial that we fully understand the evolution of deuterium after BBN, in order to correctly infer the primordial abundance from observations in the $z \ll 10^{10}$ universe.
Key to D is that there is no significant astrophysical production site except for the big bang, and that stars destroy D in their fully convective, pre-main sequence phase [@bodenheimer]. Together, these guarantee that any measurement of D is a solid lower bound on the primordial abundance, and that in sufficiently primitive environments D should be essentially primordial. The lack of astrophysical D production was established in the classic paper by @els [hereafter ELS], who considered all known sites in which [*any*]{} nucleosynthesis occurs, and demonstrated that none of these produce D in significant quantities.
Deuterium is observed in diverse astrophysical settings. In high-redshift ($z \sim 3$) QSO absorption systems, neutral D is observed. The 5 best systems [@bt98a; @bt98b; @omeara; @kirkman; @pb] give $({\rm D/H})_{\rm QSOALS} = (2.78 \pm 0.29) \times 10^{-5}$. The D abundance in the solar nebula is inferred from solar wind observations of 3 (which measure pre-solar D+3) minus meteoritic determinations of 3 alone. These give [@gg] a value $({\rm D/H})_{\rm pre-\odot} = (2.1 \pm 0.5) \times 10^{-5}$, which probes proto-solar material 4.6 Gyr ago, or at $z \sim 0.4$. D/H is also observed [@ry; @linsky] in the local interstellar medium (ISM); recent FUSE observations [@moos] give $({\rm D/H})_{\rm ISM} = (1.52 \pm 0.08) \times 10^{-5}$ in the Local Bubble today, at $z=0$.
The central ELS argument–that the big bang is the sole source of D–has stood the test of time, borne out by the drop in D/H from its high-redshift to pre-solar to local ISM abundance. But given the crucial role of D in cosmology, it is important to carefully examine the assumptions made, and to identify any possible loopholes. Moreover, such an effort becomes crucial in light of growing evidence that D/H has large variations over short distances in the local ISM. Recent FUSE observations [@hoopes] add weight to earlier suggestions [@alfred and refs. therein] that D/H can vary by as much as a factor of 2 over different lines of sight, pointing to inhomogeneities on scales $\ga 100 \ \rm pc$.[^1]
Such a re-analysis of ELS was recently carried out by @ml [hereafter ML]. Specifically, they discussed D production by suprathermal energetic particles in flare stars. Although ELS had considered flare production, ML note that ELS had neglected flare production of neutrons, which can then go on to produce D via radiative capture onto a proton n + p d + with the emission of a photon with energy $E_\gamma = 2.223$ MeV. ML do not make a detailed calculation of $n$ yields from spallation reactions, but note that $n$ production can be large compared to direct $d$ production, depending on the energetic particle spectra. ML then suggest that D created in such a way can escape from the flare site into the stellar wind. Also, because of its low mass D might be preferentially ejected relative to the heavier Li, Be, an B which would otherwise contaminate the ISM. ML thus raise the possibility that flares could be a significant source of D in the Galaxy. From the point of view of Galactic chemical evolution, this mechanism might then explain some or all of the ISM variation in D. Moreover, from the point of view of cosmology, the chance of significant D production would call into question the ELS argument which underlies the recent spectacular agreement between high-redshift D/H measurements and the predictions of BBN with the WMAP baryon density.
Thus, the ML scenario has important implications and deserves careful further examination. In this paper, we expand the analysis of ELS to include the $n$ channel for D production suggested by ML. Our approach is to identify general nuclear physics constraints that follow from detailed calculations of $n$, $d$, and $\gamma$-ray production by energetic particles. The result is to close this loophole–i.e., we show that flare D production is not sufficient to offset the monotonic decline of D with time. However, flare D production should occur at [*some*]{} level, and we find that $\gamma$-ray observations by the recently launched INTEGRAL mission may be able to probe this process.
Light Element Production by Flares
==================================
Flares are violent releases of magnetic energy during which matter reaches temperatures of tens or hundreds of million Kelvin, and particles get accelerated to very high energies. They are associated with active stellar regions [@flares]. Nuclear reactions then occur between flare-accelerated particles and the ambient stellar atmosphere. In this section we will discuss spallation reactions that result in production of neutrons, deuterium and lithium. We will show that, as a result of spallation, $1 \la n/d \la 10$ for most spectral indices, which means that ML channel is indeed [*more*]{} important than channels considered by ELS, confirming and quantifying the suggestion of ML. However, we will see that even including this channel for $d$ production, the concomitant Li production remains large and thus severely constrains the possibility that flares are an important source of D.
Formalism
---------
Nuclide production in flares depends on the initial spectrum of the flare, its modification as it interacts with the stellar atmosphere, and the compositions of the flare and the atmosphere. In general, the nucleosynthesis yields in flares have a complicated time dependence. Fortunately, there are two limiting approximations for energetic particles in flares: the thin-target and thick-target models [@Ramaty75]. In the thin-target model, particle production is assumed to occur before the spectrum of accelerated particles is modified by ionization energy losses. On the other hand, in the thick-target model, energy losses are taken to be large, and particles slow due to ionization losses as they move downward form the flare region, prior to nuclear interactions.
But are both of these scenarios equally important for flare processes? A simple analysis of the mean free path of projectile particles sheds light on this question. The mean free path of projectiles against nuclear interactions is $$\begin{aligned}
\lambda = \frac{1}{n \sigma}\end{aligned}$$ where $n$ is the target number density and $\sigma$ is the reaction cross section, while mean free path of a flare particle (charge $Z$, mass number $A$) that is losing energy due to its interaction with the surrounding medium is given by $$\begin{aligned}
\lambda_{\epsilon} = \frac{A}{Z^2} \frac{R_p(\epsilon)}{n m_p}\end{aligned}$$ Here, $\epsilon$ is the projectile energy per nucleon, $n$ stands for the number density of the medium, $\sigma $ is the cross-section for a reaction of interest, $m_p$ is the proton mass, and $R_p$ is the ionization energy loss range of protons in units of $\rm g/cm^2$. In the thin-target limit $\lambda \ll
\lambda_{\epsilon} $ which puts a lower limit on the projectile range $$\begin{aligned}
R_p (\epsilon) \gg 167 \ \frac{Z^2}{A} \
{\left( \frac{\sigma}{10 \rm mb} \right) }^{-1} \,\,\,\,\,\,
\frac{ \rm g}{\rm cm^2}
\label{eq:R-lim}\end{aligned}$$ For low energies, $R_p(\epsilon) \simeq 4 \times 10^{-4} \ {\rm g/cm^2} \
(\epsilon/{\rm MeV})^2$, so that eq. (\[eq:R-lim\]) implies that the thin-target approximation holds for energies 600 MeV/nucleon This energy is much larger than those of typical flare particles. Thus, we conclude that the thick-target approximation is well-satisfied and is the appropriate one for flare processes. This result, based on the physics of particle propagation, is in agreement with the empirical result of @thick who found that flare data are best described by thick-target rather than thin-target models.
In the thick target model, the production of secondary particles of species $l$ from reaction $i+j \rightarrow l+ \cdots$ is given by [@Ramaty75]: Q\_l = y\_i y\_j \_[\_[th]{}]{}\^ d \_\^ N\_p (’) d ’ where $Q_l$ is the total number of secondary particles produced per incident flare particle, and $\epsilon_{\rm th}$ is the threshold energy for a particular reaction. The ambient number density of target-species $j$ is $n_j$ and $\sigma_{ij}^l$ is the cross section for the reaction of interest. Projectile and target abundances are given as numbers relative to hydrogen: $y_i = n_i/n_p$; these are used to compute the mean target mass $\langle m \rangle \simeq m_p \sum_i A_i y_i/\sum_j y_j = 1.4 m_p$.
The projectile and target compositions are poorly known for flare stars, but one can imagine several possibilities. Since solar flares show enhanced metals and helium relative to hydrogen [@mrk], one might expect similar enrichment for flare stars. On the other hand, one expects M dwarfs in general to have metallicities similar to those of G or K dwarfs [@wg; @kfcm], and thus on average to be subsolar in metallicity. Thus we will examine several possible compositions and find the impact of composition on spallation production. For a fiducial case, we will assume solar abundances [@abundances] for both projectiles an targets. We then will examine the effect of more extreme variations: a five-fold increase in metals and helium as in solar flares, and a metal-free primordial case.
The spectrum $N_i(\epsilon) \ d\epsilon$ measures the total number of projectile particles with energy in $(\epsilon,\epsilon+d\epsilon)$. Following @Ramaty75, we adopt the power-law form $$\begin{aligned}
\label{eq:spec}
N_p(\epsilon ) &=& k_p \epsilon^{-s} \\
N_i(\epsilon ) &=& y_i N_p(\epsilon )\end{aligned}$$ where $k_p$ is the constant determined by normalizing $N_p$ to 1 proton of energy greater than $\epsilon_0=30$ MeV and $s$ is the spectral index which is assumed to be the same for all accelerated particles. The choice of this particular spectrum normalization is purely conventional (see e.g. @Ramaty75), but in any case it will not matter in our analysis since we are only interested in ratios of productions of different particle species, and the normalization will drop out.
In addition to the particle compositions, the spectral index $s$ is the other key parameter for the results we present. We will present results for a range of $s$, but some observations exist which constrain its value. For solar flares, @rmk find that $\gamma$-ray line ratios in an ensemble of flares point to a range of spectral indices of about $s = 4 \pm 1$. For flare stars themselves, information on the spectra of the ions is not available, but radio data from electron synchrotron emission points to [*electron*]{} spectra with $s_e = 2$ to 3. Whether the ions should have the same indices is, however, unclear. In the case of the Sun, @lmh find that electron spectral indices (for $E_e < 100$ keV) are correlated with proton indices, but with $s_e = 2$ corresponding to $s \simeq 4$; they find no significant correlation for $E_e > 200$ keV. These results hint that the ion spectral indices in flare stars may also prefer $s \simeq 4$, but this parameter remains uncertain.
The particle ionization energy loss is measured by $$\frac{d \epsilon}{dR} = \frac{|d\epsilon/dt|}{\rho v}$$ with units of \[MeV/(${\rm g \ cm^{-2}}$)\] where $v$ is the velocity of incident particle. We adopt the usual Bethe-Bloch energy formula [@bethe] $$\begin{aligned}
\frac{{d \epsilon}}{dR} = \frac{4 \pi z Z^2 e^4}{A \langle m \rangle m_e v^2}
\left[ \ln \left( \frac{2 \gamma ^2 m_e v^2}{I} \right) - \frac{v^2}{c^2} \right]\end{aligned}$$ In this equation $Z$ is the charge of projectile, $z$ is number of electrons per atom which is approximately equal to 1 in the solar atmosphere, $A$ is the number of nucleons of projectile, $m_e$ is electron mass, and the mean excitation potential is the value for hydrogen, $I=13.6$ eV.
Table 1 lists the reactions we have included, their lab-frame threshold energies, and references for the cross-sections. We note here that cross sections for deuterium production via $\alpha + \alpha
\rightarrow D + \cdots$ are unavailable. To make an estimate of deuterium production via $ \alpha \alpha$ reactions we use exponential fit of @Mercer to the cross sections for $\alpha + \alpha \rightarrow d + \li6$, the channel with the lowest threshold, and ignore other possible final states ($4d$, $dd\he4$). Also, the only existing cross sections for $\alpha +$C and $\alpha +$O production of deuterium are given for the single energy $E_{\alpha} =58 {\rm MeV}$, or $\epsilon_{\alpha} =14.5$ MeV/nucleon [@bpk]. In this case we approximated the cross sections with: $$\sigma(\epsilon) =
\left\{
\begin{array}{cc}
\sigma_0 \ e^{-2 \pi \eta} & \epsilon > \epsilon_{\rm th} \\
0 & \epsilon \le \epsilon_{\rm th}
\end{array}
\right.$$ where $\sigma_0$ is normalized to experimental values [@bpk], and the Gamow factor $\eta = {z_i z_j e^2}/{\hbar c \beta}$ accounts for the Coulomb barrier.
[ccc]{} Reaction & $\epsilon_{\rm th}$ & Reference\
& \[MeV/nucleon\] &\
p+p $\rightarrow$ n+x & 410.20 & @Ramaty75\
p+ $\alpha \rightarrow$ n+x & 22.54 & @Ramaty75\
p+CNO $\rightarrow$ n+x & 2.43 & @Ramaty75\
$\alpha + \alpha \rightarrow $ n+x & 9.44 & @Ramaty75\
$\alpha $ +CNO $\rightarrow $ n+x & 0.49 & @Ramaty75\
&&\
p+ $\alpha \rightarrow $ d+x & 23.07 & @meyer\
p+CNO $\rightarrow $ d+x & 17.50 & @meyer\
$\alpha + \alpha \rightarrow $ d+$^6$Li & 15.38 & @Mercer\
$\alpha $ + C $\rightarrow $ d+x & 4.53 & @bpk\
$\alpha $ + O $\rightarrow $ d+x & 5.10 & @bpk\
&&\
p+C $\rightarrow ^6$Li+x & 24.45 & @rv\
p+N $\rightarrow ^6$Li+x & 17.52 & @rv\
p+O $\rightarrow ^6$Li+x & 23.57 & @rv\
$\alpha + \alpha \rightarrow ^6$Li +x & 11.19 & @Mercer\
$\alpha $+C $\rightarrow ^6$Li +x & 7.91 & @rv\
$\alpha $+N $\rightarrow ^6$Li +x & 2.83 & @rv\
$\alpha $+O $\rightarrow ^6$Li +x & 6.02 & @rv\
Results
-------
Before we present our final results let us first make analytic estimates for couple of reactions for this model. In order to perform analytic estimates some approximations will be needed. To analytically solve thick-target case we approximate energy losses via: $$\frac{dR}{d \epsilon}
= 0.14
\frac{A}{2 \pi Z^2 z}
\frac{m_e \langle m \rangle}{m_p e^4} \ \epsilon$$ Let us also assume that a cross-section for a particular reaction is flat and that we can neglect relativistic corrections which is a valid assumption for low projectile energies. Therefore, equation (6) now has the following form: Q\_l = 5.7 10\^[-4]{} ( ) [ ( ) ]{}\^2 [ ( ) ]{}\^[1-s]{}
In Figure \[fig:check\] we present our analytic estimates along with numerical results for deuterium production by $p \alpha$ ,6 production by $\alpha \alpha $ and neutron production by $p \alpha$ reactions. The analytic result is calculated for the case of $s=4$ which is the best choice given the assumptions that were made, while numerical results are functions of spectral index $s$. Note that our results also include reactions where target and projectile species are interchanged (initial-state particle species can also serve as targets, since we assume that flares have the same composition as the ambient medium). Although we made some rough estimates we see that they are in good agreement with our numerical predictions, which gives us confidence in our code.
Our numerical results for production of neutrons, deuterium and 6 are shown in Figure \[fig:results\] . Results are represented in the form of ratios of total numbers of particles produced in spallation as a function of the spectral index $s$. The purpose of this was to see how neutron production compares to production of deuterium, since the key idea of ML paper was that a significant amount of neutrons can be produced via spallation which can then undergo radiative capture to make a non-negligible amount of deuterium. We see that indeed, $n/d \ga 1$, so that neutron production does in fact dominate direct $d$ production for all spectral indices $s$. More specifically, $n/d \sim 1$ around $s = 4$, the best-fit value for solar flares, and can be as high as $n/p \sim 200$ for an extreme value of $s = 7$. These calculations thus confirm the suggestion of ML that in fact $n$ production is significant compared to deuteron production, and thus radiative capture of the neutrons offers an important channel for deuterium synthesis which had been neglected by ELS.
With the neutron production in hand, we can now update the ELS argument to address the ML loophole. The key point here is that spallation production of $d$ and $n$ is also inevitably accompanied by production of other light elements, notably lithium. In particular, since 6 is uniquely produced by spallative processes [@fo; @elisa], it offers the strongest constraint, as follows. Assume that all neutrons made in flares undergo radiative capture onto protons, rather than suffering decay or non-radiative capture on 3. We also assume that all 6 in the ISM comes from flares, which maximizes the flare contribution. Then by using our $(n+d)/\li6$ ratio for the entire range of spectral indices, $20 \la (n+d)/\li6 \la 1000$, combined with the solar $\li6/{\rm H} = 1.5 \times 10^{-10}$, we infer a flare-produced deuterium abundance in the range $3 \times 10^{-9} \la {\rm D/H} \la 1.5 \times 10^{-7}$, much smaller than observed deuterium abundance. This limit would be further strengthened if we also note that a large fraction ($\ga 50\%$) of neutrons escape the Sun, or are captured onto 3 rather than on protons [@Reuven]. If $f_{np} \la 0.5$ is the fraction of $n$ which do capture on protons, thus the correct ratio for D/6 is $(f_{np} n+d)/\li6 = (f_{np}n/d+1)d/\li6$, which lowers the total deuterium production by an additional $\sim 25\%$ for $n/d \sim 1$ and $f_{np} = 1/2$.
We can also go the other way. Assume that just 10% of observed ISM deuterium abundance comes from flares. Then from our $(n+d)/\li6$ ratio for all spectral indices we get that 6/H abundance is in the range $1.5 \times 10^{-9} \la \li6/{\rm H} \la 7.5 \times 10^{-8}$, which is between 1 and 2.5 orders of magnitude larger than the observed solar abundance of 6 (where overproduction of about $300$ corresponds to the spectral index most favored for solar flares). Thus we conclude that then we can rule out the ML loophole, if flare spallation products escape with equal probability. This result updates and reaffirms the same conclusion by ELS.
However, many flare stars will have fewer metals, and thus a different composition. For example, disk G and K dwarfs are known [@wg; @kfcm] to have a mean metallicity that is subsolar by about a factor of 2. What effect will this have on the production ratios and ultimately the revised ELS constraints? To get a sense for this, we follow ELS and repeat our calculation for the extreme limit of a [*primordial*]{} composition–i.e., where the flare and the ambient medium contain only 4 (with mass fraction 24.8%) and the balance is H. The difference here is that we exclude reactions involving CNO, but since Li can be made by $\alpha \alpha$ fusion, the 6 constraint remains. In this case the $(n+d)/\li6$ ratio is in the range of $10 \la (n+d)/\li6 \la 30000$, as shown in Figure \[fig:altabs\]a, thus placing the upper limit (for very low spectral indices) on the expected deuterium abundance due to flares about an order of magnitude higher. Therefore we see that even in primordial environment the ELS constraint holds, especially since thick-target production ratios that we are discussing change only slightly around most favorable spectral index of $s \approx 4$.
Now, although our fiducial calculation is based on the solar abundances, the Sun itself shows a different and more enriched composition in flare projectiles and possibly targets. In solar flares, 4/H can be as much as 5 times higher than in mean solar matter–in which case protons and $\alpha$ particles can be within a factor of two of each other by number; abundances of other heavy elements can also be similarly enhanced in flares [@mrk]. Therefore, one might wonder how will our results change in that case? Figure \[fig:altabs\]b plots our results obtained as before, but this time for abundances that are 5 times greater than solar. However, even in this case our main conclusions stay the same: ([*i*]{}) the neutron channel for D production can dominate over spallation production of D, but not enough to be a significant source of non-primordial D, and ([*ii*]{}) the ELS constraint on 6/D holds, and indeed is the strongest in this case, since overproduction of 6 goes between 1 and 3 orders of magnitude. Therefore we see that by using a solar composition to describe flare-processes, our constraints were in fact generous compared to this case.
Of course, ML note that it is possible that D escapes more readily than does Li, due to its small mass. Clearly, the question of transport is complex, involving competing effects such as convection and mass loss. These difficult issues in magnetohydrodynamics are beyond the scope of this paper, but we can at least quantify the bias needed to avoid our updated ELS constraint. We saw that if interstellar D is due to flares, then 6 is overproduced by a factor that is about between 10 and 250. Thus, 6 mass loss must be suppressed relative to D by at least this factor in order for the ML loophole to remain open. While it is difficult to rule this possibility out (or in!), clearly this offers a strong quantitative constraint on the particle loss mechanism.
Thus, we strongly constrain the ML loophole, which now requires a strong bias against Li escape. But if such a bias could be created, is the ML loophole viable? To further explore that scenario, we turn to the 2.22 MeV $\gamma$-ray line as an additional constraint on neutron captures in the Galaxy.
Gamma-Ray Line Constraints on Flare Production of Deuterium
===========================================================
In the previous section we have seen that analysis of particle production through spallation reactions does not completely answer the question of the amount of deuterium produced in stellar flares. For that reason we will now approach this problem from a different angle. The reaction of radiative capture $ n + p \rightarrow d + \gamma $ which was proposed to be potentially significant source of deuterium in ML paper, has as a result, besides deuterium, a $2.22$ MeV $\gamma$-ray line. Our idea is to predict the Galactic $\gamma$-ray intensity of that line under the assumption that flares produce significant amounts of deuterium. That way, without going into details about mechanisms that can transport produced deuterium into the ISM, we can place an upper limit on non-primordial deuterium production via radiative capture.
If we denote number of 2.22 MeV $\gamma$-ray lines produced by a single flare with $N_{\gamma}$, and use $N_d^{\rm tot}$ to denote number of deuterium produced by the same flare in both radiative capture and spallation processes then we can write $$\begin{aligned}
\frac{N_d^{\rm tot}}{N_{\gamma}} = \frac{1}{f_{2.2}} \frac{N_d^{\rm tot}}{N_n} \end{aligned}$$ where $N_n$ is the number of neutrons produced by the same star via spallation. The factor $f_{2.2}$ is the efficiency for neutron-to-2.22 MeV photon conversion, averaged over neutron energy spectrum [@Reuven]; this includes stopping of the $\gamma$ rays, as well as neutron escape and neutron “poisoning” by capture onto 3. Since deuterium is made via radiative capture as well as in spallation, we have $N_d^{\rm tot} = N_d^{\rm spall} + N_d^{\rm rc} $. Let us further assume that all neutrons made in spallation reactions, $N_n$, go into making of deuterium by radiative capture, that is, assume that $ N_d^{\rm rc}= N_n$. Thus the 2.22 MeV $\gamma$-ray intensity estimated with this assumption corresponds to the upper limit of Galactic flare production of deuterium. We can now rewrite equation (14) and obtain $
\ndot_d^{\rm tot} $, the rate of total deuterium production by a single star, as a function of deuterium-to-neutron spallation production ratio \_d\^[tot]{} =
What we actually want is the Galactic deuterium production rate by mass, $ {\dot{M}}_d^{gal} $, which is a function of total deuterium production rate ${\ndot}_d^{\rm tot}$ and total number of flare stars in the galaxy $\ndme$: $$\begin{aligned}
{\dot{M}}_d^{\rm gal} &=& m_d {\ndot}_d^{\rm tot} {\cal N}_{*}^{\rm gal} \\
&=& \frac{{\ndot}_{\gamma}}{f_{2.2}}
\left[ 1+ {\left( \frac{d}{n} \right)}_{\rm spall} \right]
m_d {\cal N}_{*}^{\rm gal}
\label{eq:Mdot}\end{aligned}$$ where ${\cal N}_{*}$ is the number of flare stars in the Galaxy, and $m_d$ is the deuteron mass.
We wish to find the observable Galactic 2.22 MeV $\gamma$-ray intensity $I_{\gamma} $, in terms of the 2.22 MeV emissivity $ {q}_{\gamma}^{\rm gal}$, the total Galactic production rate of 2.22 MeV $\gamma$ rays per volume per steradian \[eq:emissivity\] [q]{}\_\^[gal]{} = where $n_{*}(\vec{r})$ is the number density of flare stars. The intensity is simply a line-of-sight integral: I\_ = \_[los]{} [q]{}\_\^[gal]{} dl = where $N_{\rm los} = \int_{\rm los} n(\vec{r}) \ dl$ is the column density of flare stars along the line of sight $l$.
Therefore, we can link Galactic deuterium production rate by mass as a function of Galactic 2.22 MeV $\gamma$-ray intensity by combining eqs. \[eq:Mdot\] and \[eq:emissivity\] $$\begin{aligned}
\label{eq:D2gamma}
{\dot{M}}_d^{\rm gal}
= \frac{4 \pi}{f_{2.2}}
\left[ 1 + {\left( \frac{d}{n} \right)}_{\rm spall} \right]
\frac{m_d {\cal N}_{*}}{N_{\rm los}} I_{\gamma}\end{aligned}$$ or \[eq:gamma2D\] I\_ = \^[-1]{} \_d\^[gal]{} Equation (\[eq:D2gamma\]) allows us to estimate the most modest Galactic 2.22 MeV $\gamma$-ray intensity that we expect if the flare stars are producing significant amounts of deuterium. In order to predict the lower intensity limit, let us take $d/n = 1$ (although from Fig. 2 we see that this ratio is always smaller then 1). We also adopt the $f_{2.2}=0.1$ neutron-to-$\gamma$ conversion efficiency calculated by @Reuven; this value is dominated by escape (followed by decay), and then the neutron poisoning due to the large $n$-capture cross section for 3.
The Galactic distribution $n(\vec{r})$ of flare stars (i.e., M dwarfs) controls both the column density and flare star number which appear in eqs. (\[eq:gamma2D\]) and (\[eq:D2gamma\]). We have verified that disk M dwarfs dominate the calculation, while spheroid (“stellar halo”) M dwarfs [@gfb] do not contribute significant numbers or column of stars. For the disk M dwarfs, we adopt the [@zheng] distribution \[eq:Mdist\] n(r,z) & = & n\_0 (-)\
& & @zheng use [*Hubble Space Telescope*]{} star counts to determine the initial mass function in the M dwarf range, and find the parameters in eq. (\[eq:Mdist\]) to be: local M dwarf density $n_0 = \rho_0/\langle m_{\rm M \, dwarf} \rangle
= 5.2\times 10^{-2} \ {\rm pc^{-3}}$, radial scale length $H = 2.75\ {\rm kpc}$, solar Galactocentric distance $R_0 = 8 \ {\rm kpc}$, and vertical scale parameters $h_1 = 156 \ {\rm pc}$, $h_2 = 439 \ {\rm pc}$, and $\beta = 0.381$. With these, we find the number of Galactic M dwarfs to be $\ndme = 2.4 \times 10^{10}$, and a column density towards the Galactic center of $N_{\rm los} = 5.1 \times 10^3 \ {\rm pc^{-2}}$.
The deuterium production rate that we would consider to be significant is one that is comparable to the Galactic destruction rate of deuterium due to its cycling through stars (“astration”), which burn D during the pre-main sequence phase. The present rate at which deuterium is lost is \_[d,[astrate]{}]{} = -X\_d [E]{} = -2.5 10\^[-5]{} M\_/ [yr]{} where $X_d = 2.1 \times 10^{-5}$ is the present ISM mass fraction of deuterium, and ${\cal E}$ is the current rate at which mass is ejected from dying stars. This is roughly given by ${\cal E} = R \psi$, where $R \simeq 0.3$ is a conservative estimate of the “return fraction” of mass from stars [@pagel e.g.], and we adopt a current Galactic star formation rate $\psi \approx 4 \, M_{\odot}/{\rm yr}$ following, e.g., @focv.
We then can place a lower limit on the 2.22 MeV Galactic $\gamma$-ray intensity that corresponds to flare production of deuterium equal to the amount of deuterium that is destroyed per year in the Galaxy: $$\begin{aligned}
\label{eq:MLgamma}
I_{\gamma}
= 4 \times 10^{-2} \ {\rm cm}^{-2} \ {\rm sec}^{-1} \ {\rm sr^{-1}}\end{aligned}$$
We can compare this prediction with 2.22 MeV all-sky map done by COMPTEL [@Comptel]. This map contains a single possible point source but no diffuse emission is found, down to a level estimated (M. McConnell, private communication) to be I\_\^[obs]{} < 5 10\^[-4]{} [cm\^[-2]{} sec\^[-1]{} sr\^[-1]{}]{} This limit is about 1% of what we would expect if a significant amount of deuterium originates in flares (eq. \[eq:MLgamma\]). Or to go other way around, we can use eq. (\[eq:gamma2D\]) to translate the COMPTEL upper limit into an upper limit for deuterium production by flares: $$\begin{aligned}
\label{eq:maxMdot}
{\dot{M}}_d^{\rm gal} \la 3.0 \times 10^{-7} \ M_{\odot}/{\rm yr}
\simeq 1 \times 10^{-2} | {\dot{M}}_{d,{\rm astrate}} |\end{aligned}$$ Thus we see that the maximum possible D production rate is at most $1\%$ of the destruction rate, so that the net effect is that D is indeed destroyed and thus decreases with time. If we were only to look at the $n$ channel of deuterium production, as it was proposed by ML, we would get even lower production rate (by about a factor of two). Therefore on the basis of COMPTEL observations and our numerical analysis of production rates we can conclude that stellar flares are [*not*]{} significant sources of non-primordial deuterium on the Galactic scale, and thus do not spoil the monotonic decline of D with time.
Although we have estimated the diffuse 2.22 MeV radiation from flare stars, our calculation in fact is generally applicable to any Galactic sources of deuterium via neutron radiative capture. The $\gamma$-ray constraints are thus stronger than the abundance-based constraints of the previous section. Furthermore, further limits (or observations) at 2.22 MeV will constrain [*any*]{} source of D production from neutrons. We encourage INTEGRAL observations to be made to tighten the constraints we present here.
While $\gamma$-ray observations rule out flare D production at a level which would overwhelm destruction, the fact remains that neutron production is a significant and inevitable result of flare activity, as emphasized by ML. Thus, a $\gamma$-ray signature of this process [*must*]{} exist at some level. We can make a crude prediction of diffuse 2.22 MeV $\gamma$-ray intensity that INTEGRAL might observe. The rate at which 2.22 MeV $\gamma$-ray are produced in a single star is given by \[eq:star\_lum\] \_ = f\_[2.2]{} L\_[flare]{} where $L_{\rm flare}$ is the time-averaged luminosity in the flare state, and = \_i N\_i() d = k\_p \^[-s]{} d\_[i]{} y\_i A\_i measures the flare energy going into accelerated particles. Using the spectral index $s=4$ favored by solar flares, and integrating from the lowest threshold energy among reactions involved in neutron production (note that because of assumed power law spectrum of flare particles it is not possible to extrapolate to zero energy) we find that the number of neutrons produced per flare unit energy is ${Q_n}/{\langle \epsilon \rangle}=0.015 \ \rm atoms \ erg^{-1}$.
For the total number of neutrons produced per proton (above 30 MeV), we used our thick-target $s=4$ numerical result $Q_n=5.53 \times 10^{-3}$. Then by taking a bolometric flare luminosity of dM5e star to be $L_{\rm flare}=1.9 \times 10^{29}$ erg/sec [@mdwarfs] and neutron-to-photon efficiency factor to be $f_{2.2}=0.1$ it follows from equation (19) and (20) that \[eq:flare2.2\] I\_ \~1.2 10\^[-8]{} cm\^[-2]{} s\^[-1]{} sr\^[-1]{} In this calculation we took $V_* = 10^{12} \ \rm pc^3$ and $n_{*}=0.86 \ \rm pc^{-3}$ [@weistrop], and we took the a line of sight length $l=20$ kpc towards the Galactic center. Finally, we note that using the same estimate of flare activity, (i.e., scaling to the flare star luminosity), we can also arrive at an estimate of the flare D production rate, namely ${\dot{M}}_d^{\rm flare} \sim 2.5 \times 10^{-10} \ M_{\odot}/{\rm yr}$, which is completely negligible compared to the D astration rate.
The emission in eq. (\[eq:flare2.2\]) is low, and in particular is too dim to be observed by INTEGRAL. On the other hand, our estimate is crude, and while detection of flare stars would be a surprise, it would also provide unique new information about the particle content and energetics in stellar flares. Also, as we have noted, simply tightening the limits on diffuse 2.22 MeV radiation will strengthen the case that D production is negligible, and can go further to constrain (or probe) flares as a contributor to the D variation in the ISM [@hoopes]. For these reasons, we encourage INTEGRAL observations at 2.22 MeV, particularly towards the Galactic center where the emission should be the strongest.[^2]
Constraints on Localized Deuterium Production
=============================================
It is important to note, however, that this conclusion applies to the Galaxy as a whole. While the global D production rate due to flares is small, it is a separate question whether D production could be sufficient to create D variations on smaller scales.
To constrain local enrichment, we rely on the Galactic average deuterium production rate $\dot{M}_d$, which we have constrained in the previous subsection on the basis of gamma-ray observations, and estimated using flare energetics. With the global production rate in hand, we can find the average D production rate for one star. The limit from eq. (\[eq:maxMdot\]) gives $\dot{m}_d = \dot{M}_d/\ndme \sim 10^{-17} \ \msol/{\rm yr}$. Using this, let us find the needed D production to pollute a parcel of gas at a level of $\delta {\rm D/D} \sim \delta M_D/M_D \sim 1$, i.e., to product a factor of 2 variation in D/H over the current ISM level. To pollute a gaseous region of total mass $M_g$ given the entire age of the universe requires that $M_{\rm M \, dwarf} \sim 70 M_g$, i.e., a gas fraction of no more than $1/70$, that is, the M dwarfs must outweigh the ambient gas by a large factor. On the other hand, a clump of M dwarfs in the hot and diffuse ISM could achieve this level. If we instead adopt the deuterium production rate $\dot{M}_d \sim 10^{-10} \ \msol/{\rm yr}$, based on energetics, we find a needed gas fraction $< 10^{-5}$. This becomes harder to achieve, but is difficult to completely exclude.
Discussion and Conclusions
==========================
For more than a quarter century, the ELS argument that all deuterium is primordial has played a central role in cosmology. The importance of this argument demands that its assumptions be carefully checked. In this context, Mullan & Linsky’s discovery that D production by neutron capture in flares can evade the ELS constraints is thus very important and demands further investigation.
In this paper, we have examined and constrained the ML scenario in two ways. First, we have made a detailed calculations of spallation yields in flares. These calculations show that flares have $1 \la n/d \la 200$, confirming the ML suggestion that neutron production is at least as significant as direct deuteron production, and considerably more for some spectral indices (albeit ones atypical of solar flares). We have thus updated the ELS constraints on flares, now taking into account neutron production (all which we assume leads to deuterium formation by radiative capture). Specifically, we consider 6 production which accompanies $n$ and $d$ synthesis. The ELS argument is quantitatively changed, because the $\li6/(n+d)$ ratio is lower than the former $\li6/d$ ratio, but the qualitative conclusion remains, that the solar 6 abundance forbids a significant $d$ component from flares. The only way that flare nucleosynthesis could avoid this bound is for $d$ to escape the star preferentially with respect to 6, the needed bias being a factor around 1000 in escape efficiency.
Second, we considered the implications of the 2.22 MeV $\gamma$-ray line produced in the radiative capture. If D is currently produced in the Galaxy via this mechanism (either in flare stars or elsewhere) then there inevitably is an observable 2.22 MeV line signature. In the case of flare stars, the emission would be unresolved and would lead to a diffuse intensity tracing the Galactic plane. The observed limits on 2.22 MeV emission from COMPTEL immediately translate into a constraint on D production by radiative capture, and rule out this mechanism as a significant source of D at the global Galactic scale.
It is more difficult to constrain flare production of D as the source of the possible scatter in local D abundances as reported recently by FUSE [@hoopes]. However, arguments based on energetics and on gamma-ray fluxes do demand that the mean D production rate per star is small. This means either that (1) to produce significant fluctuations requires a high local concentration of M dwarfs relative to the gas they pollute, or (2) that flare D production is dominated by a very small fraction of M dwarfs, which dominate the global mean production and can thus create local anomalies. In either case, the nearest local sources may be detectable through their gamma-ray lines.
Another constraint on the ML scenario comes from observations of deuterated molecules towards the Galactic center. @lubowich detect DCN in a molecular cloud 10 pc away from the Galactic center, and on the basis of a molecular chemistry model estimate that ${\rm D/H} = (1.7 \pm 0.3) \times 10^{-6}$ in this cloud. Unfortunately, the fractionation corrections here are large and thus the results are somewhat model-dependent. With this caveat, the important point here is that D/H is found to be [*lower*]{} towards the Galactic center, suggesting that D indeed has a positive Galactocentric gradient, which argues against [*any*]{} significant stellar source of D, including flares.
Finally, we note that while flare star radiative capture synthesis of D is insufficient to alter the conclusions of ELS, it is certain that the process does occur at some level–the flares exist, and must produce neutrons. Thus, the 2.22 MeV signature of this process must exist at some level. We have made a simple estimate of the surface brightness towards the Galactic center. We find this to be below the sensitivity of INTEGRAL, but given the crudeness of our estimate, it is worth investigation, as limits (or detection!) of this line has immediate implications for stellar flares, neutron capture processes, and deuterium evolution in our Galaxy.
We thank Don York for discussions which stimulated this work. We are particularly grateful to Mark McConnell and Ed Chupp for very helpful discussions of the COMPTEL 2.22 MeV results, and to Jeffrey Linsky and Dermott Mullan for constructive comments on an earlier version of this paper. This material is based upon work supported by the National Science Foundation under Grant No. AST-0092939.
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[^1]: More observations are needed to firmly establish the nature and degree of D variations, and their (anti-)correlations with metallicity [@gary]. Nevertheless, it is already clear that FUSE is opening new windows on the Galactic and local evolution of deuterium; this in turn motivates a careful re-examination of the basic ELS assumptions which have guided our thinking on these issues.
[^2]: We note that COMPTEL [@Comptel; @Comptel2] found a single point source at 2.22 MeV, with flux $\phi_\gamma = (3.3 \pm 0.9) \times 10^{-5} \ {\rm cm^{-2} \ s^{-1}}$. A flare star with luminosity $\ndot_\gamma$ (eq. \[eq:star\_lum\]) at distance $r$ has a time-averaged flux $\langle \phi_\gamma \rangle \simeq 3 \times 10^{-12} (r/1 \, {\rm pc})^{-2}
\ {\rm cm^{-2} \ s^{-1}}$. Thus, the point source is too bright to be even a nearby flare star unless COMPTEL happened to see a brief, intense burst, from a nearby star with flare luminosity much larger than the average dMe. INTEGRAL observations of this source can test the flare star hypothesis by looking for 2.22 MeV time variability in coincidence with, e.g., H$\alpha$ and other line emission, as well as X-ray and UV signatures.
|
---
abstract: 'We report new precision measurements of the properties of our Galaxy’s supermassive black hole. Based on astrometric (1995-2007) and radial velocity (2000-2007) measurements from the W. M. Keck 10-meter telescopes, a fully unconstrained Keplerian orbit for the short period star S0-2 provides values for the distance (R$_0$) of 8.0 $\pm$ 0.6 kpc, the enclosed mass (M$_{bh}$) of 4.1 $\pm$ 0.6 $\times$ $10^6 M_{\odot}$, and the black hole’s radial velocity, which is consistent with zero with 30 km/s uncertainty. If the black hole is assumed to be at rest with respect to the Galaxy (e.g., has no massive companion to induce motion), we can further constrain the fit and obtain R$_0$ = 8.4 $\pm$ 0.4 kpc and M$_{bh}$ = 4.5 $\pm$ 0.4 $\times$ $10^6 M_{\odot}$. More complex models constrain the extended dark mass distribution to be less than 3-4 $\times$ $10^5 M_{\odot}$ within 0.01 pc, $\sim$100x higher than predictions from stellar and stellar remnant models. For all models, we identify transient astrometric shifts from source confusion (up to 5x the astrometric error) and the assumptions regarding the black hole’s radial motion as previously unrecognized limitations on orbital accuracy and the usefulness of fainter stars. Future astrometric and RV observations will remedy these effects. Our estimates of R$_0$ and the Galaxy’s local rotation speed, which it is derived from combining R$_0$ with the apparent proper motion of Sgr A\*, ($\theta_0$ = 229 $\pm$ 18 km s$^{-1}$), are compatible with measurements made using other methods. The increased black hole mass found in this study, compared to that determined using projected mass estimators, implies a longer period for the innermost stable orbit, longer resonant relaxation timescales for stars in the vicinity of the black hole and a better agreement with the M$_{bh}$-$\sigma$ relation.'
author:
- 'A. M. Ghez, S. Salim, N. N. Weinberg, J. R. Lu, T. Do, J. K. Dunn, K. Matthews, M. Morris, S. Yelda, E. E. Becklin, T. Kremenek, M. Milosavljevic, J. Naiman'
title: 'Measuring Distance and Properties of the Milky Way’s Central Supermassive Black Hole with Stellar Orbits'
---
INTRODUCTION {#sec:intro}
============
Ever since the discovery of fast moving (v $>$ 1000 km s$^{-1}$) stars within 0.$\tt''$3 (0.01 pc) of our Galaxy’s central supermassive black hole (Eckart & Genzel 1997; Ghez et al. 1998), the prospect of using stellar orbits to make precision measurements of the black hole’s mass (M$_{bh}$) and kinematics, the distance to the Galactic center (R$_0$) and, more ambitiously, to measure post-Newtonian effects has been anticipated (Jaroszynski 1998, 1999; Salim & Gould 1999; Fragile & Mathews 2000; Rubilar & Eckart 2001; Weinberg, Milosavlejic & Ghez 2005; Zucker & Alexander 2007; Kraniotis 2007; Will 2008). An accurate measurement of the Galaxy’s central black hole mass is useful for putting the Milky Way in context with other galaxies through the apparent relationship between the mass of the central black hole and the velocity dispersion, $\sigma$, of the host galaxy (e.g., Ferrarese & Merrit 2000; Gebhardt et al. 2000; Tremaine et al. 2002). It can also be used as a test of this scaling, as the Milky Way has the most convincing case for a supermassive black hole of any galaxy used to define this relationship. Accurate estimates of R$_0$ impact a wide range of issues associated with the mass and structure of the Milky Way, including possible constraints on the shape of the dark matter halo and the possibility that the Milky Way is a lopsided spiral (e.g., Reid 1993; Olling & Merrifield 2000; Majewski et al. 2006). Furthermore, if measured with sufficient accuracy ($\sim$1%), the distance to the Galactic center could influence the calibration of standard candles, such as RR Lyrae stars, Cepheid variables and giants, used in establishing the extragalactic distance scale. In addition to estimates of $M_{bh}$ and R$_0$, precision measurements of stellar kinematics offer the exciting possibility of detecting deviations from a Keplerian orbit. This would allow an exploration of a possible cluster of stellar remnants surrounding the central black hole, suggested by Morris (1993), Miralda-Escud[é]{} & Gould(2000), and Freitag et al. (2006). Estimates for the mass of the remnant cluster range from $10^4 - 10^5 M_{\odot}$ within a few tenths of a parsec of the central black hole. Absence of such a remnant cluster would be interesting in view of the hypothesis that the inspiral of intermediate-mass black holes by dynamical friction could deplete any centrally concentrated cluster of remnants. Likewise, measurements of post-newtonian effects would provide a test of general relativity, and, ultimately, could probe the spin of the central black hole.
Tremendous observational progress has been made over the last decade towards obtaining accurate estimates of the orbital parameters of the fast moving stars at the Galactic center. Patience alone permitted new astrometric measurements that yielded the first accelerations (Ghez et al. 2000; Eckart et al. 2002), which suggested that the orbital period of the best characterized star, S0-2, could be as short as 15 years. The passage of more time then led to full astrometric orbital solutions (Schödel et al. 2002, 2003; Ghez et al. 2003, 2005a), which increased the implied dark mass densities by a factor of $10^4$ compared to earlier velocity dispersion work and thereby solidified the case for a supermassive black hole. The advent of adaptive optics enabled radial velocity measurements of these stars (Ghez et al. 2003), which permitted the first estimates of the distance to the Galactic center from stellar orbits (Eisenhauer et al. 2003, 2005). In this paper, we present new orbital models for S0-2. These provide the first estimates of the distance to the Galactic center and limits on the extended mass distribution based on data collected with the W. M. Keck telescopes. The ability to probe the properties of the Galaxy’s central supermassive black hole has benefitted from several advancments since our previous report (Ghez et al. 2005). First, new astrometric and radial velocity measurements have been collected between 2004 and 2007, increasing the quantity of kinematic data available. Second, the majority of the new data was obtained with the laser guide star adaptive optics system at Keck, improving the quality of the measurements (Ghez et al. 2005b; Hornstein et al. 2007). These new data sets are presented in §\[sec:obs\]. Lastly, new data analysis has improved our ability to extract radial velocity estimates from past spectroscopic measurements, allowing us to extend the radial velocity curve back in time by two years, as described in §\[sec:data\_analysis\]. The orbital analysis, described in §\[sec:orbit\], identifies several sources of previously unrecognized biases and the implications of our results are discussed in §\[sec:disc\].
OBSERVATIONS & DATA SETS {#sec:obs}
========================
High Angular Resolution Imaging: Speckle and Adaptive Optics
------------------------------------------------------------
For the first eleven years of this experiment (1995-2005), the proper motions of stars orbiting the center of our Galaxy were obtained from $K$\[2.2 $\mu m$\]-band speckle observations of the central stellar cluster with the W. M. Keck I 10-meter telescope and its facility near-infrared camera, NIRC (Matthews & Soifer 1994; Matthews et al. 1996). A total of 27 epochs of speckle observations are included in the analysis conducted in this paper, of which 22 have been reported in earlier papers by our group (Ghez et al. 1998, 2000, 2005a). Five new speckle observations, between 2004 April and 2005 June, were conducted in a similar manner. In summary, during each observing run, $\sim$10,000 short ($t_{exp}$ = 0.1 sec) exposure frames were obtained with NIRC in its fine plate scale mode, which has a scale of 20.46 $\pm$ 0.01 mas pixel$^{-1}$ (see Appendix B) and a corresponding field of view of 52 $\times$ 52. Interleaved with these observations were similar sequences on a dark patch of sky. From these data, we produce images that are diffraction-limited ($\theta$ = 005) and have Strehl ratios of $\sim$0.05.
With the advent of laser guide star adaptive optics (LGSAO) in 2004 on the 10 m W. M. Keck II telescope (Wizinowich et al. 2006; van Dam et al. 2006), we have made measurements of the Galaxy’s central stellar cluster with much higher Strehl ratios (Ghez et al. 2005b). Between 2004 and 2007, nine LGSAO data sets were taken using the W. M. Keck II facility near-infrared camera, NIRC2 (P.I. K. Matthews), which has an average plate scale of 9.963 $\pm$ 0.006 mas pixel$^{-1}$ (see Appendix C) and a field of view of 102 $\times$ 102. All but one of the observations were obtained through a K’ ($\lambda_0$=2.12 $\mu$m, $\Delta \lambda$=0.35 $\mu$m) band-pass filter, with the remaining one obtained through narrow band filters (CO: $\lambda_0$ = 2.278 $\mu$m, $\Delta \lambda$ = 0.048 $\mu$m and Kcont: $\lambda_0$ = 2.27 $\mu$m, $\Delta \lambda$ = 0.030 $\mu$m). During these observations, the laser guide star’s position was fixed to the center of the camera’s field of view and therefore moved when the telescope was dithered. While the laser guide star is used to correct most of the important atmospheric aberrations, it does not provide information on the tip-tilt term, which, for all our LGSAO observations (imaging and spectroscopy), was obtained from visible observations of USNO 0600-28577051 (R = 13.7 mag and $\Delta r_{SgrA*}$ = 19$\arcsec$). Details of the observing setup for 2004 July 26, 2005 June 30, and 2005 July 31 are described in detail in Ghez et al. (2005b), Lu et al. (2008), and Hornstein et al. (2007), respectively. While each of these early LGSAO observations had a slightly different setup and dither pattern, the more recent, deeper, LGSAO measurements (2006-2007) were obtained with nearly identical setups. Specifically, we used a 20 position dither pattern with randomly distributed (but repeatable) positions in a 07 $\times$ 07 box and an initial position that placed IRS 16NE on pixel (229, 720) at a sky PA set to 0.0. This setup keeps the brightest star in the region, IRS 7 (K=6.4), off the field of view at all times. At each position, three exposures, each composed of 10 coadded 2.8 sec integrations, were obtained; the integration time was set with the aim of keeping the detector’s response linear beyond the full width at half maximum (FWHM) point for the brightest (K=9.0) star in the field of view; the number of images per position was chosen to provide the minimum elapsed time needed to allow the LGSAO system’s optimization algorithm to converge ($\sim$3 min.) before dithering. Table \[tbl\_img\] summarizes all the new imaging data sets.
Adaptive Optics Spectroscopy
----------------------------
To monitor the line-of-sight motions of stars orbiting the center of our Galaxy between the years 2000 and 2007, high angular resolution spectroscopic observations of stars in the Sgr A\* stellar cluster were taken with both the natural guide star adaptive optics (NGSAO; Wizinowich et al. 2000) system (2000-2004) and the LGSAO system (2005-2007) on the W. M. Keck II 10 m telescope. The NGSAO atmospheric corrections and the LGSAO tip-tilt corrections were made on the basis of visible observations of USNO 0600-28579500 (R = 13.2 mag and $\Delta r$ $\sim$ 30$\arcsec$) and USNO 0600-28577051 (R = 13.7 mag and $\Delta r$ $\sim$ 19$\arcsec$), respectively. While the angular resolution of the NGSAO spectra was typically 2-3 times the diffraction limit ($\theta_{diff}$ = 54 mas), a point spread function (PSF) FWHM of $\sim$ 70 mas at 2 $\micron$ was achieved for the LGSAO long exposure spectra.
Three different spectrometers have been used over the course of this study. Our earliest measurements were obtained in 2000 June with NIRSPEC (McLean et al. 1998, 2000) in its low resolution slit spectrometer mode (R $\sim$ 2600). It was not originally designed to go behind the adaptive optics system and therefore had inefficient throughput in its AO mode; it was, however, the only spectrometer available behind the AO system in 2000. While the resulting low signal to noise data set yielded no line detections in the initial analysis of S0-2 (Gezari et al. 2002), we now have the advantage of knowing what type of lines are present in the spectra and have therefore included this data set in our analysis by retroactively identifying the Br$\gamma$ line, which is used to measure radial velocities (see §3.2) Between 2002 and 2005, NIRC2 (P.I. K. Matthews) was used in its spectroscopic R $\sim$ 4000 mode, which is generated with a 20 mas pixel scale, a medium-resolution grism and a 2 pixel slit. In 2002, this produced the first line detection in S0-2 (Ghez et al. 2003) and, since then, three new NIRC2 measurements (2 with NGSAO and 1 with LGSAO) have been obtained. Since 2005, OSIRIS, which is an integral field spectrograph with a 2 $\micron$ spectral resolution of $\sim$ 3600 (Larkin et al. 2006), has been used. The field of view of this spectrograph depends on the pixel scale and filter. Most of the OSIRIS observations were taken using the 35 mas pixel scale and the narrow band filter Kn3 (2.121 to 2.229 $\micron$; includes Br$\gamma$), which results in a field of view of $1.\arcsec 12 \times 2.\arcsec 24$, and were centered on S0-2. All of the OSIRIS observations were obtained with the LGSAO system. Table \[tbl\_spec\] summarizes the details of the 10 new spectroscopic measurements of S0-2 that were made between the years 2003 and 2007 (see Gezari et al 2002 & Ghez et al. 2003 for details of the 2000-2002 measurements).
DATA EXTRACTION {#sec:data_analysis}
===============
Image Analysis & Astrometry
---------------------------
The individual speckle and adaptive optics data frames are processed in two steps to create a final average image for each of the 34 imaging observing runs. First, each frame is sky-subtracted, flat fielded, bad-pixel-corrected, corrected for distortion effects and, in the case of the speckle data, resampled by a factor of two; the distortion correction applied to the NIRC2/LGSAO data is from the NIRC2 pre-ship review results (<http://www2.keck.hawaii.edu/inst/nirc2/preship_testing.pdf>) and those applied to the speckle data sets are the combined transformations given in Ghez et al. (1998) and Lu et al. (2008). The frames are then registered on the basis of the position of IRS 16C, for the speckle images, and a crosss-correlation of the entire image, for the LGSAO image, and combined. For the adaptive optics data sets, the frames whose PSF has a FWHM $<$ 1.25 x FWHM$_{min}$, where FWHM$_{min}$ is the minimum observed FWHM for each epoch and which typically includes $\sim$70% of the measured frames, are combined with a weighted average with weights set equal to their strehl ratios. To increase the signal to noise ratio of the 2005 June data set, the data taken through the two narrow-band filters are averaged together. For the speckle data set, only the best $\sim$ 2,000 frames from each observing run are combined using a weighted “Shift-and-Add" technique described by Hornstein (2007). The selected frames from each observing run (speckle and LGSAO) are also divided into three independent subsets from which three subset images are created in a similar manner to the average images; these subset images are used to assess photometric and astrometric measurement uncertainties. Figure \[fig\_aoVsp\] shows examples of the final average LGSAO and speckle images. While all the images sets have point spread function (PSF) cores that are nearly diffraction-limited ($\theta$ $\sim$ 006 vs. $\theta_{diff.~lim}$ = 005), the LGSAO images have much higher image quality than the speckle images, with median Strehl ratios of $\sim$0.3 and 0.07, for the LGSAO and speckle images, respectively.
Point sources are identified and characterized in each of the images using the PSF fitting program StarFinder (Diolaiti et al. 2000) on both the average images and the subset images. StarFinder iteratively generates a PSF based on user selected point sources in the image and identifies additional sources in the image by cross-correlating the resulting PSF with the image. The initial source list for each image is composed only of sources detected in the average images with correlation values above 0.8 and in all three subset images with correlation values above 0.6. Eleven bright (K$<$14 mag), non-variable sources establish the photometric zero points for each list based on measurements made by Rafelski et al. (2007; IRS 16C, IRS 16SW-E, S2-17, S1-23, S1-3, S1-4, S2-22, S2-5, S1-68, S0-13, S1-25). As shown in Figure \[speck\_ao\_klimit\], the deep LGSAO images (K$_{lim} \sim $19 mag) are three magnitudes more sensitive than the speckle images (K$_{lim} \sim $16 mag), which results in roughly three times more sources being detected in the LGSAO images than the speckle images over a comparable region. Because of the higher signal to noise, as shown in Figure \[speck\_ao\_astro\], the centroiding uncertainties ($\delta X', \delta Y'$), which are estimated from the RMS error of the measurements in the three subset images, are a factor of 6 more precise for the deep LGSAO data sets (0.17 mas) than the speckle data sets (1.1 mas), for bright stars (K$<$13 mag); the plateau observed in the relative centroiding uncertainties for the brighter stars (K$<$13) in the LGSAO images is likely caused by the combined effects of differential tip-tilt jitter and residual optical distortions across the field of view.
The sources identified each night are matched across multiple epochs and their positions are transformed to a common coordinate system that will be referred to as the [*cluster reference frame*]{}. As detailed in Appendix A, the transformation for each epoch is derived by minimizing the net displacement of a set of “coordinate reference" stars, allowing for proper motions, relative to their positions in a common reference image, which, in this case, is the 2004 July LGSAO image. This procedure attempts to ensure that in the cluster reference frame the coordinate reference stars are at rest (i.e., no net translation, rotation, expansion, or skew). A total of $\sim$470 and $\sim$120 stars serve as coordinate reference stars in the LGSAO and speckle epochs, respectively. These stars are selected based on the following criteria: (1) high detection correlations ($>$0.9), ensuring good positional accuracy, (2) located more than 05 from Sgr A\* to avoid sources with measurable non-linear motions (i.e., accelerations in the plane of the sky $> \sim$8 km/s/yr), (3) low velocities ($<$ 15 mas/yr, or equivalently $\sim$600 km s$^{-1}$), which eliminates possible coordinate reference sources that have been mismatched across epochs, and (4) lack of spectroscopic identification as a young star from Paumard et al. (2006) to eliminate the known net rotation of the young stars in the cluster reference frame. Positional uncertainties from this transformation process, which are characterized by a half sample bootstrap applied to the coordinate reference stars, are a factor of $\sim$1.5 (speckle) to 6 (LGSAO) smaller than the centroiding uncertainties and grow by less than a factor of 2 between the center of the field of view (minimum) and a radius of 3.
An additional source of positional error originates from residual optical distortion in NIRC2. While the residual distortion in NIRC2 is small, the extremely precise centroid measurements in the deep LGSAO images make it a significant effect. The presence of such a systematic error is established by examining the distribution of positional residuals, normalized by measurement (centroiding plus alignment) uncertainties, to the linear proper motion fits for the coordinate reference stars. The speckle data sets do not show large, measurable biases; the speckle measurements, on average, are only 1$\sigma$ off from the linear proper motion fit. In contrast, the much more precise deep-LGSAO astrometric measurements are, on average, 5$\sigma$ off from these fits. As described in Appendix B, we account for this effect at two stages of our analysis. First, 0.88 mas is added in quadrature to the positional uncertainties of the coordinate reference stars to account for systematic errors in the coordinate transformations. Second, a local correction, in the coordinate reference frame, is derived and applied to the positions of the short period stars that were made with LGSAO setups that differ from that of the reference image. This procedure ensures that residuals from both linear proper motion fits to the coordinate reference stars (see Appendix A & B) and from orbit fits to S0-2 (see §4) are consistent with a normal distribution.
Source confusion can introduce positional biases that can be comparable to and, at certain times, larger than the statistical errors caused by background or detector noise. This occurs when two stars are sufficiently close to each other that only one source, rather than two, is identified in our analysis with a brightness that includes flux from both sources and a position that corresponds roughly to the photocenter of the two stars. We divide the problem of handling source confusion in our data set into the following two cases: (1) the impact of unresolved, underlying stars that are known sources, because they were sufficiently well separated at other times, and bright enough, to be independently detected, and (2) the impact of unresolved, underlying stars that are not identified by this study at another time. Because the sources are moving so rapidly, instances of the former case are easily identified and are typically blended for one year. An underlying source that is comparably bright to the source of interest can have a significant impact on the astrometry; to quantify this effect, we examine the idealized, noise-free case of a perfectly known PSF by using our empirical PSFs to generate idealized binary stars and running StarFinder on these simulated images, inputting the known PSF. In this case, the astrometric bias is zero once the two components are detected. As Figure \[bias\_binary\] shows, when the sources are blended, the resulting astrometric biases can be easily as large as 10 mas, which is much larger than our centroiding uncertainties. Such a large astrometric bias occurs when the underlying source is at least half as bright as the primary source and has a projected, although unresolved, separation of $\sim$ 40 mas. We conservatively choose to eliminate all astrometric measurements that are known to be the blend of two sources from the orbital analysis; specifically, if the predicted positions of two known sources are separated by less than 60 mas and only one of them is detected, then that measurement is removed from our analysis. For S0-2 (K=14.2 mag), the eliminated data points are those made in 1998, due to confusion with S0-19 (K=15.6 mag), in 2002, due to overlap with SgrA\*-IR (K$_{median}$ = 16.4 mag, but can be as bright as 14 mag; see Do et al. 2008), and in May 2007, due to superposition with S0-20 (K=15.9). The impact of these overlapping sources, in the first two cases, can be seen in the photometric measurements (see Figure \[s02\_phot\]).
Source confusion from unknown sources is a smaller effect than that from known sources, since the unknown stars, in general, are fainter than the known sources. Given the long time-baseline of the speckle imaging experiment, knowledge of sources in this region is most likely complete down to K= 16.0 mag. While sources as faint as K = 19 mag have been detected in this region with LGSAO, crowding and the short time baseline of these deeper observations limit the census of these sources. Therefore, source confusion from unknown sources can give rise to astrometric biases for S0-2 as large as 3 mas (from a K=16 mag source), but are typically significantly smaller since underlying sources will generally be fainter than K = 16 mag. To characterize the expected astrometric bias from the undetected source distribution, a Monte Carlo simulation was performed by generating multiple images with all known stars plus a random stellar distribution that, in total, follows the K luminosity function and radial profile from Schödel et al. (2007). By running these simulated images through our data analysis prodedure, we estimate that the astrometric error from unknown sources for S0-2 is, on average, 0.5 mas and 1.2 mas for the LGSAO and speckle images, respectively, and that it scales roughly with the photometric bias and galacto-centric distance. However, it should be noted that the exact value of this bias is model dependent. While the photometric bias may be detected in the speckle data toward closest approach (see Figure \[s02\_phot\]), the estimated astrometric biases are smaller than other sources of positional uncertainty already included for the majority of the S0-2 data points. We therefore do not incorpate them into the reported positional uncertainties. Confusion with unknown sources gives rise to larger astrometric biases for S0-16, S0-19, and S0-20, since these sources are fainter than S0-2. Given the velocity dispersion in this region and the angular resolution of the data sets, the expected timescale associated with biases from source confusion is $\sim$1-2 years.
As a final step, the relative astrometric positions are placed in an absolute coordinate reference frame using the positions of seven SiO masers (Reid et al. 2003, 2007). Infrared observations of these masers with the Keck II LGSAO/NIRC2 system between 2005 and 2007 were obtained with the same camera (i.e., plate scale) used for the precision astrometry measurements described above, but with a nine position box pattern and a 6$\arcsec$ dither offset to create a 22$\arcsec \times$22$\arcsec$ mosaic of these masers (see Appendix C for details). A comparison of the maser positions measured in this infrared mosaic to the predicted radio positions at this epoch from Reid et al. (2003) establishes that the mosaic has an average pixel scale of 9.963 $\pm$ 0.005 mas/pixel and a position angle of north with respect to the NIRC2 columns of 0.$^o$13 $\pm$ 0.$^o$02. This same analysis localizes the radio position of Sgr A\* in the infrared mosaic to within 5 mas in the east-west and north-south directions. By aligning the infrared stars detected in both the larger infrared mosaic and the precision astrometry image taken during the same observing run, we have the necessary coordinate transformations to convert our relative astrometric position measurements into an absolute reference frame. For the orbit analysis described in §4, the uncertainties in this transformation are applied only after model orbits have been fit to the relative astrometry and are a negligible source of uncertainty in the final mass and R$_o$ estimates.
Spectral Analysis & Radial Velocities
-------------------------------------
In the analysis of the spectral data, we accomplish the initial basic data processing steps using standard IRAF procedures, for NIRC2 and NIRSPEC, and a facility IDL data extraction pipeline for OSIRIS. Specifically, each data set is first (1) flat fielded, (2) dark subtracted, (3) bad pixel and cosmic ray corrected, (4) spatially dewarped, and (5) wavelength calibrated. Wavelength calibration is performed by identifying OH emission lines from sky spectra and fitting a low-order polynomial function to the location of the lines. For the NIRSPEC spectra, neon emission lines from arc lamps provide the wavelength calibration. The accuracy of the wavelength calibration is $\sim$9 km s$^{-1}$ or less for NIRC2 and OSIRIS as measured by the dispersion of the residuals to the fit. Next, the one-dimensional stellar spectra are extracted using a spatial window that covers $\sim 0\farcs 1$ for the two dimensional spectral data sets from NIRC2 and NIRSPEC. For the three dimensional spectral data set from OSIRIS, an extraction box 014 $\times$ 014 was used. To correct for atmospheric telluric absorption features, each spectrum is divided by the spectrum of an A-type star. Prior to this step, the A-type star’s strong intrinsic Br${\gamma}$ feature is removed. In the case of the NIRC2 and OSIRIS observations, this correction is done with observations of a G2V star, which is divided by a model solar spectrum. The Br$\gamma$ corrected region in the G star is then substituted into the same region of the A star (Hanson et al. 1996). In the case of the NIRSPEC observations, the A-type star’s [Br${\gamma}$ ]{}feature is corrected with a model spectrum of Vega[^1] rebinned to the resolution of the A-type star’s spectrum and convolved with a Gaussian to match the spectral resolution of the observations. The resulting stellar spectra are corrected for all telluric absorption features; however, they are still contaminated by background emission due to the gas around the Galactic center. The local background is estimated and removed by subtracting spectra extracted from regions that are $\sim$01 away. Finally, all the spectra within each night of observation are combined in an average, weighted by the signal to noise ratio.
Radial velocity estimates are determined for each spectrum on the basis of the location of the Br$\gamma$ line. While a few of our spectra with broader spectral coverage also show a weaker He I triplet at 2.116 $\micron$, we do not incorporate measurements from this line, as it is a blend of transitions that can bias the resulting radial velocities (see Figure \[spec\_avg\]). A Gaussian model is fit to each of the Br$\gamma$ line profiles and the wavelength of the best fit peak, is compared to the rest wavelength of $\lambda_{vacuum}$ = 2.1661 $\micron$ to derive an observed radial velocity. To obtain radial velocities in the local standard of rest (LSR) reference frame, each observed radial velocity is corrected for the Earth’s rotation, its motion around the Sun, and the Sun’s peculiar motion with respect to the LSR (U = 10 km s$^{-1}$, radially inwards; Dehnen & Binney 1998). Since the LSR is defined as the velocity of an object in circular orbit at the radius of the sun, the Sun’s peculiar motion with respect to the average velocity of stars in its vicinity should give the Sun’s motion toward the center of the Galaxy. The uncertainties in the final radial velocities are obtained from the rms of the fits to the line profile measurements from at least three independent subsets of the original data set. Figure \[spec\_brg\] shows how S0-2’s Br$\gamma$ line has shifted over time and how the measurement of this line has improved by a factor of 5 with improved instrumentation. For the deep LGSAO spectroscopic observations, the radial velocity uncertainties for S0-2 are typically $\sim$20-25 km s$^{-1}$.
ORBITAL ANALYSIS & RESULTS {#sec:orbit}
==========================
Point Mass Only Analysis
------------------------
To derive the black hole’s properties, we assume that the stars are responding to the gravitational potential of a point mass. In this analysis, the 7 properties of the central black hole that are fitted are its mass ($M$), distance (R$_0$), location on the plane of the sky ($X_0$, $Y_0$) and motion ($V_x$, $V_y$, $V_z$). In addition to these common free parameters, there are the following 6 additional free parameters for each star: period ($P$), eccentricity ($e$), time of periapse passage ($T_0$), inclination (i), position angle of the ascending node ($\Omega$), and the longitude of periapse ($\omega$). Using a conjugate gradient $\chi^2$ minimization routine that simultaneously fits the astrometric and radial velocity measurements, we fit this model to measurements that are given in Tables \[tbl\_pos\] & \[tbl\_rv\], which includes 27 epochs of astrometric measurements and 11 epochs of radial velocity (RV) measurements, as well as 5 additional epochs of radial velocity measurements reported in the literature (Eisenhauer et al. 2003, 2005). This excludes all the astrometric measurements of S0-2 that are confused with another known source (see §3.1). While the 2002 astrometric data are eliminated due to confusion with SgrA\*, the 2002 RV points are not, since SgrA\* is featureless and therefore does not bias the measurement of RV from S0-2’s Br$\gamma$ absorption line. In total, there are 38 astrometric data points and 5 RV measurements. All values reported for each parameter are the best fit values obtained from minimizing the total $\chi^2$, which is the sum of the $\chi^2$ from each data type (i.e., $\chi^2_{tot} = \chi^2_{ast} + \chi^2_{RV}$).
The uncertainties on the fitted parameters are estimated using a Monte Carlo simulation, which is a robust approach when performing a fit with many correlated parameters. We created $10^5$ artificial datasets ($N_{sim}$) containing as many points as the observed dataset (astrometry and radial velocities), in which each point is randomly drawn from a Gaussian distribution centered on the actual measurement and whose 1 $\sigma$ width is given by the associated uncertainty, and run the $\chi^2$ minimization routine for each realization. $N_{sim}$ was set to $10^5$ in order to achieve $\sim$6% accuracy in the resulting estimates of the a 99.73% confidence limits (3$\sigma$ equivalent for a gaussian distribution) of the orbital parameters. Because the $\chi^2$ function contains many local minima, each realization of the data is fit 1000 times ($N_{seed}$) with different seeds to find the global minimum. The resulting distribution of $10^5$ values of the fitted parameters from the Monte Carlo simulation, once normalized, is the a joint probability distribution function of the orbital parameters ($PDF({\vec O})$, where $\vec O$ is a vector containing all the orbital parameters,$O_i$). For each orbital parameter, $PDF({\vec O})$ is marginalized against all other orbital parameters to generate a $PDF(O_i)$. The confidence limits for each parameter are obtained by integrating each $PDF(O_i)$ from its peak[^2] outwards to a probability of 68 Compared to all other stars at the center of the Milky Way, S0-2 dominates our knowledge of the central black hole’s properties. Two facts contribute to this effect. Most importantly, it has the shortest known orbital period (P = 15 yr; Schödel et al. 2002, 2003; Ghez et al. 2003, 2005a). Furthermore, among the known short-period stars, it is the brightest star and therefore the least affected by stellar confusion (see Figure 1). Several other stars, in principle, also offer constraints on the black hole’s properties. In particular, S0-16 is the next most kinematically important star, as it is the only other star that yields an independent solution for the black hole’s properties. However, independent solutions for the black hole’s position from fits to S0-2 and S0-16 measurements differ by more than 5 $\sigma$ (see Figure \[bhPosition\]). While S0-16’s measurements in 2000 have already been omitted due to overlap with the position of SgrA\*, three independent lines of reasoning lead us to believe that some of S0-16’s remaining astrometric measurements must be significantly biased by radiation from unrecognized, underlying stars. First, as shown in Figure \[bias\_binary\], unknown sources can introduce astrometric biases as large as 9 mas for S0-16 (K=15), in contrast with only 3 mas for S0-2 (K = 14), because it is only 1 mag above the completeness limit for detection in the speckle data set (K$\sim$16 mag; see §3.1). Second, a comparison of the solution for the position of the black hole ($X_0, Y_0$) based on both the astrometric and radial velocity measurements to that based on astrometry alone (fixing the distance, which cannot be solved for without radial velocities) yields a consistent position from modeling the two cases for S0-2, but produces different results for the two cases from modeling S0-16’s measurements, with the inferred $X_0$ and $Y_0$ from astrometry alone shifting further away from that obtained from modeling S0-2’s orbit prediction and thereby increasing the discrepancy to 10$\sigma$. Third and last, while the position of the dynamical center from S0-2’s orbit is statistically consistent with SgrA\*-Radio/IR, which is the emissive source associated with the central black hole (e.g., Melia & Heino 2001; Genzel et al. 2003a; Ghez et al. 2004; 2005b; Hornstein et al. 2007), the solution from S0-16 is not (see Figure \[bhPosition\]); this difference cannot be explained by allowing the black hole to move with time or by introducing an extended mass distribution. We therefore restrict our remaining analysis to S0-2.
As shown in Figures \[s02\_orb\] & \[s02\_orb\_resid\], the astrometric and radial velocity measurements for S0-2 are well fit by a simple Keplerian model. For a 13 parameter model (right-hand side of figures), the best fit to the data produces a total $\chi^2$ of 54.8 for 57 degrees of freedom ($dof$) and a $\chi^2/dof$ of 0.961. From the Monte Carlo simulation, we derive probability distributions for the central black hole’s properties, which are shown in Figure \[massRo\] and characterized in Table \[tbl\_orb\]. These distributions give a best fit for the central black hole’s mass of M$_{bh}$ = $4.1 \pm 0.6 \times 10^6 M_{\odot}$ and distance of R$_0$ = 8.0 $\pm$ 0.6 kpc (all quoted uncertainties are 68% confidence values). The position of the black hole is confined to within $\pm$ 1 mas ($\sim$100 Schwarzschild Radii). As can be seen in Figure \[massRo\], the inferred black hole’s mass is highly correlated with its distance. Estimates from orbital modeling are expected to have a power law relationship of the form $Mass \propto M_{\odot} ~ Distance^{\alpha}$ with $\alpha$ between 1 and 3. For the case of astrometric data only, $\alpha$ should be 3 and, for the case of radial velocity data only, $\alpha$ is expected to be 1. Currently, the relationship is $M = (4.1 \pm 0.1 \times 10^6 M_{\odot})(R_0/8.0 ~ kpc)^{1.8}$, which suggests that the astrometric and radial velocity data sets are having roughly equal affect in the model fits for mass[^3].
A fit that includes the biased astrometric data points significantly alters the best fit solution for S0-2. Including both the 1998 and 2002 data points, which correspond to confusion with S0-19 and SgrA\*-IR respectively, results in a higher mass ($5.7 \times 10^6 M_{\odot}$), distance (9.4 kpc), and $\chi^2/dof$ (1.7). Including the 2002 but not the 1998 data points also produces elevated values ($5.2 \times 10^6 M_{\odot}$ and 9.1 kpc) and $\chi^2/dof$ (1.1). This demonstrates that it is important to account for the astrometric biases introduced by unresolved sources.
Formal uncertainties in mass and distance estimates from orbital fits can be reduced by adding [*a priori*]{} information. In particular, it is, in principle, possible to constrain the dynamical center to be at the position of SgrA\*-IR. However, as shown in Figure \[bhPosition\], the six measurements of SgrA\*-IR’s position in the deep LGSAO images (2005-2007), which have the most precise astrometric measurements, have an average value that differs from the position of the black hole inferred from S0-2’s orbit by 9.3 mas and a variance of 3 mas, which is a factor of 4 larger than expected from the measured positional uncertainties (0.7 mas). SgrA\*-IR is located where the underlying sources are expected to have the highest number density and velocity dispersion, which should induce time variable positional biases. SgrA\*-IR’s average K magnitude in these deep LGSAO images is 16.4, which is comparable to the completeness limit for sources in this region (see §3.1) and which is, consequently, potentially subject to large astrometric biases (see Figure \[bias\_binary\]). We therefore suspect that the measured positions of SgrA\*-IR suffer from astrometric biases from underlying sources and do not use its positions to constrain the model fits.
Another prior, which has been imposed in earlier orbital analyses of S0-2 for R$_0$ (Eisenhauer et al. 2003; 2005), is on the black hole’s motion relative to the measurements’ reference frame. Setting the three dimensional velocity to zero and fitting a 10 parameter model ($\chi^2/dof$ = 1.3; see left-hand side of Figures \[s02\_orb\] & \[s02\_orb\_resid\]) yeilds uncertainties in the black hole’s properties that are a factor of 2 smaller (R$_0$ = 8.0 $\pm$ 0.3 kpc and M$_{bh}$ = $4.4 \pm 0.3 \times 10^6 M_{\odot}$). However this assumption is not justified (see, e.g., Salim & Gould 1999; Nikiforov 2008). Introducing $V_x$ and $V_y$ (defined such that positive numbers are motions in the E and N directions, respectively) into the fit allows the dynamical center to move linearly in time in the plane of the sky with respect to the cluster reference frame. Such an apparent motion can arise from either a physical or a data analysis effect. In the case of a physical effect, the black hole could be moving with respect to the stellar cluster under the gravitational influence of a massive companion or the black hole and the cluster could be participating in a mutually opposing sloshing mode. In the case of a data analysis effect, the reference frame could be non-stationary with respect to the position of the dynamical center, which might arise if there was a systematic problem in our alignment of images. Introducing these two parameters therefore provides a way of examining possible systematic reference frame problems. Fits to a 12 parameter model ($V_z$ fixed to zero) to the data have a minimum $\chi^2/dof$ of 0.95, uncertainties in the black hole’s properties that are larger than the 10-parameter model, but smaller than the 13-parameter model (R$_0$ 8.4 $\pm$ 0.4 kpc and M$_{bh}$ = $4.5 \pm 0.4 \times 10^6 M_{\odot}$), and an estimate for the black hole’s motion relative to the central stellar cluster of $V_x$ = -0.40 $\pm$ 0.25 mas/yr (17 $\pm$ 11 km/sec) and $V_y$ = 0.39 $\pm$ 0.14 mas/yr (16 $\pm$ 6 km/sec). Since these relative velocities are comparable to the constraints on the IR reference frame’s motion with respect to SgrA\*-Radio (i.e., an absolute reference frame in which the black hole’s position is known; see Appendix C), it is important to leave $V_x$ and $V_y$ as free parameters, even for the case in which one assumes that the black hole has no intrinsic motion motion with respect to the cluster. Because the black hole is so often assumed to be at rest, we report the complete solution for the 12 parameter fit (V$_z$ fixed to zero) in Table \[tbl\_orb\].
As Figure \[RoVz\] shows, the black hole’s motion along the line of sight with respect to our assumed local standard of rest ($V_z$) dominates the uncertainties in R$_0$ in our 13 parameter model. Priors on $V_z$ therefore have a signficant impact on the resulting uncertainties. Unlike the plane of the sky, the reference frame along the line of sight is unlikely to have an instrumental systematic drift, since each of the spectra are calibrated against OH lines (see §3.2). However, it is possible that there is a residual radial velocity offset between the LSR and the S0-2 dynamical center. The Sun’s peculiar motion with respect to the LSR along the line of sight might differ from the assumed 10 km s$^{-1}$; that is, the practical realization of the LSR is not on a circular orbit around the Galactic center as might occur due the bar potential or to the spiral perturbations, so that the average velocity of stars in the solar vicinity might have a (small) net radial component. Alternatively, the dynamical center of S0-2 could differ from the dynamical center of the Galaxy as determined at the Sun’s (i.e., LSR’s) distance, as might result from the presence of an intermediate mass black hole companion. From the model fit, the implied motion of the LSR along the line-of-sight with respect to S0-2’s dynamical center is -20 $\pm$ 33 km/sec, which is consistent with no net motion. While no significant motion is detected in $V_x$, $V_y$, or $V_z$, the 3$\sigma$ upper limits for the magnitudes of all three are comparable to one another in our 13 parameter model (48, 30, and 119 km/sec, respectively). Since there are no direct contraints on these quantities that can improve these limits, we have allowed them to be fully free parameters. However, if we [*assume*]{} that the black hole is stationary with respect to the Galaxy, we also need to consider the case of Vz set to zero[^4].
Point Mass Plus Extended Mass Distribution Analysis
---------------------------------------------------
Limits on an extended mass distribution within S0-2’s orbit are derived by assuming that the gravitational potential consists of a point mass and an extended mass distribution, and allowing for a Newtonian precession of the orbits (see, e.g., Rubilar & Eckart 2001). In order to do this, we use the orbit fitting procedure described in Weinberg et al. (2005), and adopt an extended mass distribution that has a power-law density profile $\rho(r)=\rho_0(r/r_0)^{-\gamma}$. This introduces two additional parameters to the model: the normalization of the profile and its slope $\gamma$. The total enclosed mass is then given by $$M(<r) = M_{\rm BH} + M_{\rm ext}(<r_0) \left(\frac{r}{r_0}\right)^{3-\gamma},$$ where we quote values for the normalization $M_{\rm ext}(<r_0)$ at $r_0=0.01\textrm{ pc}$, corresponding to the characteristic scale of the orbit. Figure \[extended\] shows the constraint on $M_{\rm ext}(<0.01\textrm{ pc})$ and $\gamma$ from a fit to the astrometric and radial velocity measurements for S0-2. The 99.7% confidence upper-bound on the extended mass is $M_{\rm ext}(<0.01\textrm{ pc})\simeq 3-4 \times 10^5 M_\odot$ and has only a weak dependence on $\gamma$.
Mouawad et al. (2005) report a similar upper-bound on the extended mass in fits to the orbit of S0-2. Their analysis differs only slightly from that presented here in that it forces the focus to be at the inferred radio position of Sgr A\*, assumes a Plummer model mass distribution, and is based on data presented in Eisenhauer et al. (2003). Similarly, Zakharov et al. (2007) use an order of magnitude analysis to show that if the total mass of the extended matter enclosed within the S0-2 orbit is $\ga 10^5 M_\odot$, then it would produce a detectable apocenter shift $\Delta \phi \ga 10\textrm{ mas}$ (see also § 3.2 in Weinberg et al. 2005). Hall & Gondolo (2006) fit the total measured mass concentration $M(<r)$ given in Ghez et al. (2005) assuming a power-law density profile and obtain an upper bound of $\approx 10^5 M_\odot$ between $0.001-1\textrm{ pc}$.
The surface brightness of stars as a function of projected radius from Sgr A$^\ast$ is well measured down to a radius of $\sim$05 ($\sim$0.02 pc). With an assumed constant mass to light ratio, the inferred stellar mass distribution between this inner radius and an outer radius of 10$\arcsec$ is consistent with $$M_{\ast}(<r) = (6\times10^5 M_\odot) \left(\frac{r}{0.4\textrm{ pc}}\right)^{1.6}$$ (Genzel et al. 2003b; see also Schödel et al. 2007). Extrapolating this profile down to a radius of $0.01\textrm{ pc}$ gives an enclosed mass $M_{\ast}(<0.01\textrm{ pc}) \approx1-2\times10^3M_\odot$. Furthermore, theoretical estimates of the density of cold dark matter halo particles suggest that $\sim 1000 M_\odot$ of dark matter might reside in the inner $0.01 \textrm{ pc}$ of the GC (Gondolo & Silk 1999; Ullio et al. 2001; Merritt et al. 2002, Gnedin & Primack 2004). Likewise, the mass contribution from a cluster of stellar remnants, as predicted by Morris (1993) and Miralda-Escudé & Gould (2000), is expected to be $\sim 1000 M_{\odot}$ within 0.01 pc. Unfortunately, these estimates are all smaller than the current upper-bound by a factor of $\approx100$. Measurements of stellar orbits with a next generation large telescope are, however, expected to be sensitive to an extended mass distribution of magnitude $M_{\ast}(<0.01\textrm{ pc}) \approx10^3M_\odot$ (Weinberg et al. 2005).
DISCUSSION {#sec:disc}
==========
Orbit modeling of astrometric and radial velocity measurements of short period stars provides a direct estimate of the Milky Way’s central black hole mass and distance. Our analysis of S0-2’s orbit yeilds a black hole mass of M$_{bh}$ = 4.1 $\pm$ 0.6 $\times$ 10$^6 M_{\odot}$ and distance of R$_0$ = 8.0 $\pm$ 0.6 kpc, if nothing is assumed about the black hole’s intrinsic motion. If we assume that the black hole has no intrinsic motion relative to the central stellar cluster (i.e., no massive companion), but still allow for systematics in the reference frames, then we obtain M$_{bh}$ = 4.5 $\pm$ 0.4 $\times$ 10$^6 M_{\odot}$ and distance of R$_0$ = 8.4 $\pm$ 0.4 kpc. This study shows that there are three systematic errors that must be accounted for to obtain accuracy in estimates of orbital parameters and this leads to larger uncertainties than have been reported in the past. First, since a dominant source of systematic error in the data set appears to be source confusion (see §3 & 4), we use only data from the brightest short orbital period star, S0-2, and only those measurements that are not confused with other known sources. Second, the motion of the black hole relative to the measurements’ reference frame should be left as a free parameter, to account for both any possible intrinsic motion of the black hole as well as systematics in the astrometric or spectroscopic reference frames. Third, while SgrA\*-IR is detected with a precise position in deep LGSAO images, it appears to be biased; therefore, the position of the black hole should be treated as a free parameter in the fits in spite of the temptation to reduce the degrees of freedom with this detection. Because these systematics were not incoporated into earlier simultaneous estimates of M$_{bh}$ and R$_0$ from the orbit of S0-2, the uncertainties in these initial studies were significantly underestimated; Eisenhauer et al. (2003, 2005), who do not account for the first two systematics, obtain M$_{bh}$ = $3.6 \pm 0.3 \times 10^6 M_{\odot}$ and R$_0$ = 7.6 $\pm$ 0.3 kpc. Ghez et al. (2005a) used S0-2, S0-16, and S0-19 simultaneously, and allowed $V_x$ and $V_y$ to be free parameters, to derive a mass at fixed R$_o$ of 3.7 $\pm$ 0.2 $\times$ 10$^6$ (R$_0$/ 8 kpc)$^3$ M$_{\odot}$, which was pulled down by the two astrometrically-biased fainter stars, while Ghez et al. (2003) obtained a mass estimate of 4.1 $\pm$ 0.6 $\times$ 10$^6$ (R$_0$/ 8 kpc)$^3$ M$_{\odot}$ from S0-2 alone. If we ignore the first two effects in model fits to our data, as was done by Eisenhauer et al. (2003, 2005; the only other work to estimate R$_0$ from orbits), we obtain a poor quality fit ($\chi^2/dof$ = 2.0), uncertainties that are a factor of 2 smaller, and somewhat higher values than what we report in Table \[tbl\_orb\] ($M_{bh}$ = $4.7 \pm 0.3 \times 10^6 M_{\odot}$ and R$_0$ = 8.6 $\pm$ 0.2 kpc). The removal of biased astrometric points dominates the shift in the black hole’s mass and distance to lower values in our analysis. This is somewhat suprising as this would suggest that similar removal of biased points might lower the Eisenhauer et al. (2005) results. However the biases may differ, as their early astrometric data measurements were made at three times lower angular resolution. An astrometric reference frame drift could also explain this effect, since $V_x$ and $V_y$ were held fixed in their analysis. The addition of $V_z$ as a free parameter dominates the resulting uncertainties. In summary, in order to get an accurate measure of M$_{bh}$ and R$_0$ from modeling of the short period orbits at the Galactic center, it is critical to account for the three sources of systematics described above.
The black hole mass measured here from a stellar orbit is larger than the $\sim 2-3 \times 10^6 M_{\odot}$ inferred from using projected mass estimators, which rely on measured velocity dispersions (e.g, Eckart & Genzel 1997; Genzel et al 1997; Ghez et al. 1998; Genzel et al. 2000; see also Chakrabarty & Saha 2001). This difference most likely arises from the assumptions intrinsic to the use of projected mass estimators. In particular, the projected mass estimators are based on the assumption that the entire stellar cluster is measured, which is not the case for the early proper motion studies as their fields of view were quite small (r $\sim$ 0.1 pc). Such pencil beam measurements can lead to significant biases (see discussions in Haller et al. 1996; Figer et al. 2003). An additional bias can arise if there is a central depression in the stellar distribution, such as that suggested by Figer et al. (2003). These biases can introduce factors of 2 uncertainties in the values of the enclosed mass obtained from projected mass estimates and thereby account for the difference between the indirect mass estimate from the velocity dispersions and the direct mass estimate from the orbital model fit to S0-2’s kinematic data.
A higher mass for the central black hole brings our Galaxy into better agreement with the $M_{bh} - \sigma$ relation observed for nearby galaxies (e.g., Ferrarese & Merritt 2000; Gebhardt et al. 2000; Tremaine et al. 2002). For a bulge velocity dispersion that corresponds to that of the Milky Way ($\sim$103 km s$^{-1}$; Tremaine et al. 2002), the $M_{bh}-\sigma$ relationship from Tremaine et al. (2002) predicts a black hole mass of 9.4 $\times$ 10$^6$ M$_{\odot}$, which is a factor of 5 larger than the value of the Milky Way’s black hole mass used by these authors (1.8 $\times$ 10$^6$ M$_{\odot}$ from Chakrabarty & Saha 2001). The black hole mass presented here of 4.1 $\pm$ 0.6 $\times$ 10$^6 M_{\odot}$ brings the Milky Way more in line with this relationship. With one of the most accurate and lowest central black hole masses, the Milky Way is, in principle, an important anchor for the $M_{bh}-\sigma$ relationship. However, the velocity dispersion of the Milky Way is much more uncertain than that of other nearby galaxies. Therefore our revised mass has only modest impact on the coefficients of the $M_{bh}-\sigma$ relation.
Revision of the central black hole’s mass and distance can also, in principle, impact our understanding of the structure within our galaxy both on small and large scales. On the large scale, if we assume that the black hole is located at the center of our Galaxy, then its distance provides a measure of $R_0$. Its value from this study is consistent with the IAU recommended value of 8.5 kpc as well as the value of 8.0 $\pm$ 0.5 kpc suggested by Reid (1993), based on a “weighted average"[^5] of all prior indirect measurements of $R_0$. Combining the value for $R_0$ from this study with the proper motion of Sgr A\* along the direction of Galactic longitude measured with VLBA in the radio quasar reference frame (Reid & Brunthaler 2004; $\mu_{SgrA*,
long}$ = -6.379 $\pm$ 0.026 mas yr$^{-1}$) and the Sun’s deviation from a circular orbit (Cox 2000; 12 km s$^{-1}$) in the direction of Galactic rotation, we obtain an estimate of the local rotation speed, $\theta_0$, of 229 $\pm$ 18 km s$^{-1}$, which is statistically consistent with other measurements; these include a value of 222 $\pm$ 20 km s$^{-1}$ from the review of Kerr & Lynden-Bell (1986) and 270 km s$^{-1}$ derived by Méndez et al. (1999) from the absolute proper motions of $\sim$30,000 stars in the Southern Proper-Motion survey. As two of the fundamental Galactic constants, $R_0$ and $\theta_0$ are critical parameters for axisymmetric models of the Milky Way. Under the assumption that the stellar and gas kinematics within our Galaxy are well measured, the values of $R_0$ and $\theta_0$ determine the mass and shape of the Milky Way (Olling & Merrifield 2000; Olling & Merrifield 2001). Of particular interest is the value of the short-to-long axis ratio of the dark matter halo, q, as it offers a valuable opportunity to distinguish between different cosmological models. As Olling & Merrifield (2001) demonstrate, the uncertainty in q for the Milky Way is dominated by the large uncertainties in $R_0$ and $\theta_0$. While our uncertainties in R$_0$ are currently too large to constrain q, future precision measurements of R$_0$ through stellar orbits may be able to do so and could thereby possibly distinguigh between various dark matter candidates (Olling & Merrifield 2001). Closer to the black hole, knowing its mass and distance from the Sun improves our ability to study the kinematics of stars within its sphere of influence. Much less kinematic information is needed to determine the orbital parameters for stars whose motion is dominated by the gravitational influence of the central black hole; for instance, with only measurements of a star’s position, velocity, and accelaration in the plane of the sky along with a single line of sight velocity, a complete orbital solution can be derived once the black hole’s mass and distance are well contrained. Improved constraints on the central black hole’s properties and their degeneracies, as presented here, along with improved astrometry, has allowed us to derive orbital information for individual stars at much larger galacto-centric distances. With these measurements, in Lu et al. (2006, 2008), we test for the existence and properties of the young stellar disk(s), proposed by Levin & Beloborodov (2003) and Genzel et al. (2003b) from a statistical analysis of velocities alone. The direct use of individual stellar orbits out beyond a radius of 1$\tt''$ reveals only one, relatively thin, disk of young stars (Lu et al. 2008).
On an even smaller scale, the mass and distance of the black hole set the magnitude and time-scale for various relativistic effects. Given estimated Keplerian orbital elements for stars at the Galactic center, we expect to be able to measure their stellar orbits with sufficient precision in upcoming years to detect the Roemer time delay, the special relativistic transverse Doppler shift, the general relativistic gravitational red-shift, and the prograde motion of periapse (e.g., Weinberg et al. 2005; Zucker & Alexander 2007). These effects will most likely be measured with S0-2 first, as it has the shortest orbital period (P=15 yr), is quite eccentric (e=0.89) and, as one of the brighter stars ($K_{S0-2}$ = 14 mag), it can be measured with the greatest astrometric and spectroscopic accuracy. The radial velocity signatures of the first three effects are expected to be comparable to each other and will impart a $\sim$200 km/s deviation at closest approach (Zucker & Alexander 2007), when the star is predicted to have a line of sight velocity of -2500 km/s based on our updated Keplerian model. This effect is large compared to the radial velocity precision ($\sim$ 20 km/sec). Likewise, the expected apoapse center shift for S0-2, $\Delta s = \frac{6 \pi G M_{bh}}{R_0 (1-e) c^2} = 0.9$ mas (see e.g., Weinberg 1972; Weinberg et al. 2005), is an order of magnitude larger than our current measurement precision ($\sigma_{pos}$ $\sim$ 0.1 mas). Improved adaptive optics systems on existing telescopes and larger telescopes (see Weinberg et al. 2005) will improve the sensitivity to the predicted apocenter shift. To put this measurement into context with existing tests of general relativity, it is useful to note that one of the strongest constraints on general relativity to date comes from the Hulse-Taylor binary pulsar, PSR 1913+16, which has a relativistic parameter at periapse, $\Gamma = r_{sch} / r_{periapse}$, of only $5 \times 10^{-6}$, $\simeq$ 3 orders of magnitude smaller than that of S0-2 (Taylor & Weisberg 1989; Zucker & Alexander 2007). The stars at the Galactic center are therefore probing an unexplored regime of gravity in terms of the relativistic object’s mass scale and compactness.
Precession from general relativistic effects also influences the timescale for resonant relaxation processes close to the black hole (see, e.g, Rauch & Tremaine 1996; Hopman & Alexander 2006). When precession from general relativity dominates over that from the extended mass distribution, the resonant relaxation timescale is proportional to $M_{bh}^2 \times (J_{LSO}/J)^2 \times P$, where $J$ and $J_{LSO}$ are the orbital angular momenta for the orbit of interest and at the last stable circular orbit around the black hole, respectively, and P is the orbital period. For a given semi-major axis and accounting for the linear mass dependence of $(J_{LSO}/J)^2$, this results in a $M_{bh}^{5/2}$ dependency. Thus the higher black hole mass inferred from this study increases the timescale over which the black hole’s loss cone would be replenished in the regime where general relativity dominates. For the regime where the extended mass distribution dominates, the resonant relaxation timescale scales only as $M_{bh}^{1/2}$. A higher black hole mass also implies a longer period for the innermost stable circular orbit. If the central black hole is non-spinning, the innermost stable circular orbit has a period of 31 $\frac{M_{bh}}{4.1 \times 10^6 M_{\odot}}$ min. Periodicities on shorter timescales, such as the putative quasi-periodic oscillation (QPO) at $\sim$ 20 min (Genzel. et al. 2003a; Eckart et al. 2006; Bélanger et al. 2006) have been interpreted as arising from the innermost stable circular orbit of a spinning black hole. At the present mass, the spin would have to be 0.6 of its maximal rate to be consistent with the possible periodicity. However, it is important to caution that other mechanisms can give rise to such short periodicities, such as a standing wave pattern recently suggested by Tagger & Melia (2006). Furthermore, claims of a QPO in SgrA\* have been called into question; Do et al. (2008) find that the near-IR temporal power spectrum of SgrA\* is statistically consistent with pure red noise, such as might be caused by disk instabilities or intermittent jet fluctuations, and Belanger et al. (in preparation) reach a similar conclusion for the X-rays variations.
CONCLUSIONS
===========
The short orbital period star S0-2 has been intensively studied astrometrically (1995-2007) and spectroscopically (2000- 2007) with the W. M. Keck 10 meter telescopes. Fits of a Keplerian orbit model to these data sets, after removing data adversely affected by source confusion, result in estimates of the black hole’s mass and distance of $4.1 \pm 0.6 \times 10^6 M_{\odot}$ and 8.0 $\pm$ 0.6 kpc, respectively. While the current analysis is dominated by 11 years of astrometric measurements that have $\sim$ 1.2 mas uncertainties, the LGSAO over the last 3 years have positional uncertainties that are an order of magnitude smaller (100-200 $\mu$as). With higher strehl ratios and more sensitivity, LGSAO measurements are also less affected by source confusion; this is especially important for the closest approach measurements, which have to contend with source confusion from the variable source SgrA\*-IR. Following S0-2 for another 10 years should result in the measurement of the Sun’s peculiar motion in the direction of the Galactic center from the orbit of S0-2 with a precision of a few km s$^{-1}$ and 1% measurement of $R_0$. At this precision, the measurement of $R_0$ is of particular interest because it could reduce the uncertainty in the cosmic distance ladder. We thank the staff of the Keck observatory, especially Joel Aycock, Randy Campbell, Al Conrad, Jim Lyke, David LeMignant, Chuck Sorensen, Marcos Van Dam, Peter Wizinowich, and director Taft Armandroff, for all their help in obtaining the new observations. We also thank Brad Hanson, Leo Meyer, and Clovis Hopmann for their constructive comments on the manuscript, and the referee, Rainer Schodel, for his helpful suggestions. Support for this work was provided by NSF grant AST-0406816 and the NSF Science & Technology Center for AO, managed by UCSC (AST-9876783), and the Levine-Leichtman Family Foundation. The W. M. Keck Observatory, is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.
Cluster Reference Frame {#app:cluster_ref_frame}
=======================
All positional measurements from the individual images ($X', Y'$) are transformed with a full first order polynomial to a common reference system ($X, Y$), which we refer to as the cluster reference frame (see Ghez et al. 1998, 2000, 2005a; Lu et al. 2008). The transformations are derived by minimizing the net displacements, allowing for proper motions, of all the coordinate reference stars (see §3.1) relative to their positions in a common reference image ($ref$), which for this study is the 2004 Jul LSGAO image. Specifically, we minimize the following sum over the coordinate reference stars ([*s*]{}): $$D = \displaystyle\sum_{s}^{N_{stars}} (\Delta X_{s, e}^2 + \Delta Y_{s,e}^2) / W_{s,e},$$ where $\Delta X_{s,e} = X_{s,ref} + V_{x_s} \times (t_e - t_{ref}) - X_{s,e}$, $\Delta Y_{s,e} = Y_{s,ref} + V_{y_s} \times (t_e - t_{ref}) - Y_{s,e}$, and $W_{s,e} = \sigma_{\Delta X'_{s,e}} + \sigma_{\Delta Y'_{s,e}}$, and where $X$ and $Y$ are expressed as the following function of the measured positions $X'$ and $Y'$ for for each epoch ($e$) $$X_e = a_{0_e} + a_{1_e} \times X'_e + a_{2_e} \times Y'_e,$$ $$Y_e = b_{0_e} + b_{1_e} \times X'_e + b_{2_e} \times Y'_e.$$ The coefficients for the reference epoch are fixed to $a_{1_{ref}} = b_{2_{ref}} = 1$ and $a_{0_{ref}} = a_{2_{ref}} = b_{0_{ref}} = b_{1_{ref}} = 0$ and the coefficients for the remaining epochs ($a_{0_e}$, $a_{1_e}$, $a_{2_e}$, $b_{0_e}$, $b_{1_e}$, and $b_{2_e}$) come from the minimization of $D$. Because of degeneracies between coordinate transformations and proper motions of the coordinate reference stars, the net displacment is minimized in two steps. First, $D$ is minimized with the proper motions ($V_x$ and $V_y$) of the coordinate reference stars set to zero in order to obtain preliminary transformation coefficients. Using these initial coeffficients, we transform all the positional measurements to a common coordinate system and fit a linear motion model to them in order to derive a first pass estimate of the proper motions. Second, $D$ is minimized again, using the preliminary proper motions and holding them fixed, while the final transformation coefficients are derived. This procedure produces proper motions for the coordinate reference stars that have no significant mean motion. We therefore conclude that the resulting cluster reference frame is stable and free of significant systematics.
This procedure is also used to check the stability of the combined effects of the camera systems and the coordinate reference stars. By carrying out transformations that allow for only translation, rotation, and a scale change, we examine the apparent stability of the camera’s pixel scale and angle relative to that recorded in the header. Figure \[trans\] shows, on the left-hand side, that the relative pixel scales for the cameras are stable to within 0.053% (rms) over the time baseline of this study and that the uncertainty in the angle relative to the header PA is dominated by inaccuracies in the header value (most of the jumps correspond to times when the camera is known to have been opened for engineering purposes). This also provides a measure of the resampled NIRC pixel scale relative to the NIRC2 pixel scale (1.0269 $\pm$ 0.0005) and an absolute NIRC pixel scale of 20.46 $\pm$ 0.01 mas pix$^{-1}$ when combined with the absolute NIRC2 pixel scale from Appendix C. On the right-hand side, Figure \[trans\] displays the results of the same excerise but using a set of coordinate reference stars that includes the known young stars; the clear systematic trend in the relative pixel scales demonstrates the importance of removing this set of stars with known net rotation from the coordinate reference star list.
NIRC2 Geometric Optical Distortions {#app:nirc2_distortion}
===================================
Relative stellar positions from the deep LGSAO images have accuracies ($\sim$0.2 mas) that are an order of magnitude smaller than the currently available optical distortion map for NIRC2 (<http://www2.keck.hawaii.edu/inst/nirc2/preship_testing.pdf>). Since LGSAO/NIRC2 data was obtained with four different setups (e.g., centerings and/or position angles on the sky), imperfections in the optical distortion corrections can introduce 1-2 mas systematics, if unaccounted for, into the relative positions of S0-2 (and S0-16). We therefore introduce two steps into our analysis to correct for this effect. First, we add, in quadrature, an additional 0.88 mas to all the LGSAO positional measurements of the coordinate reference stars, such that the proper motions and hence coordinate transformations are not biased. The magnitude of this term is derived by finding the value that reduced the average offset of these LGSAO points from the linear proper motion fits, which exclude these points, from 5$\sigma$ to 1$\sigma$. Second, we derive explicit correction terms for the local optical distortions for S0-2 and S0-16 positions in each of LGSAO epochs not obtained with the same set up as the reference image (2004 July), using the orbits of 5 “calibration" stars (S0-3, S0-7, S0-19, S0-26, and S0-27) that are within 05 of S0-2. These terms are obtained by first using only the speckle data, which are distortion calibrated with respect to the reference image (2004 July/LGSAO; see Lu et al. 2008), the reference image (taken with setup\#1), and the one other LGSAO image taken with the same setup as the reference image to solve for the orbits of the 5 calibration stars. For each LGSAO epoch not included in these fits, the average offsets of these five stars’ aligned measurements from their predicted location is used to characterize the residual distortions for that image (relative to the reference image) at the position of S0-2 and S0-16 and the standard deviation of the offsets provide an estimate of the uncertainties in these values. Setup \#3 is the only LGSAO observational configuration, other than that used for the reference image, used in multiple epochs. From the measurements with setup \#3, it can be seen that the rms of their estimated bias terms (0.24 mas) is smaller than the uncertainty in each bias term estimated from the rms of the 5 stars ($\sim$ 0.67 mas). This suggests that the bias terms are relatively static (see also Appendix A) and that their uncertainties are dominated by our uncertainties in the stellar orbits (and possible structure in the distortion on scales $<$05). We therefore derive an average bias correction value and uncertainty for each setup. The final bias terms, which range in value between 1.6 and 2.6 mas, are added to the LGSAO positional measurements made with setups \#2-4 in the analysis presented in §\[sec:orbit\] and their uncertainties are added in quadrature with the uncertainties associated with centroiding and coordinate transformation; this bias term as already been incorporated into the values and uncertainties reported in Table \[tbl\_pos\]. Correlations in the bias corrections for setup \#3 are applied and accounted for in the Monte Carlo simulations described in §4.
Absolute Astrometry {#app:absolute_astrometry}
===================
An absolute astrometric reference frame for the Galactic center was established from radio observations of seven SiO masers (Reid et al. 2003, 2007). Relative measurements in the infrared were tied to the absolute frame by observing, in the infrared, the red-giant stars that are the source of the maser emission (Figure \[fig:masers\] & Table \[tab:masers\]). Observations were taken in 2005 June, 2006 May, and 2007 August using LGSAO/NIRC2 (see §\[sec:obs\]) with $10.86 ~ \textrm{s}$ integrations in the K’ band, each composed of 60 co-added 0.181 s exposures in order to avoid saturating the bright masers. A nine position dither box pattern was used to construct a 22$\arcsec\times$22$\arcsec$ mosaic with two exposures at each position for the 2005 mosaic and three exposures at each position for the 2006 and 2007 mosaics. The individual frames for each data set were cleaned, undistorted, and then registered and mosaicked using the IRAF [*xregister*]{} and [*drizzle*]{} routines. Subset-mosaics were also created with only 1 exposure at each position and were used to derive centroiding uncertainties. StarFinder was run on the resulting mosaicked images to extract stellar positions and uncertainties from the RMS error of the subset-mosaics. Centroiding errors were typically on the order of 1.4 mas. This yields an IR starlist for each epoch with positions in NIRC2 pixel coordinates.
The radio maser positions were propagated forward using velocities from Reid et al. 2007 to create a radio maser starlist at the epoch of each of the above IR mosaics. Uncertainties in these propogated radio positions are, on average, $\sim$1.4 mas. For each epoch, the IR maser starlist was aligned to the Radio mosaic starlist, which resulted in a new IR mosaic starlist in the absolute astrometric reference frame with Sgr A\*-Radio at the origin. This alignment process used only four independent parameters (a global pixel scale, a rotation, and an origin in the x and y directions) to transform between the NIRC2 coordinate system of the IR mosaics to the absolute coordinate system of the radio masers. While using higher order polynomial transformations reduce the residual offsets positions from SgrA\* between the infrared and radio measurements, we conservatively chose to use this low order transformation to capture within the uncertainties the possible impact of systematics, such as uncorrected residual camera distortions and differential atmospheric refraction. This is particularly important given the sparse sampling of masers across the rgion of interest (see Figure \[fig:masers\]). Uncertainties in the transformation to absolute coordinates, which were determined with a half-sample bootstrap Monte Carlo simulation of 100 iterations where each iteration uses only half the stars in each starlist, were added in quadrature to the infrared centroiding uncertainties to produce a final uncertainites in the infrared absolute positions of the masers. After the transformation to absolute coordinates, the absolute value of the offsets between the positions of the masers relative to SgrA\* measured in the infrared and radio are on average 0.8$\sigma$ and 0.8 $\sigma$, or equivalenly, 5.7 mas and 5.7 mas in the x and y direction, respectively (see Table \[tab:masers\]); we take this to be our uncertainty in the position of Sgr A\*-Radio in the infrared maser mosaic. Likewise, the transformations between the infrared and radio reference frame yields a plate scale of 9.963 $\pm$ 0.005 mas/pix and a position angle offset for NIRC2 of 0.13$^\circ \pm$ 0.02$^\circ$. Each of the three infrared maser mosaics yields comparable results (see Table \[tab:nirc2astrometry\]). Uncertainties in the absolute positions in the infrared reference frame are dominated by residual optical distortions, which are amplified by the large dithers necessary to construct the mosaics.
A comparison of the maser’s proper motions as measured in the radio and the infrared provides an estimate of how accurately we can transform our relative measurements into a reference frame in which SgrA\*-Radio is at rest and the orientation is set by background quasars (Reid et al. 2007). The absolute infrared proper motions of the masers, as well as all other stars detected in the infrared maser mosaics, were derived by fitting a linear model to the positions as a function of time from the three IR maser starlists that were separately aligned to the radio reference frame. Because the alignment uncertainties are dominated by residual distortion and therefore correlated across epochs for a given maser, this source of uncertainty is not included in the linear proper motion modeling. The differences in the proper motions measured in the radio and in the infrared have an average value of 0.6 $\pm$ 0.4 mas/yr and -0.9 $\pm$ 0.6 mas/yr in the x and y directions, respectively, where the uncertainties are the standard deviation of the mean. Therefore, at present, it is not possible to use these measurements to eliminate possible drifts in the cluster reference frame as the source of any apparent $V_x$ or $V_y$ from the orbital fits of S0-2 (see §4.1).
The relative astrometry measurements presented in §3 were transformed into this absolute reference frame through a set of infrared stars we designated as [*infrared absolute astrometric standards*]{}. Absolute astrometric standards were defined to be those stars that are 1) detected in all three IR mosaics (2005, 2006, 2007), 2) outside the central arcsecond (r$>$05), 3) have velocities less than 15 mas/yr and velocity errors less than 5 mas/yr, 4) have reasonable velocity fits ($\chi^2/dof < 4$), and 5) are brighter than K=15. With absolute kinematics for 158 stars within 5”, we solve for a 4 parameter transformation model by comparing the relative positions in the reference epoch image, which are in instrumental pixel coordinates, and the estimated absolute coordinates for that epoch, which are in arcsec relative to the position of SgrA\*. Since all other epochs are aligned to this reference epoch, positional measurements for all stars in all epochs are easily transformed into absolute coordinates. While uncertainties in the absolute infrared reference frame dominate the final absolute positional uncertainties relative to SgrA\*-Radio, they are a negligible source of uncertainty for the orbital analysis.
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[^1]: Model taken from the 1993 Kurucz Stellar Atmospheres Atlas (<ftp://ftp.stsci.edu/cdbs/cdbs2/grid/k93models/standards/vega_c95.fits>)
[^2]: While the best values from minimizing $\chi^2$ can differ slightly (but well within the uncertainties) from the peak of the $PDF(O_i)$ values, this has negligible impact on the reported uncertainties.
[^3]: The uncertainty in the mass scaling relationship is obtained for the case in which R$_0$ is fixed to 8.0 kpc and therefore does not include the uncertainty in R$_0$.
[^4]: Allowing for the uncertainty in the LSR in $V_z$ ($\pm$ 2 km/sec; Gould 2004) produces results that are not distinguishable from those reported for the $V_z$ = 0 case.
[^5]: consensus value with consensus errors
|
---
abstract: |
Gross substitutability is a central concept in Economics and is connected to important notions in Discrete Convex Analysis, Number Theory and the analysis of Greedy algorithms in Computer Science. Many different characterizations are known for this class, but providing a constructive description remains a major open problem. The construction problem asks how to construct all gross substitutes from a class of simpler functions using a set of operations. Since gross substitutes are a natural generalization of matroids to real-valued functions, matroid rank functions form a desirable such class of simpler functions.
Shioura proved that a rich class of gross substitutes can be expressed as sums of matroid rank functions, but it is open whether all gross substitutes can be constructed this way. Our main result is a negative answer showing that some gross substitutes cannot be expressed as positive linear combinations of matroid rank functions. En route, we provide necessary and sufficient conditions for the sum to preserve substitutability, uncover a new operation preserving substitutability and fully describe all substitutes with at most $4$ items.
author:
- |
Eric Balkanski\
Harvard University
- |
Renato Paes Leme\
Google Research
bibliography:
- 'sigproc.bib'
title: On the Construction of Substitutes
---
Introduction
============
The concept of gross substitutes (GS) occupies a central place in different areas such as economics, discrete mathematics, number theory and has been rediscovered in many contexts. In Economics it was proposed by @KelsoCrawford as a sufficient (and in some sense necessary [@GulStachetti]) condition for Walrasian equilibrium to exist in economies with indivisible goods. The notion also appears in the existence of stable matchings in two sided markets [@HatfieldMilgrom; @Roth84], to design combinatorial auctions [@ausubel2002ascending], in the study of trading networks [@hatfield2013stability; @ikebe2015stability], among others. In fact, @hatfield2015hidden show that many tractable classes of preferences with complementarities have a hidden substitutability structure. The phenomenon of substitutes being embedded in more complex settings is also present in [@sun2009double; @ostrovsky2008stability].
In discrete mathematics, @MurotaShioura define the concept of $M^\natural$-concave function which ports the concept of convex functions from continuous domains to the discrete lattice, carrying over various strong (Fenchel-type) duality properties (see [@murota2016discrete] for a recent survey on discrete convex analysis). In number theory, @DressWenzel defined the concept of valuated matroids to generalize the Grassmann-Plücker relations in $p$-adic analysis. Those correspond to the same class of functions as shown by @FujishigeYang.
Finally, gross substitutes have a special role in computer science, since they correspond to the class of set function $v:2^{[n]} \rightarrow {\mathbb{R}}$ for which the optimization problem $\max_{S \subseteq [n]} v(S) - \sum_{i \in S} p_i$ can be solved by the natural greedy algorithm for all $p \in {\mathbb{R}}^n$.
Given the central role that GS plays in many different fields, understanding its structure is an important problem. There have been many equivalent characterizations of GS through time: @KelsoCrawford, @Murota96, @MurotaShioura, @DressTerhalle_WellLayered [@DressTerhalle_Rewarding], @ausubel2002ascending, @ReijnierseGellekomPotters, @LehmannLehmannNisan, @ben2017walrasian. All the previous characterizations define GS as the class of functions satisfying a certain property. Yet, providing a constructive description of GS remains an elusive open problem. This is in sharp contrast with submodular functions, a more complex class in many respects, but one that has a simple constructive description.
#### The construction problem.
A constructive description consists of a base class of simpler functions (e.g. unit demand or matroid rank functions) together with a set of operations (e.g. sum, convolution, endowment, affine transformations) such that all GS functions can be constructed from the base class by applying such operations.
The first version of this question was asked by @HatfieldMilgrom. They noted that most examples of substitutes arising in practical applications could be described as valuations that are built from assignment valuations (which are convolutions of unit-demand valuations) and the endowment operation. They called this class *Endowed Assignment Valuations* (EAV) and asked whether EAV exhaust all gross substitutes. @ostrovskypaesleme2014 provided a negative answer to this question showing that some matroid rank functions cannot be constructed using those operations. They do so by adapting a result of @brualdi1969comments on the structure of transversal matroids to the theory of gross substitutes. The main insight in [@ostrovskypaesleme2014] is that unit-demand valuations are not strong enough as a base class. They propose a class called *Matroid Based Valuations* (MBV) which are constructed from weighted matroid rank functions. It is still unknown whether MBV exhausts the whole class of substitutes or not.
Another important construction is due to @shioura2012matroid, who provides a construction of a rich class of valuations called *Matroid Rank Sums* (MRS) and shows that this class contains many important examples of gross substitutes. Matroid rank sums are positive linear combinations of matroid rank functions such that the matroids satisfy a *strong quotient* property. In general positive linear combinations of matroids are not GS but the strong quotient property provides a sufficient condition for this to be true.
#### GS and matroids.
Gross substitutes are similar to matroids in many respects. First, matroids can be described as the subset systems that can be optimized via greedy algorithms while GS is the collection of real-valued set functions that can be optimized using these same greedy algorithms. Another similarity is that both classes can be described via the exchange property. Finally, when we restrict our attention to GS functions where marginals are in $\{0,1\}$ we obtain exactly the set of matroid rank functions. These similarities, among others, explain why GS are seen as the natural extension of matroids to real-valued functions.
The lack of constructive characterization for GS, combined with GS naturally generalizing matroids, begs the following question.
*Can gross substitutes be constructed from matroids?*
In other words, do all GS functions look like matroids or does the lift from subset systems to real-valued functions produce significantly different functions? Similarly as for Shioura’s construction, we focus on positive linear combinations and ask the question of whether all gross substitutes are positive linear combinations of matroids.
#### Our results.
Our main result is a negative answer to the question above. We show that not all gross substitute functions can be constructed via positive linear combinations from matroid rank functions. This implies in particular that MRS does not exhaust the class of GS functions.
The proof consists of exhibiting a GS valuation function over $5$ elements that cannot be expressed as a positive linear combination of matroids. We note that there are $406$ matroids over $5$ elements ($38$ up to isomorphism) and that a GS function over $5$ elements is defined by $40$ non-linear conditions. The search space is huge, continuous and non-convex, so solving it by enumeration is infeasible even for $5$ elements.
Our techniques involve building a combinatorial and polyhedral understanding of GS functions. In fact, we prove that for $3$ and $4$ elements, all GS functions can be written as convex combinations of matroids. In this process, we provide a polyhedral understanding of the set of GS functions and give necessary and sufficient conditions for the positive combination of two GS functions to be GS. We obtain the counter-example for $n=5$ by carefully understanding where the techniques used for proving the $n=3$ and $4$ fail.
Upon obtaining an example for which the proof technique fails, we still need to argue that it is not in the convex combination of matroids. One way to do it is to enumerate over the $406$ matroids over $5$ elements and solve a large linear program. That would be a valid approach, but one that would make verifying correctness a much more complicated task. Instead, we use a combination of linear algebra and combinatorial facts about matroids to provide a complete mathematical proof that our counter-example is indeed not a convex combination of matroids.
#### Paper organization.
We begin with preliminaries in Section \[sec:prelim\]. In Section \[sec:building\_block\], we discuss the main question of obtaining a constructive characterization of GS. We prove our main result in Section \[sec:counterexample\], that there exists a GS function that is not a positive linear combination of matroid rank functions. In Section \[sec:tree\], we present the tree-concordant-sum operation, show that it preserves substitutability, use it to show a positive answer to our main question for $n \leq 4$, and discuss how it helped in finding the negative instance for the main result. The conclusion is in Section \[sec:conclusion\].
Preliminaries {#sec:prelim}
=============
#### Valuation functions
A valuation function is a map $v:2^{[n]} \rightarrow {\mathbb{R}}$. We restrict ourselves to functions defined on the hypercube $2^{[n]}$ although the notions studied here generalize to the integer lattice ${\mathbb{Z}}^n$. Given a vector $p \in {\mathbb{R}}^n$ we define $v_p$ as the function $$v_p(S) = v(S) - \sum_{i \in S} p_i.$$
#### Affine transformations and normalized valuations
We say that a valuation $\tilde{v}$ is an *affine transformation* of $v$ if there is a vector $p \in {\mathbb{R}}^n$ and a constant $c \in {\mathbb{R}}$ such that $\tilde{v} =
c + v_p$. We say that a valuation function $v$ is normalized if $v(\emptyset) = 0$ and $v(\{i\}) = 0$ for every $i \in [n]$. Every valuation function can be obtained from a normalized valuation function via an affine transformation. Unless otherwise specified, we consider matroid rank functions in their normalized form.
#### Marginals and discrete derivatives
Given sets $S, T \subseteq
[n]$ we define the marginal contribution of $S$ to $T$ as $v(S \vert T) = v(S \cup T) - v(T)$. We omit parenthesis when clear from context and often replace $v(\{i,j\}\vert S)$ and $S \cup
\{j\}$ by $v(i,j\vert S)$ and $S \cup j$ respectively.
Given a function $v:2^{[n]} \rightarrow {\mathbb{R}}$ and $i \in [n]$ we define the derivative with respect to element $i$ as the function $\partial_i v:2^{[n]\setminus i} \rightarrow {\mathbb{R}}$ where $\partial_iv(S) = v(S \cup i) - v(S)$. The first derivative is simply the marginal $v(i \vert S)$. Applying the operator twice we obtain the second derivative: $$\partial_{ij} v(S) = \partial_j [\partial_i v(S)] =
\partial_i v(S \cup j) - \partial_i v(S) = v(S\cup ij) - v(S \cup i) -
v(S\cup j) + v(S)$$ There is a nice economic interpretation of the second derivatives as a measure of the degree of substitutability of two goods. $\partial_{ij} v(S)$ represents the difference between the value of the bundle $\{i, j\}$ and sum of values of the two goods $i$ and $j$ separately. For example, if $\partial_{ij} v(S)= 0$, it means that having good $i$ does not affect the value for good $j$ conditioned on having a bundle $S$.
#### Functions as vectors
We often view valuation functions $v:2^{[n]} \rightarrow {\mathbb{R}}$ as a vector in ${\mathbb{R}}^{2^n}$ with coordinates indexed by $S \subseteq [n]$. This allows us to view a class of valuation functions as a subset of ${\mathbb{R}}^{2^n}$. We define the inner product between two valuations in the usual way: $$\langle v, w \rangle = \sum_{S \subseteq [n]} v(S) w(S).$$
Substitutability
----------------
There are several equivalent ways to define gross substitutes, often also called GS, discrete concave functions or simply substitutes. The definition that is most convenient to work with for our theorems is the definition via discrete derivatives due to @ReijnierseGellekomPotters . The set ${\ensuremath{\mathbf G}}^n$ of gross substitutes is defined by: $${\ensuremath{\mathbf G}}^n = \{ v:2^{[n]} \rightarrow {\mathbb{R}}; \text{ } \partial_{ij} v(S) \leq \max
[\partial_{ik}v(S), \partial_{jk}v(S)] \leq 0, \text{ } \forall S \subseteq [n],
\forall i,j,k \notin S \}$$ We note that the definition does not require monotonicity.
We also state some of the most common ways to define substitutes below. We refer to [@leme2017gross] for a proof that they are equivalent to the definition above as well as other formulations.
- [ **No price complementarity.**]{} In economics, substitutes was originally formulated as the condition that an increase in price for a certain good, cannot decrease the demand for other goods. Formally, given a vector $p \in
R^n$, let $D(v;p) = \operatorname{argmax}_{S \subseteq [n] } v_p(S)$ be the demand correspondence. Then $v \in {\ensuremath{\mathbf G}}^n$ iff for all vectors $p \leq p'$ and $S \in D(v;p)$ there is $S' \in D(v;p')$ such that $S \cap \{j; p_j = p'_j \} \subseteq S'$.
- [**Discrete concavity.**]{} In discrete mathematics, substitutes are a natural notion of concavity for functions defined in the hypercube. We say that a function over the reals $f : {\mathbb{R}}^n \rightarrow {\mathbb{R}}$ is concave if for every vector $p \in {\mathbb{R}}^n$, every local minimum of $f_p(x) = f(x) - \sum_{i} p_i x_i$ is also a global minimum[^1]. This definition naturally extends to the hypercube: $v \in {\ensuremath{\mathbf G}}^n$ iff for every $p \in {\mathbb{R}}^n$, if $S$ is a local minimum of $v_p$, i.e. $$v_p(S) \geq v_p(S \cup i) \quad v_p(S) \geq v_p(S \setminus j) \quad
v_p(S) \geq v_p(S \cup i \setminus j), \forall i \notin S, j \in S$$ then $S$ is also a global minimum, i.e. $S \in D(v;p)$.
- [**Matroidality.**]{} Substitutes can also be defined in terms of greedy algorithms: $v \in {\ensuremath{\mathbf G}}^n$ iff for every $p \in {\mathbb{R}}^n$, the greedy algorithm always computes the maximum of $v_p$, i.e., if we start with $S = \emptyset$ and keeps adding the element $i \in \operatorname{argmax}_{i} v_p(i \vert S)$ with largest marginal contribution while it is positive, we obtain $S \in D(v;p)$.
Matroids
--------
Many of those definitions strikingly resemble the definition of matroids. In fact some of the early appearances of gross substitutes were attempts to generalize matroids from collections of subsets to real-valued functions. @DressTerhalle_WellLayered [@DressTerhalle_Rewarding] called their definition *matroidal maps* and Dress and Wenzel called their notion *valuated matroids*.
A subset collection over $[n]$ is simply a subset ${\mathcal{M}}\subseteq 2^{[n]}$. This subset collection is a matroid if it satisfies one of the following equivalent properties:
- [**Greedy optimization.**]{} The collection ${\mathcal{M}}$ is a matroid if for every vector $p \in {\mathbb{R}}^n$, the set $S \in {\mathcal{M}}$ maximizing $\sum_{i \in S} p_i$ can be obtained by the greedy algorithm that starts with the empty set $S = \emptyset$ and keeps adding $i \in
\operatorname{argmax}_{i; S\cup i \in {\mathcal{M}}} p_i$ to $S$ while $p_i$ is positive.
- [**Exchange property.**]{} We say that collection ${\mathcal{M}}$ is a matroid if $T \subseteq S \in {\mathcal{M}}$ then $T \in {\mathcal{M}}$ and for every $S, T \in {\mathcal{M}}$ with ${\left\vert{S}\right\vert} < {\left\vert{T}\right\vert}$, there is $i \in T \setminus
S$ such that $S\cup i \in {\mathcal{M}}$.
Given any subset system ${\mathcal{M}}$, we can define its rank function $r_{\mathcal{M}}:2^{[n]} \rightarrow {\mathbb{Z}}_+$ as $r_{\mathcal{M}}(S) = \max \{ {\left\vert{T}\right\vert}; T \subseteq S \text{ and } T \in {\mathcal{M}}\}$. This allows us to define the set of matroid rank functions as: $${\ensuremath{\mathbf M}}^n = \{ r_{\mathcal{M}}; {\mathcal{M}}\text{ is matroid over } [n] \}$$ When we translate the subset system ${\mathcal{M}}$ to a rank function $r_{\mathcal{M}}$, the exchange property becomes exactly the discrete differential equation in the definition of ${\ensuremath{\mathbf G}}^n$. This observation implies that matroid rank functions are exactly the gross substitutes functions with $\{0,1\}$-marginals: $${\ensuremath{\mathbf M}}^n = \{v \in {\ensuremath{\mathbf G}}^n; v(\emptyset) = 0; \partial_i v(S) \in \{0,1\}, \forall S
\subseteq [n], {i \notin S} \}$$ For completeness, we provide a proof of this result in Theorem \[thm:matroid\_submodular\] in the appendix.
Relation between classes of functions {#sec:otherfunctions}
-------------------------------------
Another class that is important for us is submodular functions: $${\mathbf{S}}^n = \{v : 2^{[n]} \rightarrow {\mathbb{R}}; \partial_{ij}v(S) \leq 0, \forall S
\subseteq [n], i,j \notin S \}$$ The following relation between the classes hold: ${\ensuremath{\mathbf M}}^n \subseteq {\ensuremath{\mathbf G}}^n \subseteq {\mathbf{S}}^n$. The classes ${\ensuremath{\mathbf G}}^n$ and ${\mathbf{S}}^n$ are defined in terms of second order discrete derivatives, which are invariant under affine transformations, i.e, if $\tilde{v}$ is obtained from $v$ via an affine transformation then $\partial_{ij}v(S) = \partial_{ij}\tilde{v}(S)$. In particular this means that gross substitutes is invariant under affine transformations. If we want to understand ${\ensuremath{\mathbf G}}^n$ it is enough to understand the class ${\ensuremath{\mathbf G}}^n_0$ of normalized gross substitutes: $${\ensuremath{\mathbf G}}_0^n = \{v \in {\ensuremath{\mathbf G}}^n; v(\emptyset) = v(i) = 0, \text{ } \forall i \in [n]
\}$$ since we can describe ${\ensuremath{\mathbf G}}^n = {\ensuremath{\mathbf G}}_0^n + {\ensuremath{\mathbf E}}^n$ where ${\ensuremath{\mathbf E}}^n$ is the class of affine valuations functions: $$\textstyle {\ensuremath{\mathbf E}}^n = \{ v : 2^{[n]} \rightarrow {\mathbb{R}}; v(S) = c + \sum_{i \in S} p_i; c, p_i \in {\mathbb{R}}\}$$ It will also be convenient to define the notion of normalized matroid rank functions: $$\textstyle {\ensuremath{\mathbf M}}^n_0 = \{v : 2^{[n]} \rightarrow {\mathbb{R}}; \exists {\mathcal{M}}\text{ matroid s.t. }
v(S) = r_{\mathcal{M}}(S) - \sum_{i \in S} r_{\mathcal{M}}(i) \},$$ which are exactly the normalized gross substitutes functions ${\ensuremath{\mathbf G}}_0^n$ with $\{-1, 0\}$-marginals.
What are the building blocks of Substitutes ? {#sec:building_block}
=============================================
A major open question in the theory of gross substitutes is how to find a constructive description of the class. A constructive description has two parts: a base class of simpler functions and a set of operations that allow us to build complex functions from simpler ones. There are a number of operations that are known to preserve substitutability. We mention the two that are most relevant for this paper here and discuss additional operations in Section \[sec:conclusion\].
- [**Affine transformations.** ]{} If $v : 2^{[n]} \rightarrow {\mathbb{R}}$ satisfies gross substitutes and $p \in {\mathbb{R}}^{[n]}$ is a vector and $u_0 \in {\mathbb{R}}$ then we can build the affine transformation $\tilde{v} : 2^{[n]} \rightarrow {\mathbb{R}}$ as $\tilde{v}(S) = v(S) + \sum_{i \in S} p_i + u_0$.
- [**Strong Quotient Sum [@shioura2012matroid].**]{} Give two valuations $v,w : 2^{[n]} \rightarrow {\mathbb{R}}$ and $\alpha_1, \alpha_2 \geq 0$ , we say that $v$ is a strong quotient of $w$ if $v(S \vert T) \leq w(S \vert T)$ for all $S, T \subseteq [n]$. Given two valuations $v$ and $w$ such that $v$ is a strong quotient of $w$ and $w$ is a matroid rank function, we define the strong quotient sum $\tilde{v} : 2^{[n]} \rightarrow {\mathbb{R}}$ as $\tilde{v}(S) = \alpha_1 v(S) + \alpha_2
w(S)$.
Those operations are known to preserve gross substitutability. A major open question is whether all substitutes can be built from matroid rank functions using those operations (or perhaps a larger class of simpler operations). We focus here on affine transformations and positive linear combinations (which is a strict generalization of the strong quotient sum operation).
Are all gross substitutes positive combinations of matroid rank functions modulo an affine transformation? Formally, given $v \in {\ensuremath{\mathbf G}}^n$, is there an affine transformation $\tilde{v} \in {\ensuremath{\mathbf G}}^n$, matroid rank functions $r_i \in
{\ensuremath{\mathbf M}}^n$ and positive constants $\alpha_i \in {\mathbb{R}}_+$ such that: $$\textstyle\tilde{v} = \sum_i \alpha_i r_i.$$
An equivalent way to ask this question is via the normalized classes: this allows us to ignore the affine transformations. Given $v \in {\ensuremath{\mathbf G}}^n_0$, are there $r_i \in {\ensuremath{\mathbf M}}^n_0$ and $\alpha_i \geq 0$ such that $v = \sum_i \alpha_i
r_i$?
Building blocks for submodular functions
----------------------------------------
Before we go into our results, we would like to mention a simple constructive description for submodular functions ${\mathbf{S}}^n$ having the set of matroid rank functions as base. This will serve as a warm up for the study of substitute valuations. Besides affine transformations, the following operations preserve submodularity:[^2]
- [**Positive linear combination.** ]{} If $v_1, v_2:2^{[n]} \rightarrow {\mathbb{R}}$ are submodular and $\alpha_1, \alpha_2 \geq 0$ then $\tilde{v} =
\alpha_1 v_1 + \alpha_2 v_2$ is also submodular.
- [**Item grouping.**]{} If $v:2^{[n]} \rightarrow {\mathbb{R}}$ is submodular and $S_1,
\hdots, S_k$ is a partition of $[n]$ then the function $w : 2^{[k]}
\rightarrow {\mathbb{R}}$ defined as $w(T) = v(\cup_{t \in T} S_t)$ is also submodular.
It turns out those operations are sufficient to build any submodular function starting from the set of matroid rank functions. A proof is provided in Appendix \[appendix:submodular\_constr\].
\[thm:submodular\_construction\] Any submodular function can be obtained starting from the set of matroid rank functions and applying the operations of affine transformations, positive linear combination and item grouping.
GS is not in the cone of matroids {#sec:counterexample}
=================================
In this section, we exhibit a specific GS function and show that it cannot be expressed as a positive linear combination of matroid rank functions. This section is devoted to a (non-computational) proof of that fact. We defer to Section \[sec:tree\] a discussion on how we found such function. At a high level, the analysis uses duality and Farkas’ lemma. Farkas’s conditions require the existence of a certificate whose inner product with all matroid rank functions has non-negative sign and the inner product with the candidate function is strictly negative (Section \[sec:certificate\]). The core of the proof consists in showing in a non-computational manner that the certificate satisfies the desired conditions (Section \[sec:conditions\]). This is done with a simple lemma about the local structure of matroid rank functions and a non-trivial partition of the collection of sets into local groups that can be analyzed individually with that lemma.
The GS function and the certificate {#sec:certificate}
-----------------------------------
The GS function that we consider for the remaining in this section is over five elements $[5] = \{1, 2, 3, 4, 5\}$. [This function $v$ is described in its *normalized form*[^3] in [Figure]{} \[tab:function\]. Since the function is normalized, it is enough to define it for ${\left\vert{S}\right\vert} \geq 2$. Checking that the function satisfies the GS conditions ($v \in {\ensuremath{\mathbf G}}^5_0$) amounts to checking the inequalities: $\partial_{ij}v(S) \leq \max[\partial_{ik}v(S), \partial_{kj}v(S)] \leq 0$ for all $S \subseteq [5]$. There are $40$ such inequalities. It is a tedious but short verification, which can be found in Appendix \[sec:checkcond\].]{}
Now that we established that $v \in {\ensuremath{\mathbf G}}^5_0$, we want to prove that there do not exist (normalized) matroid rank functions[^4] $r_i \in {\ensuremath{\mathbf M}}^5_0$ and $\alpha_i \geq 0$ such that $v = \sum_{i=1}^n \alpha_i r_i$.
Sets of size 2 value Sets of size 3 value Sets of size 4 value
---------------- ------- ---------------- ------- ---------------- -------
{1,2} -1 {1,2,3} -2 {1,2,3,4} -3
{1,3} -1 {1,2,4} -2 {1,2,3,5} -3
{1,4} 0 {1,2,5} -2 {1,2,4,5} -3
{1,5} 0 {1,3,4} -1 {1,3,4,5} -2
{2,3} -1 {1,3,5} -1 {2,3,4,5} -2
{2,4} 0 {1,4,5} -1
{2,5} 0 {2,3,4} -1
{3,4} 0 {2,3,5} -1 Sets of size 5 value
{3,5} 0 {2,4,5} -1 {1,2,3,4,5} -4
{4,5} 0 {3,4,5} -1
Let $M \in {\mathbb{R}}^{m\times n}$ and $v \in {\mathbb{R}}^m$. Then exactly one of the following two statements is true:
- There exists $\alpha \in {\mathbb{R}}^n$ such that $v = M \alpha $ and $\alpha \geq 0$
- There exists $y \in {\mathbb{R}}^m$ such that ${M}^{\intercal} y \geq 0$ and $\langle v, y\rangle < 0$.
We immediately obtain the following corollary which gives two conditions such that, if satisfied, we obtain the desired negative result.
\[cor:farkas\] Let $v$ be a function over five elements. If there exists $y \in {\mathbb{R}}^{32}$ such that $\langle v, y\rangle < 0$ and $ \langle r, y\rangle \geq 0$ for all normalized matroid rank functions $r$ over five elements, then $v$ cannot be expressed as a positive linear combination of normalized matroid rank functions.
Farkas’ conditions {#sec:conditions}
------------------
We consider the certificate $y$ given in Figure \[tab:certificate\] and show that the two conditions for Corollary \[cor:farkas\] hold for that particular certificate $y$ and the gross substitute function $v$. The first and second conditions are shown in Lemma \[lem:farkas1\] and Lemma \[lem:farkas2\] respectively. The first condition is trivial to verify.
Sets of size 2 value Sets of size 3 value Sets of size 4 value
---------------- ------- ---------------- ------- ---------------- -------
{1,2} -1 {1,2,3} -1 {1,2,3,4} 1
{1,3} 1 {1,2,4} 1 {1,2,3,5} 1
{1,4} -1 {1,2,5} 1 {1,2,4,5} -1
{1,5} -1 {1,3,4} -1 {1,3,4,5} -1
{2,3} 1 {1,3,5} 1 {2,3,4,5} -1
{2,4} -1 {1,4,5} 1
{2,5} -1 {2,3,4} -1
{3,4} -1 {2,3,5} 1 Sets of size 5 value
{3,5} -1 {2,4,5} 1 {1,2,3,4,5} -1
{4,5} -1 {3,4,5} 1
\[lem:farkas1\] Let $v$ be the gross substitute function given in Figure \[tab:function\] and $y$ be the certificate given in Figure \[tab:certificate\], then $\langle y, v\rangle < 0$.
This is a simple summation and we get $\langle y, v\rangle = -1 < 0$.\
The interesting condition to show is $\langle y, r\rangle \geq 0$ for all matroid rank functions $r$. We first give a simple lemma about the local structure of matroid rank functions which will motivate the approach for the analysis of this second condition.
\[lem:matroid\] Let $r$ be a matroid rank function. For any set $S$ and elements $a_1, a_2 \not \in S$,
- if $r(S \cup a_1) - r(S) = - 1$ and $ r(S \cup a_2) - r(S) = -1$, then $r(S \cup {\{a_1, a_2\}}) - r(S) = - 2$;
- if $r(S \cup a_1) - r(S) = -1 $ and $ r(S \cup a_2) - r(S) = 0$, then $r(S \cup {\{a_1 , a_2\}}) - r(S) = - 1$.
We first decompose the quantity of interest in two terms, $$\begin{aligned}
r(S \cup a_1 , a_2) - r(S) = \left(r(S \cup a_1 , a_2) - r(S \cup a_2)\right) +
\left(r(S \cup a_2) - r(S)\right).\end{aligned}$$ Since $r$ is a matroid rank function, marginal contributions are either $-1$ or $0$ and $r(S \cup a_1 , a_2) - r(S \cup a_2) \in \{-1, 0\}$. Next, by submodularity and the assumption for both cases, we get $r(S \cup a_1 , a_2) - r(S \cup a_2) \leq r(S \cup a_1) - r(S)
= -1$. Thus, $r(S \cup a_1 , a_2) - r(S \cup
a_2) = -1$ and $$r(S \cup a_1 , a_2) - r(S) = -1 + r(S \cup a_2) - r(S),$$ which concludes the proof.
The main idea to show that $\langle y, r \rangle \geq 0$ for all matroid rank functions $r$ is to first partition the sets into six local groups $G_1, \ldots, G_6$, described in Figure \[tab:partition\], and then use Lemma \[lem:matroid\] to argue about the value $\langle y_G, r_G\rangle$ for each local group $G$, where $v_G$ denotes the subvector of a vector $v$ of length $|G|$ induced by the indices corresponding to sets in group $G$.
Group 1 value Group 2 value Group 4 value Group 5 value
--------- ------- --------- ------- ----------- ------- ------------- -------
{3,4} -1 {1,3} 1 {1,5} -1 {1,2} -1
{4,5} -1 {1,4} -1 {2,5} -1 {1,2,4} 1
{1,3,4} -1 {3,5} -1 {1,2,5} 1
{1,4,5} 1 {1,2,4,5} -1
Group 3 value {2,4,5} 1
{2,3} 1 {3,4,5} 1 Group 6 value
{2,4} -1 {2,3,5} 1 {1,2,3} -1
{2,3,4} -1 {2,3,4,5} -1 {1,2,3,4} 1
{1,3,5} 1 {1,2,3,5} 1
{1,3,4,5} -1 {1,2,3,4,5} -1
\[lem:farkas2\] Let $y$ be the certificate. Then for all matroid rank functions $r$, $\langle y, r\rangle \geq 0$.
We first show that for all $G_i$ such that $i \neq 4$, $\langle y_{G_i},
r_{G_i}\rangle \geq 0$ for all matroid rank functions $r$. Then, we show that $\langle y_{G_4}, r_{G_4}\rangle \geq -1$ and that if $\langle y_{G_4},
r_{G_4}\rangle = -1$, then for at least one other group $G$, $\langle y_{G},
r_{G} \rangle \geq 1$. Since $\langle y, r \rangle = \sum_{i =1}^6 \langle
y_{G_i}, r_{G_i}\rangle$, we then obtain $\langle y, r \rangle \geq 0$ (recall that the empty set and singletons are normalized to have value $0$ [and that function values are nonpositive for all subsets]{}).
We first show that for all $G_i$ such that $i \neq 4$, $\langle y_{G_i}, r_{G_i}\rangle \geq 0$ for all matroid rank functions $r$. This is trivial for group $G_1$ since $r(S) \leq 0$ for all sets $S$. For group $G_2$, recall that we have $$y(1, 3) = 1 \ \ \ \ y(1, 4) = -1 \ \ \ \ y(1, 3, 4) = -1.$$ Since $r(1) = 0$ and the marginal contributions of matroid rank functions [are either $0$ or $-1$,]{} $r(1, 3), r(1, 4) \in \{-1,0\}$ and we consider the following three cases:
- If $r(1, 3)= -1$ and $r(1, 4) = -1$, then by Lemma \[lem:matroid\], $r(1,3,4) = -2$. We get $\langle y_{G_2}, r_{G_2}\rangle = 2$.
- If $r(1, 3) = -1$ and $ r(1, 4) = 0$, then by Lemma \[lem:matroid\], $r(1,3,4) = -1$ and we get $\langle y_{G_2}, r_{G_2}\rangle = 0$.
- If $r(1, 3) = 0$, then $\langle y_{G_2}, r_{G_2}\rangle \geq 0$.
Group $G_3$ follows similarly as for group $G_2$. For group $G_5$, recall that we have $$y(1, 2) = -1 \ \ \ \ y(1, 2,4) = 1 \ \ \ \ y(1, 2,5)=1 \ \ \ \ y(1, 2,4,5) = -1.$$ Since $r$ is submodular, $r(S) + r(T) \geq r(S \cap T) + r(S \cup T)$ for any sets $S,T$. Thus, $-r(1, 2) + r(1, 2,4) + r(1, 2,5) - r(1, 2,4,5) \geq 0$ and this implies that $\langle y_{G_5}, r_{G_5}\rangle \geq 0$. Group $G_6$ follows similarly as for group $G_5$.
It remains to show that $\langle y_{G_4}, r_{G_4}\rangle \geq -1$ and that if $\langle y_{G_4}, r_{G_4}\rangle = -1$, then for at least one other group $G$, $\langle y_{G}, r_{G}\rangle \geq 1$. First, consider the following three possible partitions of $G_4$. $$\begin{aligned}
\left( \{3, 5\}, \{3,4,5\}, \{1, 3,5\}, \{1,3,4,5\}\right), \left(\{ 2, 5\}, \{2, 4,5\}, \{2, 3,5\}, \{2, 3, 4,5\}\right), \left(\{1, 5\}, \{1,4,5\}\right) \\
\left( \{3, 5\}, \{3,4,5\}, \{2, 3,5\}, \{2,3,4,5\}\right),
\left(\{ 1, 5\}, \{1, 4,5\}, \{1, 3,5\}, \{1, 3, 4,5\}\right), \left(\{2, 5\}, \{2,4,5\}\right) \\
\left(\{2, 5\}, \{2,4,5\}, \{2, 3,5\}, \{2,3,4,5\}\right),
\left(\{ 1, 5\}, \{1, 4,5\}, \{1, 3,5\}, \{1, 3, 4,5\}\right), \left(\{3, 5\}, \{3,4,5\}\right) \end{aligned}$$ For the first two parts $G_4'$ and $G_4''$ of each possible partition, similarly as for $G_5$, we obtain that $\langle y_{G_4'}, r_{G_4'}\rangle \geq 0$ and $\langle y_{G_4''}, r_{G_4''}\rangle \geq 0$. For the last part $G'''_5$, it is easy to see that $\langle y_{G_4'''}, r_{G_4'''}\rangle \geq -1$. Thus, we obtain that $\langle y_{G_4}, r_{G_4}\rangle \geq -1$.
Next, consider $$r(1,4,5) - r(1, 5) \ \ \ \ r(2,4,5) - r(2, 5) \ \ \ \ r(3,4,5) - r(3, 5).$$
If one of these three difference is $0$, then by considering the corresponding above partition where the two terms in the difference form the last part $G_4'''$ of the partition, we get that $\langle y_{G_4}, r_{G_4}\rangle \geq 0$.
Next, consider the case where these three differences are all equal to $-1$ and $\langle y_{G_4}, r_{G_4}\rangle = -1$. If $r(4,5) = - 1$, then $y(4,5) r(4,5) =
1$ compensates for $\langle y_{G_4}, r_{G_4}\rangle = -1$ and $\langle y, r
\rangle \geq 0$. Otherwise, $r(4,5) = 0$. This implies [ that $r(1,4,5) = r(2,4,5) = r(3,4,5) = -1$ and $r(1,5) = r(2,5) = r(3,5) = 0$]{} since the above differences are all $-1$. It must also be the case that $r(2,3,4,5) = -2$ by submodularity since $a_2$ has marginal contribution $-1$ to $\{a_4, a_5\}$ and since $ r(3,4,5) = -1$. Similarly, $r(1,3,4,5) = -2$.
Next, we focus on $G_2$ and recall that we have $$y(1, 3) = 1 \ \ \ \ y(1, 4) = -1 \ \ \ \ y(1, 3, 4) = -1.$$ Note that $r(1,3,4) \leq -1$ since $r(1,3,4,5) = -2$. Since $r(1,3) \geq r(1,3,4)$, it must be the case that $r(1,4) = 0$, $r(1,3) { = } r(1,3,4) = -1$ for $\langle y_{G_2}, r_{G_2}\rangle = 0$. If that is not the case, then $\langle y_{G_2}, r_{G_2}\rangle \geq 1$ and that compensates for $G_4$ and we get $\langle y, r \rangle \geq 0$.
Thus, in the remaining case $r(1,3) = -1$. Similarly for $G_3$ with $r(2,3,4,5) = -2$, the remaining case is if $r(2,4) = 0$ and $r(2,3) = -1$. Next, with $r(2,3) = r(1,3) = -1$, then, by Lemma \[lem:matroid\], $r(1,2,3) = -2$, which in turn implies that $r(1,2) = -1$.
Next, we consider $G_5$. Recall that $$y(1, 2) = -1 \ \ \ \ y(1, 2,4) = 1 \ \ \ \ y(1, 2,5)=1 \ \ \ \ y(1, 2,4,5) = -1.$$ Since $r(4,5) = 0$, $r(1,4,5) = -1$ and $r(2,4,5) = -1$, by Lemma \[lem:matroid\], $r(1,2,4,5) = -2$. If $r(1,2,5) { = } r(1,2,4) = - 1$, then with $r(1,2) = -1$, $\langle y_{G_5}, r_{G_5}\rangle \geq 1$ and we are done. Otherwise, $r(1,2,5)$ or $r(1,2,4)$ is equal to $- 2$. But this is impossible since we are in a case where $r(1,5) = 0$ and $r(1,4) = 0$. Thus, if $\langle y_{G_4}, r_{G_4}\rangle = -1$, then for at least one other group $G$, $\langle y_{G}, r_{G}\rangle \geq 1$ and we get $\langle y, r \rangle \geq 0$.
Combining Corollary \[cor:farkas\], Lemma \[lem:farkas1\] and Lemma \[lem:farkas2\], we obtain the main result.
\[thm:main\] The gross substitute function $v$ cannot be expressed as a positive linear combination of normalized matroid rank functions. [In particular, no affine transformation of $v$ can be expressed as a positive linear combination of matroid rank functions.]{}
In particular, this implies that the strong quotient sum and [ tree-concordant-sum]{} operations are not sufficient to construct all gross substitutes from matroid rank functions and that MRS valuations do not exhaust gross substitutes. We note that even though $v$ is non-monotone, the main result holds for monotone gross substitute functions since there exists some affine transformation of $v$ that is monotone. We also extend the negative result from matroid rank functions to weighted matroid rank functions. This follows from the fact that weighted matroid rank functions can be expressed as a positive linear combination of unweighted matroid rank functions, which was proven in [@shioura2012matroid] and we give a proof in Appendix \[sec:appcounterexample\] for completeness as Lemma \[lem:weighted\].
[No affine transformation of function $v$ can]{} be expressed as a positive linear combination of weighted matroid rank functions.
The Tree-Concordant-Sum Operation {#sec:tree}
=================================
In this section, we define the notion of a tree representation which abstracts the combinatorial structure of *tree-form Hessians* of @hira2004m. We first show that this representation has the following nice property: the condition that two functions have such a tree representation that is *compatible* is necessary and sufficient for the summation operation to preserve substitutability. We call this new operation preserving substitutability tree-concordant-sum and show that it also provides a polyhedral characterization of gross substitutes.
We then use this representation to give a positive answer to our main question for $n \leq 4$: a GS over at most $4$ elements can be written as a positive linear combination of matroid rank functions. This implies that at least $5$ elements are necessary to obtain the negative answer from the previous section.
Finally, we discuss how the polyhedral understanding of gross substitutes based on this tree representation combined with computational techniques led to the discovery of the counterexample for $n= 5$.
Substitution Trees
------------------
The tree representation of GS is best explained using the discrete derivative property introduced in Section \[sec:prelim\]. Since it will be convenient to work with non-negative numbers, we introduce the notation: $$\Delta_{ij}^S(v) = -\partial_{ij} v(S)$$ omitting $v$ when clear from context. In this notation, we can write the GS condition as: $$\Delta^S_{ij} \geq \min(\Delta^S_{ik}, \Delta^S_{jk}) \geq 0$$ If we permute the identities of $i,j,k$ such that the symbols are sorted, we have the following triangle property: $\Delta^{S}_{ij} = \Delta^S_{ik} \leq \Delta^S_{jk}$. @hira2004m and @BingLehmannMilgrom note that this resembles the definition of *ultra-metrics*, which admit tree-like representations. This enables similar tree-like representations for GS. Below we describe the notion of @hira2004m following the presentation in [@leme2017gross]{}.
\[thm:hirai\] A function $v$ satisfies the GS condition iff for every subset $S$ there is a tree $T(v)^S$ having the elements of $[n] \setminus S$ in the leaves and non-negative real number labels in the internal nodes such that:
- The label of each internal node is larger than or equal to the label of its parent.
- For every $i,j \notin S$, $\Delta^S_{ij}$ corresponds to the label of the lowest common ancestor $({\mathsf{lca}})$ of the leaves corresponding to $i$ and $j$.
We observe that the representation of Murota and Hirai has two components: a purely combinatorial structure, which is the collection of trees and a numerical component, which are the values of the labels. If we abstract the numerical component, we obtain what we call a tree structure:
A tree structure corresponds to a collection of trees $\{T^S\}_S$ indexed by subsets $S \subseteq [n]$ such that the leaves of tree $T^S$ correspond to the elements of $[n] \setminus S$.
We say that a valuation function $v$ admits a tree structure $\{T^S\}_S$ if we can represent $v$ in the sense of Theorem \[thm:hirai\] using those trees. The tree structure might not be unique. For example, if for a certain $v$ and $S$, $\Delta_{ij}^S$ is given by the matrix in the left of Figure \[fig:treepartition\], then the two trees in the figure are valid structures for $\Delta_{ij}^S$. There is therefore some flexibility in the choice of the tree structure, which allow us to define the notion of tree-concordant:
We say that two GS valuations are *tree concordant* if they admit the same tree structure $\{T^S\}_S$.
A clean way to check when two functions are tree concordant is via the concept of minimal representation. We say that a tree representation for valuation $v$ is *minimal* if no node has the same label as its parent. The tree on the left in Figure \[fig:treepartition\] is minimal, for example while the one on the right is not. By the definition of minimal, it is clear that each GS valuation has an unique minimal representation. It can be obtained by starting from any representation and collapsing tree edges connecting internal nodes with the same label.
[.25]{} $\left[ \begin{aligned}
& * && 2 && 1 && 1 && 1 \\
& 2 && * && 1 && 1 && 1 \\
& 1 && 1 && * && 3 && 3 \\
& 1 && 1 && 3 && * && 3 \\
& 1 && 1 && 3 && 3 && * \\
\end{aligned} \right] \quad$
[.35]{}
=\[sibling distance=30mm\] =\[sibling distance=15mm\] child [node\[circle, draw=black\] [$2$]{} child [node [$a$]{}]{} child [node [$b$]{}]{} ]{} child [node\[circle, draw=black\] [$3$]{} child [node [$c$]{}]{} child [node [$d$]{}]{} child [node [$e$]{}]{} ]{};
[.35]{}
=\[sibling distance=30mm\] =\[sibling distance=15mm\] =\[sibling distance=30mm\] =\[sibling distance=15mm\] child [node\[circle, draw=black\] [$2$]{} child [node [$a$]{}]{} child [node [$b$]{}]{} ]{} child [node\[circle, draw=black\] [$3$]{} child [node\[circle, draw=black\] [$3$]{} child [node [$c$]{}]{} child [node [$d$]{}]{} ]{} child [node [$e$]{}]{} ]{};
To check when there is one tree structure that two functions simultaneously admit, it is enough to look at the minimal representations. To see that, it is useful to view a tree as a laminar family. Given a tree $T^S$ with elements $[n]
\setminus S$ in the leaves, we can represent it by a family of subsets $L^S$ constructed as follows: a set $X \subseteq [n] \setminus S$ is in $L^S$ [if and only if]{} there is an internal node [v]{} in $T^S$ such that $X$ is the set of leaves below $v$. Such subset collection is what is called a laminar family:
[A collection $L$ of subsets ]{} is called a laminar family if for every $X, Y \in L$ either: (i) $X \cap Y = \emptyset$; or (ii) $X \subseteq Y$ or (iii) $Y \subseteq X$.
Now, we can check if two functions $u$ and $v$ are tree-concordant as follows:
\[lemma:laminar\] If $u$ and $v$ are GS functions and $\{T^S(u)\}$ and $\{T^S(v)\}$ are its minimal tree structures and $\{L^S(u)\}$ and $\{L^S(v)\}$ are its corresponding laminar family representations, then $u$ and $v$ are tree concordant iff for every $S$, $L^S(u) \cup L^S(v)$ is also a laminar family.
The proof is straightforward from definitions and the fact that there is natural one-to-one mapping between trees and laminar families. What is interesting about tree concordance is that it provides necessary and sufficient conditions for the sum of two GS functions to be GS:
Let $u$ and $v$ be two gross substitute functions. The function $u + v$ is a gross substitute function if and only if $u$ and $v$ are tree-concordant.
We first show that if $u$ and $v$ are tree-concordant, then $u+v$ is a gross-substitute. Consider the tree representation $T$ that has the structure of $u$ and $v$, which is identical since they are tree-concordant, but with numerical values at each internal node that is the sum of the values at that node for $u$ and $v$. Since second order derivatives are linear: $$\Delta_{ij}^S(u+v) = \Delta_{ij}^S(u) + \Delta_{ij}^S(v)$$ and it clearly admits the same tree structure $T$, so by Theorem \[thm:hirai\], $u+v$ is in GS.
Next, we show that if $u$ and $v$ are not tree-concordant, then $u+v$ is not a gross substitute function. Assume $u$ and $v$ are not tree-concordant then by Lemma \[lemma:laminar\] there is a set $S$ such that for the laminar representations $L^S(u)$ and $L^S(v)$ there are sets with non-trivial intersection, i.e., there are $X \in L^S(u)$ and $Y \in L^S(v)$ such that we can find $i \in X \setminus Y$, $j \in X \cap Y$ and $k \in Y \setminus X$. Therefore: $$\Delta^S_{ij}(u) > \Delta^S_{ik}(u) = \Delta^S_{kj}(u)$$ $$\Delta^S_{ij}(v) = \Delta^S_{ik}(v) < \Delta^S_{kj}(v)$$ therefore, it must be the case that: $$\Delta^S_{ik}(u+v) < \min[ \Delta^S_{ij}(u+v), \Delta^S_{kj}(u+v) ]$$ which is a violation of gross substitutability.
Since it is a necessary and sufficient condition, this includes the strong-quotient-sum property in [@shioura2012matroid] as a special case:
If a GS function $u$ and a matroid rank function $v$ satisfy the strong-quotient-sum property, then they are tree-concordant.
Polyhedral description of ${\ensuremath{\mathbf G}}^n$
------------------------------------------------------
The concept of tree structure and tree concordance provide a good tool for describing the geometry of ${\ensuremath{\mathbf G}}^n$. Viewing valuations as vectors in ${\mathbb{R}}^{2^n}$, we can view ${\ensuremath{\mathbf G}}^n$ as a subset of that space. Lehmann, Lehmann and Nisan [@LehmannLehmannNisan] observe that the set is a non-convex and has zero measure.
The concepts developed earlier in this section allow us to decompose the space in finitely many convex cones. Since a tree structure is a finite combinatorial object, there are finitely many such structures. Fix a tree structure $\{T^S\}_S$ and consider the subset of ${\ensuremath{\mathbf G}}^n$ with functions that admit $\{T^S\}_S$. All functions in this subset are tree concordant, so the set is closed under positive linear combination, forming a *convex cone*.
What we would like to do next is to understand how those cones look like. For that we will fix a tree structure and assign labels to internal nodes. The labels must be non-negative and the label of a node cannot be smaller than the label of its parent, but those conditions alone are not sufficient for the existence of a valuation function producing those $\Delta_{ij}^S$. An extra condition that is required is what we call *integrability*[^5][(we simplify the notation for $S \cup \{i\}$ with $S+i$ for the remainder of the paper)]{}:
$$\label{eq:int}\tag{Int}
\Delta^{{S + k}}_{ij} - \Delta^S_{ij} = \Delta^{{S + j}}_{ik} - \Delta^S_{ik}
= \Delta^{{S + i}}_{jk} - \Delta^S_{jk}.$$
Given values $\Delta^S_{ij}$ [for all $i, j, S$ such that $i, j \not \in S$,]{} there is a valuation $v$ such that $\partial_{ij}v(S) = - \Delta^S_{ij}$ [if and only if]{} the integrability conditions (\[eq:int\]) hold.
In Appendix \[appendix:integrability\] we provide a discussion on integrability conditions for discrete functions. The lemma above follows directly from Lemma \[lemma:int:2\] in the appendix.
Since the symbols $\Delta_{ij}^S$ have a special tree-form for gross substitutes, it is convenient to re-write the integrability conditions in the following form.
\[lem:welldefined\] An assignment of labels to the internal nodes of a tree structure corresponds to a representation of GS function [if and only if]{} the following condition holds for every $S \subseteq [n]$ and $i,j,k \notin
S$: $$\begin{aligned}
& \text{ if } & &\Delta_{ik}^S = \Delta_{jk}^S = \Delta_{ij}^S - \alpha,
{\text{ for some } \alpha \geq 0 }\\
& \text{ then } & &
\Delta_{ik}^{S+j} = \Delta_{jk}^{S+i} = \Delta_{ij}^{S+k} - \alpha
\end{aligned}$$
The following corollary (which is a rephrasing of Corollary \[cor:int:3\] in the appendix) shows how to explicitly reconstruct a function from second order derivatives. [We denote by $S_{<i}$ the set of elements smaller than $i$ according to their label, i.e., $S_{<i} = \{j : j < i\}$.]{}
\[cor:integration\] Given $\Delta_{ij}^S$ satisfying the integrability conditions (\[eq:int\]), the unique normalized function such that $\partial_{ij} v(S) =
-\Delta_{ij}^S$ is given by: $$v(S) = -\sum_{i,j \in S; i < j} \Delta_{ij}^{S_{<i}}$$ In particular, all the functions such that $\partial_{ij} v(S) =
-\Delta_{ij}^S$ are affine transformations of the function defined above.
Representations of ${\ensuremath{\mathbf G}}$
---------------------------------------------
Using the tree representation of gross substitutes, we provide a constructive characterization of gross substitutes ${\ensuremath{\mathbf G}}^n$ for $n \leq 4$ from matroids. Given a matroid ${\mathcal{M}}$ we denote its rank by $r[{\mathcal{M}}]$. We denote the uniform matroid of rank $i$ over $j$ elements by $U^i_j$.
Given two functions $v$ and $\tilde{v}$ we say that they are equivalent (up to affine transformations) if $v - \tilde{v} \in {\ensuremath{\mathbf E}}^n$. Given a normalized valuation $v \in {\ensuremath{\mathbf G}}^n_0$, we will often associate it with the equivalent function $\tilde{v}(S) = v(S) + {\left\vert{S}\right\vert}$. Often, a normalized matroid rank function $v$ is easier to recognize in its $\tilde{v}$ form.
Moreover, we write $X \simeq Y $ to denote a linear isomorphism between two sets, i.e., if there is a linear bijection $L$ such that $L(X) = Y$.
### Description of ${\ensuremath{\mathbf G}}^2$
For $n = 2$, the only constraint is $\Delta^\emptyset_{1,2} \geq 0$ so: $${{\ensuremath{\mathbf G}}^2_0 = \{(v(\emptyset) = 0, v(1) = 0, v(2) = 0, v(12)) ; v(12) \leq 0 \} \simeq {\mathbb{R}}_+.}$$ Thus, $v=(0,0,0,-1)$ is a representative of the class ${\ensuremath{\mathbf G}}_0^2$ and $\tilde{v}= (0,1,1,1)$, which is the rank function $r[U_2^1]$ of the uniform matroid of rank $1$ over $2$ elements, is a representative [of the class ${\ensuremath{\mathbf G}}^2$]{}. Since ${\ensuremath{\mathbf E}}^2 = (1,1,1,1) \cdot {\mathbb{R}}+ (0,1,0,1) \cdot
{\mathbb{R}}+ (0,0,1,1) \cdot {\mathbb{R}}\simeq
{\mathbb{R}}^3$, we have: $${\ensuremath{\mathbf G}}^2 = {\ensuremath{\mathbf E}}^2 + r[U_2^1] \cdot {\mathbb{R}}_+ \simeq {\mathbb{R}}^3 \times {\mathbb{R}}_+.$$ The set ${\ensuremath{\mathbf G}}^2$ is not very interesting since the set of gross substitutes on $2$ variables is the same as the set of submodular functions on $2$ variables, which is known to be a convex set.
### Description of ${\ensuremath{\mathbf G}}^3$
The set of gross substitutes on $3$ items is more interesting since it is not convex. We name the items $\{a,b,c\}$. For every $v \in {\ensuremath{\mathbf G}}^3_0$ and up to the renaming of the elements, there is only one possibility for the substitution tree associated to the empty set, which is depicted in Figure \[fig:G32\]. For each singleton set, there is also only one possible substitution tree. Applying Lemma \[lem:welldefined\] that is required to obtain a well-defined function, with $S = \emptyset$, we obtain the following additional necessary and sufficient constraint [for the labels in Figure \[fig:G32\]]{}: $$m_3 = m_4 = m_5 - (m_2 - m_1).$$
[.3]{}
child [node\[circle, draw=black\] [$m_2$]{} child [node [$a$]{}]{} child [node [$b$]{}]{}]{} child [node [$c$]{}]{};
[.7]{}
child [node [$b$]{}]{} child [node [$c$]{}]{}; child [node [$a$]{}]{} child [node [$c$]{}]{}; child [node [$a$]{}]{} child [node [$b$]{}]{};
We write $m_1 = x$, $m_2 - m_1 = y$ (recall $m_2 \geq m_1$) and $m_3 = z$, so we can parametrize the space of feasible $m = (m_1, m_2, m_3, m_4,m_5)$ by: $$(x,y,z) \in {\mathbb{R}}^3_+ \mapsto m = (x, x+y, z, z, y+z)$$ In other words, the space of feasible values of $m$ are a cone generated by the vectors\
$(1,1,0,0,0), (0,0,1,1,1), (0,1,0,0,1)$. It is particularly interesting to see which valuations they correspond to.
#### Vector $(1,1,0,0,0)$
By solving the equation to obtain $v$ from second derivatives $\Delta$ (Corollary \[cor:integration\]), we obtain the valuation $v \in {\ensuremath{\mathbf G}}^3_0$ such that $v(ab) = v(bc) = v(ac) = -1$ and $v(abc) = -2$. Thus, $\tilde{v}(S) = v(S) + {\left\vert{S}\right\vert} = 1$ for all $S \neq \emptyset$, which is the rank function $r[U_3^1]$.
#### Vector $(0,0,1,1,1)$
Solving the equations for $v$ we obtain $v(ab)
= v(bc) = v(ac) = 0$ and $v(abc) = -1$, so $\tilde{v}(S) = \min\{
{\left\vert{S}\right\vert}, 2 \}$ which is $r[U_3^2]$.
#### Vector $(0,1,0,0,1)$
More interestingly, by solving for $v$ we obtain: $v(ac) = v(bc) = 0$ and $v(ab) = v(abc) = -1$, so $\tilde{v}$ is such that the singletons have value $1$, $\tilde{v}(ab) =
1$ and $\tilde{v}(bc) = \tilde{v}(ac) = \tilde{v}(abc) = 2$. This is the rank function of the graphical matroid associated with the following graph:
=\[circle,fill=black,minimum size=6pt,inner sep=0pt\]; (a) at (0,0) ; (b) at (1.5,0) ; (c) at (3,0) ; (a) .. controls +(-30:20pt) and +(-150:20pt) .. node\[below\][$b$]{}(b); (a) .. controls +(+30:20pt) and +(+150:20pt) .. node\[above\][$a$]{} (b); (b) – node\[above\] [$c$]{}(c);
We note that when we describe a matroid in terms of a graph in this document we refer to the matroid where the ground set are the edges of the graph and a set is independent if the corresponding edges do not form a cycle. Call the rank of this matroid $r[M_{((ab)c)}]$. Therefore, the set of functions in ${\ensuremath{\mathbf G}}^3$ associated with the depicted $\emptyset$-tree is given by: $${\ensuremath{\mathbf E}}^3 + r[U_3^1] \cdot {\mathbb{R}}_+ + r[U_3^2] \cdot {\mathbb{R}}_+ + r[M_{((ab)c)}] \cdot {\mathbb{R}}_+$$
Since all the $\emptyset$-trees are symmetric, then all the gross substitute functions over $3$ elements are of the form: $${\ensuremath{\mathbf G}}^3 = {\ensuremath{\mathbf E}}^3 + r[U_3^1] \cdot {\mathbb{R}}_+ + r[U_3^2] \cdot {\mathbb{R}}_+ + \left(
r[M_{((ab)c)}] \cdot {\mathbb{R}}_+ \cup r[M_{((ac)b)}] \cdot {\mathbb{R}}_+ \cup r[M_{((bc)a)}]
\cdot {\mathbb{R}}_+ \right)$$
### Description of ${\ensuremath{\mathbf G}}^4$
The description of ${\ensuremath{\mathbf G}}^4$ follows similarly as for ${\ensuremath{\mathbf G}}^3$, but with more cases. It is deferred to Appendix \[sec:appG4\].
Finding the counterexample
--------------------------
The main idea used to find counterexample $v$ is to exhibit a specific tree structure over $5$ elements that is complex enough so that, unlike for $n =
1,2,3,$ and $4$, node labels cannot be decomposed into binary valued vectors which satisfy the integrability condition. We will try to mimic the same proof used for ${\ensuremath{\mathbf G}}^3$ and ${\ensuremath{\mathbf G}}^4$ and find a set of trees for which the same proof technique cannot be extended.
[.33]{}
child [node\[circle, draw=black\] [$m_2$]{} child [node [$1,2,3$]{}]{}]{} child [node [$4,5$]{}]{};
[.33]{}
child [node\[circle, draw=black\] [$m_4$]{} child [node [$1,2$]{}]{}]{} child [node [$3,4$]{}]{};
[.33]{}
child [node\[circle, draw=black\] [$m_6$]{} child [node [$2,4,5$]{}]{}]{} child [node [$3$]{}]{};
Consider the tree defined in Figure \[fig:hard\]. We can apply the integrability conditions (\[eq:int\]) to obtain the following relations among the labels:
- $m_3 - m_5 = m_2 - m_1\geq 0$ from (\[eq:int\]) with $S = \emptyset$ and $(i,j,k) = (1,3,5)$,
- $m_6 - m_3 = m_1 - m_1 = 0$ from (\[eq:int\]) with $S = \emptyset$ and $(i,j,k) = (1,4,5)$,
- $m_4 - m_6 = m_2 - m_1 \geq 0$ from (\[eq:int\]) with $S = \emptyset$ and $(i,j,k) = (1,2,5)$.
Consider a function that satisfies that tree structure and has $m_2 - m_1 =
\Delta > 0$, then $m_4 = m_6 + \Delta = m_3 + \Delta = m_5 + 2 \Delta$. Therefore, any parametrization of the space of functions sharing those trees obtained by the same method as used for ${\ensuremath{\mathbf G}}^3$ and ${\ensuremath{\mathbf G}}^4$ will have a non-binary coefficient and hence does not decompose the space in combinations of matroid rank functions.
This is yet not a proof, since we do not know yet if there is GS functions with that tree structure and if there is a parametrization obtained by other methods that decomposes the space. Next, we complete this tree description and (computationally) solve a linear program to find a function satisfying that tree structure. Once we get a candidate functions that is the output of this program, we write a second program that tries to write it as a convex combination of matroid rank functions by explicitly enumerating over the set of all matroid rank functions and creating one variable for each in the linear program. Next, we verify that the program is infeasible and obtain a Farkas’ type certificate. Finally, since we do not want to rely on the correctness of the enumeration and the computational steps, we give a human readable proof.
Conclusion {#sec:conclusion}
==========
The class of gross substitutes is a well-studied family of valuation functions that has many different characterizations, but for which we do not know a constructive description. Our main result shows that gross substitutes cannot be constructed via positive linear combinations of matroid rank functions. We also give a new operation, called tree-concordant-sum, which provides a necessary and sufficient condition for the sum operation to preserve substitutability and which is used to find the counterexample for the main result.
In addition to affine transformations, strong quotient sum, and tree-concordant-sum, other operations are known to preserve substitutability. Two important examples are endowment or restriction [@HatfieldMilgrom] and convolution [@Murota96] or OR [@LehmannLehmannNisan]. It remains an important open question whether there is a collection of substitutability-preserving operations that allow constructing all gross substitutes from matroid rank functions, or another simple class of functions.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Kazuo Murota and Akiyoshi Shioura for their comments on an earlier version of this manuscript and their various suggestions to improve the presentation. We would also like to thank the mantainers of the database of matroids [@databasematroid] which was in invaluable resource in this project.
Eric is supported by a Google PhD Fellowship.
Appendix {#appendix .unnumbered}
========
Missing Proofs from Section \[sec:prelim\]
==========================================
\[thm:matroid\_submodular\] ${\ensuremath{\mathbf M}}^n = \{v \in {\ensuremath{\mathbf G}}^n; v(\emptyset) = 0; \partial_i v(S) \in \{0,1\}, \forall S
\subseteq [n] \} $.
It follows from the following characterization of matroid rank functions that can be found in Section 39.7 of Schrijver [@schrijver2003combinatorial] (rephrased in the language of discrete derivatives):
Let $r$ be a valuation function, then $r \in {\ensuremath{\mathbf M}}^n$ if and only if $r(\emptyset) = 0$, $\partial_i r(S) \in \{0,1\}$ and $\partial_{ij} r(S) \leq 0$.
This means that matroid rank functions are exactly the submodular functions that have $\{0,1\}$-marginals. Using this lemma, we can now prove Theorem \[thm:matroid\_submodular\]:
The inclusion $\supseteq$ follows directly from the previous lemma and the fact that every GS function is submodular.
For the inclusion ${\ensuremath{\mathbf M}}^n \subseteq {\ensuremath{\mathbf G}}^n$, we first note that if $r \in {\ensuremath{\mathbf M}}^n$, then $\partial_{ij}r(S) = \partial_i r(S\cup j) - \partial_i r(S) \in
\{0,-1\}$ since the first derivatives are in $\{0,1\}$ and the second derivatives are non-positive. Moreover, $\partial_{ij}r(S) = -1$ if and only if $\partial_i r(S\cup j) = \partial_j r(S\cup i) = 0$ and $\partial_i r(S) =
\partial_j r(S) = 1$. Therefore the condition: $$\partial_{ij} v(S) \leq \max[ \partial_{ik} v(S), \partial_{kj} v(S) ]$$ is violated only when $\partial_{ij} v(S) = 0$ and $\partial_{ik} v(S) =
\partial_{kj} v(S) = -1$. This implies that $\partial_i v(S) = \partial_j v(S)
= \partial_k v(S) = 1$. But since $\partial_{ij} v(S) = 0$ we must have $v(S \cup ijk) \geq v(S\cup ij) = v(S) + 2$. This implies [ $\partial_{ij}v(S\cup k) > 0$ which contradicts submodularity.]{}
Missing Proofs from Section \[sec:building\_block\] {#appendix:submodular_constr}
===================================================
The main ingredient in Theorem \[thm:submodular\_construction\] is the following characterization of monotone integer valued submodular functions:
Every monotone integer valued submodular function that evaluates to zero at the empty set can be obtained from a matroid rank function by item grouping. Formally: if $v:2^{[n]} \rightarrow
{\mathbb{Z}}$ is a monotone submodular function (where monotone means that $\partial_i
v(S) \geq 0$) with $v(\emptyset) = 0$, then there is a matroid ${\mathcal{M}}$ defined on a set $U$ and a partition $(X_1, \hdots, X_n)$ of $U$ such that $v(S) =
r_{\mathcal{M}}(\cup_{s \in S} X_s)$.
Therefore we only need to show that all submodular functions can be constructed from monotone integer valued submodular functions using positive linear combinations and affine transformations. To show this we start by viewing each submodular function as a vector in ${\mathbb{R}}^{2^n}$ indexed by the subsets of $[n]$. From this perspective, the set of submodular functions ${\mathbf{S}}^n$ correspond to the set of ${\mathbb{R}}^{2^n}$-points satisfying the linear inequalities given by $\partial_{ij}
v(S) \leq 0$, which is a system of homogeneous linear inequalities. The set of solutions of such system is usually called a *polyhedral cone*. A classic result in convex analysis (see [@rockafellar2015convex] for example) says that every polyhedral cone is finitely generated, i.e., every point can be written as a positive combination of a finite set of points. In other words, there is a finite set $v_1, \hdots, v_k \in {\mathbf{S}}^n$ such that: $${\mathbf{S}}^n = \left\{ \textstyle \sum_{i=1}^k \alpha_i v_i ; \alpha_i \geq 0 \right\}$$ When this set is minimal, those are called extremal rays of the cone. Also from convex analysis, if the constraints have rational coefficients then there is a set of extremal rays with integer coefficients.
Finally, observe that we can construct general integer value submodular functions from monotone ones by applying an affine transformation. For each $v_i$, let $M = \min_{S
\subseteq [n], j \notin S} \partial_j v_i(S)$ and then define a normalized version of $v_i$ as: $$\bar v_i(S) = v_i(S) - M {\left\vert{S}\right\vert} - v_i(\emptyset)$$ It is simple to see that $\bar v_i(S)$ is monotone and evaluates to zero at the empty set and that $v_i$ can be constructed from $\bar v_i$ using an affine transformation.
#### Extremal submodular functions
One can ask whether the *item grouping* operation is necessary. Equivalently, are all the extremal rays matroid rank functions? For ${\mathbf{S}}^2$ and ${\mathbf{S}}^3$ all submodular functions are convex combinations of matroid rank functions (modulo affine transformations). For $n=4$, however, the following submodular function is extremal and is not a matroid rank function: define $f(S)$ over $\{a,b,c,d\}$ such that $f(S) = 0$ for ${\left\vert{S}\right\vert} \leq 1$, $f(ab) = f(bd) = {f(bc)} = f(acd) = -1$, $f(ac) = f(ad) = f(cd) = 0$, $f(abc) = f(abd) = f(bcd) = -2$ and $f(abcd) = -3$.
In general the set of extremal submodular functions can be obtained using the standard technique of converting between the $H$-representation and $V$-representation of a cone. See @ziegler2012lectures for a complete discussion and the LRS package [@lrs] for an implementation of such algorithms.
Missing Proofs from Section \[sec:counterexample\] {#sec:appcounterexample}
==================================================
Checking ${\ensuremath{\mathbf G}}^5_0$-conditions for candidate {#sec:checkcond}
----------------------------------------------------------------
Below we check conditions $\partial_{ij}v(S) \leq \max[\partial_{ik}v(S),
\partial_{kj}v(S)] \leq 0$ for the candidate function in Table \[tab:function\]. There are $40$ inequalities to be checked, which we do below.
$$\begin{aligned}
&-1 = \partial_{3,2}v(\emptyset) \leq \max(\partial_{3,1}v(\emptyset),\partial_{1,2}v(\emptyset)) = \max(-1,-1) = -1 \leq 0\\
&0 = \partial_{4,2}v(\emptyset) \leq \max(\partial_{4,1}v(\emptyset),\partial_{1,2}v(\emptyset)) = \max(0,-1) = 0 \leq 0\\
&0 = \partial_{4,3}v(\emptyset) \leq \max(\partial_{4,1}v(\emptyset),\partial_{1,3}v(\emptyset)) = \max(0,-1) = 0 \leq 0\\
&0 = \partial_{4,3}v(\emptyset) \leq \max(\partial_{4,2}v(\emptyset),\partial_{2,3}v(\emptyset)) = \max(0,-1) = 0 \leq 0\\
&0 = \partial_{5,2}v(\emptyset) \leq \max(\partial_{5,1}v(\emptyset),\partial_{1,2}v(\emptyset)) = \max(0,-1) = 0 \leq 0\\
&0 = \partial_{5,3}v(\emptyset) \leq \max(\partial_{5,1}v(\emptyset),\partial_{1,3}v(\emptyset)) = \max(0,-1) = 0 \leq 0\\
&0 = \partial_{5,3}v(\emptyset) \leq \max(\partial_{5,2}v(\emptyset),\partial_{2,3}v(\emptyset)) = \max(0,-1) = 0 \leq 0\\
&0 = \partial_{5,4}v(\emptyset) \leq \max(\partial_{5,1}v(\emptyset),\partial_{1,4}v(\emptyset)) = \max(0,0) = 0 \leq 0\\
&0 = \partial_{5,4}v(\emptyset) \leq \max(\partial_{5,2}v(\emptyset),\partial_{2,4}v(\emptyset)) = \max(0,0) = 0 \leq 0\\
&0 = \partial_{5,4}v(\emptyset) \leq \max(\partial_{5,3}v(\emptyset),\partial_{3,4}v(\emptyset)) = \max(0,0) = 0 \leq 0\\
&0 = \partial_{4,3}v(1) \leq \max(\partial_{4,2}v(1),\partial_{2,3}v(1)) = \max(-1,0) = 0 \leq 0\\
&0 = \partial_{5,3}v(1) \leq \max(\partial_{5,2}v(1),\partial_{2,3}v(1)) = \max(-1,0) = 0 \leq 0\\
&-1 = \partial_{5,4}v(1) \leq \max(\partial_{5,2}v(1),\partial_{2,4}v(1)) = \max(-1,-1) = -1 \leq 0\\
&-1 = \partial_{5,4}v(1) \leq \max(\partial_{5,3}v(1),\partial_{3,4}v(1)) = \max(0,0) = 0 \leq 0\\
&0 = \partial_{4,3}v(2) \leq \max(\partial_{4,1}v(2),\partial_{1,3}v(2)) = \max(-1,0) = 0 \leq 0\\
&0 = \partial_{5,3}v(2) \leq \max(\partial_{5,1}v(2),\partial_{1,3}v(2)) = \max(-1,0) = 0 \leq 0\\
&-1 = \partial_{5,4}v(2) \leq \max(\partial_{5,1}v(2),\partial_{1,4}v(2)) = \max(-1,-1) = -1 \leq 0\\
&-1 = \partial_{5,4}v(2) \leq \max(\partial_{5,3}v(2),\partial_{3,4}v(2)) = \max(0,0) = 0 \leq 0\\
&0 = \partial_{5,4}v(1,2) \leq \max(\partial_{5,3}v(1,2),\partial_{3,4}v(1,2)) = \max(0,0) = 0 \leq 0\\
&0 = \partial_{4,2}v(3) \leq \max(\partial_{4,1}v(3),\partial_{1,2}v(3)) = \max(0,0) = 0 \leq 0\\
\end{aligned}$$
$$\begin{aligned}
&0 = \partial_{5,2}v(3) \leq \max(\partial_{5,1}v(3),\partial_{1,2}v(3)) = \max(0,0) = 0 \leq 0\\
&-1 = \partial_{5,4}v(3) \leq \max(\partial_{5,1}v(3),\partial_{1,4}v(3)) = \max(0,0) = 0 \leq 0\\
&-1 = \partial_{5,4}v(3) \leq \max(\partial_{5,2}v(3),\partial_{2,4}v(3)) = \max(0,0) = 0 \leq 0\\
&-1 = \partial_{5,4}v(1,3) \leq \max(\partial_{5,2}v(1,3),\partial_{2,4}v(1,3)) = \max(-1,-1) = -1 \leq 0\\
&-1 = \partial_{5,4}v(2,3) \leq \max(\partial_{5,1}v(2,3),\partial_{1,4}v(2,3)) = \max(-1,-1) = -1 \leq 0\\
&-1 = \partial_{3,2}v(4) \leq \max(\partial_{3,1}v(4),\partial_{1,2}v(4)) = \max(-1,-2) = -1 \leq 0\\
&-1 = \partial_{5,2}v(4) \leq \max(\partial_{5,1}v(4),\partial_{1,2}v(4)) = \max(-1,-2) = -1 \leq 0\\
&-1 = \partial_{5,3}v(4) \leq \max(\partial_{5,1}v(4),\partial_{1,3}v(4)) = \max(-1,-1) = -1 \leq 0\\
&-1 = \partial_{5,3}v(4) \leq \max(\partial_{5,2}v(4),\partial_{2,3}v(4)) = \max(-1,-1) = -1 \leq 0\\
&0 = \partial_{5,3}v(1,4) \leq \max(\partial_{5,2}v(1,4),\partial_{2,3}v(1,4)) = \max(0,0) = 0 \leq 0\\
&0 = \partial_{5,3}v(2,4) \leq \max(\partial_{5,1}v(2,4),\partial_{1,3}v(2,4)) = \max(0,0) = 0 \leq 0\\
&0 = \partial_{5,2}v(3,4) \leq \max(\partial_{5,1}v(3,4),\partial_{1,2}v(3,4)) = \max(0,-1) = 0 \leq 0\\
&-1 = \partial_{3,2}v(5) \leq \max(\partial_{3,1}v(5),\partial_{1,2}v(5)) = \max(-1,-2) = -1 \leq 0\\
&-1 = \partial_{4,2}v(5) \leq \max(\partial_{4,1}v(5),\partial_{1,2}v(5)) = \max(-1,-2) = -1 \leq 0\\
&-1 = \partial_{4,3}v(5) \leq \max(\partial_{4,1}v(5),\partial_{1,3}v(5)) = \max(-1,-1) = -1 \leq 0\\
&-1 = \partial_{4,3}v(5) \leq \max(\partial_{4,2}v(5),\partial_{2,3}v(5)) = \max(-1,-1) = -1 \leq 0\\
&0 = \partial_{4,3}v(1,5) \leq \max(\partial_{4,2}v(1,5),\partial_{2,3}v(1,5)) = \max(0,0) = 0 \leq 0\\
&0 = \partial_{4,3}v(2,5) \leq \max(\partial_{4,1}v(2,5),\partial_{1,3}v(2,5)) = \max(0,0) = 0 \leq 0\\
&0 = \partial_{4,2}v(3,5) \leq \max(\partial_{4,1}v(3,5),\partial_{1,2}v(3,5)) = \max(0,-1) = 0 \leq 0\\
&0 = \partial_{3,2}v(4,5) \leq \max(\partial_{3,1}v(4,5),\partial_{1,2}v(4,5)) = \max(0,-1) = 0 \leq 0\\
\end{aligned}$$
Weighted matroids
-----------------
\[lem:weighted\] Any weighted matroid rank functions can be written as a positive linear combination of unweighted matroid rank functions.
Let $w_1 \geq \ldots \geq w_n \geq 0$ be the weights of the $n$ elements $[n] := \{1,
\ldots, n\}$ of a weighted matroid rank function $v$ associated to matroid ${\mathcal{M}}$. Let ${\mathcal{M}}_i$ be the matroid ${\mathcal{M}}$ restricted to elements $[i]$ and $r_i$ be the unweighted matroid rank function over elements $[i]$ associated with matroid ${\mathcal{M}}_i$. Since the greedy algorithm finds a maximum weight base of a weighted matroid, we have $$\begin{aligned}
v(S) & = \sum_{i \in [n]} w_i \left( r(S \cap [i]) - r(S \cap [i-1]) \right) \\
& = \sum_{i \in [n]} \sum_{j = i}^{n} (w_j - w_{j+1}) \left( r(S \cap [i]) - r(S \cap [i-1]) \right) \\
& = \sum_{ j \in [n]} (w_j - w_{j+1}) \cdot {r(S \cap \{j\})} \\
& = \sum_{j \in [n]} (w_j - w_{j+1}) \cdot {r_j(S)}. \end{aligned}$$
Integrability Conditions {#appendix:integrability}
========================
Given differentiable functions $b_i : {\mathbb{R}}^n \rightarrow {\mathbb{R}}$ for $i = 1, \ldots,
n$, it is well known that there is a function $f$ such that $b_i(x) = \partial f(x) /
\partial x_i$ if and only if the functions $b_i$ satisfy the conditions $\partial b_i /
\partial x_j = \partial b_j / \partial x_i$. [In physics, those correspond to the necessary and sufficient conditions for a field to be a conservative field. We refer to Section 10.16 of [@apostol1969calculus] for a complete discussion. Those conditions can also be derived as a special case of Stoke’s Theorem.]{} The exact same condition provides integrability over the hypercube:
\[lemma:int:1\] Given functions $\beta_i : 2^{[n] \setminus i} \rightarrow {\mathbb{R}}$ for $i \in [n]$, then there exists a function $v$ such that $\beta_i = \partial_i v$ for all $i$ if and only if $\partial_i \beta_j = \partial_j \beta_i$ for all $i \neq j$.
Given $\beta_i$ satisfying $\partial_i \beta_j = \partial_j \beta_i$, fix any order among the elements in $[n]$ and let $S_{<i} = \{j \in S; j < i\}$. Now, define $v$ as follows: $$v(S) = \sum_{i \in S} \beta_i(S_{<i})$$ First we argue that the definition is order independent, i.e., for any ordering of the elements, we construct the same function $v$. To see that, start for an arbitrary order and swap a pair of adjacent elements $i<j$. Then if only one is in $S$, this doesn’t change $v(S)$. If both are in $S$, we change the definition of $v(S)$ from: $$\hdots + \beta_i( T ) + \beta_j (T\cup i) + \hdots$$ to the following (where the terms in $\hdots$ are left unchanged): $$\hdots + \beta_j( T ) + \beta_i (T\cup j) + \hdots$$ Since $\partial_j \beta_i(T) = \partial_i \beta_j(T)$ we have $\beta_i( T ) +
\beta_j (T\cup i) = \beta_j( T ) + \beta_i (T\cup j)$. This is equivalent to the notion of *path-independence* for continuous functions [(see Section 10.17 of [@apostol1969calculus])]{}.
Now, fixed $j$, we can assume without loss of generality that $j$ is placed in the end of the ordering. Therefore by the definition of $v$ we have $v(S
\cup j) = v(S) + \beta_j(S)$ so $\beta_j(S) = \partial_j v(S)$.
We can now obtain second order integrability conditions from the first order ones easily:
\[lemma:int:2\] Given functions $\alpha_{ij} : 2^{[n] \setminus ij} \rightarrow {\mathbb{R}}$ for $i \neq j$, then there exists a function $v$ such that $\alpha_{ij} =
\partial_{ij} v$ for all $i,j$ if and only if $\alpha_{ij} = \alpha_{ji}$ and $$\partial_i \alpha_{jk} = \partial_j \alpha_{ik} = \partial_k \alpha_{ij}
\quad \forall \text{ distinct } i,j,k.$$
First observe that by first order integrability conditions (Lemma \[lemma:int:1\]) we can find for each $i$, a function $\beta_i$ such that $\alpha_{ij} = \partial_j \beta_i$ since $\partial_k \alpha_{ij} = \partial_j
\alpha_{ik}$. Now observe that the functions $\beta_1, \hdots, \beta_n$ constructed satisfy first order integrability conditions, since $\partial_j
\beta_i = \alpha_{ij} = \partial_i \beta_j$ so there is $v$ such that $\beta_i
= \partial_i v$.
\[cor:int:3\] Given functions $\alpha_{ij} : 2^{[n] \setminus ij}
\rightarrow {\mathbb{R}}$ satisfying integrability conditions in the previous lemma, all $v : 2^{[n]} \rightarrow {\mathbb{R}}$ are affine transformations of the function: $$v(S) = \sum_{i<j; i,j \in S} \alpha_{ij}(S_{<i})$$
Using Lemma \[lemma:int:1\] we can reconstruct the functions $\beta_i$ as: $$\beta_i(S) = \beta_i(\emptyset) + \sum_{j \in S} \alpha_{ij}(S_{<j})$$ Applying the same process for reconstructing $v$ from $\beta_i$, we get: $$\begin{aligned}
v(S) & = v(\emptyset) + \sum_{i \in S} \beta_i(S_{<i}) = v(\emptyset) +
\sum_{i \in S} \left[ \beta_i(\emptyset) + \sum_{j \in S_{<i}}
\alpha_{ij}(S_{<j}) \right] \\ & = v(\emptyset) + \sum_{i \in S}
\beta_i(\emptyset) + \sum_{j < i; i,j \in S} \alpha_{ij}(S_{<j})
\end{aligned}$$
Description of ${\ensuremath{\mathbf G}}^4$ {#sec:appG4}
===========================================
For the case of ${\ensuremath{\mathbf G}}^4$, we start by observing that up to renaming the items, the $\emptyset$-tree must have one of two forms, which we will refer as the shallow tree and the deep tree respectively.
child [node\[circle, draw=black\] [$m_2$]{} child [node [$a$]{}]{} child [node [$b$]{}]{}]{} child [node\[circle, draw=black\] [$m_3$]{} child [node [$c$]{}]{} child [node [$d$]{}]{}]{}; ; child \[sibling distance = 15mm\][node [$a$]{}]{} child \[sibling distance = 15mm\][node\[circle, draw=black\] [$m_2$]{} child [node [$b$]{}]{} child [node\[circle, draw=black\] [$m_3$]{} child \[sibling distance = 15mm\] [node [$c$]{}]{} child \[sibling distance = 15mm\] [node [$d$]{}]{}]{}]{};
#### Shallow Tree Case
Assume we have a valuation function $v \in {\ensuremath{\mathbf G}}^4_0$ whose $\emptyset$-tree is shallow (i.e., is like the left diagram above). We know $m_1 \leq m_2$ and $m_1 \leq m_3$. For convenience of notation we will refer to $m_1 = x$, $m_2 = x + y$ and $m_3 = x + z$ for $x,y,z \geq 0$. Applying Lemma \[lem:welldefined\] with $S = \emptyset$ and for every triple of elements, we get:
$$\begin{aligned}
w_1 =: \Delta_{ac}^d = \Delta_{ad}^c = \Delta_{cd}^a - z\\
w_2 =: \Delta_{bc}^d = \Delta_{bd}^c = \Delta_{cd}^b - z\\
w_3 =: \Delta_{ac}^b = \Delta_{bc}^a = \Delta_{ab}^c - y\\
w_4 =: \Delta_{ad}^b = \Delta_{bd}^a = \Delta_{ab}^d - y
\end{aligned}$$
If $w_3 \neq w_4$ we can wlog (up to permuting the identity of the items) assume that $w_3 < w_4$. In such case, observe that: $\Delta_{ac}^b = w_3$, $\Delta_{ad}^b = w_4$, and $\Delta_{cd}^b = w_2 + z$. Since the minimum value is repeated among $\Delta_{ac}^b$, $\Delta_{ad}^b$ and $\Delta_{cd}^b$ we must have $w_3 = w_2 + z$. Now, looking at the substitution symbols for the $\{a\}$-tree, we get: $\Delta_{bc}^a = w_3$, $\Delta_{bd}^a = w_4$, and $\Delta_{cd}^a = w_1 + z$; so we must have by the same argument: $w_3 = w_1 + z$. In particular we will get: $w_1 = w_2 = w_3 - z = w_4 - z - {q}$ for some ${q}\geq 0$. This means that we can write: $$\label{eq:shallow_case}\tag{S}
w_1 = w \qquad w_2 = w \qquad w_3 = w + z \qquad w_4 = w + z + {q}.$$
If $w_3 = w_4$, then there are two cases. If $w_1 \neq w_2,$ then up to permuting the identity of items this case is identical to $w_3 \neq w_4$ and we can obtain with indices permuted. If $w_1 = w_2$ and $w_3 = w_4,$ [then up to permuting the identity of items this case is a special case of with $ {q}= 0$.]{} Therefore it is w.l.o.g. to focus on the setup in .
The values of $w_1$, $w_2$, $w_3$ and $w_4$ define
the $\{i\}$-trees for all $i = a,b,c,d$. It is possible now to reconstruct $v(S)$ for $S \neq \emptyset$ with Corollary \[cor:integration\]: $$\begin{aligned}
&v(abc) = -\Delta_{ab}^\emptyset -\Delta_{ac}^\emptyset - \Delta_{bc}^a = -2x
- y - w - z\\
&v(abd) = -\Delta_{ab}^\emptyset -\Delta_{ad}^\emptyset - \Delta_{bd}^a = -2x -
y - w - z - {q}\\
&v(acd) = -\Delta_{ac}^\emptyset -\Delta_{ad}^\emptyset - \Delta_{cd}^a = -2x -
w - z\\
&v(bcd) = -\Delta_{bc}^\emptyset -\Delta_{bd}^\emptyset - \Delta_{cd}^b = -2x
- z - w \\
& v(abcd) = -3x - y - 2z - 2w - {q}- t
\end{aligned}$$ for some $t \geq 0$. This gives us a valuation $v$ in gross substitutes parametrized by $(x,y,z,w,u,t) \in {\mathbb{R}}^6_+$. Therefore, every valuation $v$ in ${\ensuremath{\mathbf G}}^4_0$ whose $\emptyset$-tree is shallow can be written as a non-negative combination of $6$ *extremal* valuation functions. The extremal valuations are obtained when we set one of the coefficients to one and all coefficients to zero. It is instructive to see which valuations are those.
- $x = 1$ and all other coefficients are zero. We obtain a valuation function such that $v(S) = 0$ for ${\left\vert{S}\right\vert} \leq 1$, $v(S) = -1$ for ${\left\vert{S}\right\vert} = 2$, $v(S) = -2$ for ${\left\vert{S}\right\vert} = 3$ and $v(S) = -3$ for ${\left\vert{S}\right\vert} = 4$. We get $\tilde{v}(S) = 0$ for $S
= \emptyset$ and $\tilde{v}(S) = 1$ for $S \neq \emptyset$, which is $r[U^1_4]$.
- $y=1$ and all other coefficients are zero. Then $v(S) = -1$ if $\{a,b\}
\subseteq S$ and $v(S) = 0$ otherwise, and $\tilde{v}$ is the rank function of the following matroid:
=\[circle,fill=black,minimum size=5pt,inner sep=0pt\]; (a) at (0,0) ; (b) at (1.5,0) ; (c) at (3,0) ; (d) at (4.5,0) ; (a) .. controls +(-30:20pt) and +(-150:20pt) .. node\[below\][$b$]{}(b); (a) .. controls +(+30:20pt) and +(+150:20pt) .. node\[above\][$a$]{} (b); (b) – node\[above\] [$c$]{}(c); (c) – node\[above\] [$d$]{}(d);
- $z = 1$ and all other coefficients are zero. Then $v(S) = 0$ for ${\left\vert{S}\right\vert} \leq 1$, $v(cd) = -1$, $v(S) = 0$ for all other $S$ with ${\left\vert{S}\right\vert} =
2$, $v(S) = -1$ for all $S$ with ${\left\vert{S}\right\vert} = 3$, and finally $v(abcd) = -2$. So $\tilde{v}$ is the rank function of the matroid:
=\[circle,fill=black,minimum size=5pt,inner sep=0pt\]; (a) at (0,0) ; (b) at (2,0) ; (c) at (1,1.5) ; (a) .. controls +(-30:20pt) and +(-150:20pt) .. node\[below\][$c$]{}(b); (a) .. controls +(+30:20pt) and +(+150:20pt) .. node\[above\][$d$]{} (b); (a) – node\[above\] [$a$]{}(c); (b) – node\[above\] [$b$]{}(c);
- $w = 1$ and all other coefficients are zero, then $v(S) = 0$ for ${\left\vert{S}\right\vert} \leq 2$, $v(S) = -1$ for ${\left\vert{S}\right\vert} = 3$ and $v(S) = -2$ for ${\left\vert{S}\right\vert}
= 4$, and $\tilde{v}$ is $r[U_4^2]$.
- ${q}= 1$ and all other coefficients are zero, then $v(S) = -1$ if $\{a,b,d\} \subseteq S$ and $v(S) = 0$ otherwise. So, $\tilde{v}(S) = v(S)
+ {\left\vert{S}\right\vert}$ is the rank function of the matroid:
=\[circle,fill=black,minimum size=5pt,inner sep=0pt\]; (a) at (0,0) ; (b) at (2,0) ; (c) at (1,1.5) ; (d) at (4,0) ; (a) – node\[below\] [$a$]{}(b); (a) – node\[above\] [$b$]{}(c); (b) – node\[above\] [$d$]{}(c); (b) – node\[above\] [$c$]{}(d);
- if $t = 1$ and all other coefficients are zero, then $v(S) = 0$ if ${\left\vert{S}\right\vert} \leq 3$ and $v(S) = -1$ for ${\left\vert{S}\right\vert} = 4$, therefore, $\tilde{v}(S)$ is $r[U_4^3]$.
Therefore we identified the following cone which is a subset of ${\ensuremath{\mathbf G}}^4$: $${\ensuremath{\mathbf E}}^4 + \sum_{j=1}^6 r_j \cdot {\mathbb{R}}_+$$ where $r_1, \hdots, r_6$ are the rank functions of the matroids identified in the previous items. Also, since ${\ensuremath{\mathbf G}}^4$ is symmetric with respect to permutations of the identities of the items, we can obtain $11$ other cones by permuting the identities of the items.
#### Deep Tree Case.
Assume now that we have a valuation $v \in {\ensuremath{\mathbf G}}^4_0$ whose $\emptyset$-tree is deep. We know $m_1 \leq m_2 \leq m_3$, so we will refer for convenience to $m_1
= x$, $m_2 = x+y$ and $m_3 = x+y+z$ for $x,y,z \geq 0$. By applying again Lemma \[lem:welldefined\] we obtain:
$$\begin{aligned}
w_1 =: \Delta_{ab}^c = \Delta_{ac}^b = \Delta_{bc}^a - y\\
w_2 =: \Delta_{ab}^d = \Delta_{ad}^b = \Delta_{bd}^a - y\\
w_3 =: \Delta_{ac}^d = \Delta_{ad}^c = \Delta_{cd}^a - y - z\\
w_4 =: \Delta_{bc}^d = \Delta_{bd}^c = \Delta_{cd}^b - z
\end{aligned}$$
It is instructive to re-write the substitution symbols to write together the symbols for the same tree:
$$\begin{aligned}
& a : \\
& \Delta_{bc}^a = w_1 + y \\
& \Delta_{bd}^a = w_2 + y \\
& \Delta_{cd}^a = w_3 + y + z
\end{aligned}
\qquad
\begin{aligned}
& b : \\
& \Delta_{ac}^b = w_1 \\
& \Delta_{ad}^b = w_2 \\
& \Delta_{cd}^b = w_4 + z
\end{aligned}
\qquad
\begin{aligned}
& c : \\
& \Delta_{ab}^c = w_1 \\
& \Delta_{ad}^c = w_3 \\
& \Delta_{bd}^c = w_4
\end{aligned}
\qquad
\begin{aligned}
& d : \\
& \Delta_{ab}^d = w_2 \\
& \Delta_{ac}^d = w_3 \\
& \Delta_{bc}^d = w_4
\end{aligned}$$
In each column, at least two values are equal and the third value is at least as large as the two equal values. Let’s look at the symbols for the $\{c\}$-tree. There are four possibilities:
1. $w_1 = w_3 < w_4$. In such case, by looking at the $\{d\}$-tree, since $w_3 < w_4$, it must be the case that $w_2 = w_3$ since the minimum substitution symbol in the $\{d\}$-tree must be repeated. Therefore we must have $w_1 = w_2 = w_3 = w$ and $w_4 = w + {q}$. It is simple to see that if this condition is true, for every $\{i\}$-tree, the minimum value is repeated.
2. $w_1 = w_4 < w_3$. In such case, by looking at the $\{d\}$-tree, since [$w_4 < w_3$,]{} it must be the case that $w_2 = w_4$ since the minimum substitution symbol in the $\{d\}$-tree must be repeated. Therefore we must have $w_1 = w_2 = w_4 = w$ and $w_3 = w + {q}$.
3. $w_3 = w_4 < w_1$. Now, we consider other three possibilities for the $\{b\}$-tree:
1. $w_1 = w_4 + z \leq w_2$, so we have $w_1 = w + z$, $w_2 = w + z + {q}$ and $w_3 = w_4 = w$,
2. $w_2 = w_4 + z < w_1$, so we have: $w_1 = w+z+u$, $w_2 = w + z$, and $w_3 = w_4 = w$.
3. $w_1 = w_2 < w_4 + z$, so we have $ w_1 = w + {q}$, $w_2 = w + {q}$, $w_3 = w_4 = w$, and $ z = {q}+ {s} $.
4. $w_1 = w_3 = w_4$. By inspecting the $\{b\}$-tree, we must have $w_2 =
w_1$. Since this is a special case of the previous cases, we ignore this case from now on.\
We note that in either case, we have:
$$\begin{aligned}
&v(abc) = -\Delta_{ab}^\emptyset -\Delta_{ac}^\emptyset - \Delta_{bc}^a = -2x
- y - w_1\\
&v(abd) = -\Delta_{ab}^\emptyset -\Delta_{ad}^\emptyset - \Delta_{bd}^a = -2x -
y - w_2 \\
&v(acd) = -\Delta_{ac}^\emptyset -\Delta_{ad}^\emptyset - \Delta_{cd}^a = -2x -
y - z - w_3\\
&v(bcd) = -\Delta_{bc}^\emptyset -\Delta_{bd}^\emptyset - \Delta_{cd}^b = -2x
- 2y - z - w_4
\end{aligned}$$
[By Corollary \[cor:integration\], we have $$v(abcd) =
-\Delta^{\emptyset}_{ab} - \Delta^{\emptyset}_{ac} - \Delta^{\emptyset}_{ad} -
\Delta_{bc}^a - \Delta_{bd}^a - \Delta_{cd}^{ab} = −3x − 2y − w_1 − w_2 − \Delta_{cd}^{ab}.$$ Next, we get $$\Delta^{ab}_{cd} = \begin{cases} t + {q}+ z & \text{case 1 and case 2} \\ t & \text{case 3a and case 3b} \\ t + {s} & \text{case 3c} \end{cases}$$ for some $t \geq 0$ where case 1 is by Lemma \[lem:welldefined\] with $S = \{b\}$ and triplet $acd$, case 2 and case 3c are by Lemma \[lem:welldefined\] with $S = \{a\}$ and triplet $bcd$, and case 3a and case 3b are since we simply have $\Delta^{ab}_{cd} \geq 0$. We obtain $$v(abcd) = \begin{cases} −3x − 2y − 2w - z - {q}- t & \text{case 1 and case 2} \\ −3x − 2y − 2w -2z -{q}- t & \text{case 3a and case 3b} \\ −3x − 2y − 2w - 2{q} - {s} - t& \text{case 3c} \end{cases}$$]{}
Now, following the same procedure used in the previous section, we have that in each case we have:
- Case 1: $v \in {\ensuremath{\mathbf E}}^4 + \sum_{j=1}^6 r_j \cdot {\mathbb{R}}_+$ where $r_1, \hdots,
r_6$ are the ranks of the following matroids [which respectively correspond to the variables $x, y, z, w, {q}, t$]{}: $${ U^1_4 },
\begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (4,0) {};
\draw (a) .. controls +(-60:40pt) and +(-120:40pt) .. node[below]{$b$}(b);
\draw (a) .. controls +(+60:40pt) and +(+120:40pt) .. node[above]{$c$} (b);
\draw (a) -- node[below] {$d$}(b);
\draw (b) -- node[above] {$a$}(c);
\end{tikzpicture},
\begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (4,0) {};
\node[vertex] (d) at (-2,0) {};
\draw (a) .. controls +(-30:20pt) and +(-150:20pt) .. node[below]{$c$}(b);
\draw (a) .. controls +(+30:20pt) and +(+150:20pt) .. node[above]{$d$}(b);
\draw (a) -- node[below] {$a$}(d);
\draw (b) -- node[above] {$b$}(c);
\end{tikzpicture}, {U_4^2},
\begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (1,1.2) {};
\node[vertex] (d) at (4,0) {};
\draw (a) -- node[above] {$b$}(c);
\draw (b) -- node[above] {$c$}(c);
\draw (a) -- node[below] {$d$}(b);
\draw (b) -- node[above] {$a$}(d);
\end{tikzpicture}, { U_4^3 }$$
- Case 2: Same as before but with the rank functions of the following matroids [which respectively correspond to the variables $x, y, z, w, {q}, t$]{}: $${U^1_4},
\begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (4,0) {};
\draw (a) .. controls +(-60:40pt) and +(-120:40pt) .. node[below]{$b$}(b);
\draw (a) .. controls +(+60:40pt) and +(+120:40pt) .. node[above]{$c$} (b);
\draw (a) -- node[below] {$d$}(b);
\draw (b) -- node[above] {$a$}(c);
\end{tikzpicture},
\begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (4,0) {};
\node[vertex] (d) at (-2,0) {};
\draw (a) .. controls +(-30:20pt) and +(-150:20pt) .. node[below]{$c$}(b);
\draw (a) .. controls +(+30:20pt) and +(+150:20pt) .. node[above]{$d$}(b);
\draw (a) -- node[below] {$a$}(d);
\draw (b) -- node[above] {$b$}(c);
\end{tikzpicture},{ U_4^2},
\begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (1,1.2) {};
\node[vertex] (d) at (4,0) {};
\draw (a) -- node[above] {$a$}(c);
\draw (b) -- node[above] {$c$}(c);
\draw (a) -- node[below] {$d$}(b);
\draw (b) -- node[above] {$b$}(d);
\end{tikzpicture}, {U_4^3}$$
- Case 3a: Same as before but with the rank functions of the following matroids [which respectively correspond to the variables $x, y, z, w, {q}, t$]{}: $${U^1_4},
\begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (4,0) {};
\draw (a) .. controls +(-60:40pt) and +(-120:40pt) .. node[below]{$b$}(b);
\draw (a) .. controls +(+60:40pt) and +(+120:40pt) .. node[above]{$c$} (b);
\draw (a) -- node[below] {$d$}(b);
\draw (b) -- node[above] {$a$}(c);
\end{tikzpicture},
\begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (1,1.2) {};
\draw (a) .. controls +(-30:20pt) and +(-150:20pt) .. node[below]{$c$}(b);
\draw (a) .. controls +(+30:20pt) and +(+150:20pt) .. node[above]{$d$}(b);
\draw (a) -- node[above] {$a$}(c);
\draw (b) -- node[above] {$b$}(c);
\end{tikzpicture}, {U_4^2},
\begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (1,1.2) {};
\node[vertex] (d) at (4,0) {};
\draw (a) -- node[above] {$a$}(c);
\draw (b) -- node[above] {$b$}(c);
\draw (a) -- node[below] {$d$}(b);
\draw (b) -- node[above] {$c$}(d);
\end{tikzpicture}, {U_4^3}$$
- Case 3b: Same as before but with the rank functions of the following matroids [which respectively correspond to the variables $x, y, z, w, {q}, t$]{}: $${U^1_4},
\begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (4,0) {};
\draw (a) .. controls +(-60:40pt) and +(-120:40pt) .. node[below]{$b$}(b);
\draw (a) .. controls +(+60:40pt) and +(+120:40pt) .. node[above]{$c$} (b);
\draw (a) -- node[below] {$d$}(b);
\draw (b) -- node[above] {$a$}(c);
\end{tikzpicture},
\begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (1,1.2) {};
\draw (a) .. controls +(-30:20pt) and +(-150:20pt) .. node[below]{$c$}(b);
\draw (a) .. controls +(+30:20pt) and +(+150:20pt) .. node[above]{$d$}(b);
\draw (a) -- node[above] {$a$}(c);
\draw (b) -- node[above] {$b$}(c);
\end{tikzpicture}, {U_4^2},
\begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (1,1.2) {};
\node[vertex] (d) at (4,0) {};
\draw (a) -- node[above] {$a$}(c);
\draw (b) -- node[above] {$b$}(c);
\draw (a) -- node[below] {$c$}(b);
\draw (b) -- node[above] {$d$}(d);
\end{tikzpicture}, {U_4^3}$$
- Case 3c: Same as before but with the rank functions of the following matroids [which respectively correspond to the variables $x, y, {q}, w, {s}, t$]{}: $${U^1_4},
\begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (4,0) {};
\draw (a) .. controls +(-60:40pt) and +(-120:40pt) .. node[below]{$b$}(b);
\draw (a) .. controls +(+60:40pt) and +(+120:40pt) .. node[above]{$c$} (b);
\draw (a) -- node[below] {$d$}(b);
\draw (b) -- node[above] {$a$}(c);
\end{tikzpicture},
{
\begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (1,1.2) {};
\draw (a) .. controls +(-30:20pt) and +(-150:20pt) .. node[below]{$c$}(b);
\draw (a) .. controls +(+30:20pt) and +(+150:20pt) .. node[above]{$d$}(b);
\draw (a) -- node[above] {$a$}(c);
\draw (b) -- node[above] {$b$}(c);
\end{tikzpicture},} {U_4^2},
{ \begin{tikzpicture}[baseline=0ex, scale=.6]
\tikzstyle{vertex}=[circle,fill=black,minimum size=5pt,inner sep=0pt];
\node[vertex] (a) at (0,0) {};
\node[vertex] (b) at (2,0) {};
\node[vertex] (c) at (4,0) {};
\node[vertex] (d) at (-2,0) {};
\draw (a) .. controls +(-30:20pt) and +(-150:20pt) .. node[below]{$c$}(b);
\draw (a) .. controls +(+30:20pt) and +(+150:20pt) .. node[above]{$d$}(b);
\draw (a) -- node[below] {$a$}(d);
\draw (b) -- node[above] {$b$}(c);
\end{tikzpicture}}, {U_4^3}.$$
[^1]: That is perhaps not the most common definition of concavity but it is completely equivalent to the condition $f(tx +
(1-t)y) \geq tf(x) + (1-t) f(y), \forall t \in [0,1]$.
[^2]: We note that even though positive linear combination and item grouping preserve submodularity, neither of them preserves substitutability in general.
[^3]: Recall from Section \[sec:prelim\] that a valuation function is normalized if $v(\emptyset) = v(i) = 0$ for all $i \in
[5]$.
[^4]: Recall the definition of a normalized matroid rank function in Section \[sec:otherfunctions\].
[^5]: We thank Kazuo Murota for his suggestion on a earlier version of this manuscript to phrase this analysis in terms of integrability condition.
|
---
abstract: 'Elastic-scattering phase shifts for four-nucleon systems are studied in an $ab$-$initio$ type cluster model in order to clarify the role of the tensor force and to investigate cluster distortions in low energy $d$+$d$ and $t$+$p$ scattering. In the present method, the description of the cluster wave function is extended from a simple (0$s$) harmonic-oscillator shell model to a few-body model with a realistic interaction, in which the wave function of the subsystems are determined with the Stochastic Variational Method. In order to calculate the matrix elements of the four-body system, we have developed a Triple Global Vector Representation method for the correlated Gaussian basis functions. To compare effects of the cluster distortion with realistic and effective interactions, we employ the AV8$^{\prime}$ potential as a realistic interaction and the Minnesota potential as an effective interaction. Especially for $^1S_0$, the calculated phase shifts show that the $t$+$p$ and $h$+$n$ channels are strongly coupled to the $d$+$d$ channel for the case of the realistic interaction. On the contrary, the coupling of these channels plays a relatively minor role for the case of the effective interaction. This difference between both potentials originates from the tensor term in the realistic interaction. Furthermore, the tensor interaction makes the energy splitting of the negative parity states of $^4$He consistent with experiments. No such splitting is however reproduced with the effective interaction.'
author:
- 'S. Aoyama, K. Arai, Y. Suzuki, P. Descouvemont, D. Baye'
title: ' Four-nucleon scattering with a correlated Gaussian basis method'
---
Introduction {#sect.1}
============
The microscopic cluster model is one of the successful models to study the structure and reactions of light nuclei [@tang77]. In the conventional cluster model, one assumes that the nucleus is composed of several simple clusters with $A \le 4$ which are described by (0$s$) harmonic-oscillator shell model functions, and use an effective $N$-$N$ interaction which is appropriate for such a model space. However, it is well known that the ground states of the typical clusters $d$, $t$, $h$ and $^4$He have non-negligible admixtures of $D$-wave component due to the tensor interaction. Since the conventional cluster model does not directly treat the $D$-wave component, the strong attraction of the nucleon-nucleon interaction due to the tensor term is assumed to be renormalized into the central term of the effective interaction.
Recently, $ab$-$initio$ structure calculations [@benchmark] have been successfully developed: Stochastic Variational Method (SVM) [@varga94; @varga97; @book; @vs95], Global Vector Representation method (GVR) [@DGVR; @GVR], Green’s function Monte Carlo method [@carlson98], no core shell model [@navratil00], correlated hyperspherical harmonics method [@viviani98r], unitary correlation operator method [@ucom], and so on. Although the application of $ab$-$initio$ reaction calculations with a realistic interaction are restricted so much in comparison with structure calculations, it has been intensively applied to the four-nucleon systems $t$+$n$ and $h$+$p$ [@arai10; @hofmann01; @deltuva07; @navratil09; @viviani98; @lazauskas05; @fisher06]. Especially $d$+$d$ scattering states, which couple to $t$+$p$ and $h$+$n$ channels, have attracted much attention, because the $d$+$d$ radiative capture is one of the mechanisms making $^4$He through electro-magnetic transitions [@arriaga91; @sabourov04] and also have posed intriguing puzzles for analyzing powers [@hofmann08; @hofmann97; @deltuva08; @lazauskas04; @ciesielski99], which are motivated by the famous $A_y$ problem in the three-nucleon system.
Furthermore, the $d$+$d$ elastic-scattering phase shifts are interesting because the astrophysical S-factor of the $d$($d$,$\gamma$)$^4$He reaction is not explained by any calculation using an effective interaction that contains no tensor term, and is expected to be contributed by the $D$-wave components of the clusters through $E2$ transitions [@langanke87; @fowler67].
Also, thanks to recent developments of the microscopic cluster model, the simple model using the (0$s$) harmonic-oscillator wave function with an effective interaction is not mandatory any more, at least, in light nuclei. We can use a kind of $ab$-$initio$ cluster model which employs more realistic cluster wave functions with realistic interactions. Therefore, it is interesting to see the difference between the $ab$-$initio$ reaction calculations with a tensor term and the conventional cluster model calculations without a tensor term in few-body systems. The microscopic $R$-matrix method (MRM) with a cluster model (GCM or RGM) has been applied to studies of many nuclei [@baye77; @kanada85; @arai01; @desc10]. It is now used in $ab$ $initio$ descriptions of collisions [@navratil09]. We have also applied the MRM to the $h$+$p$ scattering problem with more realistic cluster wave functions by using a realistic interaction [@arai10]. The Gaussian basis functions for the expansion of the cluster wave functions are chosen by a technique of the SVM [@book]. In the MRM, as will be shown later, the relative wave function between clusters ($a$ and $b$) is connected to the boundary condition at a channel radius. The problem is how to calculate the matrix elements. In this paper, we develop a method called the Triple Global Vector Representation method (TGVR), by which we calculate the matrix elements in a unified way. Although we restricted ourselves to four nucleon systems in the present paper, the formulation of the TGVR itself can be applied to more than four-body systems as in the previous studies of the Global Vector Representation methods (GVR) [@GVR]. Furthermore, for scattering problems, the TGVR can deal with more complicated systems than the double (or single) global vector which was given in the previous papers [@DGVR; @GVR], because we need three representative orbital angular momenta, the total internal orbital momenta of both clusters and the orbital momentum of their relative motion, in order to reasonably describe the scattering states. In other words, the first global vector represents the angular momentum of cluster $a$, the second global vector represents the angular momentum of cluster $b$, and the third global vector represents the relative angular momentum between the clusters.
In this paper, we will investigate the effect of the distortion of clusters on the $d$+$d$ elastic-scattering by comparing the phase shifts calculated with a realistic and an effective interaction. In section 2, we explain the MRM in brief. In section 3, the correlated Gaussian (CG) method with the TGVR, which has newly been developed for the present analysis, will be presented. In section 4, we will explain how to calculate the matrix elements with TGVR basis functions. The typical matrix elements are also given in the appendix. In section 5, we will present and discuss the calculated scattering phase shifts in detail. Finally, summary and conclusions are given in section 6.
Microscopic $R$-matrix method {#sect.2}
=============================
In the present study we calculate $d$+$d$ and $t$+$p$ (and $h$+$n$) elastic scattering phase shifts with the microscopic $R$-matrix method. Though the method is well documented in e.g. Refs. [@baye77; @kanada85; @desc10], we briefly explain it below in order to present definitions and equations needed in the subsequent sections. Since our interest is on low-energy scattering, we consider only two-body channels. A channel $\alpha$ is specified by the two nuclei (clusters) $a,
b$, their angular momenta, $I_a, I_b$, the channel spin $I$ that is a resultant of the coupling of $I_a$ and $I_b$, and the orbital angular momentum $\ell$ for the relative motion of $a$ and $b$. The wave function of channel $\alpha$ with the total angular momentum $J$, its projection $M$, and the parity $\pi$ takes the form $$\begin{aligned}
\Psi^{JM\pi}_{\alpha}=
{\cal A}
\left[\left[\Phi^{a}_{I_a}\Phi^{b}_{I_b}\right]_I
\chi_{\alpha}
(\mbox{\boldmath$\rho$}_{\alpha})\right]_{JM},
\label{wf-ch1}\end{aligned}$$ where $\Phi^{a}_{I_a}$ and $\Phi^{b}_{I_b}$ are respectively antisymmetrized intrinsic wave functions of $a$ and $b$, and ${\cal A}$ is an operator that antisymmetrizes between the clusters. The square bracket $[{I_a} \ {I_b}]_I$ denotes the angular momentum coupling. The coordinate ${{\mbox{\boldmath $\rho$}}}_\alpha$ in the relative motion function $\chi^J_{\alpha}$ is the relative distance vector of the clusters. The channel spin $I$ and the relative angular momentum $\ell$ in $\alpha$ are coupled to give the total angular momentum $J$. The relative-motion functions $\chi_{\alpha}$ also depend on $J$ and $\pi$. For simplicity, this dependence is not displayed explicitly in the notation for $\chi_{\alpha}$ as well as for some other quantities below.
The configuration space is divided into two regions, internal and external, by the channel radius $a$. In the internal region ($\rho_\alpha\le a$), the total wave function may be expressed in terms of a combination of various $\Psi_{\alpha}^{JM\pi}$s $$\begin{aligned}
\Psi^{JM\pi}_{\rm int}&=&\sum_{\alpha} \Psi^{JM\pi}_{\alpha}\nonumber \\
&=&\sum_{\alpha}\sum_{n}f_{\alpha n} {\cal A}
u_{\alpha n}(\rho_\alpha) \phi^{JM\pi}_{\alpha},
\label{wf.int}\end{aligned}$$ with $$\begin{aligned}
\phi^{JM\pi}_{\alpha} = \frac{1}{\sqrt{(1+\delta_{I_aI_b}\delta_{ab})(1+\delta_{ab})}} \left\{
\left[\left[\Phi^{a}_{I_a}\Phi^{b}_{I_b}\right]_I Y_{\ell}(\widehat{{\mbox{\boldmath $\rho$}}}_\alpha)\right]_{JM}
\right. \nonumber \\ \left.
+(-1)^{A_a+I_a+I_b-I+\ell}
\left[\left[\Phi^{b}_{I_b}\Phi^{a}_{I_a}\right]_I Y_{\ell}(\widehat{{\mbox{\boldmath $\rho$}}}_\alpha)\right]_{JM} \delta_{ab}
\right\},
\label{wf.channel}\end{aligned}$$ where $A_a$ is the number of nucleons in cluster $a$, $\delta_{ab}$ is unity if $a$ and $b$ are identical clusters and zero otherwise, and $\delta_{I_aI_b}$ is unity if the clusters are in identical states and zero otherwise. In the second line of Eq. (\[wf.int\]), the relative motion functions of Eq. (\[wf-ch1\]) are expanded in terms of some basis functions as $$\begin{aligned}
\chi_{\alpha m}({{\mbox{\boldmath $\rho$}}}_\alpha) = \sum_n f_{\alpha n}
u_{\alpha n}(\rho_\alpha)Y_{\ell m}(\widehat{{\mbox{\boldmath $\rho$}}}_\alpha).
\label{chi.expansion}\end{aligned}$$ In what follows we take $$\begin{aligned}
u_{\alpha n}(\rho_\alpha)=\rho_\alpha^{\ell}\exp(-\frac{1}{2}\lambda_n \rho_\alpha^2)
\label{u.functions}\end{aligned}$$ with a suitable set of $\lambda_n$s.
In the external region ($\rho_\alpha\ge a$), the total wave function takes the form $$\begin{aligned}
\Psi^{JM\pi}_{\rm ext}=\sum_{\alpha} g_{\alpha}(\rho_\alpha)
\phi^{JM\pi}_{\alpha}
\label{wf.ext}.\end{aligned}$$ Note that the antisymmetrization between the clusters is dropped in the external region under the condition that the channel radius $a$ is large enough. The function $g_{\alpha}(\rho_\alpha)$ of Eq. (\[wf.ext\]) is a solution of the equation $$\begin{aligned}
\left[-\frac{\hbar^2}{2\mu_{\alpha}}\left( \frac{d^2}{d\rho_\alpha^2}
+\frac{2}{\rho_\alpha}\,\frac{d}{d\rho_\alpha}-\frac{\ell(\ell+1)}{\rho_\alpha^2} \right)+\frac{Z_aZ_be^2}{\rho_\alpha}\right] g_{\alpha}(\rho_\alpha) =
E_{\alpha} g_{\alpha}(\rho_\alpha),\end{aligned}$$ where $\mu_{\alpha}$ is the reduced mass for the relative motion in channel $\alpha$, $Z_ae$ and $Z_be$ are the charges of $a$ and $b$, and $E_{\alpha}=E-E_a-E_b$ is the energy for the relative motion, where $E$ is the total energy, and $E_a$ and $E_b$ are the internal energies for the clusters $a$ and $b$, respectively. For the scattering initiated through the channel $\alpha_0$, the asymptotic form of $g_{\alpha}$ for the open channel $\alpha$ $(E_{\alpha} \geq 0)$ is $$\begin{aligned}
g_{\alpha}(\rho_\alpha) = v_\alpha^{-1/2} \rho_\alpha^{-1}
[I_{\alpha}(k_\alpha \rho_\alpha)\delta_{\alpha\,
\alpha_0}-S_{\alpha \, \alpha_0}^{J\pi} O_{\alpha}(k_\alpha \rho_\alpha)],\end{aligned}$$ where $k_\alpha=\sqrt{2\mu_\alpha |E_\alpha |}/\hbar$, $v_\alpha=\hbar k_\alpha/\mu_\alpha$ and $S_{\alpha \, \alpha_0}^{J\pi}$ is an element of the $S$-matrix (or collision matrix) to be determined. Here $I_{\alpha}(k_\alpha \rho_\alpha)$ and $O_{\alpha}(k_\alpha \rho_\alpha)$ are the incoming and outgoing waves defined by $$\begin{aligned}
I_{\alpha}(k_\alpha \rho_\alpha)=O_{\alpha}(k_\alpha \rho_\alpha)^*=
G_\ell(\eta_{\alpha}, k_\alpha \rho_\alpha)-iF_\ell(\eta_{\alpha}, k_\alpha \rho_\alpha),\end{aligned}$$ with the regular and irregular Coulomb functions $F_\ell$ and $G_\ell$. The Sommerfeld parameter $\eta_{\alpha}$ is $\mu_\alpha Z_a Z_b e^2/\hbar^2
k_\alpha$. For a closed channel $\alpha$ $(E_\alpha < 0)$, the asymptotic form of $g_\alpha$ is given by the Whittaker function $$\begin{aligned}
g_{\alpha}(\rho_\alpha) \propto \rho_\alpha^{-1}
W_{-\eta_{\alpha},\ell+1/2}(2k_\alpha\rho_\alpha).\end{aligned}$$
The matrix elements $S_{\alpha \alpha_0}^{J\pi}$ are determined by solving a Schrödinger equation with a microscopic Hamiltonian $H$ involving the $A_a + A_b$ nucleons, $$\begin{aligned}
(H+{\cal L} -E)\Psi_{\rm int}^{JM\pi}={\cal L}\Psi_{\rm ext}^{JM\pi},\end{aligned}$$ with the Bloch operator ${\cal L}$ $$\begin{aligned}
{\cal L}=\sum_{\alpha}\frac{\hbar^2}{2\mu_\alpha a}
\vert \phi_{\alpha}^{JM \pi} \rangle \delta(\rho_\alpha-a)
\left(\frac{\partial}{\partial \rho_\alpha}-\frac{b_\alpha}{\rho_\alpha} \right)\rho_\alpha
\langle \phi_{\alpha}^{JM\pi}\vert,
\label{Bloch}\end{aligned}$$ where the channel radius $a$ is chosen to be the same for all channels, and the $b_\alpha$ are arbitrary constants. Here, we choose $b_\alpha=0$ for the open channels and $b_\alpha=2k_\alpha a W'_{-\eta_\alpha,\ell+1/2}(2k_\alpha a)/W_{-\eta_\alpha,\ell+1/2}(2k_\alpha a)$ for the closed channels. The results do not depend on the choices for $b_\alpha$ but these values simplify the calculations. Notice that the projector on $|\phi_{\alpha}^{JM \pi}\rangle$ in Eq. (\[Bloch\]) is not essential in a microscopic calculation and can be dropped since the various channels are orthogonal at the channel radius.
The Bloch operator ensures that the logarithmic derivative of the wave function is continuous at the channel radius. In addition, $\Psi_{\rm int}^{JM\pi}$ must be equal to $\Psi_{\rm ext}^{JM\pi}$ at $\rho_\alpha=a$. Projecting the Schrödinger equation on a basis state, one obtains $$\begin{aligned}
\sum_{\alpha n} C_{\alpha' n', \alpha n} \, f_{\alpha n}
= \langle \Phi_{\alpha' n'}^{JM\pi} \vert {\cal L} \vert \Psi_{\rm ext}^{JM\pi} \rangle
\label{SEq}\end{aligned}$$ with $$\begin{aligned}
C_{\alpha' n', \alpha n} =
\langle \Phi_{\alpha' n'}^{JM\pi} \vert H+{\cal L} -E
\vert {\cal A}\Phi_{\alpha n}^{JM\pi} \rangle_{\rm int},\end{aligned}$$ and $$\Phi_{\alpha n}^{JM\pi} = u_{\alpha n}(\rho_\alpha)\phi_{\alpha}^{JM\pi}.$$ Here $\langle\vert {\cal O} \vert\rangle_{\rm int}$ indicates that the integration with respect to $\rho_\alpha$ is to be carried out in the internal region. Actually $\langle\vert {\cal O} \vert\rangle_{\rm int}$ is obtained by calculating the matrix element $\langle\vert {\cal O} \vert\rangle$ in the entire space and subtracting the corresponding external matrix element $\langle\vert {\cal O} \vert\rangle_{\rm ext}$ that is easily obtained because no intercluster antisymmetrization is needed. The $R$-matrix and $Z$-matrix are defined by $$\begin{aligned}
&&{\cal R}_{\alpha' \alpha} \equiv \frac{\hbar^2 a}{2}
\left(\frac{k_{\alpha'}}{\mu_{\alpha'}\mu_{\alpha}k_\alpha}\right)^\frac{1}{2}
\sum_{n' n}u_{\alpha' n'}(a)(C^{-1})_{\alpha' n', \alpha n}u_{\alpha n}(a),
\\
&&{\cal Z}_{\alpha' \alpha} \equiv I_{\alpha} (k_\alpha a) \delta_{\alpha' \alpha}-
{\cal R}_{\alpha' \alpha}k_\alpha a I'_{\alpha}(k_\alpha a).\end{aligned}$$ The $S$-matrix is finally obtained as $$\begin{aligned}
S^{J\pi}=({\cal Z}^*)^{-1}{\cal Z}.\end{aligned}$$ In this paper we focus on the elastic phase shifts $\delta_{\alpha}^{J\pi}$ that are defined by the diagonal elements of the $S$-matrix, $$\begin{aligned}
S_{\alpha \alpha}^{J\pi}=\eta_\alpha^{J\pi} e^{2i\delta_\alpha^{J\pi}}.\end{aligned}$$
We study four-nucleon scattering involving the $d$+$d$, $t$+$p$ and $h$+$n$ channels in the energy region around and below the $d$+$d$ threshold. In Table \[chan0\] we list all possible labels $^{2I+1}\ell_J$ of physical channels for $J^{\pi}=0^{\pm}$, $1^{\pm}$, and $2^{\pm}$, assuming $\ell \leq 2$. Here “physical” means that the channels involve the cluster bound states that appear in the external region as well. Non-physical channels involving excited pseudo states will also be included in most calculations. Note that the $d$+$d$ channel must satisfy the condition of $I+\ell$ even (see Eq. (\[wf.channel\])). The channel spin $I=0$ or 2 can couple with only even $\ell$, but $I=1$ with only odd $\ell$. It is noted that the relative motion for the $d$+$d$ scattering can have $\ell=0$ only when $J^{\pi}$ is equal to $0^+$ and $2^+$.
0$^+$ 1$^+$ 2$^+$ 0$^-$ 1$^-$ 2$^-$
------------------------------------------------------------------------------ --------- --------- --------- --------- --------- ---------
$d(1^+)$+$d(1^+)$ $^1S_0$ $^5D_1$ $^5S_2$ $^3P_0$ $^3P_1$ $^3P_2$
$^5D_0$ $^1D_2$
$^5D_2$
$t(\frac{1}{2}^+)$+$p(\frac{1}{2}^+), \;h(\frac{1}{2}^+)$+$n(\frac{1}{2}^+)$ $^1S_0$ $^3S_1$ $^1D_2$ $^3P_0$ $^1P_1$ $^3P_2$
$^3D_1$ $^3D_2$ $^3P_1$
: Channel spins ($^{2I+1}\ell_J$) of physical $d$+$d$, $t$+$p$, and $h$+$n$ channels for $J \le 2$ and $\ell \le 2$.
\[chan0\]
Because one of our purposes in this investigation is to understand the role of the tensor force played in the four-nucleon dynamics, we want to compare the phase shifts obtained with two Hamiltonians that differ in the type of $NN$ interactions. One is a realistic interaction called the AV8$^{\prime}$ potential [@av8p] that includes central, tensor and spin-orbit components. We also add an effective three-nucleon force (TNF) in order to reproduce reasonably the binding energies of $t$, $h$ and $^4$He [@hiyama04], which makes reasonable thresholds. In the present calculation, the TNF is included in all calculations for AV8$^{\prime}$. Another is an effective central interaction called the Minnesota (MN) potential [@thompson77], which reproduce reasonably the binding energies of $t$, $h$ and $^4$He, though it has central terms alone (with an exchange parameter $u=1$). The Coulomb potential is included for both potentials.
The intrinsic wave function $\Phi^{k}_{I_k}$ of cluster $k$ $(k=a,\ b)$ is described with a combination of $N_k$ basis functions with different $L_k$ and $S_k$ values $$\begin{aligned}
\Phi_{I_{k}M_{I_k}}^k=\sum^{N_k} {\cal A} \left[
\psi_{L_k}^{(\rm{space})}\psi_{S_k}^{(\rm{spin})}
\right]_{I_k M_{I_k}}
\psi^{(\rm{isospin})}_{T_k M_{T_k}},
\label{wf.cluster}\end{aligned}$$ where $\psi_{L_k}^{(\rm{space})}$, $\psi_{S_k}^{(\rm{spin})}$ and $\psi_{T_k M_{T_k} }^{(\rm{isospin})}$ denote the space, spin and isospin parts of the cluster wave function. In the case of the AV8$^{\prime}$ potential, the $t$ (or $h$) wave function is approximated with thirty Gaussian basis functions that include $L_k \leq 2$, and $S_k=\frac{1}{2}$ and $\frac{3}{2}$. The deuteron wave function is also approximated with Gaussian basis functions, four terms both in the $S$- and $D$-waves, respectively. The falloff parameters of the Gaussian functions are selected using the SVM [@book] and the expansion coefficients are determined by diagonalizing the intrinsic cluster Hamiltonian. A similar procedure is applied to the case of the MN potential.
The calculated energies $E$, root-mean-square (rms) radii $R^{\rm rms}$ and $D$ state probabilities $P_D$ are given in the fourth to sixth columns in Table \[sub1\]. We use the truncated basis in order to obtain the phase shifts in reasonable computer times, they slightly deviate from more elaborate calculations, which are given in the last three columns. Fortunately, except for the small shift of the threshold energy, the phase shifts are not very sensitive to the details of the cluster wave functions because they are determined by the change of the relative motion function of the clusters. The $N_k$ values in parenthesis for $^4$He are the number of $J^{\pi}=0^+$ basis functions in the major multi-channel calculation. The energy of $^4$He calculated in Table 2 with the multi-channel calculation is thus not optimized but found to be very close to that of the more extensive calculation. It is noted that the calculated $R^{\rm rms}$ value for the deuteron is smaller than in other calculations. This is due to the restricted choice of the length parameters of the basis functions, which permits us to use a relatively small channel radius of $a \sim 15$ fm. We have checked that the phase shifts for a $d+d$ single channel calculation do not change even when more extended deuteron wave functions are employed.
---------------- -------------------- -------- ---------- --------------- -------- ---------- --------------- ---------
potential cluster
$N_k$ $E$ $R^{\rm rms}$ $P_D$ $E$ $R^{\rm rms}$ $P_{D}$
(MeV) (fm) ($\%$) (MeV) (fm) ($\%$)
$d(1^+)$ 8 $-$2.18 1.79 5.9 $-$2.24 1.96 5.8
AV8$^{\prime}$ $t(\frac{1}{2}^+)$ 30 $-$8.22 1.69 8.4 $-$8.41 - -
(with TNF) $h(\frac{1}{2}^+)$ 30 $-$7.55 1.71 8.3 $-$7.74 - -
$^4$He$(0^+)$ (2370) $-$27.99 1.46 13.8 $-$28.44 - 14.1
$d(1^+)$ 4 $-$2.10 1.63 0 $-$2.20 1.95 0
MN $t(\frac{1}{2}^+)$ 15 $-$8.38 1.70 0 $-$8.38 1.71 0
$h(\frac{1}{2}^+)$ 15 $-$7.70 1.72 0 $-$7.71 1.74 0
$^4$He$(0^+)$ (1140) $-$29.94 1.41 0 $-$29.94 1.41 0
---------------- -------------------- -------- ---------- --------------- -------- ---------- --------------- ---------
: Energies $E$, rms radii $R^{\rm rms}$ and $D$-state probabilities $P_D$ of the clusters that appear in four-nucleon scattering and $^4$He with the AV8$^{\prime}$ (with TNF) and MN potentials. $N_k$ is the number of basis functions used to approximate the wave function of cluster $k$. The values in the last three columns for three- and four-body systems are taken from Ref. [@hiyama04] for AV8$^{\prime}$ and Ref. [@DGVR] for MN.
\[sub1\]
Correlated Gaussian function with triple global vectors {#sect.3}
=======================================================
As explained in the previous section, the calculation of the $S$-matrix reduces to that of the Hamiltonian and overlap matrix elements with the functions defined by (\[wf-ch1\]) and (\[wf.cluster\]), and it is conveniently performed by transforming that wave function into an $LS$-coupled form, $$\begin{aligned}
&&{\cal A}
\left[\left[\left[\psi^{(\rm space)}_{L_a}\psi^{(\rm space)}_{L_b}\right]_{L_{ab}}
\chi_\alpha({{\mbox{\boldmath $\rho$}}}_\alpha)\right]_L
\left[\psi^{(\rm spin)}_{S_a} \psi^{(\rm spin)}_{S_b}\right]_{S}
\right]_{JM}.\end{aligned}$$ The transformation can be done as $$\begin{aligned}
& &{\hspace{-10mm}}
{\cal A}\left[ \left[
[\psi_{L_a}^{(\rm{space})}\psi_{S_a}^{(\rm{spin})}]_{I_a}
[\psi_{L_b}^{(\rm{space})}\psi_{S_b}^{(\rm{spin})}]_{I_b}
\right]_I \chi_\alpha({{\mbox{\boldmath $\rho$}}}_\alpha)\right]_{JM}
\psi^{(\rm{isospin})}_{T_a M_{T_a}}\psi^{(\rm{isospin})}_{T_b M_{T_b}}
\nonumber \\
&=&\sum_{L_{ab}LS}
\left[
\begin{array}{ccc}
L_a&S_a&I_a\\
L_b&S_b&I_b\\
L_{ab}&S&I\\
\end{array}\right]
(-1)^{L_{ab}+J-I-L}U(S L_{ab} J \ell; IL)\nonumber\\
&&\times
{\cal A}\left[
\psi_{L_a L_b (L_{ab}) \ell L}^{(\rm space)}
\left[\psi^{(\rm spin)}_{S_a} \psi^{(\rm spin)}_{S_b}\right]_{S}
\right]_{JM}
\psi^{(\rm{isospin})}_{T_a M_{T_a}}\psi^{(\rm{isospin})}_{T_b M_{T_b}}\end{aligned}$$ with $$\begin{aligned}
\psi_{L_a L_b (L_{ab}) \ell L}^{(\rm space)}
=\left[\left[\psi^{(\rm space)}_{L_a}\psi^{(\rm space)}_{L_b}\right]_{L_{ab}}
\chi_\alpha({{\mbox{\boldmath $\rho$}}}_\alpha)\right]_L,
\label{space.part}\end{aligned}$$ where $U$ and $[ \ \ ]$ are Racah and 9$j$ coefficients in unitary form [@book].
The evaluation of the matrix element can be done in the spatial, spin, and isospin parts separately. The spin and isospin parts are obtained straightforwardly. In the following we concentrate on the spatial matrix element. The spatial part (\[space.part\]) of the total wave function is given as a product of the cluster intrinsic parts and their relative motion part. The coordinates used to describe the $2N$+$2N$ channel are depicted in Fig. \[fig.1\](a) with ${{\mbox{\boldmath $\rho$}}}_\alpha={{\mbox{\boldmath $x$}}}_3$, whereas the coordinates suitable for the $t$+$p$ and $h$+$n$ channels are shown in Fig. \[fig.1\](b) with ${{\mbox{\boldmath $\rho$}}}_\alpha={{\mbox{\boldmath $x$}}}_3'$. These coordinate sets are often called H-type and K-type. Therefore the calculation of the spatial matrix element requires a coordinate transformation involving the angular momenta $L_a, L_b,
\ell$ and $L_{ab}$. Moreover the permutation operator in ${\cal A}$ causes a complicated coordinate transformation. All these complexities are treated elegantly by introducing a correlated Gaussian [@boys60; @vs95; @book], provided each part of $\psi_{L_a L_b (L_{ab}) \ell L}^{(\rm space)}$ is given in terms of (a combination of) Gaussian functions as in the present case. In what follows we will demonstrate how it is performed. Because the formulation with the correlated Gaussian is not restricted to four nucleons but can be applied to a system including more particles, the number of nucleons is assumed to be $N$ in this and next sections as well as in Appendices B and C unless otherwise mentioned.
The relative and center of mass coordinates of the $N$ nucleons, ${{\mbox{\boldmath $x$}}}_i\, (i=1,\ldots,N)$, and the single-particle coordinates, ${{\mbox{\boldmath $r$}}}_i\, (i=1,\ldots,N)$, are mutually related by a linear transformation matrix $U$ and its inverse $U^{-1}$ as follows: $${{\mbox{\boldmath $x$}}}_i=\sum_{j=1}^{N}U_{ij}{{\mbox{\boldmath $r$}}}_j,\ \ \ \ \
{{\mbox{\boldmath $r$}}}_i=\sum_{j=1}^{N}(U^{-1})_{ij}{{\mbox{\boldmath $x$}}}_j.
\label{def.matU}$$ We use a matrix notation as much as possible in order to simplify formulas and expressions. Let ${{\mbox{\boldmath $x$}}}$ denote an $(N\!-\!1)$-dimensional column vector comprising all ${{\mbox{\boldmath $x$}}}_i$ but the center of mass coordinate ${{\mbox{\boldmath $x$}}}_N$. Its transpose is a row vector and it is expressed as $$\widetilde{{{\mbox{\boldmath $x$}}}}=({{\mbox{\boldmath $x$}}}_1,{{\mbox{\boldmath $x$}}}_2,...,{{\mbox{\boldmath $x$}}}_{N-1}).$$ The choice for ${{\mbox{\boldmath $x$}}}$ is not unique but a set of Jacobi coordinates is conveniently employed. For the four-body system, the Jacobi set is identical to the K-type coordinate, and the corresponding matrix $U$ is given by $$\begin{aligned}
U_K=
\left(\begin{array}{rrrr}
1&-1&0&0\\
\frac{1}{2}&\frac{1}{2}&-1&0\\
\frac{1}{3}&\frac{1}{3}&\frac{1}{3}&-1\\
\frac{1}{4}&\frac{1}{4}&\frac{1}{4}&\frac{1}{4}\\
\end{array}\right).\end{aligned}$$ The transformation matrix for the H-type coordinate reads $$\begin{aligned}
U_H=
\left(\begin{array}{rrrr}
1&-1&0&0\\
0&0&1&-1\\
\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\\
\frac{1}{4}&\frac{1}{4}&\frac{1}{4}&\frac{1}{4}\\
\end{array}\right).\end{aligned}$$ The K-type coordinate is obtained directly from the H-type coordinate by a transformation matrix $U_K U_H^{-1}$ $$U_KU_H^{-1}=
\left(
\begin{array}{rrrr}
1&0&0&0\\
0&-\frac{1}{2}&1&0\\
0&\frac{2}{3}&\frac{2}{3}&0\\
0&0&0&1\\
\end{array}
\right)
=\left(
\begin{array}{cc}
U_{KH}&0\\
0& 1\\
\end{array}
\right),
\label{HtoK}$$ where $U_{KH}$ is a 3$\times$3 sub-matrix of $U_K U_H^{-1}$.
Each coordinate set emphasizes particular correlations among the nucleons. As mentioned above, the H-coordinate is natural to describe the $d$+$d$ channel, whereas the K-coordinate is suited for a description of the 3$N$+$N$ partition. It is of crucial importance to include both types of motion in order to fully describe the four-nucleon dynamics [@benchmark]. In order to develop a unified method that can incorporate both types of coordinates on an equal footing, we extend the explicitly correlated Gaussian function [@suzuki00; @DGVR] to include triple global vectors $$\begin{aligned}
&&F_{L_1 L_2 (L_{12})L_3 L M}(u_1,u_2,u_3,A,{{\mbox{\boldmath $x$}}})\nonumber\\
&&\quad ={\rm exp}\left(-{\frac{1}{2}}{\widetilde{{{\mbox{\boldmath $x$}}}}} A {{\mbox{\boldmath $x$}}}\right)
\left[[{\cal Y}_{L_1}(\widetilde{u_1}{{\mbox{\boldmath $x$}}})
{\cal Y}_{L_2}(\widetilde{u_2}{{\mbox{\boldmath $x$}}})]_{L_{12}}
{\cal Y}_{L_3}(\widetilde{u_3}{{\mbox{\boldmath $x$}}})\right]_{LM},
\label{cgtgv}\end{aligned}$$ where $${\cal Y}_{L_iM_i}(\widetilde{u_i}{{\mbox{\boldmath $x$}}})=
|\widetilde{u_i}{{\mbox{\boldmath $x$}}}|^{L_i}Y_{L_iM_i}(\widehat{\widetilde{u_i}{{\mbox{\boldmath $x$}}}})$$ is a solid spherical harmonics and its argument, $\widetilde{u_i}{{\mbox{\boldmath $x$}}}$, what we call a global vector, is a vector defined through an $(N-1)$-dimensional column vector $u_i$ and ${{\mbox{\boldmath $x$}}}$ as $$\widetilde{u_i}{{\mbox{\boldmath $x$}}}=\sum_{j=1}^{N-1}(u_i)_j{{\mbox{\boldmath $x$}}}_j,
\label{def.ux}$$ where $(u_i)_j$ is the $j$th element of $u_i$. In Eq. (\[cgtgv\]) $A$ is an $(N-1)\times(N-1)$ real and symmetric matrix, and it must be positive-definite for the function $F$ to have a finite norm, but otherwise may be arbitrary. Non-diagonal elements of $A$ can be nonzero.
The matrix $A$ and the vectors $u_1, u_2, u_3$ are parameters to characterize the “shape” of the correlated Gaussian function. The Gaussian function including $A$ describes a spherical motion of the system, while the global vectors are responsible for a rotational motion. The spatial function (\[space.part\]) is found to reduce to the general form (\[cgtgv\]). Suppose that ${{\mbox{\boldmath $x$}}}$ stands for the H-type coordinate. Then a choice of $\widetilde{u_1}$=(1,0,0), $\widetilde{u_2}$=(0,1,0) and $\widetilde{u_3}$=(0,0,1) together with a diagonal matrix $A$ provides us with the basis function (\[space.part\]) employed to represent the configurations of the $2N$+$2N$ channel. On the other hand, the K-type basis function looks like $${\rm exp}\left(-{\frac{1}{2}}{\widetilde{{{\mbox{\boldmath $x$}}}'}} A_K {{\mbox{\boldmath $x$}}}'\right)
\left[[{\cal Y}_{L_1}({{\mbox{\boldmath $x$}}}'_1)
{\cal Y}_{L_2}({{\mbox{\boldmath $x$}}}'_2)]_{L_{12}}
{\cal Y}_{L_3}({{\mbox{\boldmath $x$}}}'_3)\right]_{LM},
\label{K.basis}$$ where $\widetilde{{{\mbox{\boldmath $x$}}}'}=({{\mbox{\boldmath $x$}}}'_1,{{\mbox{\boldmath $x$}}}'_2,{{\mbox{\boldmath $x$}}}'_3)$ is the K-coordinate set (see Fig. 1(b)) and $A_K$ is a 3$\times$3 diagonal matrix. Noting that ${{\mbox{\boldmath $x$}}}'$ is equal to ${{\mbox{\boldmath $x$}}}'=U_{KH}{{\mbox{\boldmath $x$}}}$, we observe that the basis function (\[K.basis\]) is obtained from Eq. (\[cgtgv\]) by a particular choice of parameters, that is, $\widetilde{u_1}$=(1,0,0), $\widetilde{u_2}$=(0,$-\frac{1}{2}$,$1$) and $\widetilde{u_3}$=(0,$\frac{2}{3}$,$\frac{2}{3}$), and the matrix $A$ is related to $A_K$ by $$\begin{aligned}
A=(u_1 u_2 u_3)A_K \left(
\begin{array}{c}
\widetilde{u_1}\\
\widetilde{u_2}\\
\widetilde{u_3}\\
\end{array}
\right)
=\widetilde{U_{KH}}A_K U_{KH}.\end{aligned}$$ Thus the form of the $F$-function remains unchanged under the transformation of relative coordinates.
Note that $A$ is no longer diagonal. The choice of a different set of coordinates ends up only choosing appropriate parameters for $A$, $u_1$, $u_2$, and $u_3$.
It is also noted that the triple global vectors in Eq. (\[cgtgv\]) are a minimum number of vectors to provide all possible spatial functions with arbitrary $L$ and parity $\pi$. A natural parity state with $\pi=(-1)^L$ can be described by only one global vector, that is, using e.g., $L_1=L$, $L_2=0$, $L_{12}=L$, $L_3=0$ [@vs95; @suzuki98; @GVR]. To describe an unnatural parity state with $\pi=(-1)^{L+1}$ except for $0^-$ case, we need at least two global vectors, say, $L_1=L$, $L_2=1$, $L_{12}=L$, $L_3=0$ [@suzuki00; @DGVR]. The simplest choice for the $0^-$ state is to use three global vectors with $L_1=L_2=L_{12}=L_3=1$ [@suzuki00]. In this way, the basis function (\[cgtgv\]) can be versatile enough to describe bound states of not only four- but also more-particle systems with arbitrary $L$ and $\pi$.
To assure the permutation symmetry of the wave function, we have to operate a permutation $P$ on $F$. Since $P$ induces a linear transformation of the coordinate set, a new set of the permuted coordinates, ${{\mbox{\boldmath $x$}}}_P$, is related to the original coordinate set ${{\mbox{\boldmath $x$}}}$ as ${{\mbox{\boldmath $x$}}}_P\!=\!{\cal P}{{\mbox{\boldmath $x$}}}$ with an $(N-1)\times (N-1)$ matrix ${\cal P}$. As before, this permutation does not change the form of the $F$-function: $$\begin{aligned}
& &PF_{L_1 L_2(L_{12}) L_3 LM}(u_1,u_2,u_3,A,{{\mbox{\boldmath $x$}}})
\nonumber \\
&&\quad =F_{L_1 L_2 (L_{12})L_3 LM}(u_1,u_2,u_3,A,{{\mbox{\boldmath $x$}}}_P)
\nonumber \\
&&\quad =F_{L_1 L_2(L_{12}) L_3 LM}(\widetilde{\cal P}u_1,
\widetilde{\cal P}u_2,\widetilde{\cal P}u_3,\widetilde{\cal P}A{\cal P},
{{\mbox{\boldmath $x$}}}).
\label{trans.cg}\end{aligned}$$ The fact that the functional form of $F$ remains unchanged under the permutation as well as the transformation of coordinates enables one to unify the method of calculating the matrix elements. This unique property is one of the most notable points in the present method.
Calculation of matrix elements {#sect.4}
==============================
Calculations of matrix elements with the correlated Gaussian $F$ are greatly facilitated with the aid of the generating function $g$ [@vs95; @book] $$\begin{aligned}
g({{\mbox{\boldmath $s$}}}; A, {{\mbox{\boldmath $x$}}})=
\exp\Big(-{\frac{1}{2}}{\widetilde{{\mbox{\boldmath $x$}}}}A{{\mbox{\boldmath $x$}}}+
\widetilde{{\mbox{\boldmath $s$}}}{{\mbox{\boldmath $x$}}}\Big),\end{aligned}$$ with $\widetilde{{\mbox{\boldmath $s$}}}=({{\mbox{\boldmath $s$}}}_1,{{\mbox{\boldmath $s$}}}_2,\ldots,{{\mbox{\boldmath $s$}}}_{N-1})$, where ${{\mbox{\boldmath $s$}}}_i=\sum_{j=1}^3\lambda_j(u_j)_i{{\mbox{\boldmath $e$}}}_j$, ${{\mbox{\boldmath $e$}}}_j$ is a 3-dimensional unit vector (${{\mbox{\boldmath $e$}}}_j\cdot{{\mbox{\boldmath $e$}}}_j=1$), and $\lambda_j$ is a scalar parameter. More explicitly $$\widetilde{{\mbox{\boldmath $s$}}}{{\mbox{\boldmath $x$}}}=\sum_{i=1}^{N-1}{{\mbox{\boldmath $s$}}}_i\cdot{{\mbox{\boldmath $x$}}}_i=
\sum_{i=1}^{N-1}\sum_{j=1}^3\lambda_j
(u_{j})_i {{\mbox{\boldmath $e$}}}_j \cdot{{\mbox{\boldmath $x$}}}_i
=\sum_{j=1}^3 \lambda_j{{\mbox{\boldmath $e$}}}_j\cdot (\widetilde{u_j}{{\mbox{\boldmath $x$}}}).$$ The correlated Gaussian $F$ is generated as follows: $$\begin{aligned}
&&F_{L_1 L_2(L_{12}) L_3 LM}(u_1,u_2,u_3,A,{{\mbox{\boldmath $x$}}})\nonumber\\
&&\quad =\left(\prod_{i=1}^3 \frac{B_{L_i}}{L_i !}\int d{{\mbox{\boldmath $e$}}}_i\right)
\left[\left[Y_{L_1}({{\mbox{\boldmath $e$}}}_1)Y_{L_2}({{\mbox{\boldmath $e$}}}_2)\right]_{L_{12}}
Y_{L_3}({{\mbox{\boldmath $e$}}}_3)\right]_{LM}\nonumber\\
&&\quad \times \left(\frac{\partial^{L_1+L_2+L_3}}
{\partial \lambda_1^{L_1} \partial \lambda_2^{L_2} \partial \lambda_3^{L_3} }
\,g({{\mbox{\boldmath $s$}}};A,{{\mbox{\boldmath $x$}}})\right)\Bigg\vert_{\lambda_1=\lambda_2=\lambda_3=0},\label{gfn}\end{aligned}$$ where $$\begin{aligned}
\hspace*{-2cm} B_L&=&\frac{(2L+1)!!}{4\pi}.\end{aligned}$$ When $g({{\mbox{\boldmath $s$}}}; A,{{\mbox{\boldmath $x$}}})$ is expanded in powers of $\lambda_1$, only the term of degree $\lambda_{1}^{L_1}$ contributes in Eq. (\[gfn\]), and this term contains the $L_1$th degree ${{\mbox{\boldmath $e$}}}_1$ because $\lambda_1$ and ${{\mbox{\boldmath $e$}}}_1$ always appear simultaneously. In order for the term to contribute to the integration over ${{\mbox{\boldmath $e$}}}_1$, these $L_1$ vectors ${{\mbox{\boldmath $e$}}}_1$ must couple to the angular momentum $L_1$ because of the orthonormality of the spherical harmonics $Y_{L_1M_1}({{\mbox{\boldmath $e$}}}_1)$, that is, they are uniquely coupled to the maximum possible angular momentum. The same rule applies to $\lambda_2$, ${{\mbox{\boldmath $e$}}}_2$ and $\lambda_3$, ${{\mbox{\boldmath $e$}}}_3$ as well.
We outline a method of calculating the matrix element for an operator ${\cal O}$ $$\langle F_{L_4 L_5(L_{45}) L_6 L'M'}(u_4, u_5, u_6, A',{{\mbox{\boldmath $x$}}})\vert
{\cal O}\vert F_{L_1 L_2(L_{12}) L_3 LM}(u_1, u_2, u_3, A,{{\mbox{\boldmath $x$}}})\rangle.
\label{meofop}$$ In what follows this matrix element is abbreviated as $\langle F'\vert {\cal O}\vert F\rangle $. Using Eq. (\[gfn\]) in Eq. (\[meofop\]) enables one to relate the matrix element to that between the generating functions: $$\begin{aligned}
\left<F'\vert {\cal O} \vert F \right>
&\!=\!&\left(\prod_{i=1}^6 \frac{B_{L_i}}{L_i !}\int d{{\mbox{\boldmath $e$}}}_i\right)
\left[\left[Y_{L_4}({{\mbox{\boldmath $e$}}}_4)Y_{L_5}({{\mbox{\boldmath $e$}}}_5)\right]_{L_{45}}
Y_{L_6}({{\mbox{\boldmath $e$}}}_6)\right]_{L'M'}^* \nonumber\\
&\!\times\!&\left[\left[Y_{L_1}({{\mbox{\boldmath $e$}}}_1)Y_{L_2}({{\mbox{\boldmath $e$}}}_2)\right]_{L_{12}}
Y_{L_3}({{\mbox{\boldmath $e$}}}_3)\right]_{LM}
\nonumber\\
&\!\times\!&\left(\prod_{i=1}^6
\frac{\partial^{L_i}}{\partial \lambda_i^{L_i}}\right)
\left<g({{\mbox{\boldmath $s$}}}',A',{{\mbox{\boldmath $x$}}}')\vert {\cal O} \vert
g({{\mbox{\boldmath $s$}}},A,{{\mbox{\boldmath $x$}}})\right>\Big\vert_{\lambda_i=0},
\label{grandformula}\end{aligned}$$ with $${{\mbox{\boldmath $s$}}}={\lambda}_1u_1{{\mbox{\boldmath $e$}}}_1 \!+\!{\lambda}_2u_2{{\mbox{\boldmath $e$}}}_2
\!+\!{\lambda}_3u_3{{\mbox{\boldmath $e$}}}_3,\hspace*{1cm}
{{\mbox{\boldmath $s$}}}'={\lambda}_4u_4{{\mbox{\boldmath $e$}}}_4 \!+\!{\lambda}_5u_5{{\mbox{\boldmath $e$}}}_5
\!+\!{\lambda}_6u_6{{\mbox{\boldmath $e$}}}_6.$$ The calculation of the matrix element consists of three stages: (1) Evaluate the matrix element between the generating functions, $\left<g({{\mbox{\boldmath $s$}}}',A',{{\mbox{\boldmath $x$}}}')\vert {\cal O} \vert
g({{\mbox{\boldmath $s$}}},A,{{\mbox{\boldmath $x$}}})\right>$. (2) Expand that matrix element in powers of $\lambda_i$ and keep only those terms of degree $L_i$ for each $i$. (3) Recouple the vectors ${{\mbox{\boldmath $e$}}}_i$ and integrate over the angle coordinates. In the second stage the remaining terms should contain ${{\mbox{\boldmath $e$}}}_i$s of degree $L_i$ as well. Hence any term with $\lambda_i^2 {{\mbox{\boldmath $e$}}}_i\!\cdot\!{{\mbox{\boldmath $e$}}}_i\!
=\!\lambda_i^2$ etc. can be omitted because the degree of ${{\mbox{\boldmath $e$}}}_i$ becomes smaller than that of $\lambda_i$.
We will explain the above procedures for the case of an overlap matrix element. The matrix element between the generating functions is $$\begin{aligned}
\left<g({{\mbox{\boldmath $s$}}}',A',{{\mbox{\boldmath $x$}}}')\vert
g({{\mbox{\boldmath $s$}}},A,{{\mbox{\boldmath $x$}}})\right>
=\left(\frac{(2\pi)^{N-1}}{\mbox{det} B}\right)^{3/2}
\exp\left(\frac{1}{2}\widetilde{{{\mbox{\boldmath $z$}}}}B^{-1}{{\mbox{\boldmath $z$}}}\right)\end{aligned}$$ with $$\begin{aligned}
B=A'+A,\qquad {{\mbox{\boldmath $z$}}}={{\mbox{\boldmath $s$}}}+{{\mbox{\boldmath $s$}}}'=\sum_{i=1}^6\lambda_i{{\mbox{\boldmath $e$}}}_iu_i.\end{aligned}$$ To perform the operation in the second stage we note that $$\frac{1}{2}\widetilde{{{\mbox{\boldmath $z$}}}}B^{-1}{{\mbox{\boldmath $z$}}}=\frac{1}{2}\sum_{i,j=1}^6
\rho_{ij}\lambda_i\lambda_j {{\mbox{\boldmath $e$}}}_i\cdot{{\mbox{\boldmath $e$}}}_j$$ with $$\rho_{ij}=\widetilde{u_i}B^{-1}u_j.
\label{def.rho}$$ As mentioned above, here we can drop the diagonal terms, $\lambda_i^2
{{\mbox{\boldmath $e$}}}_i\cdot{{\mbox{\boldmath $e$}}}_i$, and we get $$\begin{aligned}
&&\left(\prod_{i=1}^6\frac{\partial^{L_i}}{\partial \lambda_i^{L_i}}\right)
\left<g({{\mbox{\boldmath $s$}}}',A',{{\mbox{\boldmath $x$}}}')\vert
g({{\mbox{\boldmath $s$}}},A,{{\mbox{\boldmath $x$}}})\right>\Big\vert_{\lambda_i=0} \nonumber\\
&&\quad =\left(\frac{(2\pi)^{N-1}}{\mbox{det} B}\right)^{3/2}\prod_{i=1}^6L_i!
\prod_{i<j}^6\frac{\left(\rho_{ij}{{\mbox{\boldmath $e$}}}_i\cdot{{\mbox{\boldmath $e$}}}_j\right)^{n_{ij}}}{n_{ij}!}.\end{aligned}$$ Here the non-negative integers $n_{ij}$ must satisfy the following equations in order to assure the degree $L_i$ for ${{\mbox{\boldmath $e$}}}_i$ in the different terms, $$\begin{aligned}
&&n_{12}+n_{13}+n_{14}+n_{15}+n_{16}=L_1, \nonumber\\
&&n_{12}+n_{23}+n_{24}+n_{25}+n_{26}=L_2, \nonumber\\
&&n_{13}+n_{23}+n_{34}+n_{35}+n_{36}=L_3, \nonumber\\
&&n_{14}+n_{24}+n_{34}+n_{45}+n_{46}=L_4, \nonumber\\
&&n_{15}+n_{25}+n_{35}+n_{45}+n_{56}=L_5, \nonumber\\
&&n_{16}+n_{26}+n_{36}+n_{46}+n_{56}=L_6.
\label{nij}\end{aligned}$$
The last step is to recouple the angular momenta arising from the various terms. Since we have to couple ${{\mbox{\boldmath $e$}}}_i$s to the angular momentum $L_i$ from the terms of degree $L_i$, we may replace the term $\left(\rho_{ij}{{\mbox{\boldmath $e$}}}_i\cdot{{\mbox{\boldmath $e$}}}_j\right)^{n_{ij}}$ with just one piece $$\frac{ (-\rho_{ij})^{n_{ij}} n_{ij}! \sqrt{2n_{ij}+1}}{B_{n_{ij}}}
\left[Y_{n_{ij}}({{\mbox{\boldmath $e$}}}_i)Y_{n_{ij}}({{\mbox{\boldmath $e$}}}_j)\right]_{00}.$$ Other pieces like $\left[Y_{\kappa}({{\mbox{\boldmath $e$}}}_i)Y_{\kappa}({{\mbox{\boldmath $e$}}}_j)\right]_{00}$ with $\kappa < n_{ij}$ do not contribute to the matrix element. We thus have a product of 15 terms of $\left[Y_{n_{ij}}({{\mbox{\boldmath $e$}}}_i)Y_{n_{ij}}({{\mbox{\boldmath $e$}}}_j)\right]_{00}$. The coupling of these terms is done by defining various coefficients that are all expressed in terms of Clebsch-Gordan, Racah, and 9$j$ coefficients. For example, we make use of the formulas $$\begin{aligned}
& &{\hspace{-1cm}}[Y_a({{\mbox{\boldmath $e$}}}_1)Y_a({{\mbox{\boldmath $e$}}}_2)]_{00}\ [Y_b({{\mbox{\boldmath $e$}}}_1)Y_b({{\mbox{\boldmath $e$}}}_3)]_{00}\
[Y_c({{\mbox{\boldmath $e$}}}_2)Y_c({{\mbox{\boldmath $e$}}}_3)]_{00}
\nonumber \\
& \to& X(abc)\ [[Y_{a+b}({{\mbox{\boldmath $e$}}}_1)Y_{a+c}({{\mbox{\boldmath $e$}}}_2)]_{b+c}Y_{b+c}({{\mbox{\boldmath $e$}}}_3)]_{00},\\
\nonumber\\
& &{\hspace{-1cm}}[Y_a({{\mbox{\boldmath $e$}}}_1)Y_a({{\mbox{\boldmath $e$}}}_4)]_{00}\ [Y_b({{\mbox{\boldmath $e$}}}_1)Y_b({{\mbox{\boldmath $e$}}}_5)]_{00}\
[Y_c({{\mbox{\boldmath $e$}}}_1)Y_c({{\mbox{\boldmath $e$}}}_6)]_{00}
\nonumber \\
&\to& R_3(abc)\ [Y_{a+b+c}({{\mbox{\boldmath $e$}}}_1)\ [[Y_{a}({{\mbox{\boldmath $e$}}}_4)Y_{b}({{\mbox{\boldmath $e$}}}_5)]_{a+b}Y_c({{\mbox{\boldmath $e$}}}_6)]_{a+b+c}]_{00}.\end{aligned}$$ Here the symbol $\to$ indicates that no other terms arising from the left hand side of the equation contribute to the integration over the angles ${{\mbox{\boldmath $e$}}}_i$s, so that only the term on the right hand side has to be retained. Another coefficient is $$\begin{aligned}
&&{\hspace{-1cm}}[[[Y_a({{\mbox{\boldmath $e$}}}_4)Y_b({{\mbox{\boldmath $e$}}}_5)]_qY_c({{\mbox{\boldmath $e$}}}_6)]_Q\
[[Y_{a'}({{\mbox{\boldmath $e$}}}_4)Y_{b'}({{\mbox{\boldmath $e$}}}_5)]_{q'}Y_{c'}({{\mbox{\boldmath $e$}}}_6)]_{Q'}]_{\ell}
\nonumber \\
&\to& \sum_{\ell'}W(abcqQ,a'b'c'q'Q',\ell \ell')
[[Y_{a+a'}({{\mbox{\boldmath $e$}}}_4)Y_{b+b'}({{\mbox{\boldmath $e$}}}_5)]_{\ell'}Y_{c+c'}({{\mbox{\boldmath $e$}}}_6)]_{\ell}.\end{aligned}$$ Expressions for the coefficients, $X, R_3, W$, are given in Appendix A. Performing the integration of the six unit vectors, ${{\mbox{\boldmath $e$}}}_i$s, as prescribed in Eq. (\[grandformula\]) leads to the overlap matrix element $$\begin{aligned}
& &{\hspace{-1cm}}\left<F'\vert F \right>
\nonumber \\
&=&\left(\frac{(2\pi)^{N-1}}{\mbox{det} B}\right)^{3/2}
\left(\prod_{i=1}^6 B_{L_i}\right)
\frac{(-1)^{L_1+L_2+L_3}}{\sqrt{2L+1}}\delta_{LL'}\delta_{MM'}\nonumber\\
&\times & \hspace*{-0.3cm}\sum_{n_{ij}}\left(\prod_{i<j}^6 (-\rho_{ij})^{n_{ij}}
\frac{ \sqrt{2n_{ij}+1}}{B_{n_{ij}}}\right)
O(n_{ij}; L_1L_2L_3L_4L_5L_6,L_{12}L_{45}L)\label{eq6a},
\label{me.overlap}\end{aligned}$$ with $$\begin{aligned}
&&{\hspace{-1cm}}O(n_{ij}; L_1L_2L_3L_4L_5L_6,L_{12}L_{45}L)\nonumber\\
&&{\hspace{-1cm}}=X(n_{12}n_{13}n_{23})R_3(n_{14}n_{15}n_{16})R_3(n_{24}n_{25}n_{26})
R_3(n_{34}n_{35}n_{36})X(n_{45}n_{46}n_{56})\nonumber\\
&&{\hspace{-1cm}}\times Z(n_{12}\!+\!n_{13}\ L_1\!-\!n_{12}\!-\!n_{13})
Z(n_{12}\!+\!n_{23}\ L_2\!-\!n_{12}\!-\!n_{23})
Z(n_{13}\!+\!n_{23}\ L_3\!-\!n_{13}\!-\!n_{23})\nonumber\\
&&{\hspace{-1cm}}\times \sum_{\ell_1 \ell_2 \ell_3}
\left[\begin{array}{ccc}
L_1&L_1\!-\!n_{12}\!-\!n_{13}&n_{12}\!+\!n_{13}\\
L_2&L_2\!-\!n_{12}\!-\!n_{23}&n_{12}\!+\!n_{23}\\
L_{12}&\ell_1&n_{13}\!+\!n_{23}\\
\end{array}\right]
\left[\begin{array}{ccc}
L_{12}&\ell_1&n_{13}\!+\!n_{23}\\
L_3&L_3\!-\!n_{13}\!-\!n_{23}&n_{13}\!+\!n_{23}\\
L&L&0\\
\end{array}\right]
\nonumber\\
&&{\hspace{-1cm}}\times W(n_{14}n_{15}n_{16}\ n_{14}\!+\!n_{15}\ L_1\!-\!n_{12}\!-\!n_{13},
n_{24}n_{25}n_{26}\ n_{24}\!+\!n_{25}\ L_2\!-\!n_{12}\!-\!n_{23}, \ell_1 \ell_2)\nonumber\\
&&{\hspace{-1cm}}\times W(n_{14}\!+\!n_{24}\ n_{15}\!+\!n_{25}\ n_{16}\!+\!n_{26}\ \ell_2\ \ell_1,
n_{34}n_{35}n_{36}\ n_{34}\!+\!n_{35}\ L_3\!-\!n_{13}\!-\!n_{23}, L\ell_3) \nonumber\\
&&{\hspace{-1cm}}\times W(L_4\!-\!n_{45}\!-\!n_{46}\ L_5\!-\!n_{45}\!-\!n_{56}\ L_6\!-\!n_{46}\!-\!n_{56}\ \ell_3\ L, \nonumber\\
&&\hspace*{3cm}
n_{45}\!+\!n_{46}\ n_{45}\!+\!n_{56}\ n_{46}\!+\!n_{56}\ n_{46}\!+\!n_{56}\ 0, L L_{45}),\end{aligned}$$ where $Z$ is the coefficient given in Eq. (\[def.z\]). The summation in Eq. (\[me.overlap\]) extends over all possible sets of $n_{ij}$ that satisfy Eq. (\[nij\]). In most cases the values of $L_i$ are limited up to 2, so that the number of terms to be evaluated is not so large and the calculation of the matrix element is fast.
Expressions for the Hamiltonian matrix elements are collected in Appendix B. One advantage of our method is that the calculation of matrix elements can be done analytically. In addition we do not need to do angular momentum and parity projections because the correlated Gaussian function (\[cgtgv\]) already preserves those quantum numbers.
The Fourier transform of the correlated Gaussian function $F$ is a momentum space function and it becomes a useful tool to calculate various matrix elements that depend on the momentum operators [@DGVR]. For example, the distribution of the relative momentum is obtained by the expectation value of $\delta({{\mbox{\boldmath $p$}}}_i-{{\mbox{\boldmath $p$}}}_j-{{\mbox{\boldmath $p$}}})$, where ${{\mbox{\boldmath $p$}}}_j$ is the momentum of the $j$th particle. It is obviously much easier to calculate the distribution using the momentum space function rather than the coordinate space function. We show in Appendix C that the Fourier transform of $F$ reduces to a linear combination of $F$s in the momentum space.
![ Two-body thresholds calculated with the AV8$^{\prime}$ (left) and MN (middle) potentials. The solid lines are physical channels and the dashed lines are pseudo channels. We also plot experimental two-body thresholds for physical channels (right). The dotted line is the $p$+$p$+$n$+$n$ threshold.[]{data-label="fig:threshold"}](ene_thr.eps){width="9.0" height="8.0"}
Results
=======
$2N$+$2N$ and $3N$+$N$ channels
-------------------------------
In Table \[chan0\], we gave the physical channels, $d$+$d$, $t$+$p$, and $h$+$n$. Fig. \[fig:threshold\] displays two-body decay thresholds in the $d$+$d$ threshold energy region. The three physical channels are the main channels that describe the scattering around the three lowest thresholds ($d$+$d$, $t$+$p$, $h$+$n$). However, the scattering wave function $\Psi^{JM\pi}_{\rm int}$ in the internal region should contain all effects that may occur when all the nucleons come close to each other. It is thus reasonable that $\Psi^{JM\pi}_{\rm int}$ may not be well described in terms of the physical channels alone. Particularly the deuteron can be easily distorted when we use realistic potentials.
We will show that some pseudo $2N$+$2N$ channels are indeed needed to simulate the distortion of the deuteron. These pseudo channels, when they are included in the phase-shift calculation, are expected to take account of the distortion of the clusters of the entrance channel [@kanada85]. Here “pseudo” means that the clusters in the pseudo channels are not physically observable but may play a significant role in the internal region. The wave functions of these $2N$ pseudo clusters are obtained by diagonalizing the intrinsic cluster Hamiltonian similarly to the case of the physical clusters. We take into account the following pseudo clusters: $d^*(1^+, T=0),
\ d^*(0^+, T=1), \ d^*(2^+, T=0), \ d^*(3^+, T=0)$, $2n^*(0^+, T=1)$, and $2p^*(0^+,T=1)$, where the upper suffix \* indicates all the excited state but the ground state of $d$. Among the pseudo clusters, the lowest energy states with 0$^+$ that are related to virtual states would be most important. We especially write them as $\bar{d}(0^+)$, 2$n(0^+)$ (di-neutron) and 2$p(0^+)$ (di-proton). Although they are not bound, they are observed as resonances or quasi-bound states with negative scattering lengths. In fact the scattering lengths are $a_s(nn)=-$16.5 fm and $a_s(pp)=-$17.9 fm, which are comparable to $a_s(np, T=1)=-$23.7 fm. The calculated thresholds of these pseudo channels are also drawn in Fig. \[fig:threshold\].
channel
------ ----------- ----- -----------------------------------------
$2N$+$2N$ I $d(1^+)$+$d(1^+)$
$d(1^+)$+$d^*(1^+)$
$d^*(1^+)$+$d^*(1^+)$
II $\bar{d}(0^+)$+$\bar{d}(0^+)$
$\bar{d}(0^+)$+$d^*(0^+)$
$d^*(0^+)$+$d^*(0^+)$
III $d^*(2^+)$+$d^*(1^+)$
$d^*(2^+)$+$d^*(2^+)$
IV $d^*(3^+)$+$d^*(1^+)$
FULL $d^*(3^+)$+$d^*(2^+)$
$d^*(3^+)$+$d^*(3^+)$
V $2n(0^+)$+$2p(0^+)$
$2n(0^+)$+$2p^*(0^+)$
$2n^*(0^+)$+$2p(0^+)$
$2n^*(0^+)$+$2p^*(0^+)$
$3N$+$N$ 1 $t(\frac{1}{2}^+)$+$p(\frac{1}{2}^+)$
$t^*(\frac{1}{2}^+)$+$p(\frac{1}{2}^+)$
2 $h(\frac{1}{2}^+)$+$n(\frac{1}{2}^+)$
$h^*(\frac{1}{2}^+)$+$n(\frac{1}{2}^+)$
: $2N$+$2N$ and $3N$+$N$ channels. The Roman and Arabic numerals correspond to sets of channels included in the calculations.
\[chan1\]
Though it is expected that the pseudo channels with low threshold energies contribute more strongly to the scattering phase shift, we take into account all of these $2N$+$2N$ channels that include a vanishing total isospin as given in Fig. \[fig:threshold\]. The total isospin of the $3N$+$N$ channel is mixed in the present calculation. Because the $T$=1 component of the scattering wave function only weakly couples to the $d(1^+, T=0)$+$d(1^+, T=0)$ elastic-channel, the channel $d(1^+, T=0)$+$\bar{d}(0^+, T=1)$ is not employed in the calculation.
We also include the excited deuteron channels that comprise the $d^*(2^+, T=0)$ and $d^*(3^+, T=0)$ clusters. The energies of these lowest thresholds are above 10 MeV. These channels are therefore expected not to be very important, but that is not always the case as will be discussed in the case of the $^1$S$_0$ $d$+$d$ phase shift.
Table \[chan1\] summarizes all the channels that are used in our calculation. The 2$N$+2$N$ channels are distinguished by Roman numerals, while the 3$N$+$N$ channels are labeled by Arabic numerals. In the following, we use an abbreviation “2$N$+2$N$” or “3$N$+$N$” to indicate calculations including all 2$N$+2$N$ channels I-V or all 3$N$+$N$ channels (1-2 in Table \[chan1\]), respectively. Here $t^*(\frac{1}{2}^+)$ and $h^*(\frac{1}{2}^+)$ are excited 3$N$ continuum states. A “FULL” calculation indicates that all the channels in the table are included to set up the $S$-matrix. In the case of the MN potential channels III and IV are not included because this potential contains no tensor force.
The relative wave functions $\chi_{\alpha m}$ are expanded with 15 basis functions. We checked the stability of the $S$-matrix against the choice of the channel radius. The channel radius employed in this calculation is about 15 fm.
Positive parity phase shifts
----------------------------
Fig. \[fig:d+d-g3\] displays the $^1S_0$ $d$+$d$ elastic-scattering phase shift obtained with the AV8$^{\prime}$ potential. The dash-dotted line is the phase shift calculated with channel I ($I_d=1^+$), and the dash-dot-dotted line is the phase shift with channels I and II ($I_d=1^+, 0^+$). The phase shifts calculated by including further excited deuterons are also plotted by the dashed and dotted lines that correspond to the channels I-III ($I_d\le2^+$) and I-IV ($I_d\le3^+$), respectively. A naive expectation that the $^1S_0$ $d$+$d$ elastic-scattering phase shift might be well described in channel I ($d(1^+)+d(1^+)$, $d(1^+)+d^*(1^+)$ and $d^*(1^+)+d^*(1^+)$) alone completely breaks down in the case of the AV8$^{\prime}$ potential.
![ $^1S_0$ $d$+$d$ elastic-scattering phase shift calculated with the AV8$^{\prime}$ potential. The phase shifts are all obtained within the $d$+$d$ channels. The set of included channels is successively increased from I to IV. See Table \[chan1\] for the deuteron states included in each channel. []{data-label="fig:d+d-g3"}](fig0+-1.eps){width="7.0" height="6.0"}
Because the deuteron has a virtual state $\bar{d}$ with $0^+$ at low excitation energy, it is reasonable that the inclusion of channel II gives rise to a considerable attractive effect of several tens of degrees on the phase shift, as shown by the dash-dot-dotted line of Fig. \[fig:d+d-g3\]. However, the phase shift exhibits no converging behavior even when the higher spin states such as $d^*$(2$^+$) and $d^*$(3$^+$) are taken into account in the calculation. The additional attractions by these channels are of the same order as that of channel II. One may conclude that the deuteron is strongly distorted even in the low energy $^1S_0$ $d$+$d$ elastic scattering but more physically we have to realize that there exist two observed $0^+$ states below the $d$+$d$ threshold. Obviously the $d$+$d$ scattering wave function is subject to the structure of those states in the internal region.
The second 0$^+$ state of $^4$He lying about 4 MeV below the $d$+$d$ threshold is known to have a $3N$+$N$ cluster structure [@horiuchi08; @hiyama04]. Thus this state together with the ground state of $^4$He cannot be described well in the $2N$+$2N$ model space alone. As seen in Table \[chan0\], the $3N$+$N$ channel contains a $^1S_0$ component, which is the dominant component of the $0^+_2$ state. Since the realistic force strongly couples the $2N$+$2N$ channel to the $3N$+$N$ channel and the $d$+$d$ scattering wave function has to be orthogonal to the main component of the underlying $0^+$ states, we expect that the deuteron in the incoming $d$+$d$ channel never remains in its ground state but has to be distorted largely due to the $3N$+$N$ channel. The phase shift for the channel I-IV (dotted line) shows a resonant pattern. This resonant state is expected to be the second 0$^+$ state because of the restricted model space within the $d$+$d$ channel.
![Comparison of the ground and second 0$^+$ state energies between calculations with the AV8$^{\prime}$ (left) and MN (middle) potentials and experiment (right). The model space for AV8$^{\prime}$ is I-IV, $3N$+$N$ and FULL and the model space for MN is I-II, $3N$+$N$ and FULL. []{data-label="fig:2nd0+"}](2nd0+.eps){width="10.0" height="8.0"}
Fig. \[fig:2nd0+\] displays the calculated ground state energy and the second 0$^+$ energy for the AV8$^{\prime}$ (left) and MN (middle) potentials. The model spaces of the calculations are I-IV, $3N$+$N$ and FULL for AV8$^{\prime}$ and I-II, $3N$+$N$ and FULL for MN. We also plot experimental energies (right) [@tilley92]. For the AV8$^{\prime}$ potential, the energies of the two lowest $0^+$ states do not change very much between the FULL and $3N$+$N$ models. But the second 0$^+$ state with the $d$+$d$ model (channels I-IV) is not bound with respect to the $d$+$d$ threshold as expected before. On the contrary, for the MN potential, the second 0$^+$ state with the $d$+$d$ model (channels I-II) is bound with respect to the $d$+$d$ threshold. We consider that this difference makes the drastic change of the $d$+$d$ phase shifts, between the AV8$^{\prime}$ and MN potentials. It is also interesting to see that the energies of the two lowest $0^+$ states for the MN potential are almost the same between the FULL and $3N$+$N$ models.
Plotted in Fig. \[fig:d+d-3N+N\] are the $^1S_0$ $d$+$d$ elastic-scattering phase shifts obtained with the AV8$^{\prime}$ potential (left) and the MN potential (right). The FULL calculation (solid line) couples all 2$N$+2$N$ and 3$N$+$N$ channels that are listed in Table \[chan1\]. The $R$-matrix analysis (crosses) [@hofmann08] is reproduced well by both the AV8$^{\prime}$ and MN potential with the FULL calculation. Compared to the uncoupled phase shift (dotted line), one clearly sees that the 3$N$+$N$ channel produces a very large effect on the $d$+$d$ elastic phase shift, especially in the case of the AV8$^{\prime}$ potential. We also verified that a calculation excluding the channels III, IV or V from the FULL channel calculation gives only negligible change in the phase shift. The slow convergence seen in Fig. \[fig:d+d-g3\] is thus attributed to the neglect of the $3N$+$N$ channel, indicating that a proper account of the $^1S_0$ $d$+$d$ elastic phase shift at low energy can be possible only when the coupled channels {$d$(1$^+$)+$d$(1$^+$)} +{$d$(0$^+$)+$d$(0$^+$)}+ {$t$(1/2$^+$)+$p(1/2^+)$} +{$h$(1/2$^+$)+$n(1/2^+)$} are considered.
Thus, the slow convergence in Fig. \[fig:d+d-g3\] suggests that the $2N$+$2N$ partition is not an economical way to include the effects of the $3N$+$N$ channel. In the case of the MN potential (right panel in Fig. \[fig:d+d-3N+N\]), the situation is very different from the AV8$^{\prime}$ case. The channel coupling effect is rather modest, and the size of the $^1S_0$ $d$+$d$ elastic phase shift is already accounted for mostly in the $d$+$d$ channel calculation. All these results are very consistent with the $0^+$ spectrum in Fig. \[fig:2nd0+\].
The large distortion effect of the deuteron clusters on the $^1S_0$ $d$+$d$ scattering phase shift is expected to appear in the 3$N$+$N$ phase shift as well because of the coupling between the 3$N$+$N$ and 2$N$+2$N$ channels. We display in Fig. \[fig:3N+N-g3mn\] the $^1S_0$ $t$+$p$ elastic-scattering phase shift at energies below the $d$+$d$ threshold. The 0$^+_2$ state of $^4$He is observed as a sharp resonance with a proton decay width of 0.5 MeV at about 0.4 MeV above the $t$+$p$ threshold. The present energies ($E_r=0.15$ MeV for AV8$^{\prime}$, $E_r=0.12$ MeV for MN) calculated with a bound state approximation are slightly smaller than the experimental value, but they are consistent with a calculation ($E_r=0.105$ MeV and $\Gamma/2=0.129$ MeV for AV18+UIX, $E_r=0.091$ MeV and $\Gamma/2=0.077$ MeV for AV18+UIX+V$^*_3$) with another realistic interaction (AV18) with three nucleon forces by Hofmann and Hale [@hofmann08]. The calculated phase shifts appear slightly larger than that in the $R$-matrix analysis (crosses in Fig. \[fig:3N+N-g3mn\]) [@hofmann08]. It is noted that the phase shift changes so much even for a small change of the 0$^+_2$ resonant pole position ($\sim$0.1 MeV) because it is very near to the threshold. The phase shifts in the FULL calculation, for both AV8$^{\prime}$ and MN cases, show a resonance pattern in a small energy interval and the overall energy dependencies of the phase shifts are similar to each other. However, the phase shifts obtained only in the $3N$+$N$ channel are quite different as indicated by the dotted lines in Fig. \[fig:3N+N-g3mn\]. In the case of the MN potential (right) the phase shift is already close to the FULL phase shift, while in the case of the AV8$^{\prime}$ potential (left) the phase shift is much smaller (by almost 90 degrees) and moreover shows no resonance pattern.
By looking into the wave functions in more detail, we argue that the large distortion effect in the $^1S_0$ $d$+$d$ and $3N$+$N$ coupled channels is really brought about by the tensor force. As shown in Table \[sub1\], the AV8$^{\prime}$ potential with TNF gives 5.8% and 8.4% (8.3%) $D$-state probability for $d$ and $t$ $(h)$, respectively. Thus the $d$+$d$ state in the $^1S_0$ state contains $L=S=0$ components (89%) as well as $L=S=2$ components (11%), where $L$ and $S$ are the total orbital and spin angular momenta of the four-nucleon system. Similarly the $3N$+$N$ state in the $^1S_0$ state contains an $L=S=0$ component (92%) and an $L=S=2$ component (8%). Thus the tensor force couples both states with $\Delta L=2$ and $\Delta S=2$ couplings, which are in fact very large compared to the central matrix element ($\Delta L=0$, $\Delta S=0$). An analysis of this type was performed for some levels of $^4$He in Refs. [@DGVR; @horiuchi08]. The MN potential contains no tensor force, so that the $d$+$d$ and $3N$+$N$ channel coupling is modest.
As listed in Table \[chan0\], there are four channels, $^5S_2$, $^1D_2$, $^3D_2$ and $^5D_2$, for $J^{\pi}=2^+$ at energies around the $d$+$d$ threshold. Among these states, we expect that the effect of the coupling between the 3$N$+$N$ and 2$N$+2$N$ channels occurs most strongly in $^1D_2$ as it appears in all physical channels. However, no sharp resonance is observed in $^4$He up to 28MeV of excitation energy, so that the coupling effect, if any, might be weaker than that observed in the $^1S_0$ case.
Fig. \[fig:2+\] displays the $^1D_2$ elastic-scattering phase shifts obtained in three types of calculations, $3N$+$N$ (dashed line), $2N$+$2N$ (dotted line), and FULL (solid line). The $t$+$p$ and $d$+$d$ phase shifts start from the $t$+$p$ ($E_{\rm c.m.}=0$) and $d$+$d$ thresholds, respectively. The phase shifts of the $3N$+$N$ and $2N$+$2N$ calculations are both slightly positive, indicating a weak attraction in the $t$+$p$ and $d$+$d$ interactions. In the FULL calculation, the $t$+$p$ phase shift becomes more attractive and the $d$+$d$ phase shift turns to be negative (repulsive). The present FULL calculation reproduces the calculation of Ref. [@hofmann08] as expected. Though the effect of the coupling is slightly larger in the AV8$^{\prime}$ potential than in the MN potential, it is much less compared to the case of the $^1S_0$ phase shift. This is understood as follows. In the $^1D_2$ state, the main component of the wave function is given by the $L=2$, $S=0$ state: Its probability is the same as that of $^1S_0$, that is, 92% in $t$+$p$ and 89% in $d$+$d$. However, the probability of finding the state with $L=0$, $S=2$, which causes a strong tensor coupling, is more than one order of magnitude smaller than in the case of $^1S_0$, namely 0.23% in $t$+$p$ and 0.44% in $d$+$d$, respectively. The reason for this small percentage is that, to obtain $L=0$, the incoming $D$-wave in the $^1D_2$ channel must couple with the $D$-components in the clusters, but this coupling leads to several fragmented components with different $L$ values. This relatively weaker coupling of the tensor force explains the phase shift behavior in Fig. \[fig:2+\].
In Fig. \[fig:2+full\] we plot the $t$+$p$ and $d$+$d$ elastic-scattering phase shifts for other channels, $^5S_2$ (solid line), $^3D_2$ (dashed line), and $^5D_2$ (dotted line). We show only the FULL result, because the phase shifts with the truncated basis do not change visibly at the scale of the figure. The obtained phase shifts are not that different between the AV8$^{\prime}$ and MN potentials, and also consistent with the previous calculation [@hofmann08]. Thus, the effect of the distortion of the clusters is very small for 2$^+$ except for $^1D_2$.
We have three channels for $J^{\pi}=1^+$, $^5D_1$, $^3D_1$ and $^3S_1$. No sharp $1^+$ resonance of $^4$He is observed experimentally up to 28 MeV of excitation energy. Another theoretical calculation neither predicts it [@horiuchi08], so that the coupling between the 2$N$+2$N$ and 3$N$+$N$ channels is expected to be weak. Fig. \[fig:1+\] exhibits the $t$+$p$ and $d$+$d$ elastic-scattering phase shifts in the FULL calculation: $^5D_1$ $d$+$d$ (solid line), $^3D_1$ $t$+$p$ (dashed line), and $^3S_1$ $t$+$p$ (dotted line). Only the FULL result is displayed because the phase shift change in other calculations is small. Both AV8$^{\prime}$ and MN potentials produce phase shifts quite similar to each other.
Negative parity phase shifts
----------------------------
As seen from Table \[chan0\], the main components of these negative parity states are considered to be $^3P_J$.
We compare in Fig. \[fig:0-\] the $^3P_0$ elastic-scattering phase shifts calculated with the AV8$^{\prime}$ (left) and MN (right) potentials. The truncated $3N$+$N$ (dashed line) and $2N$+$2N$ (dotted line) calculations are shown together with the FULL result (solid line). The $t$+$p$ phase shift of the $3N$+$N$ calculation is similar with both AV8$^{\prime}$ and MN potentials, while the $d$+$d$ phase shift of the $2N$+$2N$ calculation behaves quite differently between the two potentials: the $d$+$d$ phase shift is weakly attractive with AV8$^{\prime}$ but is very strongly attractive with MN. No typical resonance behavior shows up below the $d$+$d$ threshold, which is in contradiction to experiment. In the FULL model that combines both 3$N$+$N$ and 2$N$+2$N$ configurations, however, the two potentials predict quite different phase shifts especially in the $t$+$p$ channel. The $t$+$p$ phase shift with AV8$^{\prime}$ becomes so attractive that it crosses $\pi/2$, indicating a resonance at about 1 MeV above the $t$+$p$ threshold. The $d$+$d$ phase shift changes sign from attractive to repulsive. The result based on the AV8$^{\prime}$ potential is thus consistent with experiment. Furthermore, we reproduce the flat structure of the $^3P_0$ phase shift around several MeV above the $t$+$p$ threshold which was discussed as the coupling to the $h$+$n$ channel [@hofmann08]. On the other hand, the MN potential changes the $t$+$p$ phase shift only mildly and produces no sharp resonance behavior. The $d$+$d$ phase shift changes drastically to the repulsive side.
As seen in the above figure, the sharp 0$^-$ state appears provided a full model space with a realistic potential is employed. The mechanism to produce this resonance is unambiguously attributed to the tensor force as discussed in Ref. [@DGVR] for the realistic interaction G3RS [@tamagaki68]. According to it, the $0^-$ state consists of only two components, $L=S=1$ (95.5%) and $L=S=2$ (4.5%), ignoring a tiny component with $L=S=0$. The $L=S=2$ component arises from the coupling of the incoming $P$-wave with the $D$-states contained in the $3N$ and $d$ clusters. All the pieces of the Hamiltonian but the tensor force have no coupling matrix element between the two components. The uncoupled Hamiltonian thus gives a too high energy to accommodate a resonance. The tensor force, however, couples the two components very strongly, bringing down its energy to a right position.
The second lowest negative parity state has spin-parity 2$^-$. The physical channel for this state is only $^3P_2$ as seen in Table \[chan0\]. Fig. \[fig:2-\] compares the $^3P_2$ elastic-scattering phase shifts in a manner similar to Fig. \[fig:0-\]. The phase shift obtained with the MN potential is almost the same as the $^3P_0$ phase shift, which is consistent with the previous result [@horiuchi08] that the energies of the negative parity states calculated with the MN potential are found to be degenerate. In the case of the AV8$^{\prime}$ potential, the $^3P_2$ phase shifts grows significantly in the FULL calculation, indicating a resonant behavior. The coupling effect between the $3N$+$N$ and $2N$+$2N$ channels is however much less compared to the $0^-$ state. This is because the incoming $P$-wave coupled to the $D$-states in the clusters gives rise to several $L$ values to produce the $2^-$ state and therefore the tensor coupling does not concentrate sufficiently to produce a sharp resonance.
Fig. \[fig:1-\] displays the $^3P_1$ and $^1P_1$ elastic-scattering phase shifts calculated with the AV8$^{\prime}$ (left) and MN (right) potentials. Note that no physical $d$+$d$ channel exists in the case of the $^1P_1$ state. Because both FULL and $3N$+$N$ calculations give almost the same phase shifts, only the FULL result is shown in the figure. The $^3P_1$ phase shift calculated with the MN potential is again almost the same as those of the $^3P_0$ and $^3P_2$ cases, supporting that the three negative parity states become almost degenerate. The $^3P_1$ elastic-scattering phase shift calculated with the AV8$^{\prime}$ potential is qualitatively similar to that of $^3P_2$. The attractive nature of the $t$+$p$ phase shift becomes further weaker, and to identify a resonance appears to be very hard. Even though it is possible in some way, its width would be a few MeV, which is not in contradiction to experiment. The $^1P_1$ phase shifts are very small in both AV8$^{\prime}$ and MN cases.
For the negative parity states, the FULL model with the AV8$^{\prime}$ potential gives results that are consistent with both experiment and the theoretical calculation of Ref. [@horiuchi08]. We have pointed out that the phase shift behavior reveals the importance of the tensor force particularly in the case of $0^-$. Its effect is often masked however by the coupling between the $D$ states in the clusters and the incoming partial wave.
In this subsection, we investigate the phase shifts of the negative parity states which have dominant $T=0$ components. In Fig. \[fig.13\], we represent three experimental negative parity $T=0$ energies (left). The states are observed at $-7.29$ ($0^-$), $-6.46$ ($2^-$) and $-4.05$ ($1^-$)MeV below the four-nucleon threshold [@tilley92]. The former two are located below the $d$+$d$ threshold and their widths are 0.84 and 2.01MeV, respectively, whereas the last one is above the $d$+$d$ threshold and its width is fairly broad (6.1 MeV). Here, we calculate these energies as $-7.57$ ($0^-$), $-6.82$ ($2^-$) and $-5.95$ ($1^-$)MeV, which are approximated by the half-value position from the maximum phase shifts. The present calculation is not projected out to $T=0$, but the dominant configurations of $t$+$p$, $h$+$n$ and $d$+$d$ elastic scattering are $T=0$. Our calculated energies with AV8$^{\prime}$ reproduce the ordering of $0^-$, $2^-$ and $1^-$ (middle in Fig. \[fig.13\]). The splitting between the two lower states $0^-$ and $2-$ is reproduced, but the experimental $1^-$ energy is higher than the calculation. However, the determination of the energy for such a high energy state with large decay width (6.1 MeV) is very difficult from both experimental and theoretical side, and it usually has a large ambiguity.
This type of analysis was done by Horiuchi and Suzuki, who applied the correlated Gaussian basis with two global vectors to study the energy spectrum of $^4$He [@horiuchi08]. Because the results of Ref. [@horiuchi08] are based on approximate solutions that impose no proper resonance boundary condition, it is interesting to see how the tensor force changes the phase shifts in the negative parity states as shown in this subsection. These authors also found that the negative parity states with $T=0$ turn out to be almost degenerate when the MN potential that contains no tensor force is employed. In the present calculation, three states ($0^-$, $1^-$, $2^-$) completely degenerate at the same energy, $E=-6.64$ MeV (right), and the same phase shift pattern (solid lines in Figs. \[fig:0-\], \[fig:2-\], \[fig:1-\]). Thus we can expect to see a clear evidence for the tensor force in the scattering involving the negative parity states.
Summary and conclusion
======================
We have investigated the distortion of clusters appearing in the low-energy $d$+$d$ and $t$+$p$ elastic scattering using a microscopic cluster model with the triple global vector method. We showed that the tensor interaction changes the phase shifts very much by comparing a realistic interaction and an effective interaction. In the present $ab$-$initio$ type cluster model, the description of the cluster wave functions is extended from a simple (0$s$) harmonic-oscillator shell model to a few-body model. To compare distortion effects of the clusters with realistic and effective interactions, we employed the AV8$^{\prime}$ potential as a realistic interaction and the MN potential as an effective interaction.
For the realistic interaction, the calculated $^1S_0$ phase shift shows that the $t$+$p$ and $h$+$n$ channels strongly couple with the $d$+$d$ channel. These channels are coupled because of the tensor interaction. On the contrary, the coupling of these $3N$+$N$ channels plays a relatively minor role for the case of the effective interaction because of the absence of tensor term. In other words, the $3N$+$N$ channels strongly affect the $d$+$d$ elastic phase shift with the realistic interaction, but not with the effective interaction.
For the 2$^+$ phase shifts, there is a $^1D_2$ component in all physical channels ($d$+$d$, $t$+$p$ and $h$+$n$). The coupling of the $2N$+$2N$ and $3N$+$N$ channels in $^5D_2$ is weaker than in $^1S_2$ because of a weaker tensor coupling as discussed in section \[results\], and the calculated phase shifts are very similar for the realistic and effective potentials. For other positive parity cases, the phase shift behavior of the realistic and effective potentials are very similar, and the coupling between the $2N$+$2N$ and $3N$+$N$ channels can be neglected or is very small. Furthermore, the tensor interaction makes the energy splitting of the $0^-$, $2^-$ and $1^-$ negative parity states of $^4$He consistent with experiment. No such splitting is however reproduced with the effective interaction.
We believe that the physical picture obtained in the large model space with the realistic interaction should be close to the real physical situation. It is needless to say that $ab$-$initio$ reaction calculations are very important to understand the underlying reaction dynamics involving continuum states. Simpler calculations using effective interactions in the same framework, as carried out in the present paper, are also meaningful because we can understand more clearly the effect of the tensor force by comparing both calculations. The reaction calculations with the microscopic cluster model, whose model space and interactions are restricted, have been successfully applied to many heavier nuclei. Therefore, it is instructive to see the difference from the realistic interaction by employing a simple conventional effective interaction as MN in the few-nucleon systems.
It will be quite interesting to see the importance of the tensor force in reaction observables of four nucleons. As a direct application of the present study the radiative capture reaction $d(d,\gamma)^4$He at energies of astrophysical interest is of prime importance. It is expected to take place predominantly via $E2$ transitions [@santos85; @langanke87; @wachter88; @arriaga91; @carlson98]. As is seen from Table \[chan0\], the two deuterons can approach each other in the $S$-wave only when $J^{\pi}$ is either 0$^+$ ($^1S_0$) or 2$^+$ ($^5S_2$). The former case is excluded because a radiative capture reaction of $0^+ \to 0^+$ is forbidden in the lowest-order electromagnetic interaction, and hence the $E2$ transition should be predominant. If there were no tensor force present, the radiative capture would be suppressed near $E=0$ because neither $d$ nor $^4$He would have a $D$-wave component in contradiction with the flat behavior of the astrophysical $S$-factor [@angulo99]. The tensor force strongly changes this story because it can couple $S$- and $D$-waves, bringing a significant amount of $D$-state probability in both $^4$He and $d$. Details of this analysis will be reported elsewhere.\
\
[**Acknowledgment**]{}\
We thank Dr. R. Kamouni for helpful discussions based on his PhD thesis (in French). This work presents research results of Bilateral Joint Research Projects of the JSPS (Japan) and the FNRS (Belgium). Y. S. is supported by a Grant-in-Aid for Scientific Research (No. 21540261). This text presents research results of the Belgian program P6/23 on interuniversity attraction poles initiated by the Belgian-state Federal Services for Scientific, Technical and Cultural Affairs (FSTC). D. B. and P. D. also acknowledge travel support of the Fonds de la Recherche Scientifique Collective (FRSC). The part of computational calculations were carried out in T2K-Tsukuba.
Definitions of recoupling coefficients {#app.A}
======================================
We define an auxiliary coefficient $Z$ that appears in the coupling $$[[Y_a({{\mbox{\boldmath $e$}}}_1)[Y_b({{\mbox{\boldmath $e$}}}_1)Y_b({{\mbox{\boldmath $e$}}}_2)]_0]_a
\to Z(ab)[Y_{a+b}({{\mbox{\boldmath $e$}}}_1)Y_b({{\mbox{\boldmath $e$}}}_2)]_a.$$ By introducing a coefficient $$C(a b, c)=\sqrt{\frac{(2a+1)(2b+1)}
{4\pi(2c+1)}}\langle a\ 0\ b\ 0 \vert c \ 0\rangle \\$$ for the coupling $[Y_a({{\mbox{\boldmath $e$}}}_1)Y_b({{\mbox{\boldmath $e$}}}_1)]_{c}=C(ab, c)Y_{c}({{\mbox{\boldmath $e$}}}_1)$, we can express $Z$ as $$Z(ab)=\sqrt{\frac{2(a+b)+1}{(2a+1)(2b+1)}}C(ab,a+b)=
\frac{1}{\sqrt{4\pi}}\langle a\ 0\ b\ 0 \vert a+b \ 0\rangle.
\label{def.z}$$ Note that $C(ab,c)$ vanishes unless $a+b+c$ is even.
The coefficients that appear in Sect. \[sect.4\] are given as follows: $$\begin{aligned}
&&{\hspace{-5mm}}X(a\ b\ c)=Z(ab)Z(ac)C(bc, b+c)U(a\!+\!c\ c\ a\!+\!b\ b;\ a\ b\!+\!c),\\
&&{\hspace{-5mm}}R_3(a\ b\ c)=Z(ab)Z(a\!+\!b \ c),\\
&&{\hspace{-5mm}}W(a\ b\ c\ q\ Q,\ a'\ b'\ c'\ q'\ Q',\ \ell \ \ell')\nonumber\\
&&{\hspace{-5mm}}=
\left[\begin{array}{ccc}
q &c & Q \\
q'&c' & Q'\\
\ell'&c\!+\!c' &\ell\\
\end{array}\right]
\left[\begin{array}{ccc}
a &b & q\\
a' &b' & q'\\
a\!+\!a'&b\!+\!b' &\ell'\\
\end{array}\right]C(aa', a\!+\!a')C(bb', b\!+\!b')C(cc', c\!+\!c').\end{aligned}$$
Matrix elements for various operators
=====================================
The purpose of this appendix is to collect formulas for various matrix elements. The main procedure to derive the formulas is sketched in Sect. \[sect.4\]. More details for the case of two global vectors are given in Ref. [@DGVR].
\[app.B\]
Kinetic energy
--------------
Let ${{\mbox{\boldmath $\pi$}}}_j$ denote the momentum operator conjugate to ${{\mbox{\boldmath $x$}}}_j$, ${{\mbox{\boldmath $\pi$}}}_j=-i\hbar \frac{\partial}{\partial {{\mbox{\boldmath $x$}}}_j}$. The total kinetic energy operator for the $N$-nucleon system with its center of mass kinetic energy being subtracted takes the form $$\sum_{i=1}^N \frac{{{\mbox{\boldmath $p$}}}_i^2}{2m}-\frac{{{\mbox{\boldmath $\pi$}}}_N^2}{2Nm}
=\frac{1}{2}\widetilde{{\mbox{\boldmath $\pi$}}}\Lambda{{\mbox{\boldmath $\pi$}}},$$ where ${{\mbox{\boldmath $\pi$}}}_N=\sum_{i=1}^N{{\mbox{\boldmath $p$}}}_i$ is the total momentum, $\widetilde{{\mbox{\boldmath $\pi$}}}=({{\mbox{\boldmath $\pi$}}}_1,{{\mbox{\boldmath $\pi$}}}_2,\ldots,{{\mbox{\boldmath $\pi$}}}_{N-1})$, and $\Lambda$ is an $(N-1)\times(N-1)$ symmetric mass matrix. Defining $N-1$-dimensional column vectors $\Gamma_i$ as $$\begin{aligned}
& &\Gamma_i=A'B^{-1}u_i\ \ \ \ \ \ (i=1,2,3),\nonumber \\
& &\Gamma_i=-AB^{-1}u_i\ \ \ \ \ (i=4,5,6)
\label{def.gamma}\end{aligned}$$ and an $(N-1)\times(N-1)$ matrix $Q$ $$Q_{ij}=2\widetilde{\Gamma_i}\Lambda \Gamma_j,$$ we can calculate the matrix element for the kinetic energy through the overlap matrix element $$\begin{aligned}
\langle F'\vert \frac{1}{2}\widetilde{{\mbox{\boldmath $\pi$}}}\Lambda{{\mbox{\boldmath $\pi$}}} \vert F \rangle
= \frac{\hbar^2}{2}\left(R-\sum_{i<j}Q_{ij}\frac{\partial}{\partial \rho_{ij}}\right)
\left<F'\vert F \right>,\end{aligned}$$ where $$R=3{\rm Tr}(B^{-1}A'\Lambda A).$$ The $\rho_{ij}$ values are defined in Eq. (\[def.rho\]).
$\delta$-function
-----------------
A two-body interaction $V({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j)$ can be expressed as $$V({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j)=\int d{{\mbox{\boldmath $r$}}} V({{\mbox{\boldmath $r$}}})\ \delta({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j-{{\mbox{\boldmath $r$}}}).$$ Once the matrix element of $\delta({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j-{{\mbox{\boldmath $r$}}})$ is obtained, the matrix element of the interaction is calculated by integrating over ${{\mbox{\boldmath $r$}}}$ the $\delta$-function matrix element weighted with the form factor $V({{\mbox{\boldmath $r$}}})$. Similarly, for a one-body operator $$D({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $x$}}}_N)=\int d{{\mbox{\boldmath $r$}}} D({{\mbox{\boldmath $r$}}})\ \delta({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $x$}}}_N-{{\mbox{\boldmath $r$}}}),$$ its matrix element can be obtained from that of the $\delta$-function. Because both ${{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j$ and ${{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $x$}}}_N$ can be expressed in terms of a linear combination of the relative coordinate ${{\mbox{\boldmath $x$}}}_i$, it is enough to calculate the matrix element of $\delta(\widetilde{w}{{\mbox{\boldmath $x$}}}-{{\mbox{\boldmath $r$}}})$, where $\widetilde{w}=(w_1,w_2,\ldots,
w_{N-1})$ is a combination constant to express ${{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j$ or ${{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $x$}}}_N$.
The matrix element of the $\delta$-function is given by $$\begin{aligned}
& &{\hspace{-1cm}}\left<F'\vert \delta(\widetilde{w}{{\mbox{\boldmath $x$}}}-{{\mbox{\boldmath $r$}}}) \vert F \right>
\nonumber \\
&=&\left(\frac{(2\pi)^{N-1}}{\mbox{det} B}\right)^{3/2}
\left(\prod_{i=1}^6 B_{L_i}\right)
\frac{(-1)^{L_1+L_2+L_3}\sqrt{2L+1}}{\sqrt{2L'+1}}
\ \left(\frac{c}{2\pi}\right)^{3/2}{\rm e}^{-\frac{1}{2}cr^2} \nonumber\\
&\times & \sum_{\kappa \mu} \langle LM\kappa \mu| L'M'\rangle
Y_{\kappa \mu}^*(\widehat{{\mbox{\boldmath $r$}}})
\sum_{p_i} \left(\prod_{i=1}^6 (-c\gamma_i r)^{p_{i}}
\frac{ \sqrt{2p_{i}+1}}{B_{p_{i}}} \right)
\nonumber \\
&\times& \sum_{\ell_{12}\ell_{45}\ell \ell' \overline{L}_{12}
\overline{L}_{45} \overline{L}}
\frac{(-1)^{\ell+\ell'}}{\sqrt{(2\ell+1)(2\overline{L}+1)}}U(L\overline{L}\kappa
\ell';\ell L')\
\overline{O}(p_i ; \ell_{12} \ell_{45} \ell \ell' \kappa )
\nonumber \\
&\times&
W(p_1 p_2 p_3 \ell_{12} \ell, L_1\!-\!p_1\ L_2\!-\!p_2\ L_3\!-\!p_3\
\overline{L}_{12}\overline{L}, L L_{12})\nonumber \\
&\times&
W(p_4 p_5 p_6 \ell_{45} \ell', L_4\!-\!p_4\ L_5\!-\!p_5\ L_6\!-\!p_6\
\overline{L}_{45}\overline{L}, L' L_{45})\nonumber \\
&\times&
\sum_{n_{ij}}\left(\prod_{i<j}^6 (-\overline{\rho}_{ij})^{n_{ij}}
\frac{ \sqrt{2n_{ij}+1}}{B_{n_{ij}}}\right)
\nonumber \\
&\times&
O(n_{ij}; L_1\!-\!p_1\ L_2\!-\!p_2\ L_3\!-\!p_3\ L_4\!-\!p_4,
L_5\!-\!p_5\ L_6\!-\!p_6,\overline{L}_{12}\overline{L}_{45}\overline{L}),
\label{me.del}\end{aligned}$$ with $$\begin{aligned}
& &c=(\widetilde{w}B^{-1}w)^{-1},\ \ \ \ \ \gamma_i=\widetilde{w}B^{-1}u_i,
\ \ \ \ \ \overline{\rho}_{ij}=\rho_{ij}-c\gamma_i\gamma_j.\end{aligned}$$ The summation over non-negative integers $n_{ij}$ and $p_i$ is restricted by the following equations $$\begin{aligned}
&&n_{12}+n_{13}+n_{14}+n_{15}+n_{16}+p_1=L_1, \nonumber\\
&&n_{12}+n_{23}+n_{24}+n_{25}+n_{26}+p_2=L_2, \nonumber\\
&&n_{13}+n_{23}+n_{34}+n_{35}+n_{36}+p_3=L_3, \nonumber\\
&&n_{14}+n_{24}+n_{34}+n_{45}+n_{46}+p_4=L_4, \nonumber\\
&&n_{15}+n_{25}+n_{35}+n_{45}+n_{56}+p_5=L_5, \nonumber\\
&&n_{16}+n_{26}+n_{36}+n_{46}+n_{56}+p_6=L_6. \end{aligned}$$ Here $\overline{O}(p_i ; \ell_{12} \ell_{45} \ell \ell' \kappa )$ is defined as a coefficient that appears in the coupling of a product of six terms $$\begin{aligned}
& &{\hspace{-1cm}}\prod_{i=1}^6 [Y_{p_i}({{\mbox{\boldmath $e$}}}_i)Y_{p_i}(\widehat{{\mbox{\boldmath $r$}}})]_{00}
= \sum_{\ell_{12}\ell_{45}\ell \ell' \kappa}\overline{O}(p_i ; \ell_{12} \ell_{45} \ell \ell' \kappa)
\nonumber \\
&\times&
[[[[Y_{p_1}({{\mbox{\boldmath $e$}}}_1)Y_{p_2}({{\mbox{\boldmath $e$}}}_2)]_{\ell_{12}}Y_{p_3}({{\mbox{\boldmath $e$}}}_3)]_{\ell}\
[[Y_{p_4}({{\mbox{\boldmath $e$}}}_4)Y_{p_5}({{\mbox{\boldmath $e$}}}_5)]_{\ell_{45}}Y_{p_6}({{\mbox{\boldmath $e$}}}_6)]_{\ell'}]_{\kappa}\
Y_{\kappa}(\widehat{{\mbox{\boldmath $r$}}})]_{00},\end{aligned}$$ and it is given by $$\begin{aligned}
& &\overline{O}(p_i ; \ell_{12} \ell_{45} \ell \ell' \kappa )
\nonumber \\
& &\quad=\sqrt{\frac{2\kappa+1}{\prod_{i=1}^6(2p_i+1)}}
C(p_1 p_2,\ell_{12})C(\ell_{12} p_3,\ell)C(p_4 p_5,\ell_{45})C(\ell_{45} p_6,\ell')
C(\ell \ell', \kappa).\end{aligned}$$
The ${{\mbox{\boldmath $r$}}}$-dependence of the matrix element (\[me.del\]) is $${\rm e}^{-\frac{1}{2}cr^2} r^{p_1+p_2+p_3+p_4+p_5+p_6} Y_{\kappa \mu}^*(\widehat{{\mbox{\boldmath $r$}}}).$$ For a central interaction, $V({{\mbox{\boldmath $r$}}})$ is a scalar function, and the sum over $\kappa$ in Eq. (\[me.del\]) is limited to 0. For a tensor interaction, the angular dependence of $V({{\mbox{\boldmath $r$}}})$ is proportional to $Y_2(\hat{{\mbox{\boldmath $r$}}})$, and $\kappa$ is limited to 2. The electric multipole operator is a special case of one-body operator, so that one can make use of the formula (\[me.del\]) to calculate its matrix element. More explicitly, we give the matrix element of $V(|\widetilde{w}{{\mbox{\boldmath $x$}}}|)
Y_{\kappa \mu}(\widehat{\widetilde{w}{{\mbox{\boldmath $x$}}}})$ that includes all the cases mentioned above: $$\begin{aligned}
& &{\hspace{-1cm}}\left<F'\vert V(|\widetilde{w}{{\mbox{\boldmath $x$}}}|)
Y_{\kappa \mu}(\widehat{\widetilde{w}{{\mbox{\boldmath $x$}}}}) \vert F \right>
\nonumber \\
&=&\left(\frac{(2\pi)^{N-1}}{\mbox{det} B}\right)^{3/2}
\left(\prod_{i=1}^6 B_{L_i}\right)
\frac{(-1)^{L_1+L_2+L_3}\sqrt{2L+1}}{\sqrt{2L'+1}} \nonumber\\
&\times & \langle LM\kappa \mu| L'M'\rangle
\sum_{p_i} \left(\prod_{i=1}^6 (-\gamma_i )^{p_{i}}
\frac{ \sqrt{2p_{i}+1}}{B_{p_{i}}} \right){\cal I}^{(2)}_{p_1+p_2+p_3+p_4+p_5+p_6}(c)
\nonumber \\
&\times& \sum_{\ell_{12}\ell_{45}\ell \ell' \overline{L}_{12}
\overline{L}_{45} \overline{L}}
\frac{(-1)^{\ell+\ell'}}{\sqrt{(2\ell+1)(2\overline{L}+1)}}U(L\overline{L}\kappa
\ell';\ell L')\
\overline{O}(p_i ; \ell_{12} \ell_{45} \ell \ell' \kappa )
\nonumber \\
&\times&
W(p_1 p_2 p_3 \ell_{12} \ell, L_1\!-\!p_1\ L_2\!-\!p_2\ L_3\!-\!p_3\
\overline{L}_{12}\overline{L}, L L_{12})\nonumber \\
&\times&
W(p_4 p_5 p_6 \ell_{45} \ell', L_4\!-\!p_4\ L_5\!-\!p_5\ L_6\!-\!p_6\
\overline{L}_{45}\overline{L}, L' L_{45})\nonumber \\
&\times&
\sum_{n_{ij}}\left(\prod_{i<j}^6 (-\overline{\rho}_{ij})^{n_{ij}}
\frac{ \sqrt{2n_{ij}+1}}{B_{n_{ij}}}\right)
\nonumber \\
&\times&
O(n_{ij}; L_1\!-\!p_1\ L_2\!-\!p_2\ L_3\!-\!p_3\ L_4\!-\!p_4,
L_5\!-\!p_5\ L_6\!-\!p_6,\overline{L}_{12}\overline{L}_{45}\overline{L}),\end{aligned}$$ with the integral of the potential form factor $$\begin{aligned}
{\cal I}^{(m)}_n(c)=\left(\frac{c}{2\pi}\right)^{3/2} c^n\int_0^{\infty} dr \,r^{n+m} V(r) {\rm e}^{-\frac{1}{2}cr^2}. \end{aligned}$$ In case $V(r)$ takes the form of $r^q {\rm e}^{-\rho r^2-\rho'r}$ $(q
\geq -m)$, the integral ${\cal I}^{(m)}_n(c)$ can be obtained analytically, giving a closed form for the matrix element.
It should be noted that the matrix element for a special class of a three-body force can be evaluated with ease. For example, if the radial part of the three-body force has a form $$V_{TNF}=\exp(-\rho_1({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j)^2-
\rho_2({{\mbox{\boldmath $r$}}}_j-{{\mbox{\boldmath $r$}}}_k)^2
-\rho_3({{\mbox{\boldmath $r$}}}_k-{{\mbox{\boldmath $r$}}}_i)^2),$$ the exponent can be rewritten as $-\widetilde{{\mbox{\boldmath $x$}}}\Omega {{\mbox{\boldmath $x$}}}$ with an $(N-1)\times(N-1)$ symmetric matrix $\Omega=\rho_1 w_{ij}\widetilde{w_{ij}}+\rho_2
w_{jk}\widetilde{w_{jk}}+\rho_3 w_{ki}\widetilde{w_{ki}}$, where $w_{ij}$, $w_{jk}$ and $w_{ki}$ are defined by ${{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j=\widetilde{w_{ij}}{{\mbox{\boldmath $x$}}}$, ${{\mbox{\boldmath $r$}}}_j-{{\mbox{\boldmath $r$}}}_k=\widetilde{w_{jk}}{{\mbox{\boldmath $x$}}}$ and ${{\mbox{\boldmath $r$}}}_k-{{\mbox{\boldmath $r$}}}_i=\widetilde{w_{ki}}{{\mbox{\boldmath $x$}}}$. Thus the matrix element reduces to that of the overlap with $A$ being replaced with $A+2\Omega$ $$\langle F'\vert V_{TNF} \vert F \rangle
= \langle F'\vert F_{L_1 L_2(L_{12}) L_3 LM}(u_1, u_2, u_3,
A+2\Omega,{{\mbox{\boldmath $x$}}}) \rangle.$$
Spin-orbit potential
--------------------
The spatial form of a spin-orbit interaction reads $$V(|{{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j|)(({{\mbox{\boldmath $r$}}}_i-{{\mbox{\boldmath $r$}}}_j)\times \frac{1}{2}({{\mbox{\boldmath $p$}}}_i-{{\mbox{\boldmath $p$}}}_j))_{\mu},$$ where $({{\mbox{\boldmath $a$}}}\times{{\mbox{\boldmath $b$}}})_{\mu}$ $(\mu=0, \pm 1)$ stands for the $\mu$th component of a vector product of ${{\mbox{\boldmath $a$}}}$ and ${{\mbox{\boldmath $b$}}}$. As in the $\delta$-function matrix element, the spin-orbit potential is written as $$V(|\widetilde{w}{{\mbox{\boldmath $x$}}}|) (\widetilde{w}{{\mbox{\boldmath $x$}}}\times \widetilde{\zeta}{{\mbox{\boldmath $\pi$}}})_{\mu},$$ where $\frac{1}{2}({{\mbox{\boldmath $p$}}}_i-{{\mbox{\boldmath $p$}}}_j)$ is expressed in terms of the momentum operators ${{\mbox{\boldmath $\pi$}}}$, $\widetilde{\zeta}{{\mbox{\boldmath $\pi$}}}=\sum_{i=1}^{N-1} \zeta_i {{\mbox{\boldmath $\pi$}}}_i$.
The matrix element of the spin-orbit potential is given by $$\begin{aligned}
& &{\hspace{-1cm}}\left<F'\vert V(|\widetilde{w}{{\mbox{\boldmath $x$}}}|) (\widetilde{w}{{\mbox{\boldmath $x$}}}\times \widetilde{\zeta}{{\mbox{\boldmath $\pi$}}})_{\mu} \vert F \right>
\nonumber \\
&=&\frac{4\pi \sqrt{2} \hbar}{3}\left(\frac{(2\pi)^{N-1}}{\mbox{det} B}\right)^{3/2}
\left(\prod_{i=1}^6 B_{L_i}\right)
\frac{(-1)^{L_1+L_2+L_3}\sqrt{2L+1}}{\sqrt{2L'+1}}
\nonumber\\
&\times & \langle LM 1 \mu| L'M'\rangle
\sum_{p_i} \left(\prod_{i=1}^6 (-\gamma_i )^{p_{i}}
\frac{ \sqrt{2p_{i}+1}}{B_{p_{i}}} \right){\cal I}^{(3)}_{p_1+p_2+p_3+p_4+p_5+p_6}(c)
\nonumber \\
&\times& \sum_{\ell_{12}\ell_{45}\ell \ell' \overline{\ell}_{12}
\overline{\ell}_{45} \overline{\ell}\ \overline{\ell}'
\overline{L}_{12} \overline{L}_{45} \overline{L} }
\frac{(-1)^{\overline{\ell}+\overline{\ell}'}}
{\sqrt{(2\overline{\ell}+1)(2\overline{L}+1)}}
U(L\overline{L}1 \overline{\ell}'; \overline{\ell} L')\
\overline{O}(p_i ; \ell_{12} \ell_{45} \ell \ell' 1)
\nonumber \\
&\times&
\sum_{k=1}^6 (\widetilde{\zeta}\Gamma)_k \ T_k(p_i,\ell_{12}\ell_{45}\ell \ell', \overline{\ell}_{12} \overline{\ell}_{45} \overline{\ell}\ \overline{\ell}')
\sum_{n_{ij}}\left(\prod_{i<j}^6 (-\overline{\rho}_{ij})^{n_{ij}}
\frac{ \sqrt{2n_{ij}+1}}{B_{n_{ij}}}\right)
\nonumber \\
&\times&
O(n_{ij}; L_1\!-\!p_1^k\ L_2\!-\!p_2^k\ L_3\!-\!p_3^k\ L_4\!-\!p_4^k\
L_5\!-\!p_5^k\ L_6\!-\!p_6^k,\overline{L}_{12}\overline{L}_{45}\overline{L}),\end{aligned}$$ where ${p}_i^k$ $(k=1,2,\ldots,6)$ is $${p}_i^k=p_i+\delta_{ik},$$ and where the non-negative integers $n_{ij}$ and $p_i$ are constrained to satisfy the equations $$\begin{aligned}
&&n_{12}+n_{13}+n_{14}+n_{15}+n_{16}+p_1^k=L_1, \nonumber\\
&&n_{12}+n_{23}+n_{24}+n_{25}+n_{26}+p_2^k=L_2, \nonumber\\
&&n_{13}+n_{23}+n_{34}+n_{35}+n_{36}+p_3^k=L_3, \nonumber\\
&&n_{14}+n_{24}+n_{34}+n_{45}+n_{46}+p_4^k=L_4, \nonumber\\
&&n_{15}+n_{25}+n_{35}+n_{45}+n_{56}+p_5^k=L_5, \nonumber\\
&&n_{16}+n_{26}+n_{36}+n_{46}+n_{56}+p_6^k=L_6. \end{aligned}$$ The symbol $(\widetilde{\zeta}\Gamma)_k$ stands for the factor $$(\widetilde{\zeta}\Gamma)_k = \sum_{i=1}^6 \zeta_i (\Gamma_k)_i,$$ where $(\Gamma_k)_i$ is the $i$th element of the column vector $\Gamma_k$ defined in Eq. (\[def.gamma\]). The coefficient $T_k$ appears in the coupling $$\begin{aligned}
& &[Y_1({{\mbox{\boldmath $e$}}}_k)\ [[[Y_{p_1}({{\mbox{\boldmath $e$}}}_1)Y_{p_2}({{\mbox{\boldmath $e$}}}_2)]_{\ell_{12}}Y_{p_3}({{\mbox{\boldmath $e$}}}_3)]_{\ell}\
[[Y_{p_4}({{\mbox{\boldmath $e$}}}_4)Y_{p_5}({{\mbox{\boldmath $e$}}}_5)]_{\ell_{45}}Y_{p_6}({{\mbox{\boldmath $e$}}}_6)]_{\ell'}]_{1}]_{1\mu}
\nonumber \\
& &\to \sum_{\overline{\ell}_{12} \overline{\ell}_{45} \overline{\ell}\ \overline{\ell}'}T_k(p_i,\ell_{12}\ell_{45}\ell \ell', \overline{\ell}_{12} \overline{\ell}_{45} \overline{\ell}\ \overline{\ell}')
\nonumber \\
& &\qquad \times
[[[Y_{{p}_1^k}({{\mbox{\boldmath $e$}}}_1)Y_{{p}_2^k}({{\mbox{\boldmath $e$}}}_2)]_{\overline{\ell}_{12}}Y_{{p}_3^k}({{\mbox{\boldmath $e$}}}_3)]_{\overline{\ell}}\
[[Y_{{p}_4^k}({{\mbox{\boldmath $e$}}}_4)Y_{{p}_5^k}({{\mbox{\boldmath $e$}}}_5)]_{\overline{\ell}_{45}}Y_{{p}_6^k}({{\mbox{\boldmath $e$}}}_6)]_{\overline{\ell}'}]_{1\mu}.\end{aligned}$$ The coefficients $T_k(p_i,\ell_{12}\ell_{45}\ell \ell', \overline{\ell}_{12} \overline{\ell}_{45} \overline{\ell}\ \overline{\ell}')$ are given below: $$\begin{aligned}
& &T_1=U(1\ell 1 \ell'; \overline{\ell}1)
U(1 \ell_{12} \overline{\ell}p_3; \overline{\ell}_{12} \ell)
U(1 p_1 \overline{\ell}_{12} p_2; p_1\!+\!1\ \ell_{12})
C(1 p_1; p_1\!+\!1)
\nonumber \\
& &T_2=-(-1)^{\ell_{12}+\overline{\ell}_{12}}
U(1\ell 1 \ell'; \overline{\ell}1)
U(1\ell_{12}\overline{\ell}p_3; \overline{\ell}_{12} \ell)
U(1 p_2 \overline{\ell}_{12} p_1; p_2\!+\!1\ \ell_{12})
C(1 p_2; p_2\!+\!1)
\nonumber \\
& &T_3=-(-1)^{\ell+\overline{\ell}}
U(1\ell 1 \ell'; \overline{\ell}1)
U(1p_3 \overline{\ell} \ell_{12} ; p_3\!+\!1 \ell)
C(1 p_3; p_3\!+\!1)
\nonumber \\
& &T_4=(-1)^{\ell'+\overline{\ell}'}
U(1\ell' 1 \ell; \overline{\ell}'1)
U(1 \ell_{45} \overline{\ell}' p_4; \overline{\ell}_{45} \ell')
U(1 p_4 \overline{\ell}_{45} p_5; p_4\!+\!1 \ \ell_{45})
C(1 p_4; p_4\!+\!1)
\nonumber \\
& &T_5=-(-1)^{\ell'+\overline{\ell}'+\ell_{45}+\overline{\ell}_{45}}
U(1\ell' 1 \ell; \overline{\ell}'1)
U(1\ell_{45} \overline{\ell}' p_6; \overline{\ell}_{45}\ell')
U(1 p_5 \overline{\ell}_{45} p_4; p_5\!+\!1\ \ell_{45})
C(1 p_5; p_5\!+\!1)
\nonumber \\
& &T_6=-U(1\ell' 1 \ell; \overline{\ell}'1)
U(1 p_6 \overline{\ell}' \ell_{45} ; p_6\!+\!1\ \ell')
C(1 p_6; p_6\!+\!1).\end{aligned}$$
Momentum representation of correlated Gaussian basis {#app.C}
====================================================
The Fourier transform of the correlated Gaussian function (\[cgtgv\]) defines the corresponding basis function in momentum space. The momentum space function is useful to evaluate those matrix elements which depend on the momentum operator [@DGVR]. Suppose that we want to evaluate the matrix element of a two-body operator $V({{\mbox{\boldmath $p$}}}_i-{{\mbox{\boldmath $p$}}}_j)$ or a one-body operator $D({{\mbox{\boldmath $p$}}}_i-\frac{1}{N}{{\mbox{\boldmath $\pi$}}}_N)$. Obviously evaluating the matrix element can be done more easily in momentum space. For this purpose we need to obtain the Fourier transform of the coordinate space function. A great advantage in the correlated Gaussian function $F$ is that its Fourier transform is a linear combination of the correlated Gaussian functions in the momentum space. Thus by expressing ${{\mbox{\boldmath $p$}}}_i-{{\mbox{\boldmath $p$}}}_j$ or ${{\mbox{\boldmath $p$}}}_i-\frac{1}{N}{{\mbox{\boldmath $\pi$}}}_N$ as $\widetilde{\zeta}{{\mbox{\boldmath $\pi$}}}$, we can calculate the matrix element of the momentum-dependent operators in exactly the same way as in the coordinate space.
As in the case with two global vectors [@DGVR], the transformation from the coordinate to momentum space is achieved by a function $$\Phi({{\mbox{\boldmath $k$}}},{{\mbox{\boldmath $x$}}})=\frac{1}{(2\pi)^{\frac{3}{2}(N-1)}}\,
\exp\,(i\tilde{{\mbox{\boldmath $k$}}}{{\mbox{\boldmath $x$}}}),$$ where ${{\mbox{\boldmath $k$}}}$ is an $(N\!-\!1)$-dimensional column vector whose $i$th element is ${{\mbox{\boldmath $k$}}}_i$. With a straightforward integration together with the recoupling of angular momenta, we can show that $$\begin{aligned}
& &\langle \Phi({{\mbox{\boldmath $k$}}},{{\mbox{\boldmath $x$}}})\vert
F_{L_1 L_2 (L_{12})L_3 L M}(u_1,u_2,u_3,A,{{\mbox{\boldmath $x$}}})\rangle
\nonumber \\
& &\quad =\frac{(-i)^{L_1+L_2+L_3}} {({\rm det}A)^{3/2}}
\sum_{\ell_1 \ell_2 \ell_3 \ell_{12}}
{\cal K}(L_1L_2(L_{12})L_3 L; \ell_1 \ell_2 \ell_3 \ell_{12})
\nonumber \\
& &\quad \times F_{L_1\!-\!\ell_1\!-\!\ell_2\ L_2\!-\!\ell_1\!-\!\ell_3\
(\ell_{12})\ L_3\!-\!\ell_2\!-\!\ell_3\ L M}
(A^{-1}u_1, A^{-1}u_2, A^{-1}u_3, A^{-1},{{\mbox{\boldmath $k$}}}),\end{aligned}$$ where the coefficient ${\cal K}$ is given by $$\begin{aligned}
& &{\cal K}(L_1L_2(L_{12})L_3 L; \ell_1 \ell_2 \ell_3 \ell_{12})
\nonumber \\
& &\ =\frac{(-1)^{L-L_3+\ell_2+\ell_3-\ell_{12}}}
{\sqrt{2L\!+\!1}}
\frac{B_{L_1}B_{L_2}B_{L_3}}
{B_{\ell_1}B_{\ell_2}B_{\ell_3}B_{L_1\!-\!\ell_1\!-\!\ell_2}
B_{L_2\!-\!\ell_1\!-\!\ell_3}B_{L_3\!-\!\ell_2\!-\!\ell_3}}
\nonumber \\
& &\quad \times
\sqrt{(2\ell_1\!+\!1)(2\ell_2\!+\!1)(2\ell_3\!+\!1)
(2(L_1\!-\!\ell_1\!-\!\ell_2)+1)(2(L_2\!-\!\ell_1\!-\!\ell_3)+1)(2(L_3\!-\!\ell_2\!-\!\ell_3)+1)}
\nonumber \\
& &\quad \times X(\ell_1 \ell_2 \ell_3)
Z(L_1\!-\!\ell_1\!-\!\ell_2\ \ell_1\!+\!\ell_2)
Z(L_2\!-\!\ell_1\!-\!\ell_3\ \ell_1\!+\!\ell_3)
Z(L_3\!-\!\ell_2\!-\!\ell_3\ \ell_2\!+\!\ell_3)
\nonumber \\
& &\quad \times U(\ell_{12}\ L_3\!-\!\ell_2\!-\!\ell_3\ L_{12}\ L_3; L\
\ell_2\!+\!\ell_3)
\left[
\begin{array}{ccc}
L_1\!-\!\ell_1\!-\!\ell_2 & L_1 & \ell_1\!+\!\ell_2\\
L_2\!-\!\ell_1\!-\!\ell_3 & L_2 & \ell_1\!+\!\ell_3\\
\ell_{12} & L_{12} & \ell_2\!+\!\ell_3 \\
\end{array}
\right]
\nonumber \\
& &\quad \times
(\widetilde{u_1}A^{-1}u_2)^{\ell_1}\
(\widetilde{u_1}A^{-1}u_3)^{\ell_2}\
(\widetilde{u_2}A^{-1}u_3)^{\ell_3},\end{aligned}$$ where $Z$ and $X$ are defined in Appendix A. Non-negative integers $\ell_i$ run over all possible values that satisfy $\ell_1\!+\!\ell_2 \leq L_1,\ \ell_1\!+\!\ell_3 \leq L_2,\
\ell_2\!+\!\ell_3 \leq L_3$. The value of ${\ell}_{12}$ is restricted by the triangular relations among ($\ell_{12}, L_1\!-\!\ell_1\!-\!\ell_2, L_2\!-\!\ell_1\!-\!\ell_3$) and ($\ell_{12}, L_{12}, \ell_2\!+\!\ell_3 $).
[99]{} Wildermuth K, Tang Y C (1977) A Unified Theory of the Nucleus (Vieweg, Braunschweig). Kamada H, Nogga A, Gl[ö]{}ckle W, Hiyama E, Kamimura M [*et al.*]{} (2001) Phys Rev 64:044001 Varga K, Suzuki Y, R. G. Lovas (1994) Nucl Phys A 571:447 Varga K, Ohbayasi K, Suzuki Y (1997) Phys Lett B 396:1; Varga K, Usukura J, Suzuki Y (1998) Phys Rev Lett 80:1876; Usukura J, Varga K, Suzuki Y (1998) Phys Rev A58:1918 Suzuki Y, Varga K (1998) Stochastic variational approach to quantum-mechanical few-body problems (Lecture notes in physics, Vol. 54). Springer, Berlin Heidelberg New York Varga K, Suzuki Y (1995) Phys Rev C 52:2885 Suzuki Y, Horiuchi W, Orabi M, Arai K (2008) Few-Body Syst 42:33 Varga K, Suzuki Y, Usukura J (1998) Few-Body Syst 24:81 Carlson J, Schiavilla R (2008) Rev Mod Phys 70:743; Pudliner B.S, Pandharipande V.R, Carlson J, Pieper S.C, Wiringa R.B (1997) Phys Rev C 56:1720 Navratil P, Kamuntavicius G.P, Barrett B.R (2000) Phys Rev C 61:044001 Viviani M (1998) Few-Body Syst 25:197 Feldmeier H, Neff T, Roth R, Schnack J (1998) Nucl Phys A632:61; Neff T, Feldmeier H (2003) Nucl Phys A 713:311 Arai K, Aoyama S, Suzuki Y (2010) Phys Rev C 81:037301 Phitzinger B, Hofmann M, Hale G.M (2001) Phys Rev C 64:044003 Deltuva A, Fonseca A.C (2007) Phys Rev C 75:014005; Deltuva A, Fonseca A.C (2007) Phys Rev Lett 98:162502 Quaglioni S, Navratil P (2009) Phys Rev C 79:044606; Quaglioni S, Navratil P (2008) Phys Rev Lett 08:092501 Viviani M, Rosati S, Kievsky A (1998) Phys Rev Lett 81:1580; Viviani M, Kievsky A, Rosati S, George E.A, Knulson L.D (2001) Phys Rev Lett 86:3739;Viviani M, Kievsky A,Girlanda L, Marcucci L.E, Rosati S (2009) Few-Body Syst 45:119 Lazauskas R, Carbonell J, Fonseca A.C, Viviani M, Kievsky A, Rosati S (2005) Phys Rev C 71:034004 Fisher B.M $et$ $al.$ (2006) Phys Rev C 74:034001 Arriaga A, Pandharipande V.R, Schiavilla (1991) Phys Rev C 43:983 Sabourov K $et$ $al.$ (2004) Phys Rev C 70:064601 Hofmann H.M, Hale G.M (2008) Phys Rev C 77:044002 Hofmann H.M, Hale G.M (1997) Nucl Phys A 613:69; Hofmann H.M, Hale G.M (2003) Phys Rev C 68:021002 Deltuva A, Fonseca A.C (2007) Phys Rev C 76:021001; Deltuva A, Fonseca A.C, Sauer P.U (2008) Phys Lett B 660:471 Lazauskas R, Carbonell J (2004) Few-Body Syst 34:105 Ciesielski F, Carbonell J, Gignoux C (1999) Phys Lett B 447:199 Assenbaum H.J, Langanke K (1987) Phys Rev C 36:17 Fowler W.A, Caughlan, Zimmenrman (1967) Annu Rev Astron Astrophys 5:525 Baye D, Heenen P H, Libert-Heinemann M (1977) Nucl Phys A 291:230 Kanada H, Kaneko T, Saito S, Tang Y C (1985) Nucl Phys A 444:209 Arai K, Descouvemont P , Baye D, (2001) Phys Rev C 63:044611 Descouvemont P, Baye D (2010) Rep Prog Phys 73:036301 Pudliner B S, Pandharipande V R, Carlson J, Pieper S C, Wiringa R B (1997): Phys Rev C 56, 1720 Hiyama E, Gibson B F, Kamimura M (2003) Phys Rev C 70:031001 Thompson D R, LeMere M, Tang Y C (1977) Nucl Phys A 286:53
Boys S F (1960) Proc R Soc London Ser A 258:402 ; Singer K (1960) [*ibid.*]{} 258:412 Suzuki Y, Usukura J (2000) Nucl Inst Method B 171:67 Suzuki Y, Usukura J, Varga K (1998) J Phys B 31:31 Horiuchi W, Suzuki Y (2008) Phys Rev C 78:034305 Tilley D R, Weller H R, Hale G M (1992), Nucl Phys A 541:1
Tamagaki R (1968) Prog Theor Phys 39:91
Santos F D, Arriaga A, Eiró A M, Tostevin J A (1985) Phys Rev C 31:707 Wachter B, Mertelmeier T, Hofmann H M (1988) Phys Lett B 200:246 Angulo C, Arnould M, Rayet M, Descouvemont P, Baye D [*et al.*]{} (1999) Nucl Phys A 656:3
|
---
abstract: 'The technique used at the Sudbury Neutrino Observatory ([[SNO]{}]{}) to measure the concentration of in water is described. Water from the [[SNO]{}]{}detector is passed through a vacuum degasser (in the light water system) or a membrane contact degasser (in the heavy water system) where dissolved gases, including radon, are liberated. The degasser is connected to a vacuum system which collects the radon on a cold trap and removes most other gases, such as water vapor and N$_2$. After roughly 0.5 tonnes of [H$_2$O]{}or 6 tonnes of [D$_2$O]{}have been sampled, the accumulated radon is transferred to a Lucas cell. The cell is mounted on a photomultiplier tube which detects the $\alpha$-particles from the decay of and its progeny. The overall degassing and concentration efficiency is about 38% and the single-$\alpha$ counting efficiency is approximately 75%. The sensitivity of the radon assay system for [D$_2$O]{}is equivalent to $\sim$3$\times 10^{-15}$ g U/g water. The radon concentration in both the [H$_2$O]{}and [D$_2$O]{}is sufficiently low that the rate of background events from U-chain elements is a small fraction of the interaction rate of solar neutrinos by the neutral current reaction.'
address:
- 'Carleton University, Ottawa, Ontario K1S 5B6, Canada'
- 'Chemistry Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA'
- 'Department of Physics, Queen’s University, Kingston, Ontario K7L 3N6, Canada'
- 'Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK'
- 'Department of Physics and Astronomy, Laurentian University, Sudbury, Ontario P3E 2C6, Canada'
- 'Physics Department, University of Guelph, Guelph, Ontario N1G 2W1, Canada'
- 'Atomic Energy of Canada Limited, Chalk River Laboratories, Chalk River, Ontario K0J 1J0, Canada'
author:
- 'I. Blevis'
- 'J. Boger'
- 'E. Bonvin'
- 'B. T. Cleveland'
- 'X. Dai'
- 'F. Dalnoki-Veress'
- 'G. Doucas'
- 'J. Farine'
- 'H. Fergani'
- 'D. Grant'
- 'R. L. Hahn'
- 'A. S. Hamer'
- 'C. K. Hargrove'
- 'H. Heron'
- 'P. Jagam'
- 'N. A. Jelley'
- 'C. Jillings'
- 'A. B. Knox'
- 'H. W. Lee'
- 'I. Levine'
- 'M. Liu'
- 'S. Majerus'
- 'A. McDonald'
- 'K. McFarlane'
- 'C. Mifflin'
- 'A. J. Noble'
- 'S. Noël'
- 'V. M. Novikov'
- 'J. K. Rowley'
- 'M. Shatkay'
- 'J. J. Simpson'
- 'D. Sinclair'
- 'B. Sur'
- 'J.-X. Wang'
- 'M. Yeh'
- 'X. Zhu'
title: |
Measurement of dissolved in water\
at the Sudbury Neutrino Observatory
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radioactivity assay ,solar neutrino ,SNO ,radon
Introduction {#sec:intro}
============
The Sudbury Neutrino Observatory ([[SNO]{}]{}) is a heavy water Cherenkov detector which was built to understand why all previous solar neutrino experiments [@bib:solar_experiments; @bib:kamiokande; @bib:sage; @bib:gallex; @bib:sk; @bib:gno] have observed fewer neutrinos than are predicted by generally accepted solar models [@bib:sm1; @bib:sm2].
The [[SNO]{}]{}detector is described in detail elsewhere [@bib:SNONIM]. Briefly, [[SNO]{}]{}consists of an inner neutrino target of 1000 tonnes of ultra-pure D$_2$O contained in a 12 m diameter spherical, transparent, acrylic vessel. An array of 9438 photomultiplier tubes, mounted on an $\sim$18 m diameter stainless steel geodesic support structure, detect the Cherenkov light from electrons produced by neutrino interactions in the [D$_2$O]{}. The volume between the acrylic vessel and the tube support structure contains approximately 1700 tonnes of ultra-pure H$_2$O which shields the [D$_2$O]{}volume from high-energy $\gamma$ rays produced by radioactivity in the outer regions of the detector.
Outside the photomultipliers lie an additional 5700 tonnes of [H$_2$O]{}shielding. The two water shielding regions are separated by a nearly impermeable water seal which serves to keep the water in the outer shielding area, which has higher radon levels, isolated from the water between the photomultipliers and the acrylic vessel, where the radon level is lower. The external water shielding region is viewed by 91 outward-looking photomultiplier tubes which help to reject background from muons traversing the detector. The detector is situated in the INCO, Ltd. Creighton mine, in Sudbury, Ontario, Canada. At a depth of 6800 feet, only about 70 muons interact in the detector per day.
[[SNO]{}]{}detects solar neutrinos through three distinguishable interactions with the D$_2$O target:
\_ & + \^- + + &\
\_x & + \_x + + &\
\_x & + \^- \_x + \^- &
where $x$ denotes any of the active neutrino species $\text{e}, \mu,
\text{or } \tau$. The CC reaction is only sensitive to the flux of electron neutrinos, whereas the NC reaction is equally sensitive to all active neutrino flavors. Three techniques have been developed to observe the neutrons from the NC reaction in [[SNO]{}]{}. In the first phase of the experiment, the inner vessel was filled with pure [D$_2$O]{}. The neutrons were captured by deuterium nuclei creating 6.25-MeV $\gamma$ rays which interacted to make relativistic electrons whose Cherenkov light was detected. In the second phase, NaCl was added to the D$_2$O. Most neutrons then capture on $^{35}$Cl, an exothermic reaction that yields photons whose energies sum to 8.6 MeV. In the third phase, the salt will be removed and neutrons will be detected with low-background $^3$He-filled counters that will be installed in the [D$_2$O]{}.
The first publication [@bib:SNOP1] of [[SNO]{}]{}results indicated that the flux of electron neutrinos with energy $\ge$6.75 MeV inferred from the CC reaction is not as large as the total rate inferred from the ES reaction as measured by [[SNO]{}]{}, or with greater accuracy, by Super-Kamiokande [@bib:sk]. Since the ES reaction is mostly sensitive to the flux of electron neutrinos, but has a small contribution from the flux of other neutrino flavors, this implies that the flux of active neutrinos from the Sun is greater than the observed flux of electron neutrinos alone. As only electron neutrinos are produced in the Sun, this observation is evidence that electron neutrinos have transformed into some combination of $\mu$ and $\tau$ neutrinos by the time they reach the detector.
A much higher precision measurement of this phenomenon was obtained by a comparison at energies $\ge$5 MeV of the CC and NC rates in the [[SNO]{}]{}detector [@bib:ccnc] and their temporal variations [@bib:dn]. To make this comparison required that the radioactive backgrounds in the detector were well understood, since decays of progeny of $^{238}$U (“U-chain”) and progeny of $^{232}$Th (“Th-chain”) can mimic the neutrino interactions. The isotopes of most concern for the CC/NC comparison are in the U-chain and in the Th-chain because their decays can produce $\gamma$ rays with energies greater than 2.2 MeV. These high-energy $\gamma$ rays can photodisintegrate the deuteron, producing a free neutron, and thus mimic the NC disintegration of the deuteron.
Two techniques, which are discussed elsewhere [@bib:suf; @bib:mnox], have been developed to measure the aqueous concentration of $^{226}$Ra from the U-chain and $^{224}$Ra from the Th-chain. These give a good measurement of the concentration of radium ions in the water. Knowledge of the $^{226}$Ra concentration is, however, not sufficient to determine the U-chain radioactive background because $^{222}$Rn, the decay product of $^{226}$Ra, is a noble gas with a 3.8-d half-life. Small leaks of air and traces of radium in detector materials can introduce $^{222}$Rn, and lead to significant disequilibria between $^{226}$Ra and $^{214}$Bi. To properly understand the radioactive background, it is thus imperative to measure directly the $^{222}$Rn concentration in the water as only isotopes with short half-lives separate from the undesired $^{214}$Bi.
The underground air at the [[SNO]{}]{}laboratory contains $\sim$100 Bq/m$^3$ of (3 pCi/l or 50 Rn atoms/cm$^3$), which, if the radon were at its equilibrium concentration with the [D$_2$O]{}, would yield a dissolved level almost $10^6$ times higher than tolerable. Considerable precautions were thus taken in the design and construction of [[SNO]{}]{}to limit the leakage of into the detector. As examples of such measures, all components of the water systems were selected for low radon diffusion and emanation, the entire [D$_2$O]{}system was leak checked with a He mass spectrometer, and one of the final elements of both the [H$_2$O]{}and [D$_2$O]{}water purification systems is a degasser that has a radon removal efficiency of $>$90%. Further, the polypropylene pipes in the water system have thick walls and are made with specially selected low-radioactivity material [@bib:himont], all detachable joints are sealed with butyl rubber O-rings [@bib:butyl] which have low radon permeability and emanation, the detector cavity is lined with a membrane with low Rn permeability, and there is a “cover gas”, a continuously flowing stream of N$_2$ from the boiloff of liquid nitrogen, into the vapor space directly above the [D$_2$O]{}and [H$_2$O]{}.
Maximum allowable radon concentration
=====================================
The standard model calculation of the $^8$B neutrino flux from the Sun [@bib:sm1], together with the cross section for the NC reaction [@bib:ncxsec], predicts that 10 to 15 neutrons will be produced by solar neutrinos per day in the 1000 tonnes of [D$_2$O]{}in the [[SNO]{}]{}detector. Based on this prediction, [[SNO]{}]{}set the design goal to have no more than one neutron per day produced in the [D$_2$O]{}by U-chain contamination. As described in [@bib:mnox], this implies that the maximum allowable contamination of the D$_2$O is $3 \times
10^{-14}$ g U/g D$_2$O, assuming secular equilibrium between $^{238}$U and $^{214}$Bi. This can be translated into a maximum concentration for $^{222}$Rn in the D$_2$O of about 0.2 atoms/liter or 0.4 mBq/m$^3$. The maximum allowable radon concentrations in the [H$_2$O]{}between the acrylic vessel and the photomultipliers and in the outer bulk shielding water are given in Table \[tab:goals\]. At these levels, the background contributed by [H$_2$O]{}to the NC signal is no more than that from the [D$_2$O]{}.
[@ l @ d d @ d @]{} Region & & &\
D$_2$O & 0.2 & 0.4 & 3.0 10\^[-14]{}\
H$_2$O (I) & 3 & 6 & 4.5 10\^[-13]{}\
H$_2$O (O) & 7 & 14 & 1.1 10\^[-12]{}\
To reduce the systematic error in the NC determination that arises from the uncertainty in the radon level to less than a few percent, it is necessary to lower the radon content of the water below the maximum allowable level. Such levels can only be measured with accuracy if the ultimate sensitivity of the assay method is an order of magnitude below the design goal. As shown below, these requirements were met, thus allowing the critical neutral current measurement to be made with precision.
Assay of the H$_{\bm 2}$O {#sec:h2o}
=========================
Overview of the assay technique
-------------------------------
The assay method operates by pumping water from the detector to a degasser which extracts the radon. The radon is collected, concentrated, and then transferred to a miniature ZnS-coated scintillation counter (a Lucas cell [@bib:lucas]) for measurement. The extraction of radon is performed differently in the H$_2$O and D$_2$O systems; we describe the extraction from H$_2$O here and from D$_2$O in Sec. \[sec:d2o\]. The radon collection and counting are the same in both systems and are discussed in Sec. \[sec:rncollect\] and Sec. \[sec:counting\], respectively.
Water flow
----------
![Location of sample points for usual radon assays of H$_2$O and D$_2$O. Assay points are labelled by the valves that must be opened to extract the water. Most D$_2$O assays take water from the main purification loop shown on the left side of the acrylic vessel. This circulation normally draws water from the bottom of the vessel, and, after purification and assay, returns it at the bottom of the vessel neck. Flow in the reverse direction is also possible. Measurements of radon levels are also made occasionally at other locations within the vessel (V901, V903, and V905). Assays of the light water are usually conducted at two positions between the photomultiplier array and the acrylic vessel (V203 and V206) and at the bottom of the cavity (V202).[]{data-label="fig:sample"}](samppts.eps){width="\hsize"}
H$_2$O can be drawn from six locations in the detector. Two sample points are at the bottom of the cavity, three at the equator of the acrylic vessel, and one at the bottom of the photomultiplier array. The most commonly sampled H$_2$O and D$_2$O positions are shown in Fig. \[fig:sample\]. The distance from the sample points to the degasser is about 70 m and the piping is polypropylene with 60 mm outer diameter and 5.5 mm wall thickness. Polypropylene was chosen for its low radioactivity, low leaching of impurities in the presence of ultra-pure water, and low radon permeability. The water sample is taken by a diaphragm pump [@bib:diaphragm] whose wetted portions are made from polypropylene, except for the diaphragm which is Teflon. A typical assay samples water for 30 minutes at a flow rate of 19 liters/minute. At the maximum allowable contamination of the H$_2$O between the acrylic vessel and the photomultipliers ($4.5
\times 10^{-13}$ g U/g [H$_2$O]{}), this water would contain 1480 $^{222}$Rn atoms. The water goes through a vacuum-degassing chamber and is returned to the main H$_2$O circulation system by another diaphragm pump.
The volume of water is measured with either a rotameter flowmeter or a stroke counter attached to the diaphragm pump. The volume per stroke is 0.34 liters, and is independent of flow rate over the range of flows normally used. Both of these were calibrated by flowing water from the pump to a container on a scale. One of the larger sources of systematic uncertainty is the difficulty of accurately measuring the water flow rate from the pulsating diaphragm pump. The estimated uncertainty is 14% in the flowmeter readings and 10% in the stroke-counted readings.
The light water monitor degasser
--------------------------------
![Schematic diagram of the monitor degasser in the H$_2$O system.[]{data-label="fig:degasser"}](degasser.eps){width="0.8\hsize"}
The monitor degasser (MDG) [@bib:liu_thesis; @bib:zhu_thesis; @bib:noel_thesis] is shown in Fig. \[fig:degasser\]. It is a custom-designed stainless steel vertical cylinder 1.15 m high and 0.4 m in diameter [@bib:allweld]. The usual pressure difference between the feed water line and the degassing chamber is 45 PSI. The water enters at the top of the degasser at a temperature of 13[$\,^{\circ}$C]{}and passes through three stainless-steel full-cone spray nozzles [@bib:nozzles]. The nozzles spray upwards and produce water droplets with a diameter of $\sim$1 mm. As the drops fall, or run down the walls of the degasser chamber, the dissolved gases leave the water and are drawn off to the radon collector described in Sec. \[sec:rncollect\]. The spray and the water level are visible through two acrylic view ports mounted on flanges welded to the side of the vessel. The water level in the bottom of the degasser is maintained at a height of $\sim$0.3 m.
The degassing efficiency of the MDG was measured by bringing a small volume of water into equilibrium with radon in the air of the underground laboratory, injecting this radon-enriched water into the input of the MDG, and then extracting, concentrating, and counting the radon. The number of detected radon atoms was compared to the number expected based upon the radon solubility, the known radon activity in the air (measured by introducing a different sample of the same air into an evacuated Lucas cell and counting its activity), and the known efficiencies of collection and concentration. With H$_2$O at 13[$\,^{\circ}$C]{}and a flow rate of 19 liters/minute the degassing efficiency was measured to be $0.58\pm0.10$ [@bib:wrightson].
Assay system background {#sec:h2oback}
-----------------------
Although the materials in the assay system were selected for their low diffusion and emanation rates for radon [@bib:emanation], some radon can still enter the water being assayed through leaks, by diffusion through pipes, O-rings, etc., by emanation from contaminants in assay system components, and by emanation from surface dust, embedded dirt, etc.
The background of the degasser was measured by filling the MDG with water and flowing this water in “closed loop mode” (which sends the output water of the degasser back to its input) until it was completely degassed. Then several assays of the degassed water were conducted, each of 3-hour duration, at a flow rate of 20 liters/minute. In these experiments $19 \pm 4$ atoms of radon entered the system per hour of assay by the combination of leakage, diffusion, and emanation. For a typical 30-min assay this represents a background of 1.1% of the number of atoms that would be extracted and collected if the water were at the maximum allowable level. These measurements, however, were only of the MDG and the subsequent radon collection apparatus and did not include the assay system piping that leads to the detector. Furthermore, the background can change as a function of time. For instance, we find that vibrations may loosen the nuts on valves and flanges, for which we must compensate by periodic tightening. Thus, the background for any given assay may be higher than measured in these closed loop assays, where great care was taken to ensure that the system was tightly sealed. To take account of these additional sources of background, some of which may vary in time, we add onto the previously stated statistical error of $\pm4$ atoms/hr an additional systematic error, which we estimate to be $^{+8}_{-4}$ atoms/hr. The total assay system background is thus $19^{+9}_{-6}$ atoms/hr.
Assay of the D$_{\bm 2}$O {#sec:d2o}
=========================
The basic principles of the assay system for D$_2$O are very similar to those for H$_2$O, namely, degassing followed by radon collection, concentration, and counting. The D$_2$O system, however, must be an order of magnitude more sensitive than the H$_2$O system (see Table \[tab:goals\]), be able to function in 0.2% NaCl-D$_2$O brine, and, since the D$_2$O is so valuable, have minimal loss of D$_2$O vapor. For these reasons a polypropylene hydrophobic-membrane contact degasser was chosen, rather than a metal vacuum degasser. The system design and calibration is described in more detail in [@bib:darren].
Water flow
----------
The D$_2$O can be assayed from seven locations in the detector as well as at points within the water purification system itself. The most commonly sampled points are labeled in Fig. \[fig:sample\]. There are five assay positions within the D$_2$O on the acrylic vessel wall: at the bottom of the vessel (V901), 1/3 of the way up (not shown), 3/4 of the way up (V903), at the bottom of the neck (not shown), and 0.3 m below the surface of the water in the neck (V905). In addition, the water from the main purification circulation loop can be sampled either at the bottom of the neck (R967) or at the bottom of the vessel (V967). The pipes within the vessel are made from the same acrylic as the vessel itself; outside the vessel the pipes are polypropylene, as in the H$_2$O system. For the dedicated assay lines, a diaphragm pump identical to that used in the H$_2$O system draws water from the vessel to the D$_2$O degasser.
A typical assay samples water for 5 hours at 21 liters/min. At the maximum allowable level this water contains about 1260 atoms, which, taking into account the degassing efficiency and transfer efficiency, results in 480 atoms in the Lucas cell at the end of an assay. Since the single-$\alpha$ counting efficiency is 74% and 3 prompt alphas are emitted per radon decay (see Sec. \[sec:LCeff\]), this gives about 180 events in the first day of counting. This should be compared with the typical Lucas cell background rate of about 10 counts/d, and the background from the assay system, which contributes about 20 counts in the first day of counting. Defining the sensitivity as when the signal equals approximately three times the uncertainty of the background [@Currie], the sensitivity of the entire D$_2$O assay system in the current configuration is about one-tenth of the maximum allowable level, or about $3 \times 10^{-15}$ g U/g D$_2$O.
There is no flow meter in the D$_2$O radon assay system. Rather, the flow rate is set by adjusting the pump stroke rate to a fixed value. Since the D$_2$O system uses the same type of diaphragm pump as the H$_2$O system, we use its calibration, whose uncertainty is 10%. There may, however, be small differences in any two pumps believed to be identical, and thus we add in quadrature an assumed 5% uncertainty, giving a total flow rate uncertainty in stroke-counted experiments of 11%. Some assays were conducted before the installation of the stroke counter; for these we estimate a 17% uncertainty in the flow rate.
The heavy water monitor degasser
--------------------------------
The MDG in the D$_2$O system is a membrane contact degasser [@bib:celgard]. It consists of bundles of hollow, porous, hydrophobic polypropylene fibers woven around a hollow polypropylene water distribution tube, all of which is contained in a polypropylene housing. The distribution tube is plugged at the center, and a baffle in the containment cartridge forces the water to flow to the outside of the cartridge and pass over the tightly packed fibers. As the water flows over the fibers, the dissolved gases pass through the fiber walls to their hollow center from which a vacuum pump draws the gases into a radon collection system similar to that in the H$_2$O system. The water, on the other side of the baffle, goes back to the central water tube, exits the degasser, and is returned to the main D$_2$O purification system.
The degassing efficiency of the [D$_2$O]{}MDG for radon has not been measured directly, but we can infer its efficiency from other experiments. By measuring the radon concentration of the water that enters and exits the [D$_2$O]{}process degasser, its efficiency was found to be $83 \pm 5\%$ at a flow rate of 195 liters/min. The process degasser contains two parallel sets of three membrane degassers in series where each degasser is of identical construction to the one in the MDG, so the inferred efficiency of a single degasser cell for radon is $45 \pm 3\%$ at 97.5 liters/min. This flow rate is much higher than the rate through the MDG which is customarily 21 liters/min. To extrapolate to lower flow rate, we can use an approximate membrane degasser model [@bib:sengupta] which predicts that the efficiency $\varepsilon_{\text{degas}}$ varies with flow rate $F$ as $\ln(1 - \varepsilon_{\text{degas}}) \propto F^{-\chi}$ where $\chi$ depends on degasser module geometry but is independent of gas species. The value of $\chi$ can be found for our degasser by applying this equation to measurements of the oxygen degassing efficiency of the process degasser at different flow rates. A probe [@bib:honeywell] with ppb sensitivity in high resistivity liquids such as ultra-pure D$_2$O was used, and the measured oxygen degassing efficiency of one cell of the process degasser was $84 \pm
3\%$ at 21 liters/min and $67.9 \pm 2.4\%$ at 97.5 liters/min. From these measurements we infer $\chi = 0.31$, and, using the measured radon efficiency at 97.5 liters/min, the predicted degassing efficiency of the MDG for radon is 62% at 21 liters/min. Since several assumptions are involved in this model which may not be completely satisfied, such as independence of the degassing efficiency on the gas concentration, we assign a liberal systematic uncertainty to this estimate. The upper limit is set by noting that the efficiency for radon can be no more than that for oxygen at the same flow rate, i.e., it must be less than 84%. A firm lower bound for the radon degassing efficiency at 21 liters/min is at 45% as that was the measured efficiency at 97.5 liters/min. We consider these extreme bounds to be effective two sigma uncertainties. Our estimate for the radon degassing efficiency of the [D$_2$O]{}MDG is thus $62^{+11}_{-\phantom{1}9}\%$.
Assay system background {#sec:d2oback}
-----------------------
The emanation and leak background of the MDG and radon collector were measured under static conditions [@bib:darren] as follows: The MDG and collector were sealed and their helium leak rate was measured to be less than $10^{-8}$ cm$^3$/s. Then the system was evacuated and isolated for as long as 11 days, at the end of which the gas in each evacuated component was individually collected and counted. The most significant background was $16 \pm 1.5$ Rn atoms/hr and came from the trap used to remove water vapor from the gas stream (see Fig. \[fig:rnboard\]). The only other appreciable background was from the degasser portion of the system which contributed $1.7 \pm
0.2$ atoms/hr. Adding these two components gives a total assay system background of $18 \pm 1.5$ atoms/hr, which, for a typical assay time of 5 hours, gives a background of $90
\pm 8$ atoms on the collection trap at the end of the assay. This is much less than the $\sim$780 atoms that would be collected on the trap from the water if it were at the maximum allowable level.
The concerns regarding possible changes in background with time that were expressed for the H$_2$O system also pertain here. To account for such time-dependent changes in the background, we add, as for the [H$_2$O]{}system, a systematic uncertainty of $^{+8}_{-4}$ atoms/hr, which makes the total assay system background $18^{+8}_{-4}$ atoms/hr.
The radon collector {#sec:rncollect}
===================
{width="\hsize"}
We describe here how the radon that is extracted from the water by the monitor degasser is collected, separated from other gases, and concentrated. The equipment used for this purpose is nearly the same in both the H$_2$O and D$_2$O systems and is illustrated in Fig. \[fig:rnboard\]. It is based on apparatus developed to measure radon emanation for materials selection during the [[SNO]{}]{}design phase [@bib:emanation].
In overview, the gases extracted from the water are first drawn by a vacuum pump through a cold trap which removes water vapor. The gas stream then flows through a liquid nitrogen cooled trap filled with bronze wool which stops radon, but allows N$_2$ and O$_2$ to pass through. At the end of the sampling period the captured radon is transferred to a concentrator trap and then to a specially-developed low-background Lucas cell. These various components and their use in an assay will now be described.
Collector components and use
----------------------------
The gas stream first enters a water vapor trap [@bib:fts]. In the H$_2$O system, where the vapor load is about 10 cm$^3$/min, the trap is an acrylic cylinder with a volume of 42 liters inside of which is a refrigeration coil. The coil is held at -60[$\,^{\circ}$C]{}during an extraction. The D$_2$O system has a similar device but it is smaller as the vapor load is less by about a factor of 10.
After passing through the water trap, the dry gas enters the primary radon trap which consists of a 10 mm diameter stainless steel tube stuffed with cleaned bronze wool and bent into a ‘U’ shape. This tube, whose volume is $\sim$50 cm$^3$, is immersed in liquid nitrogen during the extraction. N$_2$, O$_2$, and Ar in the gas stream go through this trap, but radon, CO$_2$ and any residual water vapor adhere to the cold bronze wool. When a sufficient sampling time has elapsed, the valve at the inlet to the radon collector is closed. Evacuation of the trap is continued for a few minutes to remove traces of nitrogen and oxygen. Next, the valve to the vacuum pump is closed and the valve between the primary trap and the (previously evacuated) concentrator trap is opened. The concentrator trap is a coiled tube of 3 mm stainless steel tubing with 300 mm length that is also stuffed with bronze wool. The primary trap is heated with a heat gun and the concentrator trap is chilled with liquid nitrogen so the gas is cryopumped to the concentrator trap.
The cryopumping continues for about 15 min during which time a Lucas cell is attached to the system at an adjacent quick-connect port and evacuated. The concentrator trap is then isolated and heated and a valve between the Lucas cell and the concentrator trap is opened. Since the volume of the Lucas cell is larger than that of the concentrator trap and connecting tubing, most of the radon transfers to the Lucas cell by volume sharing. The pressure in the cell at the time of filling is read with a transducer and is typically $<$0.1 atm for [H$_2$O]{}assays or $<$0.5 atm for [D$_2$O]{}assays. This gas is mainly CO$_2$ which is not well separated from radon as they have similar boiling temperatures. Even at 0.5 atm, however, the $\alpha$-particle range is nearly 6 cm, considerably greater than the 1.9 cm radius of the Lucas cell. Finally, the cell is disconnected and taken to a counting facility on the surface.
Except for the water trap, all components of the radon collector are off-the-shelf stainless steel parts connected with compression fittings. Considerable care was taken in the section from the concentrator to the Lucas cell to minimize the volume by using small diameter tubing, filling all unused volumes with inert material, and selecting low dead-volume valves.
The efficiency of the primary trap for stopping radon was measured under static conditions by concentrating a large amount of radon from the air into a Lucas cell, counting the radon in the cell, injecting the radon from that cell into the input of the primary trap, and extracting the radon back into the same cell. After accounting for radon decay, radon absorption into the acrylic of the Lucas cell, and the transfer efficiency from the concentrator trap to the Lucas cell, the trapping efficiency was found to be $100.5 \pm 2.3\%$.
The efficiency of transfer from the concentrator trap to the Lucas cell was measured in the H$_2$O system by filling the concentrator trap with air at atmospheric pressure. An evacuated Lucas cell was then attached and the change in pressure gave an efficiency of $64 \pm
2\%$. For the D$_2$O system this technique yielded an efficiency of $63.8 \pm 2.0\%$.
An alternate efficiency measurement was made for the D$_2$O system using radon. A Lucas cell containing a measured amount of radon was attached to the usual Lucas cell port and connected to the concentrator trap. The concentrator trap was chilled with liquid nitrogen and the radon from the cell was drawn by a vacuum pump to the concentrator trap. After pumping all radon from the cell, the trap was heated and the radon was re-injected into the same Lucas cell by the usual volume-sharing technique. After accounting for radon decay, the transfer efficiency from the concentrator trap to the Lucas cell was measured to be $61.8 \pm 1.0\%$. This more precise value is used to infer the radon concentration in the D$_2$O.
Radon collector background
--------------------------
The background of the radon collector water trap is included in the total assay system background and was considered above in Sec. \[sec:h2oback\] and Sec. \[sec:d2oback\]. The background of the other components of the radon collector in the D$_2$O system was measured [@bib:darren] by helium leak testing to $10^{-8}$ cm$^3$/sec, evacuation of the entire collector, and then closing all valves so as to isolate the separate parts. After a seal time of a few days, the individual regions were each extracted independently into Lucas cells. Weighting each section of the collector by the time it is used in the processing of the assay gives a total background that is several times less than the assay system background. We therefore consider the background of the collector (exclusive of the water trap) to be negligible.
Data acquisition system {#sec:counting}
=======================
The Lucas cell from the assay is placed on the end of a 50-mm diameter 10- or 12-stage photomultiplier tube and counted for 8 to 10 days. The tube output is amplified, digitized, and stored in 256 channels of a 4096-channel analyzer. Every 3 hours the data stored in the analyzer are transferred to a computer and a “log” file is updated with the cumulative counting time and the cumulative number of events within a chosen region of interest. Another file is also written every three hours which contains the full cumulative energy spectrum. If desired, these spectra can be reanalyzed with the region of interest redefined and a new “log” file generated. The decay counting system contains 11 counting stations, with one or two Lucas cells assigned to each station. Since the system is in a laboratory on the surface, there is a delay of two or more hours between the end of an assay and the start of counting.
Lucas cells
-----------
The Lucas cells developed for [[SNO]{}]{}are acrylic cylinders with a hollow interior machined into a 19-mm radius hemisphere whose surface is painted with activated ZnS. One end of the cylinder has a low-volume quick-connector [@bib:swagelok] through which radon gas is admitted and the other end is sealed with a flat sheet of acrylic. When $\alpha$ particles from the decay of $^{222}$Rn or one of its $\alpha$-emitting decay products strike the ZnS coating on the acrylic surface, light is emitted. The flat acrylic end of the cell is placed atop a photomultipler tube which detects this scintillation light. The cell diameter is 5 cm and the interior gas volume is 15.5 cm$^3$. These low background devices are described in detail in [@bib:emanation].
New cells are leak checked with a He mass spectrometer and the cells in use are periodically checked by evacuating them underground, keeping them for several hours in this high radon environment, and bringing them to the surface for counting. A cell filled with underground air contains about 900 Rn atoms, compared to $\sim$480 from D$_2$O at the maximum allowable level.
One cause of leakage is improper removal of the Lucas cell from the assay system at its quick-connector at the end of an extraction. This problem appears to have occurred once in the assays of the D$_2$O. Leakage can also occur because of inadequate lubrication of, or dirt on, the connector O-rings. This problem was greatly reduced by instituting a regular program of disassembly, cleaning, and regreasing. This is essential because even small leaks can introduce a number of radon atoms comparable with the $\sim$150 atoms that are presently collected in a typical [D$_2$O]{}assay.
### Lucas cell efficiency {#sec:LCeff}
The section of the U chain that begins with is $\xrightarrow[\text{3.82 d}]{\text{5.5 MeV }\alpha}$ $\xrightarrow[\text{3.10 m}]{\text{6.0 MeV }\alpha}$ $\xrightarrow[\text{26.8 m}]{\beta}$ $\xrightarrow[\text{19.9 m}]{\beta}$ $\xrightarrow[\text{162 $\mu$s}]{\text{7.7 MeV }\alpha}$ , which has a 22-year half life. Shortly after the decay of a atom two additional $\alpha$ particles are emitted, from and . The effective efficiency of the Lucas cell for decay is thus three times its efficiency for single-$\alpha$ detection, provided counting begins a few hours or more after the cell is filled.
The efficiency of the [[SNO]{}]{}Lucas cells was measured by injecting air containing a known amount of radon into two cells at a commercial radon calibration company [@bib:pylon]. The company measured the radon concentration of the air put into the cells to be $1649 \pm 66
\text{ (stat)} \pm 12 \text{ (syst)}$ Bq/m$^3$, equivalent to about 12 000 Rn atoms in each cell. The two cells were counted at [[SNO]{}]{}about one day after the radon injection. After accounting for decay between the time of injection and the start of counting, the cells were found to have single-$\alpha$ detection efficiencies of 75% and 74% with a 0.6% statistical uncertainty and a 3.4% systematic uncertainty.
Soon after the calibration of these two cells, nine Lucas cells were taken underground, where the typical radon concentration is 100 Bq/m$^3$, evacuated, and filled with ambient air. All cells were found to have similar efficiency but with a statistics dominated 12% uncertainty in each measurement. Assuming all cells have the same efficiency, the standard deviation of efficiency from the combination of these measurements is $\pm$7%. Combining this with the known difference in efficiency from cell to cell due to their volume difference, which is $<$3%, we conclude that the single-$\alpha$ detection efficiency of cells that were not directly calibrated is $74
\pm 7\%$. The 26% loss of $\alpha$-particles is mainly due to the cell geometrical efficiency.
### Lucas cell background
When a Lucas cell is new, its background from cosmic radiation and radioactivity in the ZnS is less than 3 counts per day, but the background gradually builds up with use. This increase in background is mainly due to the 5.3-MeV $\alpha$ from 138-d which comes from the gradual accumulation of 22-year . With each use of a cell its content builds up and the following decays produce an ever increasing cell background. Each $10^4$ Rn decays increase the cell background rate by roughly one count per day. After a cell has been used extensively, it will eventually have such a high background rate that it must be retired and replaced with a new one.
The background of each Lucas cell is periodically measured by evacuating it and counting in the standard manner. The background of the cells currently in use in [D$_2$O]{}assays is about 10 counts/d and in the range of (10–20) counts/d in [H$_2$O]{}assays. These are known with a 10% statistical uncertainty.
Electronics
-----------
Spurious signals in the photomultiplier dynode chain can be a source of noise unrelated to alpha decay within a Lucas cell. The region of interest for $\alpha$ decays is set to exclude such events, which mostly occur at low energy. Nevertheless, there is some leakage into the region of interest. By counting without a Lucas cell on the photomultiplier, this noise rate was measured to be less than 0.5 counts/d. The lower limit of the region of interest will cut out some true $\alpha$ events. Measurements with a Lucas cell spiked with (which we call a “hot” cell) show that the fraction of events lost depends on the electronics in the counting station and is in the range of (0–5)%. This loss is accounted for in the counting efficiency uncertainty.
Cross-talk from adjacent counting stations is also a potential source of noise. This effect was measured by putting a “hot” cell on one station and measuring the number of counts that appeared in all other stations. Even with $\sim 10^7$ counts on the “hot” station, the average number of counts on other stations did not exceed 0.5 counts. Cross-talk is thus a negligible contributor to the systematic uncertainty.
The long term photomultiplier gain drift was measured by comparing “hot” cell counting rates over a 3-year period. During this time the rate in the station used most often for D$_2$O assays varied by $3.1 \pm 0.9\%$ and the rate in the stations used for H$_2$O assays varied by no more than $3.5 \pm 2.0\%$.
Data processing {#sec:dataproc}
===============
In this section we derive the relationship between the cumulative number of detected counts and the concentration $C$ of radon in the water that enters the degasser. We assume that $C$ is constant.
If we flow water at a constant rate of $F$ liters/min for a time interval of duration $t_{\text{assay}}$ and extract with a degassing efficiency $\epsilon_{\text{degas}}$, then the number of radon atoms from the water that are present on the first trap in the collection system (the primary radon trap) at the end of extraction is $$\label{eqn:atomsfromwater}
N_{\text{water}} = \epsilon_{\text{degas}}\epsilon_{\text{trap}}
C F (1 - e^{-\lambda t_{\text{assay}}})/\lambda,$$ where $\epsilon_{\text{trap}}$ is the radon trapping efficiency and $\lambda$ is the decay constant. The term in parentheses is due to the decay of radon during the extraction time; for short assay times this term is approximately $\lambda
t_{\text{assay}}$. To obtain the total number of radon atoms present on the trap we must add to this the background of the radon assay system. Defining $R_{\text{back}}$ as the rate of radon production by emanation, diffusion, and ingress in the degasser and the front section of the collection system, the number of radon atoms from these backgrounds on the trap at the end of extraction is $$\label{eqn:atomsfrombackground}
N_{\text{back}} = \epsilon_{\text{trap}} R_{\text{back}}
(1 - e^{-\lambda t_{\text{assay}}})/\lambda.$$
These atoms are transferred to the Lucas cell with efficiency $\epsilon_{\text{transfer}}$. As discussed in Sec. \[sec:rncollect\], the radon background of the section of the collection system used during this transfer is negligible. If the time delay between the end of extraction and the start of counting is $t_{\text{delay}}$, the number of radon atoms present in the Lucas cell at the start of counting (SOC) is $$\label{eqn:atomsincell}
N_{\text{SOC}} = \epsilon_{\text{transfer}} e^{-\lambda t_{\text{delay}}}
(N_{\text{water}} + N_{\text{back}}).$$
The counting data consist of the superposition of two components:
- the decay of . Since the initial number of atoms is $N_{\text{SOC}}$ and the number exponentially decreases in time, the count rate of this component varies with time $t$ according to $\epsilon_{\text{count}}\lambda N_{\text{SOC}} \exp(-\lambda t)$, where $\epsilon_{\text{count}}$ is the effective counting efficiency.
- a constant background rate $B_{\text{Lucas}}$. These events are mainly the decay of 22-year which has accumulated in the cell from previous assays. The value of $B_{\text{Lucas}}$ is assumed known from previous counting of this cell for background.
We add these two components to get the total count rate as a function of time $$\label{eqn:countrate}
\epsilon_{\text{count}} \lambda N_{\text{SOC}} e^{-\lambda t}
+ B_{\text{Lucas}}$$ and integrate for a time interval of duration $t_{\text{count}}$. The total number of observed counts in this time is then $$\label{eqn:totalcounts}
N_{\text{count}} = \epsilon_{\text{count}} N_{\text{SOC}}
(1 - e^{-\lambda t_{\text{count}}})
+ B_{\text{Lucas}}t_{\text{count}}.$$
Combining Eq. (\[eqn:atomsfromwater\]), (\[eqn:atomsfrombackground\]), (\[eqn:atomsincell\]), and (\[eqn:totalcounts\]), the concentration of radon in the water is given by $$\begin{aligned}
\label{eqn:conc}
C = && \frac{ 1 } { \epsilon_{\text{degas}} F } \\ \nonumber
&& \times
[
\frac{ (N_{\text{count}} - B_{\text{Lucas}} t_{\text{count}}) \lambda }
{ \epsilon (1 - e^{-\lambda t_{\text{assay}}})
e^{-\lambda t_{\text{delay}}}
(1 - e^{-\lambda t_{\text{count}}}) }
-R_{\text{back}}
],
\end{aligned}$$ where we have abbreviated $\epsilon = \epsilon_{\text{trap}} \epsilon_{\text{transfer}}
\epsilon_{\text{count}}$.
Typical values of the parameters in this equation are $\epsilon_{\text{degas}} = 0.6$, $F = 20$ liters/min, $\epsilon_{\text{trap}} = 1$, $\epsilon_{\text{transfer}} = 0.62$, $\epsilon_{\text{count}} = 3 \times 0.74$ (there are 3 prompt $\alpha$ particles emitted after each radon decay, each of which is counted with $\sim$74% efficiency), $t_{\text{assay}} = 5$ hours in the [D$_2$O]{}system and 30 min in the [H$_2$O]{}system, $t_{\text{delay}} = 2$ hours, $N_{\text{count}} = 400$ from [D$_2$O]{}and 740 from [H$_2$O]{}, $t_{\text{count}} = 8$ d, $B_{\text{Lucas}} = 20$ counts/d in the [H$_2$O]{}system and 10 counts/d in the [D$_2$O]{}system, and $R_{\text{back}} =
460$ Rn atoms/d in the [H$_2$O]{}system and 430 Rn atoms/d in the [D$_2$O]{}system. To convert $C$ from radon atoms/liter to g U/g water, multiply the right side of Eq. (\[eqn:conc\]) by the factor $1.69
\times 10^{-13}/D$ where $D$ is the water density in g/cm$^3$, 1.0 for [H$_2$O]{}and 1.1 for [D$_2$O]{}. This assumes equilibrium in the U decay chain.
To check that there is reasonable agreement between the results of this procedure and the data, the total count rate predicted from Eq. (\[eqn:countrate\]) is calculated and visually compared to the differential spectrum of number of counts versus counting time. This comparison is shown in Fig. \[fig:dataspectrum\] for an extraction from [D$_2$O]{}whose activity is slightly higher than average.
![Count rate spectrum for radon extracted from [D$_2$O]{}. The background rate for this cell was $10.0 \pm 0.5$ counts/d. Data points are indicated by horizontal lines of 1-d duration, with approximate error limits from counting statistics. Since the count rate is very low, the data from every eight 3-hour data collection intervals have been combined. The thick solid line is the count rate predicted from [Eq. (\[eqn:countrate\])]{} with the encompassing dashed band indicating the 68% confidence range from counting statistics. The 15-d data acquisition time for this spectrum is longer than customary.[]{data-label="fig:dataspectrum"}](02092416.eps){width="\hsize"}
Overall systematic uncertainties in the assays
==============================================
Most of the various terms that enter the systematic uncertainty have been given in the previous text and are summarized in Table \[tab:sys\]. This table also includes the contribution of the collector, degasser, and cell backgrounds, which will now be considered.
[@ l @ d d @]{} Source & &\
Flow rate $F$ & 14 & 17\
Flow rate $F$ (stroke counted) & 10 & 11\
Degassing efficiency $\epsilon_{\text{degas}}$ & 17 & \^[+18]{}\_[-14]{}\
Assay system background $R_{\text{back}}$ & 3 & \^[+10]{}\_[-21]{}\
Trapping efficiency $\epsilon_{\text{trap}}$ & 2 & 2\
Transport efficiency to Lucas cell $\epsilon_{\text{transfer}}$ & 3 & 2\
Cell counting efficiency $\epsilon_{\text{count}}$ & 10 & 10\
Cell background rate $B_{\text{Lucas}}$ & 7 & 2\
Electronic noise & <0.1 & <0.1\
Photomultiplier stability & 4 & 4\
Combined quadratically & 26 & \^[+29]{}\_[-32]{}\
Combined (stroke counted) & 24 & \^[+26]{}\_[-30]{}\
The systematic uncertainty from backgrounds depends upon the magnitude of the observed radon signal, which has decreased as [[SNO]{}]{}data was acquired. On average, during the pure [D$_2$O]{}phase of the experiment, the radon levels in the [H$_2$O]{}and [D$_2$O]{}were about 1/4 of the maximum allowable values. For the inner [H$_2$O]{}we thus collected roughly 100 atoms from the water. The background due to the [H$_2$O]{}MDG and the radon collector for a 30-min assay would result in about $6 \pm 3$ atoms in the Lucas cell. Thus, the uncertainty in the system background contributes a systematic error of $\sim$3%. The $^{210}$Pb background in the Lucas cells is about $20 \pm 2 $ counts/d. Since in the first four days of counting we expect $\sim$110 counts from the water sample, the uncertainty due to cell background is 8/110, or 7%.
For the [D$_2$O]{}, at 1/4 of the maximum allowable level, about 120 Rn atoms enter the Lucas cell from the water. The background from the assay system in a 5-hour assay (the usual assay time) is $56
^{+25}_{-12}$ atoms of radon in the cell. Thus, the systematic error due to uncertainty in the assay system background is estimated as $^{+10}_{-21}\%$. The $^{210}$Pb background in the Lucas cell contributes about $8.0 \pm 0.8$ counts/d background. For the first four days of counting we expect 135 counts from the radon in the [D$_2$O]{}and thus the systematic error due to uncertainty in cell background is 3/135, or 2%.
Assay results
=============
{width="0.8\hsize"}
{width="0.8\hsize"}
Weekly measurements are usually made from the three valves in [H$_2$O]{}shown in Fig. \[fig:sample\], from the [H$_2$O]{}process degasser, and from one or more of the valves in the [D$_2$O]{}. The results of water assays from 29 September 1999 through 15 May 2002 are shown in Figures \[fig:h2oassay\] and \[fig:d2oassay\]. Most of the scatter in this data is not statistical fluctuation, but is due to real changes of the radon level in the detector. For example, there was a high radon level in both the [H$_2$O]{}and [D$_2$O]{}during the early period of neutrino-data taking. This was because the initially high radon content of the water used to fill the detector was still decaying. After that time there are several intervals during which spikes of radon were unintentionally introduced. These are clearly apparent in the H$_2$O plots and were caused by the failure of the H$_2$O process degasser’s vacuum pump, which resulted in laboratory air being injected into the H$_2$O. An improved interlock system has prevented similar faults during subsequent pump failures. In the first month of neutrino data-taking, the radon levels in the D$_2$O were still elevated until the nitrogen “cover gas” protection system was improved to compensate for leaks in the seal of the D$_2$O vapor space. After that time, there were two unintentional introductions of radon into the D$_2$O, but their amplitude was not sufficient to appreciably interrupt neutrino data-taking.
As is evident by examination of the assay plots, the average radon levels in both the H$_2$O and D$_2$O are well below the maximum allowable level. The interpretation of these assay results in terms of the net U-chain contamination and the way in which this contamination influences the neutrino detection rates reported in [@bib:SNOP1; @bib:ccnc; @bib:dn] will be presented elsewhere.
Summary and Conclusions
=======================
The method that [[SNO]{}]{}has developed to determine the radon content of the water in the detector by direct assay has been described. This method is relatively simple to execute, is reliable, and has been shown to be capable of detecting a few tens of atoms of radon per tonne of water, equivalent, assuming equilibrium in the U-chain, to a concentration of a few times $10^{-15}$ g U/g water. This sensitivity is adequate to show that the contribution to the neutral current background of the [[SNO]{}]{}detector is substantially below the maximum allowable level, and thus it has been possible for [[SNO]{}]{}to determine the total flux of solar neutrinos by measurement of the neutral current interaction rate.
Should the need arise, the sensitivity of this technique could be improved by an order of magnitude by reducing the background of the [D$_2$O]{}water trap, by constructing new Lucas cells, and by doubling the assay time.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank R. Rodriguez-Jimenez for his aid in calibration and assays. We thank G. Carnes, K. Dinelle, S. Fostner, B. McPhail, T. Spreitzer, and L. Wrightson for their contributions to this work during co-operative student placements with the [[SNO]{}]{}experiment. We appreciate helpful discussions with A. Sengupta and J. Munoz regarding membrane degasser efficiency. We are grateful to R. Lange for valuable discussions and thank the [[SNO]{}]{}operations staff for their great care in carrying out the assays described here. We are grateful to the INCO, Ltd. mining company and their staff at the Creighton mine without whose help this work could not have been conducted and we greatly thank Atomic Energy of Canada, Ltd. (AECL) for the loan of the heavy water in cooperation with Ontario Power Generation. This research was supported in Canada by the Natural Sciences and Engineering Research Council, the National Research Council, Industry Canada, the Northern Ontario Heritage Fund Corporation, and the Province of Ontario; in the USA by the Department of Energy; and in the United Kingdom by the Science and Engineering Research Council and the Particle Physics and Astronomy Research Council. Further support was provided by INCO, AECL, Agra-Monenco, Canatom, the Canadian Microelectronics Corporation, AT&T Microelectronics, Northern Telecom, and British Nuclear Fuels, Ltd.
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|
---
abstract: 'The simulation of QCD on dynamical ($N_f=2$) anisotropic lattices is described. A method for nonperturbative renormalisation of the parameters in the anisotropic gauge and quark actions is presented. The precision with which this tuning can be carried out is determined in dynamical simulations on $8^3 \times 48$ and $8^3 \times 80$ lattices.'
author:
- Richie Morrin
- Alan Ó Cais
- Mike Peardon
- 'Sinéad M. Ryan'
- 'Jon-Ivar Skullerud'
bibliography:
- 'trinlat.bib'
title: Dynamical QCD simulations on anisotropic lattices
---
Introduction {#sec:intro}
============
The advantages of simulations with anisotropic lattices are well understood and the method has been used for precision determinations of an extensive range of quantities in the quenched approximation to QCD [@Karsch:1982ve; @Manke:1998qc; @Morningstar:1999rf; @Asakawa:2003re; @Hashimoto:2003fs; @Ishii:2005vc]. In general a 3+1 anisotropy is employed where the lattice spacing in the temporal direction, $a_t$, is made fine whilst keeping the spatial lattice spacing $a_s$ relatively coarse. The advantages of this approach are two-fold. The improved resolution in the temporal direction means that states whose signal to noise ratio falls rapidly can be more reliably determined. The high computational cost of this improvement is offset by savings in the coarse spatial directions.
The isotropic lattice (whose spacing in the four space-time directions is $a_x =
a_y = a_z = a_t \equiv a$) regulates QCD in a way that breaks the continuous Euclidean symmetry down to the finite group of rotations of the hypercube. Luckily the relevant operators that transform trivially under these two groups are the same and so there is no renormalisation of the speed of light on the isotropic lattice. Once an explicitly anisotropic lattice action is introduced with $a_x = a_y =
a_z \equiv a_s $ and $a_t \ne a_s$, the rotational symmetry of the theory is the cubic point group. For the gluons, there are now two distinct operators not related by rotations at dimension four: $\left\{\mbox{Tr }E^2, \mbox{Tr }B^2\right\}$; while for the quarks the set of dimension four operators $\left\{ \bar\psi \;{\slash\hspace{-2.5mm}D}\psi, \; m \bar\psi\psi \right\}$ grows to a set with three members: $\left\{ \bar\psi \gamma_i D_i \psi, \; \bar\psi \gamma_0 D_0 \psi, \; m \bar\psi\psi \right\}$. As a result, two new parameters appear in the action, and for the continuum limit to represent QCD these parameters must be determined such that a physical probe of the vacuum at scales well below the cut-off appears to have full Euclidean symmetry. The nonperturbative determination of these extra action parameters is the subject of the present paper.
In quenched QCD the anisotropy in the gauge sector, $\xi_g$, and the quark sector, $\xi_q$, can be tuned independently and [*post hoc*]{} using two separate criteria. The precision and mass-dependence of the determination of $\xi_q$ was investigated for the action we use in Ref. [@Foley:2004jf]. It was found that this parameter could be determined at the percent level from the energy-momentum dispersion relation. The mass dependence was found to be mild for quark masses in the range $m_s\leq m_q\leq m_c$ when the tuning was carried out at the strange quark mass, $m_s$. In Refs. [@Klassen:1998ua; @Alford:2000an], a determination of the gluonic parameter was made to similar precision.
We would like to use anisotropic lattices in simulations with $N_f=2$ for realistic phenomenologically-relevant calculations. In dynamical QCD the tuning procedure becomes more complicated because of the interplay between the quark and gluon sectors and the parameters must be simultaneously determined. There are several issues to resolve. Firstly, can this simultaneous tuning be accomplished; secondly, to what precision is the renormalised anisotropy determined; and thirdly, what is the mass-dependence of the renormalised anisotropy. Here we will focus on the first two issues, and leave the question of the mass dependence to a later study.
The paper is organised as follows. Section \[sec:actions\] gives the details of the gauge and quark actions used in this investigation. Section \[sec:method\] describes the tuning methodology and is followed in Section \[sec:results\] by the results for the values of the tuned bare (input) parameters $\xi^0_g$ and $\xi^0_q$. Section \[sec:conclusions\] contains our conclusions and future plans.
The action and parameters {#sec:actions}
=========================
We begin with a brief description of the anisotropic action used in this study. The details of the tuning procedure described in Section \[sec:method\] do not depend on the specific action used. Further description of the action can be found in [@Foley:2004jf] where the tuning for the same action in the quenched approximation was discussed.
The gauge action is a two-plaquette Symanzik-improved action [@Morningstar:1999dh] previously developed for high-precision glueball studies and given by $$\begin{split}
S_{G} =& \frac{\beta}{\xi^0_g} \left\{ \frac{5(1+\omega)}{3u^4_s}\Omega_s -
\frac{5\omega}{3u^8_s}\Omega^{(2t)}_s - \frac{1}{12u^6_s}\Omega^{(R)}_s \right\} \\
& +\beta\xi^0_g \left\{ \frac{4}{3u^2_s u^2_t} \Omega_{t} - \frac{1}{12u^4_s u^2_t}
\Omega^{(R)}_t \right \} ,
\end{split}$$ where $\Omega_s$ and $\Omega_t$ are spatial and temporal plaquettes. $\Omega_s^R$ and $\Omega_t^R$ are $2 \times 1$ rectangles in the $(i,j)$ and $(i,t)$ planes respectively. $\Omega_s^{2t}$ is constructed from two spatial plaquettes separated by a single temporal link. $u_s$ and $u_t$ are the mean spatial and temporal gauge link values respectively. The action has leading discretisation errors of ${\cal O}(a^3_s,a_t, \alpha_s a_s)$.
For fermions an action specifically designed for large anisotropies is used. The usual Wilson term removes doublers in the temporal direction whereas spatial doublers are removed by the addition of a Hamber–Wu term. The action has been described in detail in Ref. [@Foley:2004jf] and has leading classical discretisation errors of ${\cal O}(a_tm_q)$. In terms of continuum operators, it can be written $$S = \bar{\psi}({\slash\hspace{-2.5mm}D}+ m_0)\psi -\frac{ra_t}{2}\bar{\psi}\left ( D_0^2\right )
\psi +sa^3_s\bar{\psi}\sum_iD_i^4\psi ,
$$ which highlights the different treatment of temporal and spatial directions. $r$ is the usual Wilson coefficient which is applied in the temporal direction only in this action and is set to unity. The analagous parameter in the spatial directions is $s$, which parameterises a term that is irrelevant in the continuum limit. A precise tuning of this parameter is not necessary: in practice we choose $s=1/8$, so that the energy of a propagating quark at tree level increases monotonically across the Brillouin zone. Stout-link smearing [@Morningstar:2003gk] was used for the gauge fields in the fermion matrix. Two stoutening iterations were used, with a parameter $\rho=0.22$. This was fixed for all simulations, and chosen to approximately maximise the expectation value of the spatial plaquette on the stout links.
This study was carried out on $8^3\times 48$ and $8^3\times80$ anisotropic lattices with a spatial lattice spacing $a_s\approx0.2$fm and a target anisotropy $\xi=6$. The bare sea quark mass was set to $a_tm_q=-0.057$ in all runs. A set of gauge configurations, distributed across ten independent Markov chains, was generated for each set of input parameters ($\xi^0_g$,$\xi^0_q$). Valence quark propagators were generated with the same mass as the sea quarks.
To determine the statistical uncertainties, 1000 bootstrapped sets of configurations were taken and analysis was done on these bootstrapped sets. Both point and all-to-all propagators were used. Some preliminary results using point propagators on $8^3\times48$ lattices were presented in Ref. [@Morrin:2005tc].
Methodology {#sec:method}
===========
The bare parameters, $\xi^0_g$ and $\xi^0_q$, are renormalised by demanding that physical probes exhibit euclidean symmetry. In principle, any physical quantity can be used; however, it should be easily determined to high precision. In this study we have used the sideways potential and the pion energy-momentum dispersion relation for the gauge and fermion sectors respectively.
The gauge anisotropy $\xi_g$ is determined from the interquark potential [@Klassen:1998ua; @Alford:2000an]. The static source propagation is chosen to be along a coarse direction allowing the sources to be separated along both course and fine axes. The potential is determined at the same physical distance for these two cases. The input anisotropy is constrained so that the two calculations yield the same value of the potential, $V_s(x) = V_t(t/\xi)$ for a target anisotropy $\xi$. For a given input anisotropy $\xi^0_g$ and target anisotropy $\xi$ we can determine the mismatch parameter $c_g=V_s(x)/V_t(t/\xi)$. If $x$ is in the régime where the potential is nearly linear, the mismatch parameter is approximately related to the actual gauge anisotropy, $c_g\approx\xi_g/\xi$.
The quark anisotropy can be determined from the pseudoscalar dispersion relation. The anisotropy is inversely proportional to the square root of the slope of the dispersion relation and demanding a relativistic energy-momentum relation imposes a renormalisation condition on the bare parameter $\xi^0_q$. The ground state energy $E_0$ was determined for a range of momenta, $n^2 \in\{0,1,2,3,4,5,6\}$, where $p_n = \frac{2 \pi n}{L a_s}$ and we average over equivalent momentum values. The two-point correlator data were modelled with single exponentials and a $\chi^2$-minimisation was used to determine the best-fit ground state. These values were used to generate an energy-momentum dispersion relation.
In the quenched approximation this procedure is relatively easy since $\xi^0_g$ and $\xi^0_q$ can be determined independently. For dynamical simulations it is no longer possible to simply fix $\xi^0_g$ and then tune $\xi^0_q$ to a consistent value, since changing $\xi^0_q$ will affect the measurement of $\xi_g$. Explicitly, changing the value of $\xi^0_q$ necessitates a regeneration of the background fields with the new value of $\xi^0_q$ which in turn will change the measured anisotropy $\xi_g$ of the background fields. The solution to this problem is a simultaneous two-dimensional tuning procedure [@Peardon:2002sd].
A linear dependence on the parameters $\xi^0_g$ and $\xi^0_g$ was assumed for a small region. Three initial sets of configurations were generated and the renormalised anisotropy was determined. Planes were defined for both output values of $\xi_g$ and $\xi_q$ i.e. values $\alpha,\beta,\gamma$ were found to satisfy $\xi_a = \alpha_a\xi^0_g + \beta_a\xi^0_q + \gamma_a$ for the renormalised anisotropy $\xi_a, a=g,q$ measured for each input $(\xi^0_q,\xi^0_g)$. The intersection of these planes with the required (target) output value yields the tuned point. The statistical uncertainties were determined using bootstrap resampling, with a common bootstrap ensemble used for all measurements. When more than three simulation points were available a plane was defined using a constrained-$\chi^2$ fit.
All observables were estimated using the Monte Carlo method. An ensemble of 250 gauge field configurations divided across 10 Markov chains was generated using the Hybrid Monte Carlo (HMC) algorithm [@Duane:1987de]. Approximately 5000 CPU hours were needed in order to generate each set of configurations. The HMC algorithm can be used for these simulations without modification. One observation serves to improve performance, however. HMC adds a set of momentum variables conjugate to the gauge fields, but each conjugate momentum can be added with a different gaussian variance without changing the validity of the method. In isotropic simulations this is not a useful property, and all momentum co-ordinates are chosen to have unit variance. For the anisotropic lattice, the temporal and spatial gauge fields have different interactions, and different momenta become useful. If the HMC hamiltonian is $${\cal H} = \sum_x \left( \frac{1}{2\mu_t^2} \mbox{Tr } P_0^2(x) +
\sum_{i=1}^3 \frac{1}{2} \mbox{Tr } P_i^2(x) \right)
+S[U] ,$$ an extra tunable parameter, $\mu_t$ (the variance of the temporal link momenta), has been added to the algorithm which can be used to optimise acceptance by the Metropolis test. This is equivalent to using two distinct integration step-sizes for the spatial and temporal degrees of freedom. Some brief numerical experiments suggest that a temporal leap-frog step-size smaller by a factor $\xi$ is close to optimal, and this is borne out by considerations of free field theory.
Results {#sec:results}
=======
Run 1 2 3 4 5
----------- ------ ------- ------- ------- -------
$\beta$ 1.51 1.528 1.514 1.544 1.522
$\xi_q^0$ 6.0 7.5 7.5 8.72 8.83
$\xi_g^0$ 8.0 7.0 8.0 6.65 7.44
: Input parameters for the five dynamical simulations performed in this tuning procedure. The bare quark mass is $a_tm_q=-0.057$ for all runs.[]{data-label="tab:inputs"}
The input anisotropy parameters used are given in Table \[tab:inputs\]. We started by choosing three points (Runs 1–3) in the $(\xi^0_g,\xi^0_q)$ plane, and generated configurations at two further points as a result of the tuning procedure. The final tuning was performed on $8^3\times80$ lattices, using data from runs 1, 4 and 5 as these spanned the largest area of the plane.
Interquark Potential {#sec:sideways}
--------------------
The gluon anisotropy is determined from the static quark potential at a selected distance $R$. In practice this is done by determining the effective energy for the static quark–antiquark configuration at separation $R$ at some time $T$. It is then important to choose values for $R$ and $T$ where the potential is well determined and the value obtained for $c_g$ is stable with respect to small variations in $R$ and $T$. The same values for $R$ and $T$ must then be used for all runs in order to have a consistent procedure.
Table \[tab:gluonRT\] shows $c_g$ for different $R$ and $T$, on the $8^3\times80$ lattices. We see that the values are generally quite consistent for each run. Looking more closely at the effective potential for each $R$ as a function of $T$, we find that it has not yet reached a plateau at $T=1$, while the value for $T=3$ is consistent within errors with that for $T=2$. We choose $(T,R)=(2,3)$ as our optimal parameters, since this yields reasonably small statistical errors, while $R$ is large enough to be in the linear régime.
----- ---------- ---------- ---------- ----------- ----------- ---------
Run (1,3) (1,4) (2,3) (2,4) (3,3) (3,4)
1 0.972(2) 0.959(3) 0.972(7) 0.965(13) 0.991(25) 1.13(8)
4 0.951(2) 0.941(4) 0.945(8) 0.926(18) 0.942(34) 0.89(9)
5 0.994(2) 0.990(3) 0.991(7) 0.998(13) 0.965(25) 1.01(7)
----- ---------- ---------- ---------- ----------- ----------- ---------
: The gluon anisotropy parameter $c_g$ for different separations, $R$ and times, $T$. The final results were determined from data at $T=2$ and $R=3$.[]{data-label="tab:gluonRT"}
Dispersion relations {#sec:dispersion}
--------------------
Pseudoscalar meson correlators were computed using traditional point propagators as well as all-to-all propagators [@Foley:2005ac] with time and colour dilution and no eigenvectors.
To determine optimal fit ranges for exponential fits to the correlator data, sliding window (${t_{\text{min}}}$) plots were used: the correlation function was fitted in a range from ${t_{\text{min}}}$ to ${t_{\text{max}}}$ where ${t_{\text{max}}}$ was fixed to the largest value compatible with a good fit, and ${t_{\text{min}}}$ was varied. An example of such a plot is given in Fig. \[fig:tmin\]. The fit range was chosen so the fit would be stable with respect to small variations in ${t_{\text{min}}}$. The same fit ranges and smearing parameters were chosen for all simulation points in order to obtain a consistent determination of the dispersion relation. The final fit ranges are given in Table \[tab:fitranges\].
![A typical ${t_{\text{min}}}$ plot, showing the energy for momentum $n^2=1$ on run 1, $8^3\times 80$ lattices from fits to time ranges ${t_{\text{max}}}=40$ for various ${t_{\text{min}}}$. A stable ground state energy determination, with a good $\chi^2$, is achieved for $22\leq{t_{\text{min}}}\leq30$.[]{data-label="fig:tmin"}](slide_B1p1_40.eps){width="\colw"}
$n^2$ ${t_{\text{min}}}$ ${t_{\text{max}}}$
------- -------------------- -------------------- --
0 25 40
1 24 40
2 21 40
3 19 40
: Fit ranges.[]{data-label="tab:fitranges"}
In our initial analysis data from a $8^3\times 48$ lattice were used. However, a reliable extraction of the ground state energy proved difficult. In particular, it was observed that the energy either did not reach a plateau until near the end of the lattice or did not plateau at all. To resolve this problem the simulation was repeated on a longer, $8^3\times 80$ lattice. An immediate improvement in the quality of the fits was observed. The ground state energy was determined from fits over at least 15 timeslices and was stable with respect to changes in ${t_{\text{min}}}$. The effect of the longer lattice is illustrated in Figure \[fig:longVSshort\]. This plot also compares simulations using point and all-to-all propagators. The all-to-all propagators lead to improved precision in the fitted energies. The central values are in agreement with the energies determined using point propagators but the statistical error is smaller.
![A comparison of the dispersion relations determined from an $8^3\times 48$ lattice and an $8^3\times 80$ lattice. The solid lines are the best fits and the dotted lines are the 68% confidence levels. The figure also shows a comparison of all-to-all propagators and point propagators on the same (longer) lattice. The plot shows that the ground state energies have not reached a plateau on the shorter lattice. On the longer lattice the all-to-all and point data agree, while higher precision is achieved with all-to-all propagators.[]{data-label="fig:longVSshort"}](short_vs_long.eps){width="\colw"}
The final tuned parameters were determined using all-to-all propagators on the $8^3\times 80$ lattices. We find consistently good fits for all runs for the first four momenta considered ($n^2 = 0,1,2$ and 3). The renormalised quark anisotropy is therefore determined from fits to these momenta. Figure \[fig:dispersion\] shows the pseudoscalar dispersion relations for Runs 1, 4 and 5 which are used to determine the tuned point.
![Dispersion relations from runs 1, 4 and 5 on $8^3\times80$ lattices using all-to-all propagators. The solid line is a fit to the four points and the dotted lines are the 68% confidence levels. The quality of all three fits is very good with $\chi^2/N_{d.f.} = 2.0/2, 1.9/2, 2.0/2$ for runs 1,4 and 5 respectively.[]{data-label="fig:dispersion"}](dispersion.eps){width="\colw"}
Plane fits {#sec:planefits}
----------
Table \[tab:out\] shows the output anisotropies determined on the $8^3\times48$ and $8^3\times80$ lattices for the five simulation points.
----- ---------- --------- ---------- ---------
Run $c_g$ $\xi_q$ $c_g$ $\xi_q$
1 0.991(3) 4.98(6) 0.972(7) 5.54(6)
2 0.986(3) 6.27(4)
3 1.001(3) 5.18(6)
4 0.985(5) 6.47(5) 0.945(8) 7.08(5)
5 0.995(3) 5.80(5) 0.991(7) 6.95(8)
----- ---------- --------- ---------- ---------
: Table of measured output anisotropies at each of the run points. The errors are statistical only.[]{data-label="tab:out"}
As a check on the stability of our tuning procedure, we have repeated the calculation using different values of $R$ and $T$ in the determination of the gluon anisotropy. The results are shown in Fig. \[fig:tuning-RT\]. The plot shows that the anisotropies are insensitive to a change in $R$ but that increasing the value of $T$ from two to three leads to large statistical uncertainty, particularly in the gluon anistropy. For these reasons we choose $R=3$ and $T=2$ for our analysis.
![Tuned values of input parameters $(\xi_g^0, \xi_q^0)$ determined from the plane fit procedure on the $8^3\times 80$ lattice. The plots show the results for different values of $R$ and $T$ used to determine the gluon anisotropy. Each point corresponds to one bootstrap sample.[]{data-label="fig:tuning-RT"}](scatterRT.eps){width="\colw"}
Simulation with tuned parameters
--------------------------------
Applying the plane fit procedure of Sec. \[sec:planefits\] to a subset of configurations of Runs 1, 4 and 5 we obtained preliminary, tuned parameters $\xi^0_g=8.06{\raisebox{-0.4ex}
{$\stackrel{\scriptstyle +7}{\scriptstyle -7}$}},
\xi^0_q=7.52{\raisebox{-0.4ex}
{$\stackrel{\scriptstyle +21}{\scriptstyle -15}$}}$. 250 configurations were generated with these parameters, and $c_g$ and $\xi_q$ determined using the same values for $R$, $T$ and fit ranges as in Sections \[sec:sideways\] and \[sec:dispersion\]. We find $c_g=0.983(6), \xi_q=6.21(9)$. We see that both quark and gluon anisotropies are within 3% of the target value of 6. Although the anisotropies are not equal within statistical errors, we note that there are still systematic uncertainties at the percent level, in particular for $\xi_g$, as shown in Table \[tab:gluonRT\]. For example, if we choose $R=3, T=3$ we find $c_g=1.01(2)$.
We repeated the plane fit procedure including the new information from Run 6. Figure \[fig:scatter\] shows the resulting scatterplot determined on the $8^3\times80$ lattice from runs 1, 4, 5 and 6. The intersection points shift in a direction to move $c_g$ and $\xi_q$ even closer to the target anisotropy.
![As in Fig \[fig:tuning-RT\]. The figure shows the results from a plane fit using parameters from runs 1, 4, 5 and 6 (marked with an $\times$). The big red (gray) cross at $(\xi^0_g,\xi^0_q)=(8.42,7.43)$ indicates the result of the best fit.[]{data-label="fig:scatter"}](scatter.eps){width="\colw"}
In order to get a rough idea of the physical scales of these lattices, we compute the pion mass, the rho mass and the string tension. We find $a_tm_\pi=0.066(1)$ and $a_tm_\rho=0.120(5)$, which gives $m_\pi/m_\rho=0.54$, while a crude measurement (shown in Fig. \[fig:potential\]) of the string tension gives $a_s=0.18$fm.
![The potential between fundamental static color sources for run 6, measured from static propagation in a coarse direction. Lines show fits to the Cornell potential, and are used in a crude determination of the lattice spacing. \[fig:potential\]](potential_B6.eps){width="\colw"}
A more precise determination of the lattice spacing will be obtained from the 1P–1S splitting in charmonium [@Juge:2005nr].
Conclusions {#sec:conclusions}
===========
We have performed a first simulation of 2-flavour QCD with improved Wilson fermions on anisotropic lattices, with both quark and gluon anisotropies tuned to $\xi=6$ [^1]. The tuning was based on a linear Ansatz for the dependence of renormalised anisotropies on bare anisotropy parameters in a region of parameter space. The results from the final run demonstrate that the tuning procedure, described in Sec. \[sec:method\], works satisfactorily.
The final, tuned point was found to lie marginally outside the triangle used for the plane fit procedure, so the end result was based on an extrapolation rather than an interpolation. This increases both the statistical and systematic uncertainties of the determination. To avoid this problem, it is important to choose a large enough triangle to start with, so that successive parameter determinations are always based on interpolations.
We also found that the original ($8^3\times48$) lattices used were too short in the time direction to allow a reliable determination of ground state energies, which were found to be systematically high, in particular for higher momenta. This led in turn to systematically high values for $\xi_q$. The adoption of lattices with longer time extent was a crucial step in the procedure. As Table \[tab:fitranges\] shows, the optimal fit ranges were generally found to be beyond the range of the shorter lattice.
We were able to determine the tuned parameters $(\xi^0_g,\xi^0_g)$ with a statistical uncertainty of 1% and 3% respectively from our ensembles of 250 configurations. In addition, there are three main sources of systematic uncertainties:
1. The $R$ and $T$ values used in the determination of the sideways potential, and the fit ranges used in the determination of the pseudoscalar dispersion relation. Since the fit ranges are chosen to give stable ground state energies, we can safely assume that the latter is a small effect. The effect of varying $R$ is also small, as shown in Fig. \[fig:tuning-RT\]. There may be a systematic error arising from the choice of $T$, but this is obscured by the larger statistical uncertainties in the $T=3$ data, particularly in the $\xi^0_g$ direction.
2. Lattice sizes. The pion dispersion relation is unlikely to be strongly affected by the finite lattice volume, but the static quark potential may contain finite volume errors which affect our results. We will be performing simulations at the tuned point on larger volumes, which will show whether this is a significant issue.
3. Nonlinearities in the dependence of $(\xi_g,\xi_q)$ on $(\xi^0_g,\xi^0_q)$. Our final fit to four points shows no evidence of any significant nonlinearity. If this were found to be a serious issue in any future simulation, a two-step procedure may be adopted where a smaller triangle centred on the preliminary tuned point is used in the second step.
We have yet to verify that we get the same quark anisotropy from other hadronic probes, for example the vector meson. Differences in the anisotropies can arise from lattice artefacts and can thus be considered part of the finite lattice spacing errors.
These lattices will in the future be employed for a wide range of physics investigations, including charm physics and heavy exotics [@Juge:2005nr], spectral functions at high temperature [@Morrin:2005zq], static–light mesons and baryons [@Foley:2005af], strong decays and flavour singlets including glueballs. These studies will be carried out on larger lattice volumes. Simulations on finer lattices will necessitate a new nonperturbative tuning process like the one performed here; this will be desirable in the longer term.
This work was supported by the IITAC project, funded by the Irish Higher Education Authority under PRTLI cycle 3 of the National Development Plan and funded by IRCSET award SC/03/393Y, SFI grants 04/BRG/P0266 and 04/BRG/P0275. We are grateful to the Trinity Centre for High-Performance Computing for their support and would like to thank Colin Morningstar for generous access to computing resources in the physics department of Carnegie Mellon University in the early stages of this work.
[^1]: While this paper was in preparation the results of a dynamical anisotropic simulation using staggered fermions appeared in [@Levkova:2006gn].
|
---
author:
- 'Ryuji <span style="font-variant:small-caps;">Higashinaka</span>$^{1}$, Hideto <span style="font-variant:small-caps;">Fukazawa</span>$^{1}$[^1], Kazuhiko <span style="font-variant:small-caps;">Deguchi</span>$^{1}$ and Yoshiteru <span style="font-variant:small-caps;">Maeno</span>$^{1,2}$'
title: 'Low Temperature Specific Heat of Dy$_2$Ti$_2$O$_7$ in the Kagome Ice State'
---
Introduction
============
Geometrically frustrated systems show various prominent properties, such as spin ice, quantum spin liquid, anomalous Hall effect, etc [@Harris97; @RamirezN; @Canals98; @Taguchi01]. Among compounds realizing such systems, pyrochlore oxides $A_{2}B_{2}$O$_7$ have been extensively studied [@Review; @Snyder03]. Among pyrochlore oxides, Ho$_{2}$Ti$_{2}$O$_7$ [@Harris97], Dy$_{2}$Ti$_{2}$O$_7$ [@RamirezN] and Ho$_{2}$Sn$_{2}$O$_7$ [@Matsuhira01] are known to exhibit spin ice behavior: Without long-range ordering of the rare-earth magnetic moments, these materials maintain macroscopically degenerate state down to low temperatures [@Review].\
![\[Pyrochlore\] (a) The pyrochlore lattice: Both $A$ and $B$ sites form the pyrochlore lattice in pyrochlore oxides $A_{2}B_{2}$O$_7$. (b) The stable spin configuration (1-in 3-out) on a single tetrahedron in high field along the \[111\] direction. Circles depict $A$-site ions with Ising spins. Zeeman interaction in the \[111\] field direction competes with the spin-spin interaction that stabilizes a 2-in 2-out configuration.](fig1.eps){width="0.8\linewidth"}
In pyrochlore compounds, both $A$-site and $B$-site ions constitute a corner-shared tetrahedral network (the pyrochlore lattice (Fig. 1(a))). Because of the crystal-field effect, some of the magnetic moments of the $A$-site ions such as Dy$^{3+}$ and Ho$^{3+}$ have Ising anisotropy along the local $\langle$111$\rangle$ direction, which points to the center of a tetrahedron from each vertex. Because of the effective nearest neighbor interaction being ferromagnetic owing to the dominant dipolar interaction, the ground state of a single tetrahedron is governed by the ‘ice rule’ in which two spins point inward and the other two point outward (2-in 2-out). Such a configuration on each tetrahedron leads to a macroscopic degeneracy in the pyrochlore lattice and gives rise to Paluing’s residual entropy of (1/2)$R$ln(3/2) [@Pauling45].\
Owing to the Ising anisotropy, the spin responses to magnetic fields are very anisotropic. For polycrystalline samples, it is reported that there are specific-heat peaks at field-independent temperatures of 0.34 K, 0.47 K and 1.12 K; it was speculated that these peaks may be due to ordering of spins with their Ising axes perpendicular to the field [@RamirezN]. Moreover, owing to the difference in the stable spin configurations in fields along different directions and associated difference in the frustration dimensionality, the process of releasing the residual zero-point entropy should be qualitatively different. Magnetization measurements using single crystalline Dy$_{2}$Ti$_{2}$O$_7$ revealed such anisotropic spin responses [@Fukazawa02; @Sakakibara03]. From that study we found that the state in the field along the \[111\] direction is qualitatively different from those along the other directions \[100\] and \[110\]. In particular the state in the \[111\] field direction exhibits a new value of the residual entropy, because the lattice on which spins are frustrated changes from three-dimensional (3D) pyrochlore lattice to two-dimensional (2D) Kagome lattice with the ice rule constraint. We call this state as the “Kagome ice state” [@Matsuhira02; @Wills02].\
In the \[111\] field direction, Zeeman interaction originating from the external magnetic field favors the 1-in 3-out spin configuration on each tetrahedron (Fig. 1(b)), whereas the ice rule originating from spin-spin interaction tends to stabilize the 2-in 2-out state. One of the spins on each tetrahedron is parallel to the magnetic field and the components parallel to magnetic field of the other three spins are equivalent. The angle $\theta$ between these three spins and the magnetic field gives cos$\theta$ = 1/3; the Zeeman interaction for these spins is one third of that for the field-parallel spin. Therefore, in certain field range the direction of parallel spin is uniquely decided by the Zeeman interaction, whereas the other three still remain frustrated by competition between the Zeeman interaction and the spin-spin interaction. Viewed from the \[111\] direction, pyrochlore lattice consists of stacking of a triangular lattice and a Kagome lattice. Spins parallel to the magnetic field form the triangular lattice and the other frustrated spins form the Kagome lattice. Because the spins on the Kagome lattice are frustrated with the ice-rule constraint, this state has a different value of residual entropy from that of the spin ice state. From specific heat measurements down to 350 mK, the residual entropy of Kagome ice state was estimated as 0.44 $\pm$ 0.12 J/mol-Dy K by our group [@Higashin03] and 0.65 J/mol-Dy K by Hiroi and Matsuhira $et$ $al$. [@Matsuhira02; @Hiroi03]. From the magnetization of the latter group, it was estimated as 0.5 $\pm$ 0.15 J/mol-Dy K [@Sakakibara03]. From a theoretical point of view, the Kagome ice state maps onto a hardcore dimer model on the hexagonal lattice. The residual entropy for this model can be calculated exactly and is 0.6718 J/mol-Dy K [@Moessner01; @Udagawa02].\
Ramirez $et$ $al$. reported the specific heat of Dy$_2$Ti$_2$O$_7$ down to 0.2 K [@RamirezN]. However, their samples were polycrystals and they could not extract the properties of the Kagome ice state in the field along the \[111\] direction. To the best of our knowledge, there is no detailed report of the specific heat measurement below 0.35 K in the Kagome ice state [@HigashinHFM2003]. In this paper, we report the specific heat of single-crystalline Dy$_2$Ti$_2$O$_7$ down to 100 mK in order to examine the detailed nature of the Kagome ice state at low temperatures. In addition to the specific-heat peak at 0.98 T also present at higher temperatures, we found a new broad peak around 1.25 T emerging below 0.3 K. We discuss the origin of this additional peak.\
Experimental
============
Single crystals of Dy$_2$Ti$_2$O$_7$ used in this work were grown by a floating zone method [@Fukazawa02]. We measured the specific heat between 0.1 and 3 K and in fields up to 2 T by a relaxation method using a self-made calorimeter with a dilution refrigerator (Oxford Instrument model Kelvinox25). In order to attain accurate field alignment along the \[111\] direction, we used a single-axis sample rotator [@NishiZaki00]. The sample size was approximately $2.0 \times 2.0 \times 0.06$ mm$^3$ and the mass was 1.6 mg. The \[111\] direction lies in the surface plane of the plate-like sample in order to reduce the demagnetization effect. In fact, the demagnetization factor is estimated to be as small as $N = 0.03$; we do not make corrections in the data presented below. We evaluated the specific heat of the addenda from the specific heat measurement of a single crystal of aluminum.\
![\[TdepC\] Temperature dependence of the specific heat of Dy$_2$Ti$_2$O$_7$ at various magnetic fields along the \[111\] direction down to 100 mK. At 0 T, there is only one peak indicating spin freezing. However, there are two peaks at 0.5 and 0.75 T. The higher one (peak 1) is originating from the spins parallel to the magnetic field and the lower one (peak 2) is originating from the frustrated spins as discussed in text. The higher peak above 0.9 T exists above 3 K.](fig2.eps){width="0.8\linewidth"}
In Fig. 2, we show the temperature dependence of the specific heat in various magnetic fields. In zero field, there is a broad peak at 1.23 K, characteristic of short-range ordering in a tetrahedron by spin freezing. Below this temperature the local 2-in 2-out configuration is stabilized, but does not lead to a long-range ordering. The dipolar spin-ice model predicted that if thermalization process is efficient there should be a sharp peak originating from first-order transition at 0.18 K in the temperature dependence of specific heat at 0 T [@Melko01]. However, as reported in our previous paper, we did not detect the theoretically predicted first-order transition at 0.18 K by ac susceptibility measurement down to 60 mK [@Fukazawa02]. In the present study, we confirm the absence of a transition in specific heat at 0 T down to 100 mK.\
In this compound, the spin-spin interaction $J_{{\rm eff}} (\equiv D_{{\rm nn}} + J_{{\rm nn}})$ is estimated to be 1.11 K, where $D_{{\rm nn}}$ (= 2.35 K) and $J_{{\rm nn}}$ (= -1.24 K) represent the dipolar interaction and exchange interaction between nearest neighbors [@Hertog00]. The energy difference between in and out spins originating from spin-spin interaction is represented as 4$J_{{\rm eff}}$ per $A$ site spin surrounded by six nearest-neighbor spins. The Zeeman energy is $E_{{\rm Z}}(\theta) = g_{J}J\mu_{{\rm B}}(\mu_{0}H) \times {\rm cos}\theta$, where $\theta$ is the angle between the local Ising axis and the field direction, $g_{J}$ is Lande’s $g$ factor, $J$ is a total angular momentum, and $\mu_{{\rm B}}$ is the Bohr magneton. For a Dy$^{3+}$ spin, $g_{J}$ = 4/3 and $J$ = 15/2. The energy difference from this interaction is represented as $2E_{{\rm Z}}$ and for the frustrated spins on Kagome lattice and the field-parallel spins in the field along the \[111\] direction, these values are represented as 2$E_{{\rm Z}}^{{\rm Kagome}} = 2 E_{{\rm Z}}$(109.47) = $6.66 \mu_{{\rm B}}(\mu_{0}H)$ and 2$E_{{\rm Z}}^{{\rm parallel}} = 2 E_{{\rm Z}}$(0) = $20 \mu_{{\rm B}}(\mu_{0}H)$, respectively. Under the magnetic field (see for example data at 0.5 T in Fig. 2), the peak in the specific heat splits into two. The peak at higher temperature (peak 1) shifts to higher temperature linearly in the external field. This peak is attributable to the spins parallel to the field. Because the Zeeman interaction for the spins parallel to the field is three times greater than the others, the direction of these spins is decided first with increasing field.\
In contrast, the peak at lower temperature (peak 2) shifts to lower temperature up to 1 T. At this field, there is a sharp peak at around 400 mK. For fields greater than 1 T, the peak 2 shifts to higher temperature and becomes broad. The origin of the peak 2 is attributable to the frustrated three spins. The Zeeman interaction and the spin-spin interaction for these spins compete with each other in the \[111\] field direction because the former one stabilizes the 1-in 3-out configuration and the latter one stabilizes the 2-in 2-out configuration. At lower external field, the Zeeman interaction is smaller than the spin-spin interaction ($2E_{{\rm Z}}^{{\rm Kagome}} < 4J_{{\rm eff}}$) and the difference between these two interactions decrease with increasing field. Therefore, the characteristic peak due to this origin shifts to lower temperature with increasing field. At higher field ($2E_{{\rm Z}}^{{\rm Kagome}} > 4J_{{\rm eff}}$), the peak shifts to higher temperature. We note that the specific-heat peak at this turnover field at (0.98 T, 0.40 K) is substantially higher and sharper than those in the previous reports [@Higashin03; @Hiroi03]. This indicates that the alignment in the present experiment is more accurate than that in those reports. Thus the features presented below are considered as the intrinsic properties rather than due to field misalignment.\
![\[HdepC\] Field dependence of the specific heat of Dy$_2$Ti$_2$O$_7$ at various temperatures in the \[111\] field direction. The data at 409 mK is divided by 10.](fig3.eps){width="0.8\linewidth"}
In Fig. 3, we show the field dependence of the specific heat at various temperatures below 0.5 K. Because the specific-heat peak at 409 mK is much higher than those at other temperatures, the values obtained at 409 mK are divided by 10 in Fig. 3. In the temperature range between 230 and 409 mK, there is a temperature-independent peak at 1 T. This field agrees with the field at which the Zeeman interaction is equal to the spin-spin interaction for the frustrated three spins. For these spins, $E_{{\rm Z}}^{{\rm Kagome}}$ is estimated to be 2.24 K at 1 T; thus, the equation $4J_{{\rm eff}} = 2 E_{{\rm Z}}^{{\rm Kagome}}$ is satisfied at $\mu_{0}H$ = 0.99 T, as observed. On the other hand, below 230 mK another peak emerges around 1.25 T. The intensity of the broad 1.25 T peak increases with decreasing temperature. In these experiments, we did not observe any hysteresis.\
Discussion and Conclusion
=========================
![\[TdepC2\] Temperature dependence of the specific heat of Dy$_2$Ti$_2$O$_7$ divided by temperature attributable to contribution from frustrated spins at various fields.](fig4.eps){width="0.8\linewidth"}
In Fig. 4, we show the magnetic specific heat divided by temperature attributable to contribution from frustrated spins, defined as $C_{{\rm peak\ 2}}/T = (C_{{\rm total}} - C_{{\rm lattice}})/T - C_{{\rm peak\ 1}}/T$. We estimate the lattice contribution of specific heat using the specific heat of insulating and nonmagnetic Eu$_2$Ti$_2$O$_7$ with comparable mass per formula unit [@Higashin03]. In this temperature region, lattice contribution is of the order of $10^{-3}$ of the magnetic one and negligibly small. In the field along the accurate \[111\] direction (ideal condition), the residual entropy should depend only on the entropy of the frustrated spins. For the contribution from the field-parallel spins we approximate the higher temperature peak as a Schottky peak originating from the sum of the spin-spin interaction and the applied field with some correction, $$C_{{\rm peak\ 1}} = \frac{1}{4} \frac{N_{{\rm A}}k_{{\rm B}}}{T} \left( \frac{\Delta E}{k_{{\rm B}}T} \right)^{2} \frac{e^{\Delta E/k_{{\rm B}}T}}{(1+e^{\Delta E/k_{{\rm B}}T})^{2}},$$ where $N_{{\rm A}}$ is the Avogadro number and $k_{{\rm B}}$ is the Boltzman constant. $\Delta E$ is the energy difference between up and down spins, $\Delta E = 2E_{{\rm Z}}^{{\rm parallel}} \times a + 4J_{{\rm eff}} \times b$. Due mainly to long-range interaction, some corrections are needed for both energy terms. For simplicity, we assume that the two factors are constant and estimate that $a$ = 0.85, $b$ = 0.66 from the fitting of the peak temperatures above 1 K of specific heat measurements [@Hiroi03].\
![\[TdepS\] Temperature dependence of the entropy of Kagome ice state at various fields. The entropy difference at 3 K between the data at 0.5 T and that at 1 T represents the residual entropy of Kagome ice state. Above 1 T, the residual entropy is released below 300 mK.](fig5.eps){width="0.8\linewidth"}
In Fig. 5, we show the temperature dependence of the activation entropy $\Delta S$ attributable to frustrated spins. The entropy at 100 mK, $S$(100 mK), is estimated as $\frac{1}{2} \frac{C({\rm 100\ mK})}{T} \times {\rm 100\ mK}$ by a linear extrapolation of $C/T$ to zero at $T$ = 0. Thus the activation entropy $\Delta S(T)$ for $T$ $\textgreater$ 100 mK is estimated as $$\Delta S(T) = S({\rm 100\ mK}) + \int^{T}_{{\rm 100\ mK}}\frac{C_{{\rm peak\ 2}}}{T} dT.$$ At 1.4 T, the peak temperature of $C_{{\rm peak\ 2}}$ is higher than that at lower fields; thus, the contribution of peak 2 to the entropy at 3 K is noticeably smaller than that at lower fields. Therefore, we cannot estimate the activation entropy originating from peak 2 to be compared to that at lower fields and we do not show these data in Fig. 5. From this figure, we can estimate the difference between the entropies above 1 T and below 0.9 T. At 2.7 K, the activation entropy $\Delta S$ is 2.82 $\pm$ 0.1 (0.5 T), 2.84 $\pm$ 0.1 (0.75 T), 3.09 $\pm$ 0.1 (0.90 T) and 3.63 $\pm$ 0.1 J/K mol-Dy (1.00 T). The entropy difference between 0.5 T and 1 T is ascribable to the release of the residual entropy of Kagome ice. Around 0.5 T, the ground state retains the residual entropy of Kagome ice. On the other hand, above 1 T the residual entropy is released even below 300 mK. The residual entropy of the Kagome ice state $S_{{\rm KI}}$ is estimated as 0.81 $\pm$ 0.2 J/mol-Dy K in the present study, substantially different from the residual entropy of spin ice state $S_{{\rm SI}}$, 1.68 J/mol-Dy K. The previous results of residual entropy of Kagome ice state are 0.44 $\pm$ 0.12 (our previous one)[@Higashin03], 0.65 [@Hiroi03] and 0.5 $\pm$ 0.15 J/K mol-Dy [@Sakakibara03]. In our previous experiment, we estimated the entropy below 0.35 K by the procedure similar to that in the present study. Nevertheless, since the alignment of sample was not as good as in the present one, we cannot make a direct comparison of the two sets of the data. On the other hand, in the previous report by Matsuhira $et$ $al$. [@Matsuhira02] the definition of the entropy is different from ours. They forced the value of the entropy around 40 K as $R$ln2. However, their estimation of phonon specific heat in the fields was just the fitting $C_{{\rm phonon}} = \alpha T^3$ and in this case it was assumed that additional entropy emerges at high temperature under high magnetic field, possibly from the heat capacity of the addenda. We believe that the differences of the value of the Kagome ice residual entropy between the present and the previous reports are caused mainly by the upturn at low temperature discussed below. In the theoretical reports, only nearest neighbor ferromagnetic interaction is considered [@Udagawa02; @Moessner01]. In real system, the long-range dipolar interaction is also important, and this should tend to reduce the residual entropy. Thus the theoretical value should give the upper limit of $S_{\rm KI}$. Although not contradictory within experimental uncertainty, however, the present value is also somewhat greater than these theoretical predictions.\
![\[3Dplot\] The 3D plot of field-temperature dependence of $C/T$ of Dy$_2$Ti$_2$O$_7$ in the field along the \[111\] direction.](fig6.eps){width="\linewidth"}
We show the 3D plot of the field-temperature dependence of $C/T$ (Fig. 6), as well as the field-temperature phase diagram (Fig. 7) in the field along the \[111\] direction. In Fig. 7, at zero field below the peak temperature (Fig. 7 A) the spin ice state is realized and the ground state has the residual entropy $S_{{\rm residual}} = S_{{\rm SI}}$. Above this crossover temperature (Fig. 7 B) all the spins are thermally fluctuating and direct randomly; all-random state is realized. At a certain field range and at low temperatures (Fig. 7 C), directions of parallel spins are fixed and Kagome ice state is realized. The ground state has a different residual entropy $S_{{\rm residual}} = S_{{\rm KI}}$. At higher temperatures (Fig. 7 D) the spins on the Kagome lattice are thermally fluctuating while the spins parallel to the field are still pinned; the 1-in 3-random state is stable. The phase diagram above 350 mK reproduces the previous report [@Hiroi03].\
![\[PhaseDiagram\] The $H$-$T$ phase diagram of Dy$_2$Ti$_2$O$_7$ in the field along the \[111\] direction. The solid line represents the first-order transition and the dotted lines represent crossover line. The data points correspond to the peak in $C$. ](fig7.eps){width="0.8\linewidth"}
At $\mu_{0}H$ = 1 T, there is a transition line nearly parallel to the temperature axis and terminating at a critical end point at $(\mu_{0}H_{{\rm c}}, T_{{\rm c}}) = (0.98\ {\rm T}, 0.40\ {\rm K})$, accompanied by the sharp peak in the specific heat ascribable to the end point of the first order transition. The critical exponent $\alpha$ and $\alpha '$ are represented as $$C_{H} \propto \left( \frac{T_{{\rm c}}-T}{T_{{\rm c}}} \right) ^{-\alpha '}\ (T\ \textless \ T_{{\rm c}})\ ; \ \left( \frac{T-T_{{\rm c}}}{T_{{\rm c}}} \right) ^{-\alpha}\ (T\ \textgreater \ T_{{\rm c}}).$$ For narrow fitting ranges 0.360 K $\textless T \textless$ 0.405 K for $\alpha '$ and 0.413 K $\textless T \textless$ 0.496 K for $\alpha$, we obtain $\alpha '$ = 0.51 and $\alpha$ = 0.38. For a 3D Ising system with common Ising axis, $\alpha '$ is equal to 1/8 to 1/16 and $\alpha$ is equal to about 1/8. Since there are four different Ising axis with frustration structure in this system, we cannot compare these value simply.\
From a magnetization experiment, Sakakibara $et$ $al.$ found that the transition is of the first order [@Sakakibara03]. Our present experiment was not suitable for observing any latent heat because the phase boundary is nearly parallel to the temperature axis; measurement of magnetocaloric effect is needed to clarify this issue. The phase diagram shown in Fig. 7 is similar to the liquid-gas phase diagram [@Sakakibara03]. Because there is no change of the order parameter between the Kagome ice state (Fig. 7 C) and the 1-in 3-out ordered state (Fig. 7 F), the dotted line between Kagome ice state and 1-in 3-random state (Fig. 7 D) and that between 1-in 3-out ordered state and 1-in 3-random state represent crossover lines, not phase boundaries. In fact, we did not observe any sign of phase transitions between the regions C and D or D and F.\
In addition, a broad peak develops at 1.25 T and below 0.3 K. This suggests another state between 1 and 1.25 T (Fig. 7 E). Above 1.25 T (Fig. 7 F), the 1-in 3-out ordered state is expected to be stable. In previous specific heat measurements, the peak shape of the field dependence of the magnetic specific heat, $C_{{\rm mag}} \equiv C_{{\rm total}} - C_{{\rm lattice}}$ at 0.4 K was clearly asymmetric [@Hiroi03]. This asymmetry is consistent with the existence of multiple peaks below this temperature and indeed agrees with our present observation. This means that the residual entropy is not entirely released at the first order transition at 0.98 T, but also released around 1.25 T. One possible explanation of the state between 1 T and 1.25 T is the coexistence of some state with different spin configuration including the 2-in 2-out state (Kagome ice state), the 1-in 3-out one (ordered state) and paramagnetic one (free-spin state). Since the Zeeman interaction nearly compensates the effective nearest-neighbor spin-spin interaction in this region, spins on the Kagome lattice behave just like free-spins except for residual long-range interaction and the system may have a larger residual entropy than that in the Kagome ice state [@Zhitomersky; @Isakov04]. Since the dipolar interaction is important in this system, the state in this region may not be so simple. This conjecture is consistent with the magnetization of Kagome ice state reported by Sakakibara $et$ $al$ [@Sakakibara03]. If Kagome ice state changes to the 1-in 3-out ordered state completely by the first order transition at 0.98 T, the magnetization would have a step-function-like change at this field. The magnetizations indeed exhibit an abrupt change at the low-field side of the transition. However, just above the first order transition, magnetization does not reach the saturated moment expected in the 1-in 3-out ordered state but shows a gradual increase with the field after step-function-like change even at 50 mK [@Sakakibara03]. Because this behavior is observed at extremely low temperature 50 mK, the origin must be related to magnetic interaction, not to thermal fluctuation. This indicates that Kagome ice state changes to only an incompletely ordered state at the first-order transition and agrees with our interpretation. If the mixture of different spin configurations exists, the system would retain additional entropy and exhibit the upturn at low temperatures in the specific heat. In relation to this picture, Ogata $et$ $al$. predicted that spin ordering may exist in Kagome ice state when a small transverse field (about 0.04 T) is applied in addition to the field (almost 1 T) along the \[111\] direction [@Ogata]. Since the assumed situation is different from our experiment, their prediction does not seem to be applicable to our result.\
In conclusion, we measured the specific heat of Dy$_2$Ti$_2$O$_7$ in fields along the \[111\] direction down to 100 mK. We confirmed the residual entropy of Kagome ice state and found a new peak at 1.25 T below 0.3 K. This peak suggests the mixture of different spin configurations including the 2-in 2-out one (Kagome ice state), the 1-in 3-out state (ordered state) and paramagnetic one (free-spin state) between 1 and 1.25 T and below 400 mK. This interpretation is consistent with previous specific heat and magnetization results [@Higashin03; @Hiroi03; @Sakakibara03]. It should be helpful to examine this conjecture by neutron or NMR experiment in the field along the \[111\] direction.\
Acknowledgment {#acknowledgment .unnumbered}
==============
We acknowledge helpful discussion with M. Zhitomirsky, S. Nakatsuji, M.J.P. Gingras, H. Tsunetsugu, S. Fujimoto and M. Udagawa. This work was in part supported by the Grant in-Aid for Scientific Research (S) from the Japan Society for Promotion of Science, by the Grant-in-Aid for Scientific Research on Priority Area “Novel Quantum Phenomena in Transition Metal Oxides” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, and by a Grant-in-Aid for the 21st Century COE program “Center for Diversity and Universality in Physics” from MEXT. H.F. is grateful for the financial support from Japan Society Promotion Science.
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[^1]: Present address: Graduate School of Science and Technology, Chiba University, Chiba 263-8522, Japan
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---
abstract: 'Link prediction is a significant and challenging task in network science. The majority of known methods are similarity-based, which assign similarity indices for node pairs and assume that two nodes of larger similarity have higher probability to be connected by a link. Due to their simplicity, interpretability and high efficiency, similarity-based methods, in particular those based only on local information, have already found successful applications on disparate fields. In this research domain, an intuitive consensus is that two nodes sharing common neighbors are very likely to have a link, while some recent evidences argue that the number of 3-hop paths more accurately predicts missing links than the number of common neighbors. In this paper, we implement extensive experimental comparisons between 2-hop-based and 3-hop-based similarity indices on 128 real networks. Our results indicate that the 3-hop-based indices perform slightly better with a winning rate about 55.88%, but which index is the best one still depends on the target network. Overall speaking, the class of Cannistraci-Hebb indices performs the best among all considered candidates.'
address:
- 'CompleX Lab, University of Electronic Science and Technology of China, Chengdu 611731, P.R. China'
- 'Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 611731, P.R. China'
- 'Ant Financial Services Group, Hangzhou, 310099, People’s Republic of China'
author:
- Tao Zhou
- 'Yan-Li Lee'
- Guannan Wang
title: ' Experimental analyses on 2-hop-based and 3-hop-based link prediction algorithms'
---
`C`omplex Networks,Link Prediction,Similarity Index
Introduction
============
Link prediction is an elemental challenge in network science, which aims at estimating the existence likelihood of any nonobserved link, on the basis of observed links [@Getoor2005; @Lu2011; @Brugere2018; @Squartini2018]. Theoretically speaking, link prediction can be treated as a testing aid for mechanism models, since a model that can well explain the network formation and evolution could be in principle transferred to an accurate link prediction algorithm [@WangEPL2012; @WangEPL2014; @ZhangSR2015; @ZhangManag2017]. Practically speaking, prediction results can be used as an experimental guidance, by which we can focus on those biological interactions (e.g., regulatory interactions [@BarzelNB2013], drug-target interactions [@DingBB2013], protein-protein interactions [@ZhaoSR2015]) most likely to exist instead of blindly check all possible interactions, and thus the experimental costs can be largely reduced. Besides missing link problems, link prediction algorithms can also forecast links that may appear in the future of evolving networks, with significant commercial values in friend recommendations of online social networks [@AielloATW2012] and product recommendations in e-commercial web sites [@LuPR2012].
Many algorithms have been proposed, including similarity-based algorithms [@LibenJAIST2007; @ZhouEPJB2009], probabilistic models [@NevilleJMLR2007; @YuNIPS2007; @WangICDM2007], maximum likelihood methods [@ClausetNature2008; @GuimeraPNAS2009; @PanSR2016], and some other representatives [@LuPNAS2015; @RathaEPL2017; @BensonPNAS2018]. The probabilistic models and maximum likelihood methods are usually more accurate than similarity-based algorithms, at the same time, they suffer some intrinsic disadvantages. The probabilistic models often require information about node attributes in addition to the network structure, which highly limits their applications. Moreover, the number of parameters to be fixed are too many so that we cannot get insights about the network organization even if we have built a very accurate model. The maximum likelihood methods are highly time consuming, usually only capable to handle networks with a few thousands of nodes, while many real networks scale from millions to billions of nodes. Therefore, overall speaking, the similarity-based algorithms, in particular the ones based solely on local topological information, have found widest applications.
Generally speaking, a similarity-based algorithm will assign a similarity score to each pair of nodes, and assume that two nodes having a higher similarity score are of a larger likelihood to have a link. Therefore, all nonobserved links are ranked by their corresponding similarity scores, and the links with the highest scores are the predicted ones. Given a node pair $(i,j)$, many known local similarity indices only make use of the information contained in the 2-hop paths connecting $i$ and $j$, such as the common neighbor (CN) index [@LibenJAIST2007], the resource allocation (RA) index [@ZhouEPJB2009], the Adamic-Adar (AA) index [@AdamicSocNetw2003], and the Cannistraci resource allocation (CRA) index [@CannistraciSR2013]. Besides different mathematical details, all 2-hop-based algorithms tend to assign a larger similarity score $S_{ij}$ if $i$ and $j$ have more 2-hop paths (i.e., more common neighbors). This is in accordance with an important network organization mechanism named as homophily [@McPhersonSocio2001; @PanSR2016], that is to say, two nodes having similar attributes are likely to connect to each other. In a network where only topological information is observed, the homophily mechanism can be interpreted as the fact that two nodes sharing one or more common neighbors are likely to become direct neighbors in the future. Such mechanism has been observed in the evolving processes of many real networks [@McPhersonSocio2001; @KossinetsScience2006; @RomeroAAAI2010; @YinSIGIR2011; @MaSR2016], for example, more than 90% of new links in Twitter and Weibo are between nodes that are already connected by at least one 2-hop path [@RomeroAAAI2010; @YinSIGIR2011]. In a word, 2-hop paths are well accepted as strong evidence indicating the existence of missing link or future link between the corresponding two ends. The roles of longer paths are intuitively considered to be less significant since interacting strengths will decay along the paths [@KatzPsy1953; @ChristakisNEJM2007]. Although a certain local similarity index (named as local path index) [@ZhouEPJB2009; @LuPRE2009] has considered both contributions from 2-hop paths and 3-hop paths, the authors argued that the 2-hop paths play the leading role and the number of 3-hop paths plays a part only if the number of 2-hop paths is not sufficiently distinguishable.
Very recently, some scientists have argued that 3-hop-based similarity indices perform better than 2-hop-based indices [@RathaPA2019; @KovacsNC2019; @MuscoloniBioRxiv]. For example, Kovács *et al.* [@KovacsNC2019] proposed a degree-normalized index based on 3-hop paths and showed its remarkable advantage compared with 2-hop-based indices in predicting protein-protein interactions. Pech *et al.* [@RathaPA2019] provided a theory showing that the number of 3-hop paths is a degenerated index of a more complicated index resulted from a linear optimization. Based on experimental analysis on eight networks, they argued that even the direct count of 3-hop paths performs better than the common neighbor index (i.e., the direct count of 2-hop paths), which is to some extent counterintuitive.
To clarify this issue, this paper implements experimental analyses on 128 real networks from 16 disparate fields. Extensive comparisons between 2-hop-based and 3-hop-based similarity indices indicate that the 3-hop-based indices perform slightly better with a winning rate about 55.88%. However, given a specific target network, which index is the best choice still largely depends on the network structure. Overall speaking, the class of Cannistraci-Hebb (CH) indices [@MuscoloniBioRxiv] performs the best among all considered candidates, and the class of resource allocation (RA) indices [@ZhouEPJB2009] is the runner-up.
Methods
=======
Denote $A$ as the adjacency network of a simple network $G$, where the element $a_{ij}=1$ if $i$ and $j$ are neighboring, and $a_{ij}=0$ otherwise. The degree of node $i$ is denoted by $k_i$ and the set of neighbors of node $i$ is denoted by $\Gamma_i$. Four representative 2-hop-based indices, CN [@LibenJAIST2007], RA [@ZhouEPJB2009; @Ou2007], AA [@AdamicSocNetw2003], and CH2 [@CannistraciSR2013; @MuscoloniBioRxiv], as well as their 3-hop-based counterparts are considered in this paper. To be clear, the suffixes L2 and L3 stand for 2-hop and 3-hop, for example, RA index and its 3-hop-based counterpart will be renamed as RA-L2 and RA-L3 indices.
CN-L2 index [@LibenJAIST2007] is a structural equivalence index. Two nodes are considered to be structural equivalence if they share many common neighbors. Accordingly, the similarity score between nodes $i$ and $j$ is $$s_{ij}^{\rm{\text{CN-L2}}} = |\Gamma_i \cap \Gamma_j|.$$
![The illustration of eight similarity indices. Red nodes denote the target node pair, and green nodes denote intermediate nodes on 2-hop paths or 3-hop paths. Figure 1(a) is the illustration for indices CN-L2, AA-L2, RA-L2, where red links form 2-hop paths between nodes $i$ and $j$. Figure 1(b) is the illustration for CH2-L2, where green links and blue links denote the links counted by $c.$ and $o.$ in Eq. (\[CH2-L2\]), respectively. Figure 1(d) is the illustration for indices CN-L3, AA-L3, RA-L3, where red links form 3-hop paths between nodes $i$ and $j$. Figure 1(e) is the illustration for CH2-L3, where green links and blue links denote links counted by $\tilde{c}.$ and $\tilde{o}.$ in Eq. (\[CH2-L3\]), respectively. The corresponding similarity scores are shown in Figure 1(c) and Figure 1(f).[]{data-label="fig_IIlustration"}](Illustration2.eps){width="100.00000%"}
AA-L2 index [@AdamicSocNetw2003] weakens the contribution of large-degree common neighbors, because to be neighboring to a popular node is generally less meaningful. The corresponding similarity score between nodes $i$ and $j$ is defined as $$s_{ij}^{\rm{\text{AA-L2}}} = \sum_{x \in \Gamma_i \cap \Gamma_j}\frac{1}{\log (k_x)}.$$
RA-L2 index [@ZhouEPJB2009; @Ou2007] treats the similarity between nodes $i$ and $j$ as the resource transmitted from $i$ to $j$. Each neighbor of $i$ occupies one unit of resources, and allocates the resource equally to their neighbors. The resource received by node $j$ from $i$ is $$s_{ij}^{\rm{\text{RA-L2}}} = \sum_{x \in \Gamma_i \cap \Gamma_j}\frac{1}{k_x}.$$
CH2-L2 index [@MuscoloniBioRxiv] rewards internal links among common neighbors while penalizes links connecting common neighbors and outside. The similarity between nodes $i$ and $j$ is $$\label{CH2-L2}
s_{ij}^{\rm{\text{CH2-L2}}}= \sum_{x \in \Gamma_i \cap \Gamma_j}\frac{1+c_x}{1+o_x},$$ where $c_x$ is the number of $x$’s neighbors that are also in $\Gamma_i \cap \Gamma_j$, and $o_x$ is the number of $x$’s neighbors not in $\Gamma_i \cap \Gamma_j$, and not $i$ or $j$.
Correspondingly, CN-L3 index, AA-L3 index and RA-L3 [@KovacsNC2019] index are defined as follows, $$s_{ij}^{\rm{\text{CN-L3}}} = \sum_{x \in \Gamma_i, y \in \Gamma_j}a_{xy},$$ $$s_{ij}^{\rm{\text{AA-L3}}} = \sum_{x \in \Gamma_i, y \in \Gamma_j}\frac{a_{xy}}{\sqrt{\log(k_x)\log(k_y)}},$$ $$s_{ij}^{\rm{\text{RA-L3}}} = \sum_{x \in \Gamma_i, y \in \Gamma_j}\frac{a_{xy}}{\sqrt{k_x k_y}}.$$
CH2-L3 index [@MuscoloniBioRxiv] is defined as $$\label{CH2-L3}
s_{ij}^{\rm{\text{CH2-L3}}} = \sum_{x \in \Gamma_i, y \in \Gamma_j}\frac{a_{xy}\sqrt{(1+\tilde{c}_x)(1+\tilde{c}_y)}}{\sqrt{(1+\tilde{o}_x)(1+\tilde{o}_y)}}.$$ Similarly, $\tilde{c}_x$ is the number of links between $x$ and nodes in the set of intermediate nodes on all 3-hop paths connecting nodes $i$ and $j$, $\tilde{o}_x$ is the number of links between $x$ and nodes that are not $i$, $j$ or the intermediate nodes on any 3-hop paths connecting $i$ and $j$.
The illustration of the above eight indices are shown in figure \[fig\_IIlustration\].
Results
=======
![The winning rate of each similarity index for each network category. The 16 categories are shown in alternating white and grey for better discernibility. The 16 categories are: (1) Software [@Konect]—the networks of software components; (2) Trophic [@Gene2015; @Pajek2006]—the predation networks of biological species; (3) PPI [@KovacsNC2019; @Konect]—the protein-protein interaction networks; (4) Lexical [@Konect; @MiloScience2004]—the networks of words from natural languages; (5) GeneFun [@Gene2015]—the networks of co-functional associations of genes; (6) Computer [@Konect; @snap2014; @circuits1989]—the networks of computer-related components; (7) Citation [@Konect; @Pajek2006; @TangKDD2009]—the citation networks; (8) Animal [@Konect]—the contact networks of animals; (9) Brain [@Gene2015; @Wang2019]—the brain connection networks of cortical areas; (10) Infra. [@Konect]—the networks of physical infrastructures; (11) Hyperlink [@Konect]—the networks of web pages; (12) Social [@Konect; @snap2014; @MiloScience2004; @Hu2019]—the online social networks; (13) Comm. [@Konect; @Boguna2004]—the communication networks; (14) Coauthor [@snap2014]—the coauthorship networks; (15) Human [@Konect; @Gene2015; @GirvanPNAS2002]—the contact networks of human beings; (16) PS [@MiloScience2004]—the protein structure networks of secondary-structure elements. The best-performed index in each category is labelled above the corresponding window.[]{data-label="fig_field"}](field_performance1.eps){width="100.00000%"}
Given a simple network $G(V, E)$, where $V$ is the set of nodes and $E$ is the set of links. To test the algorithm’s accuracy, the set of links $E$ is randomly divided into two parts: (i) the training set $E^T$, which is the known information, and (ii) the probe set $E^P$, which is treated as the set of missing links. No information in $E^P$ is allowed to be used for the calculation of similarity matrix $S$. We adopt a standard metric, precision [@HerlockerATIS2004], to quantify the algorithm’s accuracy, which is defined as the ratio of the number of relevant elements in $S$ to the number of selected elements. In other words, if we select top-$L$ links as predicted links (i.e., the $L$ links with highest similarity scores in the set $E \verb|\| E^T$), among which $L_r$ links are in the probe set $E^P$, then the corresponding precision is $L_r/L$. In the experiments, for each network, the ratio of probe links to total links, say $q = |E^P|/|E|$, varies from 0.02 to 0.2, as $q = \{0.02, 0.04, \cdots ,0.2\}$. Correspondingly, we set the number of selected links $L = |E^P|$. 128 real networks from 16 disparate fields are used in the experimental comparisons, and thus 1280 comparisons among the eight similarity indices are recorded. The descriptions and topological statistics of these networks are shown in Supplementary Material Section S1.
![The overall winning rates of eight similarity indices in the 1280 comparisons.[]{data-label="fig_alg"}](alg_performance1.eps){width="100.00000%"}
Figure \[fig\_field\] shows the winning rate of each index for each network category. In a comparison, the best-performed index will get score 1 and all others get 0. If two indices are equally best, they both get score 0.5. The case with multiple winners is similar. The winning rate of an index is its total score divided by the number of comparisons. For example, in the category Hyperlink, there are three networks AB, BG and FD, and thus 30 comparisons. The index CH2-L3 wins 12 comparisons and thus has a high winning rate 40%. As shown in figure \[fig\_field\], in some categories the 3-hop-based indices are remarkably more accurate, while in some categories the 2-hop-based indices are much better. Yet for the category Social, the 2-hop-based and 3-hop-based indices exhibit nearly the same performance. Detailed results for each of the 128 networks are presented in Supplementary Material Section S2.
As shown in figure \[fig\_alg\], the overall winning rate for 3-hop-based indices in the 1280 comparisons is 55.88%, and that for 2-hop-based indices is 44.12%, namely the 3-hop-based indices perform slightly better as a whole. Together with the results reported in figure \[fig\_field\], we can conclude that there is no easy way to anticipate which category is better or which index is the best. Indeed, which index is the best choice largely depends on the specific structure of the target network. An unexpected gain from these experiments is that the class of CH2 indices perform the best among all 4 classes with a dominant winning rate 62.65%, and the class of RA indices [@ZhouEPJB2009] is the runner-up with a winning rate 24.15%, while the other two classes perform poorly, with both winning rates being about 7%. Figure \[fig\_q\] validates the robustness of the above experimental results by varying the ratio $q$. Obviously, the main findings keep unchanged for different $q$.
![The winning rates of eight similarity indices for different $q$.[]{data-label="fig_q"}](q_performance1.eps){width="100.00000%"}
Discussion
==========
Based on extensive experiments, this paper provides a direct response to a recent debate about the roles of 2-hop paths and 3-hop paths on link prediction [@RathaPA2019; @KovacsNC2019; @MuscoloniBioRxiv]. The answer is not a simple winner, but a fact that 2-hop-based and 3-hop-based indices are competitive to each other. Indeed, the 3-hop-based indices perform slightly better as a whole, while which index is the best choice still largely depends on the specific structural features of the target network.
Such experimental observations immediately raise two new questions. Firstly, can we foreknow which index or which category of indices is better for a given network by measuring some structural features (of course, the computational complexity should be lower than direct comparisons of those indices)? Secondly, how to properly make use of information contained in both 2-hop and 3-hop paths to improve the algorithm’s accuracy (at least a more subtle and effective way than the simply linear combination of CN-L2 and CN-L3 indices [@ZhouEPJB2009; @LuPRE2009])? We leave these two open questions for future studies.
The longer paths are also relevant. For example, as suggested by Pech *et al.* [@RathaPA2019], to eliminate the redundancy in 3-hop-based indices by considering the 5-hop-based paths can further improve the algorithm’s accuracy (this idea is very similar to a previous work [@ZhouNJP2009]). However, to account for longer paths is highly time-consuming while the improvement may be marginal. Intuitively, we do not think to consider longer paths is cost-efficient, however, intuition usually leads to mistakes, and thus whether our judgment is reasonable still needs further investigations.
An unexpected gain in this work is that the Cannistraci-Hebb indices [@MuscoloniBioRxiv] perform remarkably better than other indices. This is not a coincidence, but shows us an insight that the local connecting patterns in the neighborhood provide important information about the potential relationship between two nodes. In despite of the excellent performance of CH2-L3 index, it is just a naive extension of the CH2-L2 index. Once we known the value of the class of Cannistraci-Hebb indices, we are inspired to design more elegant and effective indices on the basis of local connecting patterns.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors acknowledge the valuable discussion with Dr. Carlo Vittorio Cannistraci, and Dr. Qian-Ming Zhang for providing us some of the datasets. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 61433014, 61803073 and 11975071) and by the Fundamental Research Funds for the Central Universities under Grant Nos. ZYGX2016J196.
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abstract: |
The valence of a function $f$ at a point $w$ is the number of distinct, finite solutions to $f(z) = w$. Let $f$ be a complex-valued harmonic function in an open set $R \subseteq \mathbb{C}$. Let $S$ denote the critical set of $f$ and $C(f)$ the global cluster set of $f$. We show that $f(S) \cup C(f)$ partitions the complex plane into regions of constant valence. We give some conditions such that $f(S) \cup C(f)$ has empty interior. We also show that a component $R_0 \subseteq
R \backslash f^{-1} (f(S) \cup C(f))$ is a $n_0$-fold covering of some component $\Omega_0 \subseteq
\mathbb{C} \backslash (f(S) \cup C(f))$. If $\Omega_0$ is simply connected, then $f$ is univalent on $R_0$. We explore conditions for combining adjacent components to form a larger region of univalence. Those results which hold for $C^1$ functions on open sets in $\mathbb{R}^2$ are first stated in that form and then applied to the case of planar harmonic functions. If $f$ is a light, harmonic function in the complex plane, we apply a structure theorem of Lyzzaik to gain information about the difference in valence between components of $\mathbb{C} \backslash (f(S) \cup C(f))$ sharing a common boundary arc in $f(S) \backslash C(f)$.
address: 'Department of Mathematics, University of California, Berkeley, California 94720'
author:
- Genevra Neumann
date: 'July 26, 2003; revised January 26, 2004.'
title: 'Valence of complex-valued planar harmonic functions'
---
Introduction {#sec:intro}
============
We study here complex-valued harmonic functions in the plane, which we refer to simply as harmonic functions. The behavior of such a function can be vastly different from that of a holomorphic function. Analytic polynomials take every value a finite number of times. In contrast, the range of a harmonic polynomial can exclude an open region of the complex plane. Picard’s theorem states that a transcendental entire function takes every value, with the exception of possibly one point, an infinite number of times. In contrast, there are transcendental harmonic functions that omit open regions. Also, there are transcendental harmonic functions that approach $\infty$ as $z \rightarrow \infty$ (like analytic polynomials), such that each $w \in \mathbb{C}$ has a finite number of distinct preimages (like analytic polynomials), and such that the maximum possible number of preimages is unbounded (like transcendental entire functions.)
A harmonic polynomial $f(z)$ is a harmonic function of the form $f(z) = p(z) + \overline{q(z)}$ where $p$ and $q$ are analytic polynomials in $z$. Let $n_p$ be the degree of $p$ as a polynomial in $z$ and $n_q$ be the degree of $q$. A. Wilmshurst [@W:thesis; @W:paper][^1] showed that when $f$ has a finite number of zeros, it has at most $N^2$ distinct zeros, where $N = max(n_p, n_q)$. Wilmshurst’s bound is sharp; there are examples [@BHS:maxval; @W:paper] of harmonic polynomials with $N^2$ distinct zeros when $n_p = N$ and $n_q = N-1$. For $1 \leq n_q \leq n_p - 1$, Wilmshurst conjectured that $f$ has at most $n_q (n_q - 1) + 3n_p - 2$ distinct zeros. D. Khavinson and G. Światek [@KS:3n-2] recently proved Wilmshurst’s conjecture for the case $n_q = 1$ using methods from complex dynamics.
We will be concerned with questions related to the valence of harmonic functions. The valence of a function $f$ at a given point $w$, denoted $Val(f,w)$, is the number of distinct points $z$ in the domain of $f$ such that $f(z)=w$. The valence of a function, denoted $Val(f)$, is the supremum of $Val(f,w)$.
Let $f(z) = (u(z), v(z))$ be a $C^1$ function in an open set $R \subseteq \mathbb{R}^2$. The Jacobian of $f$ is given by $J_f = u_x v_y - u_y v_x$. The inverse function theorem tells us that if $J_f(z) \neq 0$, then $f$ is a homeomorphism in some neighborhood of $z$. H. Lewy [@Lew:jacobian] proved the converse when $u$ and $v$ are both real-valued harmonic functions. Hence, if $f = u + iv$ is harmonic in an open set $R \subseteq \mathbb{C}$, then $f$ is a local homeomorphism at $z$ if and only if $J_f(z) \neq 0$. The critical set $S$ of a $C^1$ function $f$ consists of those points where the Jacobian vanishes. If $f$ is harmonic, the critical set consists of those points where $f$ is not locally 1-1.
When studying specific examples of harmonic polynomials in $\mathbb{C}$ and looking at graphs of the image of the critical set using Mathematica[^2], the author noticed that the image of the critical set partitions the complex plane into regions of constant valence. This is not true for all harmonic functions. If $\lim_{z \rightarrow \infty} |f(z)| \neq \infty$, then we need to include another set in order to partition the complex plane into regions of constant valence.
Let $C(f, \infty)$ denote those finite values $w$ such that we can find a sequence $\{z_n\}$ with $z_n \rightarrow \infty$ and $f(z_n) \rightarrow w$; $C(f, \infty)$ is the cluster set of $f$ at $\infty$. The global cluster set of a function $f$ with domain $R$ is denoted $C(f)$ and consists of all finite values which are approached on some sequence of points in $R$ which converge to a point in $\partial R \cup \{\infty\}$. (Our definitions are slightly different from those in the book of E. Collingwood and A. Lohwater [@CL:cluster], where a cluster set can include the point at infinity. We are interested in partitions of the finite complex plane and exclude the point at infinity.)
We show, under suitable conditions, that the plane can be partitioned into regions of constant valence. Since many of our results hold not only for harmonic functions but for $C^1$ functions in open subsets of $\mathbb{R}^2$, we will state the partitioning results first for $C^1$ functions defined in an open set in $\mathbb{R}^2$ and then specialize to the case where the function is harmonic in an open set in $\mathbb{C}$. We will then look at the case when $f$ is harmonic in all of $\mathbb{C}$. With each additional assumption on $f$, we can say more about the partitioning set. When $f$ is a light harmonic function, we can apply some results of A. Lyzzaik [@L:light] to show that the valences for two regions separated by an arc in the image of the critical set differ by a non-zero, even number, if that arc contains some point not in the cluster set.
Let $R_1$ be a component of $R \backslash (f^{-1}(f(S) \cup C(f)))$. We will see that $f(R_1)$ is a component of $\mathbb{R}^2 \backslash (f(S) \cup C(f))$ and that $f|_{R_1}$ is an even cover of $f(R_1)$ in the sense of Munkres [@Mun:top p. 331]. Suppose that $R_2$ is another component of this partition of $R$; suppose also that $R_2$ shares a common boundary arc with $R_1$. When $f$ is harmonic, we will look at conditions for $R_1$ and $R_2$ to be mapped to different components of $\mathbb{C} \backslash (f(S) \cup C(f))$. We will also explore the behavior of $f$ on $\partial R_1$ (including the behavior at puncture points.) If $f$ is univalent in $R_1$ and in $R_2$, can we join $R_1$ and $R_2$ along the interior of their shared boundary arc to get a larger region of univalence?
The figures below were produced from EPS files generated by Mathematica routines[^3] written by the author. Some of the figures contain alphanumeric labels; these labels were manually inserted into the EPS files.
Acknowledgements {#acknowledgements .unnumbered}
================
Many of the results in this paper appeared in the author’s thesis [@gcn:thesis]. The author is grateful to her advisor, D. Sarason, for encouraging her to study planar harmonic functions. She also wishes to thank him for his open-mindedness, patience, and kindness in helping her learn about mathematics and research. His many suggestions were essential to this work. The author is indebted to W. Hengartner for suggesting that her earlier results concerning regions of constant valence should be extendable to the case of infinite valence, and to H. Helson for suggesting that they should be extendable to the case of functions in arbitrary open regions (not just in the entire plane.) The author is also indebted to D. Khavinson for information concerning the Brelot-Choquet Lemma. The author also wishes to thank A. Bakke for adding labels to some of the EPS files generated by Mathematica.
Notation and background material {#sec:notation}
================================
Let $R$ be an open set in $\mathbb{R}^2$. Let $f: R \rightarrow \mathbb{R}^2$ be $C^1$. We can also write $f$ as $f(z) =(u(z),\ v(z))$, where $u,\ v:
R \rightarrow \mathbb{R}$ and $z=(x,y) \in \mathbb{R}^2$. Then
$$\begin{array}{rll}
J_f(z) &= &u_x(z)v_y(z)-u_y(z)v_x(z)\\
S &= &\{ z \in R: J_f(z) = 0 \}\\
C(f) &= &\{ \zeta \in \mathbb{R}^2 :
\exists \{ z_n \}_{n=1}^{\infty} \subset R
\textnormal{\ with\ }
\lim_{n \rightarrow \infty} z_n
\in \partial R \cup \{\infty\}\ $and$ \\
&&\lim_{n \rightarrow \infty} f(z_n) = \zeta\}\\
C(f, \infty) &= &\{ \zeta \in \mathbb{R}^2 :
\exists \{ z_n \}_{n=1}^{\infty} \subset \mathbb{R}^2
\textnormal{\ with\ }
\lim_{n \rightarrow \infty} |z_n| \rightarrow
\infty\ $and$ \\
&&\lim_{n \rightarrow \infty} f(z_n) = \zeta\}\\
B(w, \epsilon) &= &\{z:\ |z-w| \ < \epsilon\}\\
Val(f, w) &= &\#\{z \in R: f(z) = w\} \\
V_N(f) &= &\{w: w \notin f(S) \cup C(f)
\textnormal{\ and\ } Val(f, w)=N\}\\
Val(f, V) &= &sup_{w \in V} \{Val(f, w)\}
\end{array}$$
1. $S$ denotes the critical set of $f$.
2. $C(f)$ denotes the global cluster set in the finite plane of $f|_R$. Notice that if $R = \mathbb{R}^2$, then $C(f) = C(f, \infty)$.
3. $ \overline{f(S)} \subseteq f(S) \cup C(f) $: Let $\{w_n\}_{n = 1}^ \infty \subseteq f(S)$. Suppose that $\lim_{n \rightarrow \infty} w_n = w_0$, where $w_0$ is finite. For each $w_n$, there exists $z_n \in S$ such that $f(z_n) = w_n$. If $\{z_n\}_{n = 1} ^ \infty$ is unbounded, then $w_0 \in C(f)$ by definition. If $\{z_n\}_{n = 1} ^ \infty$ is bounded, then it has a convergent subsequence, say $\{z_m\}$, such that $z_m \rightarrow z_0 \in \mathbb{R}^2$. If $z_0 \in R$, then $z_0 \in S$ by the continuity of $J_f(z)$; hence $w_0 \in f(S)$. Otherwise, we must have that $z_0 \in \partial R$; hence $w_0 \in C(f)$.
4. It is well known that $C(f)$ is closed.
5. $ f(S) \cup C(f) $ is closed, since $\overline{f(S)} \subseteq f(S) \cup C(f)$.
6. $ f^{-1} \left( f(S) \cup C(f) \right) $ is relatively closed in $R$, since $f$ is continuous.
Lyzzaik’s local description of light harmonic functions {#sec:lyzintro}
-------------------------------------------------------
Recall that a function is said to be light if the preimage of each point is empty or totally disconnected. A. Lyzzaik [@L:light] has characterized the local behavior of a function $f$ which is light and harmonic in a simply connected, open set in $\mathbb{C}$. Lyzzaik pays special attention to the behavior of $f$ on the critical set and the behavior of $f$ in a neighborhood of the critical set. Let $S$ denote the critical set of $f$.
We first note that since $f$ is harmonic in a simply connected, open set, we can find functions $h$ and $g$ holomorphic in that set such that $f(z) = h(z) + \overline{g(z)}$. Thus, $z \in S$ if and only if $0 = J_f(z) = |h'(z)|^2 - |g'(z)|^2$. Lyzzaik defines the meromorphic function $\psi = h' / g'$ and uses this function to study the local behavior of $f$ near the critical set. Note that if $z \in S$, then either $\psi(z)$ is unimodular, $\psi(z) = 0$, or $\psi$ has a pole at $z$. Since $f$ is also light, $S$ has empty interior and $h'(z) = 0 = g'(z)$ at isolated points (see Lemma \[lem:nullpartifsnotempty\].) Hence the zeros and poles of $\psi$ which lie in $S$ are isolated. In Lyzzaik’s classification of critical points, the set of isolated critical points is denoted $N$. Since $\psi$ is unimodular at the remaining points in $S$, if $z \in S \backslash N$, then $z$ lies in an analytic arc in $S$. The set of branch points of $S$ is denoted $F_3$. Hence, if $z \in S \backslash (N \cup F_3)$, in some small neighborhood of $z$, there is a unique analytic arc in $S$ with $z$ in its interior. Lyzzaik also classifies those critical points $z \in S \backslash (N \cup F_3)$ where the the image of $f$ stops. Let $\gamma \subset S \backslash (N \cup F_3)$ be an analytic arc in $S$ such that $z \in int\ \gamma$. Suppose that $f(\gamma)$ has zero speed at $f(z)$. If the argument of the tangent to $f(\gamma)$ jumps by $\pm \pi$ at $f(z)$, then $f$ is said to have a harmonic cusp at $f(z)$ and $z \in F_1$. Otherwise, if $z \in S \backslash (N \cup F_1 \cup F_3)$ and $f'(z) = 0 = g'(z)$, then $z \in F_2$.
Lyzzaik shows that $N \cup (\cup_{j=1}^3 F_j)$ consists of isolated points, and if $\gamma \subset S \backslash (N \cup F_3)$, then $f|_\gamma$ is a local homeomorphism. Further, if $z \in S \backslash (N \cup F_1 \cup F_3)$, then a subarc of $S$ with $z$ in its interior is mapped to a convex arc[^4] by $f$. Lyzzaik also shows that if $z \in F_3$, then $f(S)$ in some neighborhood of $z$ consists of convex arcs and harmonic cusps. Lyzzaik’s classification of critical points is discussed in more detail in Section \[sec:lyzzaik\] below.
We will also use Lyzzaik’s local characterization of $f$ in a neighborhood of $z_0$ for $z_0 \in S \backslash (N \cup F_3)$. This result is given below as Theorem \[thm:lyzthm5.1\]. We now review some background for this result and explain the notation.
Let $D$ denote the unit disc centered at the origin. A function $f$ is said to be locally topologically $z^n$ at $z_0$, denoted $f_{z_0} \sim z^n$, if there exist an open neighborhood $U$ of $z_0$ and homeomorphisms $h_1: U \rightarrow D$ and $h_2: \mathbb{C} \rightarrow \mathbb{C}$ such that $h_2 \circ f(z_0) = 0$ and such that $h_2 \circ f \circ h_1^{-1}(\zeta) = \zeta^n$ for all $\zeta \in D$. A result of S. Stöilow (see [@Sto:lecons] or Lemma 1 in [@AL:local]) says that if $f$ is continuous and open in a neighborhood of $z_0$, then $f$ is locally topologically $z^n$ for some positive integer $n$. Y. Abu-Muhanna and A. Lyzzaik [@AL:local] extend Stöilow’s result to points in the boundary:
\[lem:ALlem2\] Let $G^+ = \{z: |z| < 1, Im\ z > 0\}$ be the open semi-disc and $G = \{z: |z| < 1, Im\ z \ge 0\}$ be the half-closed semi-disc. Suppose that $f: G \rightarrow \mathbb{C}$ is a continuous function open in $G^+$ and topological on $G \backslash G^+$. Then for every $z_0 \in G \backslash G^+$ there is a positive integer $n$ such that $f$ at $z_0$ is locally topologically $z^{2 n - 1}$. This result also holds for sets homeomorphic to $G^+$ and $G$.
They prove this by constructing a function $F(z)$ using an idea similar to that used in proving the Schwarz reflection principle and showing that $F$ is an open map on a disc centered at $z_0$. Then they apply Stöilow’s result to $F$. The result follows by restricting $h_1$ and $F$ to $G$.
Suppose that $f$ is a harmonic function in some open set $R \subseteq \mathbb{C}$. Given $z_0 \in R \backslash S$, $f$ is either sense-preserving ($J_f(z) > 0$) or sense-reversing ($J_f(z) < 0$) in some open neighborhood of $z_0$. Suppose that $f_{z_0} \sim z^n$. If we require $h_1$ and $h_2$ to be sense-preserving homeomorphisms, we see that $f_{z_0} \sim z^n$ if $f$ is sense-preserving in a punctured neighborhood $z_0$ and that $f_{z_0} \sim \overline{z}^n$ if $f$ is sense-reversing in a punctured neighborhood of $z_0$.
We now suppose that $f$ is a light harmonic function in some simply connected, open set. Suppose that $z_0 \in int\ \gamma_0 \subset
S \backslash (F_3 \cup N)$. Then $z_0$ can be thought of as living in a shared boundary arc of two adjacent regions. In a sufficiently small neighborhood $U$ of $z_0$, $f$ is 1-1 on $\gamma = \gamma_0 \cap U$ (recall that $f|_\gamma$ is a local homeomorphism at $z_0$) and $\gamma$ splits $U$ into a sense-preserving region $R^+$ and a sense-reversing region $R^-$. Lyzzaik [@L:light] notes that Lemma \[lem:ALlem2\] applies to $R^+ \cup \gamma$ and to $R^- \cup \gamma$. The notation $f_{z_0} \sim z^j, \overline{z}^k$ means that $f$ at $z_0$ is locally topologically $z^j$ in $R^+ \cup \gamma$ and that $f$ at $z_0$ is locally topologically $\overline{z}^k$ in $R^- \cup \gamma$. Lemma \[lem:ALlem2\] also shows that $j$ and $k$ must both be odd positive integers. A structure theorem of Lyzzaik for this case (Theorem \[thm:lyzthm5.1\] below) gives the values for $j$ and $k$ based on various conditions on $z_0$.
In particular, suppose that $z_0 \in S \backslash (N \cup
(\cup_{i=1}^3 F_i))$. Then Theorem \[thm:lyzthm5.1\] gives $f_{z_0} \sim z, \overline{z}$ since $g'(z_0) \neq 0$. We can assign a direction to $f(\gamma)$, which is part of the common boundary of $f(R^+)$ and $f(R^-)$. We see that the image of $U$ is folded over $f(\gamma)$; in other words, $f(U \backslash \gamma)$ lies to one side of $f(\gamma)$. The tangent line to $f(\gamma)$ at $f(z_0)$ lies in $f(U)$. One way to see this is to recall that $f$ is univalent and sense-preserving in $R^+$ ($f_{z_0} \sim z$ for $z \in R^+$.) Also, $f$ is locally 1-1 on $\gamma$. A result of P. Duren and D. Khavinson [@DK:concave] shows that $f(\gamma)$ is concave with respect to $f(R^+)$; hence the tangent line to $f(\gamma)$ at $f(z_0)$ lies in $f(U)$.
In Remark \[rem:sharedarc\], we note that Lyzzaik’s result implies that two components of $\mathbb{C} \backslash
f^{-1} (f(S) \cup C(f, \infty))$ which share a common boundary arc in $S$ must be mapped to the same component of $\mathbb{C} \backslash (f(S) \cup C(f, \infty))$. In Section \[sec:lyzzaik\], we use Lyzzaik’s structure theorem to compare the valence in two components of $\mathbb{C} \backslash (f(S) \cup C(f, \infty))$ when the two components share a common boundary arc in $f(S)$.
Partitioning results for $C^1$ mappings in $\mathbb{R}^2$ {#sec:c1part}
=========================================================
We show that $f(S) \cup C(f)$ partitions $\mathbb{R}^2$ into regions of constant valence when $f: R \rightarrow \mathbb{R}^2$ is $C^1$ in an open set $R \subseteq \mathbb{R}^2$. We then examine a corresponding partition of $R$.
Regions of constant valence for $\mathbf{C^1}$ mappings in $\mathbb{R}^2$
-------------------------------------------------------------------------
\[lem:c1lsc\] Let $R$ be open in $\mathbb{R}^2$. Let $f: R \rightarrow \mathbb{R}^2$ be $C^1$. Let $w_0 \in \mathbb{R}^2 \backslash \overline{f(S)}$. Suppose that $Val(f, w_0) \ge N_0 \ge 0$, where $N_0$ is finite. Then there exists an open neighborhood of $w_0$, say $W_0$, such that $Val(f, w) \ge N_0$ for all $w \in W_0$.
Trivial if $\overline{f(S)} = \mathbb{R}^2$, so suppose that $\mathbb{R}^2 \backslash
\overline{f(S)} \neq \varnothing$. The result is also trivial if $N_0 = 0$, so we will suppose that $N_0 > 0$. Since $Val(f, w_0) \ge N_0$, choose $N_0$ distinct points $z_1, ..., z_{N_0}$ in $f^{-1}(w_0)$. By assumption, these points will be in the open set $R \backslash f^{-1}(\overline{f(S)})$. By the inverse function theorem, we may find an open neighborhood of each $z_j$, say $U_j$, such that $f: U_j \rightarrow W_j$ is 1-1 and onto, where $W_j$ is an open neighborhood of $w_0$. Since $N_0$ is finite, we can choose the $U_j$ to be pairwise disjoint. Let $W_0 = \cap_{j=1}^{N_0} W_j$. Then $W_0$ is an open, non-empty set such that each $w \in W_0$ has a distinct preimage in each $U_j$. The lemma then follows.
\[lem:nvalopen\] Let $R$ be an open set in $\mathbb{R}^2$. Let $f: R \rightarrow \mathbb{R}^2$ be $C^1$. Fix $N_0 \ge 0$, finite. Then $V_{N_0}(f)$ is open in $\mathbb{R}^2$.
Let $ Q_0 = \mathbb{R}^2 \backslash (f(S) \cup
C(f)) $. Then $ Q_0 $ is open. By definition, $V_{N_0}(f) \subseteq Q_0$. If $ V_{N_0}(f) = \varnothing $, the claim is vacuously true. Suppose that $ V_{N_0}(f) \neq \varnothing $; hence we also have $Q_0 \neq \varnothing$. Choose $w_0 \in V_{N_0}(f)$. Thus, $Val(f,w_0) = N_0$ and $w_0 \in Q_0$.
We first note that there exists an open neighborhood $ \tilde{B} \subseteq Q_0$ of $w_0$ such that $Val(f,w) \ge N_0$ for all points $w \in \tilde{B}$. Since $\overline{f(S)} \subseteq f(S) \cup C(f)$, this follows from Lemma \[lem:c1lsc\].
Since $ \tilde{B}$ is open, we may choose $\epsilon_0 > 0$ such that $B(w_0, \epsilon_0) \subseteq \tilde{B}$. Suppose that we can find $\{\epsilon_j\}_{j=1}^{\infty}$ with $\lim_{j \rightarrow \infty}
\epsilon_j = 0$, such that for each $j$, $0 < \epsilon_j < \epsilon_0$ and there exists $w_j \in B(w_0, \epsilon_j)$ with $Val(f, w_j) > N_0$. By construction, $w_j \rightarrow w_0$ and $Val(f, w_j) \geq N_1$ where $N_1 = N_0 + 1$. Without loss of generality, we may suppose that the $w_j$ are distinct. We will show that $Val(f,w_0) \geq N_1 > N_0 =
Val(f,w_0)$, a contradiction.
We may find $ z_{j1}, z_{j2}, \ldots, z_{jN_1} \in R
\backslash f^{-1} \left( f(S) \cup C(f) \right) $, distinct, such that $f(z_{jk}) \\
= w_j$ for $ k = 1, 2, \ldots, N_1 $, since $w_j \notin f(S) \cup C(f)$ and $Val(f, w_j) \geq N_1$. Moreover, $ \bigcup_{k=1}^{N_1}
\{ z_{jk} \}_{j=1}^{\infty} $ consists of pairwise distinct points, since the $ w_j $ are distinct.
1. $ \bigcup_{k=1}^{N_1}
\{ z_{jk} \}_{j=1}^{\infty} $ is bounded: This is obvious if $R$ is bounded. So, suppose that $R$ is unbounded. Suppose that $\bigcup_{k=1}^{N_1}
\{ z_{jk} \}_{j=1}^{\infty} $ is unbounded. Thus, we may find $ \{ z_{j_lk_l} \}_{l=1}^{\infty} $ where $ k_l \in \{ 1, 2, \ldots, N_1 \} $ such that $ |z_{j_lk_l}| \rightarrow \infty $ as $ l \rightarrow \infty $. By the pigeonhole principle, we may find a subsequence where $ k_l $ is fixed; *i.e.*, $ \{ z_{lN} \}_{l=1}^{\infty} $ for some such that $ |z_{lN}| \rightarrow \infty $ as $ l \rightarrow \infty $. Since $ \{ z_{lN} \}_{l=1}^{\infty}
\subseteq \{ z_{jN} \}_{j=1}^{\infty} $ and $ \lim_{j \rightarrow \infty}
f(z_{jN}) = \lim_{j \rightarrow \infty}
w_j = w_0 $, $ \lim_{l \rightarrow \infty}
f(z_{lN}) = w_0 $. Since $ |z_{lN}| \rightarrow \infty $, $ w_0 \in
C(f) $. But $ C(f) \cap Q_0 = \varnothing $ and $ w_0
\in Q_0 $ by assumption. Thus $ \bigcup_{k=1}^{N_1}
\{ z_{jk} \}_{j=1}^{\infty} $ is bounded.
2. If $z$ is a finite cluster point of $ \bigcup_{k=1}^{N_1}
\{ z_{jk} \}_{j=1}^{\infty} $, then $z \in R$ and $ f(z) = w_0 $: Find $ \{ z_{j_lk_l} \}_{l=1}^{\infty} $ where $ k_l \in \{ 1, 2, \ldots, N_1 \} $ such that $ \lim_{l \rightarrow \infty}
z_{j_lk_l} = z $. As above, we may find a subsequence of this sequence with $ k_l $ constant, say $ \{ z_{lN} \}_{l=1}^{\infty}$ such that $z_{lN} \rightarrow z$. Since $ \{ z_{lN} \}_{l=1}^{\infty}
\subseteq \{ z_{jN} \}_{j=1}^{\infty} $ and $ f(z_{jN}) \rightarrow w_0 $, $ \lim_{l \rightarrow \infty}
f(z_{lN}) = w_0 $. Since $z$ is a cluster point of a subset of $R$, either $z \in R$ or $z \in \partial R$. If $z \in \partial R$, we have $w_0 \in C(f)$, a contradiction. Hence $z \in R$. Since $ z_{lN} \rightarrow z $ and $f$ is continuous at $z$, $ f(z) = w_0 $.
3. $ \bigcup_{k=1}^{N_1}
\{ z_{jk} \}_{j=1}^{\infty} $ has at least $ N_1 $ distinct cluster points in $R$: By (1), all of the cluster points are finite. By (2), all of these cluster points are in $R$. Let $z$ be a cluster point. Suppose that for each $ \epsilon > 0 $, there is a $ j>0 $ such that $ |z_{jk_1} - z| < \epsilon $ and $ |z_{jk_2} - z| < \epsilon $ for some choice of $ k_1, k_2 $ with $ k_1 \neq k_2 $. But for each such $j$, $ f(z_{jk_1}) = w_j =
f(z_{jk_2}) $, with $ z_{jk_1} \neq z_{jk_2} $, so $f$ is not locally 1-1 at $z$; hence $ z \in S $. By (2), $ f(z) = w_0 \in Q_0 $ with $ Q_0
\cap f(S) = \varnothing $, so $ z \notin S $, a contradiction. So, there exists $ \epsilon > 0 $ where for each $ j > 0 $ such that $ | z_{jk} - z | < \epsilon $ holds, it holds for exactly one value of $k$ for that choice of $j$. Thus we must have at least $ N_1 $ cluster points in $R$.
From (1)-(3), we see that $ w_0 $ has at least $ N_1 $ distinct preimages in $R$, which gives the desired contradiction. With this contradiction, we have shown that $\exists\ \epsilon > 0$ such that $Val(f, w) = N_0$ for all $w \in B(w_0, \epsilon)$. Since $w_0 \in V_{N_0}(f)$ is arbitrary, $V_{N_0}(f)$ is open in $\mathbb{R}^2$.
\[lem:infval\] Let $f$ be a $C^1$ mapping defined in an open set $R \subseteq \mathbb{R}^2$. If $Val(f, w_0) = \infty$, then $w_0 \in f(S) \cup C(f)$.
Since $Val(f, w_0) = \infty$, we may choose a sequence $\{z_n\} \subseteq
f^{-1}(w_0)$ consisting of distinct points. If $\{z_n\} \subseteq f^{-1}(w_0)$ has a bounded subsequence (which we also denote by $\{z_n\}$) converging to a point $z^* \in R$, we will show that $z^* \in S$. Given $\epsilon > 0$, we can find $N$ such that $|z_n - z^*| < \epsilon$ for all $n > N$. By continuity, $f(z^*) = w_0$. Hence $f$ is not locally 1-1 at $z^*$. By the inverse function theorem, $J_f(z^*) = 0$. Hence $z^* \in S$ and $w_0 = f(z^*) \in f(S)$.
Otherwise, we may suppose that either $\{z_n\}$ has a subsequence converging to a finite point in $\partial R$ or has an unbounded subsequence. In either case, since each point of the subsequence is mapped to $w_0$, we have $w_0 \in C(f)$.
**Comment:** It would be nice to know if the following result is similar to known results in differential topology or of other applications of similar results.
\[thm:c1part\] Let $R$ be open in $\mathbb{R}^2$. Let $f: R \rightarrow \mathbb{R}^2$ be $C^1$. Then $f(S) \cup C(f)$ partitions $\mathbb{R}^2$ into regions of constant valence.
Let $\varphi(w) = Val(f, w)$. It is enough to show that $\varphi$ is a continuous, integer-valued function at each $w \in
\mathbb{R}^2 \backslash (f(S) \cup C(f))$. Choose $w_0 \in
\mathbb{R}^2 \backslash (f(S) \cup C(f))$. By Lemma \[lem:infval\], $\varphi(w_0)$ is a finite integer, say $j$. Hence $w_0 \in V_j$. By Lemma \[lem:nvalopen\], $V_j$ is open. By definition, $V_j \subseteq \mathbb{R}^2 \backslash
(f(S) \cup C(f))$. Hence there exists $\delta > 0$ such that $B(w_0, \delta) \subseteq V_j$ and $\varphi(w) = j$ for all $w \in B(w_0, \delta)$. Thus $\varphi$ is a continuous, integer-valued function in every region off $f(S) \cup C(f)$ and the conclusion follows.
From Lemma \[lem:c1lsc\] and Theorem \[thm:c1part\], we see that $Val(f,w)$ is lower semi-continuous on $\mathbb{R}^2 \backslash \overline{f(S)}$ for points with finite valence.
\[ex:c1poly\] $f(x,y) = (x^2 + y^2, 2xy)$
Here $R = \mathbb{R}^2$, so $C(f) = C(f, \infty)$. Clearly $|f| \rightarrow \infty$ as $z \rightarrow \infty$, so $C(f, \infty) = \varnothing$. The critical set consists of the lines $y = x$ and $y = -x$. $f$ maps the critical set to the rays $y = x$ and $y = -x$ for $x \ge 0$. A calculation shows that $f(S)$ partitions $\mathbb{R}^2$ into regions of constant valence. In particular, each point with $x > 0$ and $|y| < x$ has four distinct preimages. The origin has one preimage. Each point in the image of the critical set in the right half plane has two preimages. The remaining points have no preimages. Note that the behavior of $Val(f,w)$ on the partitioning set is consistent with $Val(f,w)$ being lower semi-continuous on $\mathbb{R}^2 \backslash \overline{f(S)}$. If we rewrite $f$ as $f = u + iv$ where $z = x + iy$, it is clear that $f(z)$ is not harmonic.
We can say a little about the behavior of $Val(f,w)$ when $Val(f,w)$ is infinite.
\[lem:c1infvallsc\] Let $R$ be open in $\mathbb{R}^2$. Let $f: R \rightarrow \mathbb{R}^2$ be $C^1$. Suppose that $Val(f, w_0) = \infty$ and that $f^{-1}(w_0) \backslash S$ contains an infinite number of distinct points. Then, given $N_0 \ge 0$, finite, there exists an open neighborhood of $w_0$, say $W_0$, such that $Val(f, w) \ge N_0$ for all $w \in W_0$.
Obvious if $N_0 = 0$, so assume $N_0 > 0$. Choose $\{z_1, ..., z_{N_0}\} \subseteq
f^{-1}(w_0) \backslash S$, distinct. By the inverse function theorem, for each $z_j$, there exists an open neighborhood of $z_j$, say $B_j$, such that $f$ is 1-1 on $B_j$ and $f(B_j)$ is open. We may choose the $B_j$ so that they are pairwise disjoint. Let $W_0 = \cap f(B_j)$.
Example \[ex:flatpoly\] below demonstrates why we must require that $w_0$ have an infinite number of distinct preimages off of the critical set. In this example, the origin has infinite valence and every neighborhood of the origin contains a point with no preimages. However, the preimages of the origin all lie in the critical set.
\[thm:harmlsc\] Let $f$ be a $C^1$ mapping in an open set $R \subseteq \mathbb{R}^2$. Let $V$ be a connected component of $\mathbb{R}^2 \backslash (f(S) \cup C(f))$ and let $N_0 = Val(f, V)$. Let $w_0 \in \partial V \backslash \overline{f(S)}$. Then $Val(f, w_0) \le N_0$.
By Theorem \[thm:c1part\] and Lemma \[lem:infval\], $f$ has constant, finite valence on $V$; hence $N_0$ is finite. Suppose that $Val(f, w_0) > N_0$. Then, by Lemma \[lem:c1lsc\], there exists an open neighborhood $V_0$ of $w_0$ such that $Val(f, w) > N_0$ for all $w \in V_0$. Since $w_0 \in \partial V$, $V_0 \cap V$ is a non-empty open set. For $w \in V_0 \cap V \subseteq V$, $Val(f, w) > N_0 = Val(f, w)$, a contradiction and the theorem follows.
The preceding two results show that $Val(f,w)$ is lower semi-continuous off of $\overline{f(S)}$. Consider $f(z) = z + Re\ e^z$ in Example \[ex:transharm\] below. Since $R = \mathbb{C}$, $C(f) = C(f, \infty)$. Each $w \in f(S) \cup C(f, \infty)$ has exactly one preimage. Each point in $f(S)$ has a neighborhood containing points with no preimages and points with two preimages, so this example shows why $f(S)$ is excluded in the preceding result. If $w_0 \in C(f, \infty)$, $w_0$ lies in a horizontal line separating a region where $Val(f, w) = 2$ from a region where $Val(f, w) = 1$. Thus, for all $w$ in a sufficiently small neighborhood of $w_0$, $Val(f, w) \ge Val(f, w_0) = 1$, in accord with the result above.
\[rem:emptyint\] The results above are vacuous if the partitioning set $f(S) \cup C(f)$ fills the plane. When does the partitioning set have empty interior? If $int(f(S)) = \varnothing$, $f(S) \cup C(f)$ will have empty interior iff $int (C(f)) = \varnothing$. Why? Let $U$ be an open subset of $f(S) \cup C(f)$. Since $C(f)$ is closed, $U \backslash C(f)$ is an open subset of $f(S)$. Since $f(S)$ has empty interior, either $U = \varnothing$ or $U$ is a non-empty subset of $C(f)$.
\[thm:c1dense\] Let $R$ be open in $\mathbb{R}^2$. Let $f: R \rightarrow \mathbb{R}^2$ be a $C^1$ mapping such that $S$ is nowhere dense. Suppose that $C(f)$ has non-empty interior. Then points with infinite valence are dense in the interior of $C(f)$.
Suppose not. Then we can choose $w_0 \in $ int $C(f)$ and some $\epsilon_0 > 0$ such that $f$ has finite valence at each point in $B(w_0, \epsilon_0)$.
Since $w_0$ is a cluster point of $f$, there exist $\{z_n\} \subset R$ and $z_0 \in \partial R \cup \{\infty\}$ such that $z_n \rightarrow z_0$ and $f(z_n) \rightarrow w_0$. Since $w_0$ has finite valence, we may choose $z_N$ such that $w_0 \neq f(z_N) \in
B(w_0,\frac{\epsilon_0}{2}) \subset B(w_0, \epsilon_0)$. If $z_N \notin S$, let $\zeta_1 = z_N$. Otherwise, by the continuity of $f$ in the open set $R$, $\exists\
\delta > 0$ such that $B(z_N, \delta) \subset R$ and $f(B(z_N,\delta)) \subset
B(w_0,\frac{\epsilon_0}{2})$. Since $f$ has finite valence at each point in $B(w_0, \epsilon_0)$ and since $S$ is nowhere dense in $R$, choose $\zeta_1 \in B(z_N,\delta)
\backslash S$ such that $f(\zeta_1) \neq w_0$. Let $w_1 = f(\zeta_1)$. Since $f$ is continuous at $\zeta_1$, we may choose $0 < \epsilon_1 <
\frac{\epsilon_0}{2}$ and $0 < \delta_1 <
dist(\zeta_1, \partial R)\ /\ 2$ such that $f(B(\zeta_1, \delta_1)) \subset
\overline{B(w_1, \epsilon_1)}
\subset B(w_0, \epsilon_0)$. Since $\zeta_1 \in R \backslash S$, by the inverse function theorem, we can find non-empty open subsets $U_1 \subseteq B(\zeta_1, \delta_1)$ and $V_1 \subseteq B(w_1, \epsilon_1)$ such that $\zeta_1 \in U_1$, $w_1 \in V_1$ and $f: U_1 \rightarrow V_1$ is 1-1, onto. Moreover, $f(U_1) = V_1 \subset
B(w_0, \epsilon_0) \subset C(f)$.
Repeat the preceding argument with $w_1$ in place of $w_0$ to find $z_N \in R \backslash B(\zeta_1, \delta_1)$ such that $f(z_N) \notin \{w_0, w_1\}$ and $f(z_N) \in V_1$. This gives us $\zeta_2 \in B(z_N,\delta) \backslash S$ such that $w_2 = f(\zeta_2) \notin \{w_0, w_1\}$. Arguing as above, we may choose $0 < \epsilon_2 < \epsilon_1$ such that $\overline{B(w_2, \epsilon_2)} \subset V_1$. Similarly, we may choose $0 < \delta_2 < dist(\zeta_2, \partial R)\ /\ 2$ such that $B(\zeta_2, \delta_2) \cap B(\zeta_1, \delta_1) =
\varnothing$ and such that $f(B(\zeta_2, \delta_2)) \subseteq B(w_2, \epsilon_2)$. As above, we can find non-empty open subsets $U_2 \subseteq B(\zeta_2, \delta_2) \subset R$ and $V_2 \subseteq B(w_2, \epsilon_2)$ such that $\zeta_2 \in U_2$, $w_2 \in V_2$, and $f: U_2 \rightarrow V_2$ is 1-1, onto. Moreover, $f(U_2) = V_2 \subseteq B(w_2, \epsilon_2)
\subset V_1 \subset
B(w_1, \epsilon_1) \subset B(w_0, \epsilon_0)
\subset C(f)$. By construction, $U_1 \cap U_2 = \varnothing$.
Continue in this manner to get a sequence of nested non-empty open sets $V_n \subseteq B(w_n, \epsilon_n)$ with $\overline{B(w_n, \epsilon_n)} \subset
B(w_{n-1}, \epsilon_{n-1})$ such that $w_n \rightarrow
w^* = \cap \overline{B(w_n, \epsilon_n)}
\subset B(w_0, \epsilon_0)$. Thus, $Val(f, w^*)$ is finite. But $w^*$ has a preimage in each $U_n$ where the $U_n$ are by construction pairwise disjoint. This contradicts $w^*$ having a finite number of distinct preimages in $R$.
We are not claiming that points with infinite valence are only in the interior of $C(f)$. In Example \[ex:flatpoly\], the origin has infinite valence and is in $C(f)$. However, $C(f)$ has empty interior.
Partitioning the preimage by $f^{-1}(f(S) \cup C(f))$
-----------------------------------------------------
Suppose that $R$ is an open set in $\mathbb{R}^2$ and that $f: R \rightarrow \mathbb{R}^2$ is $C^1$. Recall that $f(S) \cup C(f)$ is closed. We have seen that $f(S) \cup C(f)$ partitions the plane into components of constant valence. What does this tell us about the behavior of $f$ in a component of $R \backslash f^{-1}(f(S) \cup C(f))$?
\[thm:preimonto\] Let $R$ be an open set in $\mathbb{R}^2$. Suppose that $f: R \rightarrow \mathbb{R}^2$ is a $C^1$ mapping. Let $R_0$ be a connected component of $R\ \backslash f^{-1}(f(S) \cup C(f))$ and choose $z_0 \in R_0$. Let $w_0 = f(z_0)$ and choose the connected component $\Omega_0 \subseteq\
\mathbb{R}^2\ \backslash (f(S) \cup C(f))$ such that $w_0 \in \Omega_0$. Then $f(R_0) = \Omega_0$.
Note that the result holds vacuously if $f^{-1}(f(S) \cup C(f)) = R$. Suppose that $R \backslash f^{-1}(f(S) \cup C(f))
\neq \varnothing$.
We first show that $f(R_0) \subseteq \Omega_0$. Since $R_0$ is connected and $f$ is continuous, $f(R_0)$ is connected. Since $w_0 \in f(R_0)$, by our choice of $\Omega_0$, $f(R_0) \subseteq \Omega_0$.
It remains to show that $\Omega_0 \backslash f(R_0)$ is empty. Suppose not. Since $R_0$ is open and $R_0 \cap S$ is empty, $f$ is an open map on $R_0$; hence $f(R_0)$ is open. Since $f(R_0)$ is open and a proper subset of the component $\Omega_0$, $\exists\ \tilde{w} \in \Omega_0 \cap
(\ \overline{f(R_0)}\ \backslash\ f(R_0))$. By Lemma \[lem:infval\], $Val(f, w_0)$ is finite. Suppose that $Val(f, w_0) = N_0$. By Theorem \[thm:c1part\], the valence of $f$ is constant in $\Omega_0$; hence $Val(f, \tilde{w}) = N_0$. Thus $\exists\ z_1, ..., z_{N_0}$ distinct in $R\ \backslash f^{-1}(f(S)
\cup C(f))$ such that $f(z_j) = \tilde{w}$ for $j=1,..., N_0$. Since $\tilde{w} \notin f(R_0)$, $\{z_1, ..., z_{N_0}\} \cap R_0 = \varnothing$.
For each $z_j$, we may find an open neighborhood of $z_j$ in $R$, say $B_j$, such that $B_j \cap S = \varnothing$, $B_j \cap R_0 = \varnothing$, and such that $f$ is 1-1 and an open map on $B_j$. We may choose the $B_j$ to be pairwise disjoint. Then $V =\ \cap_{j = 1}^{N_0} \ f(B_j)$ is open and non-empty. Also, $\tilde{B_j} = f^{-1}(V) \cap B_j$ is an open neighborhood of $z_j$ where $f$ is 1-1 and such that $\tilde{B_j} \cap R_0$ is empty. Since $\tilde{w} \in V$ and $\tilde{w} \in \overline{f(R_0)}\ $, $\exists\ w \in f(R_0)$ such that $w \in V$. Thus, each of the pairwise disjoint $\tilde{B_j}$ contains one preimage of $w$. But, $R_0$ also contains at least one preimage of $w$. Thus, $Val(f, w) \geq N_0 + 1$. However, $w \in \Omega_0$, so $Val(f, w) = N_0$, a contradiction. Thus, $f(R_0) = \Omega_0$.
\[lem:n0preims\] Let $f$, $R_0$, $\Omega_0$, and $w_0$ be as in Theorem \[thm:preimonto\]. Suppose that $w_0$ has exactly $n_0$ distinct preimages in $R_0$, where $0 < n_0 \leq\ Val(f, w_0)$. Then every $w \in \Omega_0$ has exactly $n_0$ distinct preimages in $R_0$.
This proof was suggested by D. Sarason. By Theorem \[thm:c1part\] and Lemma \[lem:infval\], $Val(f, w) = Val(f, w_0) = N_0 < \infty$ for all $w \in \Omega_0$. By Theorem \[thm:preimonto\], $1 \le Val(f|_{R_0}, w) \le N_0$ for all $w \in \Omega_0$. Let $W_j = \{w \in \Omega_0:
Val(f|_{R_0}, w) = j\}$. Then $\Omega_0 = \cup_{j = 1}^{N_0} W_j$. Clearly, the $W_j$ are pairwise disjoint. If the $W_j$ are open, then only one of the $W_j$ is nonempty since $\Omega_0$ is connected. Further, since $Val(f|_{R_0}, w_0) = n_0$, the result follows.
It remains to show that $W_j$ is open for $j > 0$. This follows from Lemma \[lem:nvalopen\] if $W_j = V_j(f|_{R_0})$. By construction, the critical set of $f|_{R_0}$ is empty; hence $V_j(f|_{R_0}) = \{w \notin C(f|_{R_0}):
Val(f|_{R_0}, w) = j\}$. We need to show that $C(f|_{R_0})$ is disjoint from $\Omega_0$. Let $\{z_n\} \subset R_0$ converge to a point $z_0 \in \partial R_0 \cup \{\infty\}$. If $z_0 \in \partial R \cup \{\infty\}$, then if $\{f|_{R_0}(z_n)\}$ has a finite cluster point, this point is in $C(f)$, hence not in $\Omega_0$. On the other hand, if $z_0 \in R$, then $f$ is continuous at $z_0 \in f^{-1}(f(S) \cup C(f))$ and $\{f|_{R_0}(z_n)\}$ converges to a point in $f(S) \cup C(f)$, which, again, is not in $\Omega_0$. Thus $V_j(f|_{R_0}) = W_j$ and $W_j$ is open.
Thus, $f^{-1}(f(S) \cup C(f))$ partitions $R$ into components, each of which is mapped onto a component of $\mathbb{R}^2 \backslash (f(S) \cup C(f))$ by $f$. We will now show that $f$ is a covering map on each component of this partition of the preimage. First, we recall a standard result (see [@Mun:top], page 341):
\[thm:munch8th4.5\] Let $p: (E, e_0) \rightarrow (B, b_0)$ be a covering map. If $E$ is path connected, then there is a surjection $\phi: \pi_1(B, b_0) \rightarrow p^{-1}(b_0)$. If $E$ is simply connected, $\phi$ is a bijection.
\[thm:preimcover\] Let $R$ be an open set in $\mathbb{R}^2$. Suppose that $f: R \rightarrow \mathbb{R}^2$ is $C^1$. Let $R_0$ be a connected component of $R \backslash f^{-1}(f(S) \cup C(f))$. Choose $z_0 \in R_0$ and let $w_0 = f(z_0)$. Suppose that $w_0$ has exactly $n_0$ distinct preimages in $R_0$. Then $R_0$ is a $n_0$-fold covering of $\Omega_0$. Moveover, if $\Omega_0$ is simply connected, then $f$ is univalent in $R_0$.
We first show that $f: R_0 \rightarrow \Omega_0$ is a covering map. By Theorem \[thm:preimonto\], $f$ is onto; $f$ is continuous by assumption. Choose $w_0 \in \Omega_0$. By Lemma \[lem:n0preims\], $w_0$ has $n_0$ distinct preimages in $R_0$. Constructing $V$ and the $\tilde{B_j}$ as in the proof of Theorem \[thm:preimonto\] (except that we choose $B_j \subset R_0$), each of the pairwise disjoint open sets $\tilde{B_j}$ is homeomorphic to $V$ since $R_0 \cap S = \varnothing$. Also, by Lemma \[lem:n0preims\], each $w \in V$ has exactly $n_0$ distinct preimages in $R_0$. Thus, $R_0 \cap f^{-1}(V) =
\cup_{j = 1}^{n_0} \tilde{B_j}$. Thus, $V$ is evenly covered by $f|_{R_0}$. Since $w_0$ is arbitrary, $f|_{R_0}$ is a covering map and $R_0$ is a covering space of $\Omega_0$. Since $f^{-1}(w)$ has $n_0$ distinct elements in $R_0$ for each $w \in \Omega_0$, $R_0$ is a $n_0$-fold covering of $\Omega_0$.
We note that $R \backslash f^{-1}(f(S) \cup C(f))$ is open in $\mathbb{R}^2$. Hence the component $R_0$ is open in $\mathbb{R}^2$. Since $R_0$ is an open, connected subset of $\mathbb{R}^2$, given any two points in $R_0$, we may find a polygonal path contained in $R_0$ that joins the two points. Hence, $R_0$ is path connected. Choose $w_0 \in \Omega_0$. By Theorem \[thm:munch8th4.5\], there is a surjection $\phi: \pi_1 (\Omega_0, w_0) \rightarrow f^{-1}(w_0)$, where we are restricting $f$ to $R_0$. If we also assume that $\Omega_0$ is simply connected, then $\pi_1 (\Omega_0, w_0)$ is the trivial group. Since $\phi$ is a surjection, $w_0$ must have exactly one preimage in $R_0$. Hence $n_0 = 1$ and $f$ is univalent on $R_0$.
\[ex:c1trans\] $f(x,y) = (x \cos y,\ y)$
Here, $R = \mathbb{R}^2$ and $C(f) = C(f, \infty)$. If we rewrite $f$ as $f = u + iv$ where $z = x + iy$, it is clear that $f(z)$ is $C^1$ but not harmonic. A calculation shows that $$\begin{array}{l}
\ S = \{(x, \frac{(2k + 1) \pi}{2}):
x \in \mathbb{R}$ and $
k \in \mathbb{Z}\}\\
\ f(S) = \{(0, \frac{(2k + 1) \pi}{2}):
k \in \mathbb{Z}\}\\
\end{array}$$ Also, $C(f, \infty)$ consists of the horizontal lines $y = \frac{(2k + 1) \pi}{2}$ where $k$ is an integer. Each $w \in \mathbb{R}^2
\backslash (f(S) \cup C(f, \infty))$ has exactly one preimage. Each $w \in f(S)$ has an infinite number of preimages. Each $w \in C(f, \infty) \backslash f(S)$ has no preimages. Note that if we fix $a \in \mathbb{R}$ and let $y_n = \cos^{-1}(a / x_n)$, then $f(x_n, y_n) \rightarrow (a,
\frac{(2k + 1) \pi}{2}) \in C(f, \infty)$ as $x_n \rightarrow \infty$, provided that we choose the branch of $\cos^{-1}$ such that $\cos^{-1}(a / x_n) \rightarrow
\frac{(2k + 1) \pi}{2}$.
The partitioning set of our domain is $f^{-1}(f(S) \cup C(f, \infty))$, which is $S$ (a collection of horizontal lines.) It is clear that if $(x, y) \notin S$, then $f(x, y) \neq (0, \frac{2 k + 1}{2} \pi)$. If we choose $w = (a, b) \in \mathbb{R}^2$ such that $w \notin f(S) \cup C(f, \infty)$, then $b \neq \frac{2 k + 1}{2} \pi$. It is clear that $w$ has exactly one preimage; namely, $(a / \cos b, b)$ and that this preimage point does not lie in $S$. Hence, $f^{-1}(f(S) \cup C(f, \infty))$ partitions $\mathbb{R}^2$ into horizontal strips where $f$ is univalent.
Adjacent components of the preimage
-----------------------------------
A non-empty, connected set is said to be degenerate if it consists of a single point. If two distinct components of our partition of $R$ share a non-degenerate common boundary arc in $R$, will $f$ map both to the same component of $\mathbb{R}^2 \backslash (f(S) \cup C(f))$?
\[lem:preimadj\] Let $R \subseteq \mathbb{R}^2$ be open. Let $f: R \rightarrow \mathbb{R}^2$ be $C^1$. Let $R_1$ and $R_2$ be distinct components of $R\ \backslash\
f^{-1} (f(S) \cup C(f))$ such that $\overline{R_1} \cap \overline{R_2} \neq \varnothing$. Suppose that $f(R_1) = f(R_2) = \Omega$. If $\partial \Omega \cap int\ \overline{\Omega}
= \varnothing$, then there exists no non-empty set $\gamma \subseteq\
(\overline{R_1}\cap\overline{R_2} \cap R)\
\backslash\ S$ such that $R_1 \cup R_2 \cup \gamma$ is open.
By contradiction. Suppose that $R_1$ and $R_2$ are disjoint components of $R\ \backslash\
f^{-1} (f(S) \cup C(f))$ such that $f(R_1) = f(R_2)$. Suppose also that there exists $\gamma \subseteq
(\overline{R_1}\cap\overline{R_2} \cap R)\
\backslash\ S$ such that $\gamma \neq \varnothing$ and such that $R_1 \cup R_2 \cup \gamma$ is open. By Theorem \[thm:preimonto\], there exist $\Omega_1, \Omega_2$, components of $\mathbb{R}^2\ \backslash\ (f(S) \cup C(f))$, such that $f(R_1)=\Omega_1$ and $f(R_2)=\Omega_2$. By assumption, $\Omega_1=\Omega_2 = \Omega$.
Since $R_1$ and $R_2$ are disjoint components of $R\ \backslash\
f^{-1} (f(S) \cup C(f))$ and $\gamma \subseteq \overline{R_1} \cap
\overline{R_2} \cap R$, $\gamma \subseteq f^{-1} (f(S) \cup C(f))$. Thus $f(\gamma) \subseteq f(S) \cup C(f)$. By continuity, $f(\gamma) \subseteq \overline{\Omega}$, so $f(\gamma) \subseteq \partial \Omega$. Now let $R_0 = R_1 \cup R_2 \cup \gamma$. By assumption, $R_0$ is open. Since $R_0 \cap S = \varnothing$, $f$ is an open map on $R_0$ by the inverse function theorem and $f(R_0)$ is open. But $f(R_0) = f(R_1) \cup f(R_2) \cup f(\gamma)
= \Omega \cup f(\gamma)$ and we have seen that $f(\gamma) \subseteq \partial \Omega$. Since $f(R_0)$ is open, $f(R_0) \subseteq int\ \overline{\Omega}$. Choose $z_0 \in \gamma$ (recall that $\gamma \neq \varnothing$.) Then $f(z_0) \in \partial \Omega \cap f(R_0)
\subseteq \partial \Omega \cap int\
\overline{\Omega}$. Thus $\partial \Omega \cap int\ \overline{\Omega}
\neq \varnothing$, a contradiction.
\[lem:harmadj\] Let $R \subseteq \mathbb{R}^2$ be open. Let $f: R \rightarrow \mathbb{R}^2$ be a light $C^1$ function. Let $R_1$ and $R_2$ be distinct components of $R\ \backslash\
f^{-1} (f(S) \cup C(f))$ such that $\overline{R_1} \cap \overline{R_2} \neq \varnothing$. Suppose that $f(R_1) = f(R_2) = \Omega$. If $\partial \Omega \cap int\ \overline{\Omega}$ consists of a finite number of points, then there exists no non-empty, non-degenerate connected set $\gamma \subseteq\
(\overline{R_1}\cap\overline{R_2} \cap R)\
\backslash\ S$ such that $R_1 \cup R_2 \cup \gamma$ is open.
By contradiction. Suppose that $R_1$ and $R_2$ are disjoint components of $R\ \backslash\
f^{-1} (f(S) \cup C(f))$ such that $f(R_1) = f(R_2)$. Suppose also that there exists $\gamma \subseteq
(\overline{R_1}\cap\overline{R_2} \cap R)\
\backslash\ S$ such that $\gamma \neq \varnothing$ and such that $R_0 = R_1 \cup R_2 \cup \gamma$ is open.
Let $P = \partial \Omega \cap int\ \overline{\Omega}$. By assumption, $P = \{w_1, w_2, ... w_n\}$. If $n = 0$, the result follows from Lemma \[lem:preimadj\]. Assume that $n > 0$. By assumption, $\gamma$ is a non-empty, non-degenerate connected set. Since $f$ is continuous, $f(\gamma)$ is connected. So, if $f(\gamma) \subseteq P$, then $f(\gamma) = \{w_j\}$ for some fixed value of $j$. This contradicts $f$ being a light mapping. Thus $f(\gamma) \backslash P \neq \varnothing$. Choose $w \in f(\gamma) \backslash P$. By the arguments used in the proof of Lemma \[lem:preimadj\], $f(\gamma) \subseteq \partial \Omega$. As in Lemma \[lem:preimadj\], $f(R_0)$ is open and $w \in int\ \overline{\Omega}$. Thus $w \in \partial \Omega \cap
int\ \overline{\Omega} = P$, a contradiction.
\[cor:planadj\] Let $f$ be a light $C^1$ function in $\mathbb{R}^2$. Let $\Omega$ be a component of $\mathbb{R}^2 \backslash (f(S) \cup C(f, \infty))$. Suppose that $R_1$ and $R_2$ are two distinct components of $\mathbb{R}^2 \backslash
f^{-1}(f(S) \cup C(f, \infty))$ such that $f(R_1) = f(R_2) = \Omega$.
1. If $\partial \Omega \cap int\ \overline{\Omega}$ consists of a finite number of points, then $(\overline{R_1} \cap \overline{R_2})
\backslash S$ contains no non-degenerate connected set $\gamma$ such that $R_1 \cup R_2 \cup \gamma$ is open.
2. Suppose that $\partial \Omega \cap int\ \overline{\Omega}$ consists of a finite number of points. Also suppose that $(int\ \overline{R_j}) \cap \partial \overline{R_j}$ consists of a finite number of points and that $\partial (int\ \overline{R_j})$ is a Jordan curve for $j = 1, 2$. If $\overline{R_1} \cap \overline{R_2}$ contains a non-degenerate Jordan arc, then this arc is in $S$.
Using the notation of the preceding two lemmas, $R = \mathbb{R}^2$. So $\overline{R_1} \cap \overline{R_2} \cap R =
\overline{R_1} \cap \overline{R_2}$. The first claim follows from Lemma \[lem:harmadj\].
The second claim follows by noting that each region $R_j$ is a Jordan region that has a finite number of puncture points. We are assuming that the boundaries intersect in a non-degenerate Jordan arc $\rho$. Since we can find subregions of the two regions, each having $\rho$ as part of its boundary, that are Jordan regions (no punctures), $R_1 \cup R_2 \cup int\ \rho$ is open. By the first claim, we must have that $(int\ \rho) \backslash S$ is totally disconnected. Since $S$ is closed, $\rho \subseteq S$.
Behavior in shared boundaries and “puncture points”
---------------------------------------------------
Let $f$ be a $C^1$ function on some open set $R \subseteq \mathbb{R}^2$. Suppose that we have a collection of components of $R \backslash f^{-1}(f(S) \cup C(f))$ where $f$ is univalent on each component. Can we combine these components to get a larger region where $f$ is univalent? The results in the preceding section (such as Corollary \[cor:planadj\]) handle components that share a common boundary arc in $R$, but say nothing about the behavior on the boundary arc.
If $R_0$ is one such component, we want to know more about the behavior of $f$ at points in $\partial R_0 \cap R$. These could be points in boundary arcs shared with other components or could be “puncture points.” A “puncture point” is an isolated point in $\partial R_0$. Does $f$ map $\partial R_0$ onto $\partial(f(R_0))$? What can we say about $Val(f|_{\partial R_0}, w)$ for $w \in \partial(f(R_0))$?
\[lem:harmbdyim0\] Let $R \subseteq \mathbb{R}^2$ be open. Let $f$ be $C^1$ in $R$. Let $R_0 \subset \mathbb{R}^2$ be a component of $R\ \backslash\
f^{-1} (f(S) \cup C(f))$. Then $f(R \cap \partial R_0) \subseteq
f(R) \cap \partial(f(R_0))$.
By Theorem \[thm:preimonto\], there exists a component of $\mathbb{R}^2\ \backslash\ (f(S) \cup C(f))$, say $\Omega_0$, such that $f(R_0) = \Omega_0$. Thus $\partial (f(R_0)) = \partial \Omega_0$. Requiring that $R_0$ is a proper subset of $\mathbb{R}^2$ guarantees that $\partial R_0 \neq \varnothing$ in the finite plane. If $\partial R_0 \subseteq \partial R$, the result holds vacuously. Suppose that $R \cap \partial R_0 \neq \varnothing$.
Since $R \cap \partial R_0 \subseteq
f^{-1} (f(S) \cup C(f))$, $f(R \cap \partial R_0) \subseteq f(S) \cup C(f)$. Fix $z_0 \in R \cap \partial R_0$. Then there exists a sequence $\{z_n\} \subset R_0$ such that $z_n \rightarrow z_0$. Since $z_0 \in R$, by the continuity of $f$ at $z_0$, $f(z_n) \rightarrow f(z_0)$. But $f(z_n) \in \Omega_0$ for all $n$ and $f(z_0) \in f(S) \cup C(f)$. Hence $f(z_0) \in
f(R) \cap \overline{\Omega_0} \cap
(f(S) \cup C(f))\ \subseteq\
f(R) \cap \partial \Omega_0$, so $f(R \cap \partial R_0) \subseteq\
f(R) \cap \partial \Omega_0$.
\[lem:harmbdyim\] Let $f$ be $C^1$ in $\mathbb{R}^2$. Let $R_0$ be a bounded component of $\mathbb{R}^2 \backslash
f^{-1} (f(S)\\ \cup C(f, \infty))$. Then $f(\partial R_0) = \partial(f(R_0))$.
By Theorem \[thm:preimonto\], there exists a component of $\mathbb{R}^2\ \backslash\ (f(S) \cup C(f, \infty))$, say $\Omega_0$, such that $f(R_0) = \Omega_0$. Thus $\partial (f(R_0)) = \partial \Omega_0$. By Lemma \[lem:harmbdyim0\], we have $f(\partial R_0) \subseteq
f(\mathbb{R}^2) \cap \partial \Omega_0$. So $f(\partial R_0) \subseteq \partial \Omega_0$.
We now show that $\partial \Omega_0 \subseteq f(\partial R_0)$. Fix $w_0 \in \partial \Omega_0$. Then there exists a sequence $\{w_n\} \subset \Omega_0$ such that $w_n \rightarrow w_0$. Since $f(R_0) = \Omega_0$, there exist $z_n \in R_0$ such that $f(z_n) = w_n$ for each $n$. By assumption $R_0$ is bounded, so $\overline{R_0}$ is compact and $\{z_n\}$ has a convergent subsequence $\{z_{n_k}\}$. Since $w_n \rightarrow w_0$, $w_{n_k} \rightarrow w_0$. Let $z_{n_k} \rightarrow\ z_0 \in \overline{R_0}$. Since $f$ is continuous in $\mathbb{R}^2$, $f(z_0) = w_0$. Since $w_0 \in f(S) \cup C(f, \infty)$, $z_0 \in f^{-1} (f(S) \cup C(f, \infty))$. Thus $z_0 \in \overline{R_0} \cap
(f^{-1} (f(S) \cup C(f, \infty)))\ =\ \partial R_0$. Since $w_0 \in \partial \Omega_0$ is arbitrary, we have $\partial \Omega_0 \subseteq f(\partial R_0)$. The result follows.
Let $\Omega \subseteq \mathbb{R}^2$ and $f$ be a function defined on $R \subseteq \mathbb{R}^2$. Recall that $Val(f|_R, \Omega)$ denotes the valence of $f|_R$ in $\Omega$. Note that if the valence of $f|_R$ is constant and finite in $\Omega$, then $Val(f|_R, \Omega) = Val(f|_R, w)$ for any choice of $w \in \Omega$.
\[thm:harmbdypreim\] Let $f$ be $C^1$ in $\mathbb{R}^2 $. Let $R$ be a component of $\ \mathbb{R}^2\ \backslash
f^{-1}(f(S) \cup C(f, \infty))$ and let $\Omega$ be a component of $\mathbb{R}^2\ \backslash
(f(S) \cup C(f,\infty))$ such that $f(z) \in \Omega$ for some $z \in R$. Choose $w \in \partial \Omega$ such that *(i)* $w \notin C(f\mid_{R},\ \infty)$, *(ii)* $f^{-1}(w)\ \cap\ \partial R\ \cap\ S
= \varnothing$, *(iii)* $f^{-1}(w)\ \cap\ \partial R
\neq \varnothing$, and such that *(iv)* for each $\zeta \in f^{-1}(w) \cap
\partial R$, there exists a neighborhood of $\zeta$, say $U_\zeta$, such that $f(U_\zeta\ \backslash\ R)\ \cap\ \Omega = \varnothing$. Then $w$ has exactly $Val(f\mid_{R}, \Omega)$ distinct preimages on $\partial R$.
By Theorem \[thm:preimonto\], $f(R) = \Omega$. Moreover, by Lemma \[lem:n0preims\], if $\eta \in \Omega$ and if $N = Val(f\mid_{R}, \eta)$, then $Val(f\mid_{R}, \Omega) = N$. Choose $w \in \partial \Omega$ satisfying the given conditions. By condition (iii), $w$ has at least one preimage on $\partial R$. Note that if $R$ is bounded, this condition follows from Lemma \[lem:harmbdyim\]. Suppose that $w$ has $M$ distinct preimages on $\partial R$.
We first show that $M \ge N$: Since $w \in \partial \Omega$, we may choose $\{w_j\} \subset \Omega$, where the $w_j$ are distinct, such that $w_j \rightarrow w$. For each $j$, there exist $z_{j1}, z_{j2}, \ldots , z_{jN}$ distinct in $R$ such that $f(z_{jk}) = w_j$. Let $Q = \bigcup_{k=1}^{N}
\{ z_{jk} \}_{j=1}^{\infty}$. So $Q \subset R$. Using arguments analogous to those used to prove Lemma \[lem:nvalopen\], we see that $Q$ has at least $N$ distinct cluster points. We also see that the cluster points of $Q$ are in $\partial R$. Hence $M \ge N$.
We now show that $M \le N$: Suppose not. Then we can find distinct values $z_1, \ldots, z_{N+1}$ such that $z_j \in \partial R$ and $f(z_j) = w$. By (ii), $z_j \notin S$. So, for each $j$, we may find some neighborhood of $z_j$, say $V_j$, where $f$ is a homeomorphism. We may choose the $V_j$ so that they are pairwise disjoint. By (iv), for each $z_j$, there exists a neighborhood $U_j$ of $z_j$ such that the preimages in $U_j$ of all points in $f(U_j)\ \cap\ \Omega$ lie in $R$. Let $B_j = U_j\ \cap\ V_j$. Then the $B_j$ are pairwise disjoint and each $B_j$ is a non-empty, open neighborhood of $z_j$ where $f$ is a homeomorphism. Since $z_j \in \partial R$, $B_j\ \cap R\ \neq\ \varnothing$; hence $f(B_j)\ \cap\ \Omega\ \neq\ \varnothing$. By construction, all preimages in $B_j$ of points in $f(B_j)\ \cap\ \Omega$ lie in $R$. Since $z_j \in B_j$ and $f(z_j) = w$, $\bigcap_{j = 0}^{N+1} f(B_j) \neq \varnothing$. Choose $\eta \in \Omega\ \cap\
(\bigcap_{j = 1}^{N + 1} f(B_j))$. Then $\eta$ has a distinct preimage in each $B_j$, giving a total of $N+1$ distinct preimages in $\bigcup_{j=1}^{N+1} B_j$. Further, each of these preimages lies in $R$. But, $\eta$ has exactly $N$ distinct preimages in $R$, a contradiction. Thus, $M = N$ and $w$ has exactly $Val(f\mid_{R}, \Omega)$ distinct preimages on $\partial R$.
We note that
1. We will apply this result to prove Lemma \[lem:puncturepoints\] below.
2. The condition $w \notin C(f\mid_{R},\ \infty)$ automatically holds if either $C(f, \infty)$ is empty or if $R$ is bounded.
\[thm:jointwoc1\] Let $f$ be a light, $C^1$ function in $\mathbb{R}^2$. Let $R_1$ and $R_2$ be disjoint bounded components of $\ \mathbb{R}^2 \backslash
f^{-1}(f(S) \cup C(f, \infty))$. Suppose that $R_1$ and $R_2$ each have at most a finite number of puncture points. Suppose that for each component $\Omega \subseteq
\mathbb{R}^2 \backslash (f(S) \cup C(f, \infty))$, $\partial \Omega \cap int\ \overline{\Omega}$ consists of a finite number of points. Suppose that $f$ is univalent on $R_1$ and on $R_2$. Let $\gamma$ be a non-degenerate, simple arc in $\partial R_1 \cap \partial R_2$. Suppose that $f(int\ \gamma)$ is a simple arc. If $S\ \cap\ int\ \gamma = \varnothing$, then
1. $f(R_1)$ and $f(R_2)$ are distinct components of $\ \mathbb{R}^2 \backslash (f(S) \cup C(f,\infty))$.
2. $f$ is a univalent on $R_1\ \cup\ R_2\ \cup\ int\ \gamma$.
By construction, the combined region, $R_1\ \cup\ R_2\ \cup\ int\ \gamma$, is open (see the proof of Corollary \[cor:planadj\].) The first claim follows from Lemma \[lem:harmadj\]. To prove the second claim, we need to show that $f$ is univalent on $int\ \gamma$. Then $f$ is univalent in the combined region (since no point in $R_1 \cup R_2$ is mapped to $f(int\ \gamma)$.) Note that $f(int\ \gamma)$ is non-degenerate since $f$ is light.
We now check that $f$ is univalent on $int\ \gamma$. Suppose not. Then we have two distinct points $\zeta_1,
\zeta_2 \in int\ \gamma$ such that $f(\zeta_1) = f(\zeta_2) = w_0
\in f(S) \cup C(f, \infty)$. Since $S \cap\ int\ \gamma = \varnothing$, we can find a neighborhood $B_1$ of $\zeta_1$ and a neighborhood $B_2$ of $\zeta_2$ where $f$ is an open map. We may assume that these neighborhoods are disjoint and are contained in the combined region. However, $f(B_1) \cap f(B_2)$ is an open neighborhood of $w_0$. So it contains a point $w \in \Omega_1$. By construction, $w$ has at least two distinct preimages in $R_1$ (one in $B_1 \cap R_1$ and another in $B_2 \cap R_1$), which contradicts the univalence of $f$ in $R_1$.
Extension of partitioning results to $C^1$ mappings in $\mathbb{R}^n$
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The notation defined at the beginning of Section \[sec:notation\] for the critical set and cluster set has obvious generalizations to the case of $C^1$ functions defined in an open set in $\mathbb{R}^n$, where $n$ is a positive integer. Moreover, the comments at the beginning of Section \[sec:notation\] concerning the properties of these sets hold. For the most part, the proofs in Section \[sec:c1part\] do not make use of special properties of $\mathbb{R}^2$. Thus, the partitioning results above can be extended to $C^1$ functions defined in open subsets of $\mathbb{R}^n$, with the exceptions of Lemma \[lem:harmadj\], Corollary \[cor:planadj\], and Theorem \[thm:jointwoc1\].
Lemma \[lem:harmadj\] and the first part of Corollary \[cor:planadj\] hold in $\mathbb{R}^n$ for $n > 1$. The hypothesis concerning the non-degenerate set $\gamma$ cannot hold for $n = 1$. The proofs of Theorem \[thm:jointwoc1\] and of the second part of Corollary \[cor:planadj\] use Jordan curves and Jordan regions; these are $n = 2$ results.
Partitioning results for harmonic mappings in an open set {#sec:harm}
=========================================================
We will apply our results about $C^1$ functions in an open subset of the real plane to the case where our function is harmonic in an open subset of the complex plane. Unless explicitly stated otherwise, a harmonic function is a complex-valued harmonic function.
\[thm:harmpart\] Let $f$ be a harmonic mapping in an open set $R \subseteq \mathbb{C}$. Then $ f(S) \cup C(f) $ partitions $\mathbb{C}$ into regions of constant valence.
Identify $\mathbb{R}^2$ with $\mathbb{C}$ by putting $x = Re\ z$ and $y = Im\ z$. Let $u = Re\ f$, $v = Im\ f$. Then apply Theorem \[thm:c1part\].
Analogs to the other results in the preceding section are shown similarly.
\[ex:transharm\] A transcendental harmonic function with finite valence. $$f(z)= z + Re\ e^z$$
This example was given by D. Bshouty, W. Hengartner and M. Naghibi-Beidokhti in [@BHN:dilat] as an example of a 2-valent, transcendental harmonic mapping in $\mathbb{C}$. Calculations show that $$C(f,\infty)=\{\zeta : Im\ \zeta = \frac{(2k + 1)\pi}
{2},\ k \in \mathbb{Z}\}$$ and $$S = \{z : Re\ e^z = -1\}$$ Letting $z=x+iy$ and fixing $\zeta=a+ib$, $f(z)=\zeta$ when $y=b$ and $x + e^x \cos b=a$. The behavior of $\varphi(x) = x + e^x \cos b$ depends on the sign of $\cos b$ and determines the number of preimages of $\zeta$. One can then check that $f(S) \cup C(f,\infty)$ partitions $\mathbb{C}$ into regions of constant valence. In particular, each $w \in f(S) \cup C(f, \infty)$ has exactly one preimage. If $Im(w)\ mod\ 2\pi \in (-\pi/2, \pi/2)$, $w$ also has exactly one preimage. If $Im(w)\ mod\ 2\pi \in (\pi/2, 3\pi/2)$, then $w$ has two distinct preimages if $w$ lies to the left of $f(S)$ and no preimages if $w$ lies to the right of $f(S)$. Note that if $\zeta = a + i\ \frac{(2 k + 1) \pi}{2}
\in C(f, \infty)$, then $f(z_n) \rightarrow \zeta$ for $z_n = x_n + i\ \cos^{-1}((a -x_n) \exp(-x_n))$ with $x_n \rightarrow \infty$ and the proper choice for the branch of $\cos^{-1}$.
\[ex:polyunbdds\] A finite valence harmonic polynomial with an unbounded critical set. $$f(z) = 2(Re(z^3) + iz)$$
One can explicity check that $$\{\zeta: Im\ \zeta = 0\} \cup
\{\zeta: Re\ \zeta = \frac{(Im\ \zeta)^3}{4} +
\frac{1}{3\ Im\ \zeta}\ \}$$ partitions $\mathbb{C}$ into regions of constant valence.
One can also check that $S = \{z: xy = -\frac{1}{6},\ $where$\ x = Re\ z\ $ and $y = Im\ z\}$ and that $C(f,\infty) = \{\zeta: Im\ \zeta = 0\}$. For example, fix $a \in \mathbb{R}$; then $f(z_n) \rightarrow a \in C(f, \infty)$ if $z_n = x_n\ +\ i\ \{a / 2\ -\ (1/ (3 x_n))\}$ and $x_n \rightarrow 0$.
A calculation shows that $f(S) = \{2 (Re(z))^3 + \frac{1}{6Re(z)} + 2i Re(z):
Re(z) \neq 0\}$, which is equivalent to $\{\zeta: Re\ \zeta = \frac{(Im\ \zeta)^3}{4} +
\frac{1}{3\ Im\ \zeta}\ \}$. Thus $f(S) \cup C(f\infty)$ partitions $\mathbb{C}$ into regions of constant valence. Each $w \in C(f, \infty)$ has exactly one preimage, as does each $w \in f(S)$. Looking at $f(S)$ in $\mathbb{R}^2$, one sees that it consists of two parabola-like curves pointing sideways. Each $w$ in the region “inside" these curves has no preimage. Outside of this region, each $w \notin (f(S) \cup
C(f, \infty))$ has two distinct preimages.
Since $C(f,\infty) \neq \varnothing$, $f$ is an example of a finite valence harmonic polynomial where $f$ does not go to $\infty$ as $z \rightarrow \infty$.
It is easy to check that $f^{-1}(f(S)) = S$. Another easy calculation shows that $f^{-1}(C(f, \infty))$ is the imaginary axis. Recall that $w \notin f(S) \cup C(f, \infty)$ has either zero or exactly two distinct preimages. Suppose we choose $w \notin f(S) \cup C(f, \infty)$ such that $Val(f, w) = 2$. Note that $Im\ w \neq 0$, since $w \notin C(f, \infty)$. Since $f(x, y) = 2 (x^3 - 3 x y^2 - y + i x)$, $Re(f^{-1}(w)) = \frac{1}{2}\ Im\ w$. Since $w \notin f(S)$, $Im(f^{-1}(w)) \neq -1\ / (3\ Im\ w)$. The vertical line $Re\ z = \frac{1}{2}\ Im\ w$ intersects $S$ at $(\frac{1}{2}\ Im\ w, -1 / (3\ Im\ w))$. The two preimages of $w$ are the endpoints of a vertical line segment bisected by $S$. From this, we see that $f^{-1}(f(S) \cup C(f, \infty))$ partitions $\mathbb{C}$ into regions where $f$ is univalent.
\[ex:flatpoly\] $f(z) = 2[Re(z^2) + i (Re(z) - Im(z))]$
One can readily check that $S = \{z: Re(z) = Im(z)\}$ and $f(S) = \{0\}$. Further, $C(f, \infty)$ is the real axis. If we rewrite $f$ letting $z = x + iy$, we see that $f(x, y) = 2 (x - y) (x + y + i)$. A calculation shows that if $w \notin
\mathbb{R}$, then $w$ has exactly one preimage. If $w$ is a non-zero real, it has no preimage. And, the origin is a point of infinite valence. Also, if we fix $a \in \mathbb{R}$ and let $y_n = x_n\ -\ (a / (4 x_n))$, then $f(z_n) \rightarrow a \in C(f, \infty)$ for $z_n = x_n + i\,y_n$ as $|x_n| \rightarrow \infty$.
\[ex:quadraticpreim\] $f(z) = z^2 + z - \overline{z}$
An easy calculation shows that $S = \{z: |z + \frac{1}{2}| = \frac{1}{2}\}$. This circle is mapped to the hypocycloid in Figure \[fig:quadratic\_imag\]. Clearly, $C(f, \infty) = \varnothing$.
D. Sarason (personal communication) has analyzed the behavior of $f$. Among many observations about $f$, he shows that $f$ is 1-1 on $S$, 4-valent in the bounded component of $\mathbb{C} \backslash f(S)$, and 2-valent on the portion of the real axis in the unbounded component. Now consider our partition of the domain (by $f^{-1}(f(S))$, since $C(f, \infty) = \varnothing$); this is shown in Figure \[fig:quadratic\_preim\]. Sarason’s analysis shows that $f$ is a 1-1 mapping of the interior of each of the bounded components onto the bounded component of $\mathbb{C} \backslash f(S)$; hence $f$ is univalent on each of the bounded components of the partition of the domain. He also showed that all $g(z) = p(z) - \overline{z}$, where $deg\ p = 2$, can be reduced to $f(z)$ by means of affine transformations of the domain and range.
By Theorem \[thm:harmpart\], $f$ must be 2-valent on the unbounded component of $\mathbb{C} \backslash f(S)$ since real values in the unbounded component of $\mathbb{C} \backslash f(S)$ are 2-valent. It follows from that Lemma \[lem:n0preims\] that $f$ must be 2:1 on the unbounded component of $\mathbb{C} \backslash f^{-1}(f(S))$.
![Image of critical set for $f(z) =
z^2 + z - \overline{z}$[]{data-label="fig:quadratic_imag"}](quadratic_imag.eps)
![Preimage of $f(S)$ for $f(z) =
z^2 + z - \overline{z}$[]{data-label="fig:quadratic_preim"}](quadratic_preim.eps)
It is nicest when our partitioning set $f(S) \cup C(f)$ has empty interior. Note that the results above are vacuous if the partitioning set fills the complex plane. For example, $f(z) = e^z$ is harmonic in $\mathbb{C}$. It is easy to see that $C(f) = \mathbb{C}$. In this case, $Val(f, w) = \infty$ if $w \neq 0$ and $Val(f, 0) = 0$.
If $z_0 \in R \subseteq \mathbb{C}$, where $R$ is open, then there exists a simply connected, open set $U \subseteq R$ such that $z_0 \in U$ (for example, $B(z_0, \delta)$ for $\delta > 0$ sufficiently small.) Suppose that $f$ is harmonic in $R$. Since $U$ is simply connected, there exist functions $h$ and $g$ holomorphic in $U$ such that $f = h + \overline{g}$ in $U$. We have $J_f = |h'|^2 - |g'|^2$ in $U$. Thus, $z \in S \cap U$ iff $|h'(z)| = |g'(z)|$. If $R$ is a simply connected, open set, then we can take $U = R$.
\[lem:nullpartifsnotempty\] Let $f$ be harmonic in a simply connected, open set $R \subseteq \mathbb{C}$. If $int\ S \neq \varnothing$, then $f(z) = \alpha\ +\ \beta u(z)$, where $u$ is a real harmonic function and $\alpha$, $\beta$ are complex constants. Moreover, $S = R$ and $Val(f, w_0) = \infty$ for every $w_0 \in f(R)$.
Since $R$ is simply connected, there exist functions $g$ and $h$ holomorphic in $R$ such that $f = h + \overline{g}$ in $R$.
First, suppose that the zeros of $g^\prime$ are not isolated in $R$. By the identity theorem, we must have $g^\prime \equiv 0$ in $R$; thus $g$ is constant in our connected set $R$. Since $int\ S \neq \varnothing$, there exists a non-empty open disc $U \subseteq S$. Since $g' \equiv 0$, $h^\prime \equiv 0$ in $U$. Since $h$ is constant in $U$, $h$ is constant in our connected set $R$. Hence, $f = h + \overline{g}$ is constant and maps $R$ to a single point; that point thus has infinite valence. The representation of $f$ follows trivially. Also note that $S = R$.
Now, suppose that the zeros of $g^\prime$ are isolated. We may then choose a non-empty open set $U \subseteq S$ such that $g^\prime(z) \neq 0$ for all $z \in U$. Then $\psi(z) = h^\prime(z) / g^\prime(z)$ is a holomorphic function on $U$ and $|\psi| \equiv 1$ on $U$. Take a non-empty closed disc $K \subset U$. By the maximum modulus principle, since $|\psi| \equiv 1$ on $K$, $\psi \equiv \lambda$ on $K$ for some unimodular constant $\lambda$. Moreover, $h^\prime - \lambda g^\prime \equiv 0$ on $int\ K$. Thus, $h^\prime - \lambda g^\prime \equiv 0$ on $R$ and $h^\prime(z) = \lambda g^\prime(z)$ for all $z \in R$. We thus have that $S = R$. Further, for some constant $\alpha \in \mathbb{C}$, $h = \alpha + \lambda g$. Choose $\tau$ such that $\tau^2 = \lambda$. Then $f = h + \overline{g} =
\alpha + 2 \tau\ Re(\tau g) =
\alpha + \beta u$ in $R$, where $\beta = 2 \tau$ and $u = Re(\tau g)$ is a real-valued harmonic function in $R$. Choose $w_0 \in f(R)$. Then we may choose $z_0 \in R$ such that $f(z_0) = w_0$. Thus, $\alpha = w_0 - \beta u(z_0)$. Consider $v(z) = u(z) - u(z_0)$. Since the zeros of a real-valued harmonic function are not isolated, $v$ has an infinite number of distinct zeros in $R$; hence $Val(f, w_0) = \infty$.
The preceding lemma can also be found in [@L:light]. If $f$ is harmonic in a simply connected, open set in $\mathbb{C}$ and if $int\ S \neq \varnothing$, then the preceding lemma implies that $f(R)$ is either a point or a subset of a line. Thus, $int(f(S) \cup C(f)) = \varnothing$. Moreover, if $w \in \mathbb{C} \backslash (f(S) \cup C(f))$, then $Val(f, w) = 0$.
\[lem:snotempty\] Let $f$ be harmonic in an open set $R \subseteq \mathbb{C}$. If $int\ S \neq \varnothing$, then there exists a point $w_0 \in f(R)$ such that $Val(f, w_0) = \infty$. Moreover, there exists an open, non-empty set $R_1$, with $R_1 \subseteq S \subseteq R$, such that $f(R_1)$ is either a point or is a subset of a line.
Choose $z_0 \in int\ S$. Then we can find $\delta > 0$ such that $B(z_0, \delta) \subseteq int\ S$. Let $R_1 = B(z_0, \delta)$. Then $R_1$ is a non-empty, simply connected, open set and we may apply Lemma \[lem:nullpartifsnotempty\] to $f|_{R_1}$. The result follows.
\[cor:goodpart\] Let $f$ be harmonic in an open set $R \subseteq \mathbb{C}$. If $Val(f, w)$ is finite for each $w \in \mathbb{C}$, then $int(f(S) \cup C(f)) = \varnothing$.
Since $f$ is $C^\infty$ on $R$, $f(S)$ has Lebesgue measure 0 by Sard’s Theorem; hence, $f(S)$ has empty interior. By Lemma \[lem:snotempty\], since we assume that $Val(f,w)$ is finite for each $w \in \mathbb{C}$, we must have $int\ S = \varnothing$. Since $S$ is relatively closed in $R$, we have that $S$ is nowhere dense in $R$. Since we have assumed that $\{w: Val(f,w) = \infty\} = \varnothing$, $int (C(f)) = \varnothing$ by Theorem \[thm:c1dense\]. The result follows from Remark \[rem:emptyint\] and Theorem \[thm:c1dense\].
The results above relate to a theorem of T. J. Suffridge and J. W. Thompson [@ST:partition]. We need a definition before stating the theorem.
\[Suffridge and Thompson\] Assume $f$ is harmonic in $\{z: 0 < |z-\alpha| < r\}$ for some $r > 0$. Define $\alpha$ to be a pole of $f$ provided $lim_{z \rightarrow \alpha} |f(z)| = \infty$.
\[Suffridge and Thompson\] \[thm:stpartition\] Let $f$ be non-constant and harmonic, except for a finite number of poles, in a simply connected region $D$ in $\mathbb{C}$. Let $C$ be a Jordan curve contained within $D$ not passing through a pole. Let $\Omega$ be the region bounded by $C$. Let $X$ be the exceptional set of $f$ in $D$. Then the image space is partitioned into open components by $f(C \cup X)$ such that all values in a given component are assumed the same number of times in $\Omega$.
The “exceptional set of $f$” consists of points in $D$ where the dilatation of $f$ (defined as $\overline{f_{\overline{z}}}/ f_z$) is unimodular. The proof of Theorem \[thm:stpartition\] in [@ST:partition] assumes that the dilatation is analytic in $D$; this occurs if $S$ has no isolated critical points. When $S$ has no isolated critical points, $S = X$. As Example \[ex:isolcrit\] below demonstrates, we must include the images of isolated critical points in the partitioning set.
We can correct this difficulty and extend Theorem \[thm:stpartition\] to the case of less restrictive conditions on $D$ and its boundary by using the partitioning results above, as follows:
\[cor:stgen\] Let $f$ be harmonic in a bounded, open set $R \subset \mathbb{C}$. Suppose that $P = int (\overline{R}) \backslash R$ is empty or consists of a finite number of poles of $f$. Suppose also that $f$ can be continuously extended to $\partial R \backslash P$. Then $f(S) \cup f(\partial R \backslash P)$ partitions $\mathbb{C}$ into regions of constant valence.
If we have a sequence $\{z_n\} \subset R$ approaching $z \in P$, then $f(z_n) \rightarrow \infty$. Thus, since $R$ is bounded, $C(f) = \{w: \exists \{z_n\} \subset R$ such that $f(z_n) \rightarrow w$ and $z_n \rightarrow
z_0 \in \partial R \backslash P\}$. Since $f$ can be continuously extended to $\partial R \backslash P$, $C(f) = f(\partial R \backslash P)$. Then apply Theorem \[thm:harmpart\].
\[ex:isolcrit\] $f(z) = \frac{1}{3}\ z^3 - \frac{1}{2}\ \overline{z}^2$
In this example, we will be looking at $f$ restricted to $R = B(0, 1)$. This example illustrates why the image ($w = 0$) of the isolated critical point ($z = 0$) must be included in the partitioning set. We will look at $f$ as a function in $\mathbb{C}$ in Example \[ex:plane\_isolcrit\] below.
Looking at $f$ in $B(0,1)$, calculations show that $S = \{0\}$ and that $C(f) = f(\{z: |z| = 1\})$. The partitioning set is graphed in Figure \[fig:isolcrit\_imaglabel\] (see Example \[ex:plane\_isolcrit\] below) where the cusps are identified by numbers. Note that the partitioning set is the same set of points as in Example \[ex:plane\_isolcrit\].
We will explicitly find the preimages of $w = 0$ (Claim 2) and show that the valence is not constant in a sufficiently small neighborhood of $w = 0$ (Claim 3.) It is helpful to rewrite $f$ as a function of $(x,y)$ where $z = x + iy$:
$f(x, y) = \frac{1}{3}x^3 - x y^2 - \frac{1}{2}x^2
+ \frac{1}{2}y^2 + i y\ (x^2 - \frac{1}{3}y^2 + x)$.
**Claim 1:** Let $y^2 = 3 x\ (x + 1)$. Then $(x, y) \in B(0,1)$ iff $0 \le x < 1/4$.
Note that $(x, y) \in R = B(0,1)$ iff $0 \le x^2 + y^2 < 1$. Using our condition on $y^2$, we require $-1 \le 4 x^2 + 3 x - 1 < 0$. We note that $\varphi(x) = 4 x^2 + 3 x - 1$ has zeros at $x = -1$, $1/4$ and attains its minimum at $x = -3/8$. So we require $-1 < x < 1/4$. Now, $\varphi(0) = -1$ and $\varphi(-3/8) < -1$. We also need $y^2 = 3 x\ (x + 1) \ge 0$ and that forces us to require $x \ge 0$.
**Claim 2:** $Val(f|_R, 0) = 1$.
To satisfy $Im\ f(x, y)\ =\ 0$, we must have $y = 0$ or $y^2 = 3 x\ (x + 1)$. We must also satisfy $Re\ f = 0$. We can rewrite this as $(3 - 6 x)\ y^2 + x^2\ (2 x -3) = 0$. It is easy to check that this equation has no solutions when $x = \frac{1}{2}$. Thus, we must have $y^2 = \{x^2\ (2 x - 3)\} / (6 x - 3)$. To get $f = 0$ when $y = 0$, we must have $x = 0$ or $\frac{3}{2}$. To get $f = 0$ when $y^2 = 3 x\ (x + 1)$ , we must have $x = \frac{3}{8}(-1 \pm \sqrt{5})$; neither value of $x$ satisfies Claim 1. Also note that the corresponding values of $y^2$ for both choices of $x$ are positive, so these correspond to preimages of $0$ outside of $R$. So, $w = 0$ has six distinct preimages in $\mathbb{C}$, but only one preimage $R = B(0,1)$; namely, $z = 0$.
**Claim 3:** Choose $|\epsilon| > 0$, sufficiently small. Then $Val(f|_R, \epsilon) = 2$.
As in Claim 2 above, $Im\ f = 0$ requires $y = 0$ or $y^2 = 3 x\ (x + 1)$. We also require that $Re\ f = \epsilon$.
First, let $y = 0$. Then $Re\ f = \epsilon$ becomes $\varphi(x) = 0$, where $\varphi(x) =
\frac{1}{3}x^3 - \frac{1}{2}x^2 - \epsilon$ and $-1 < x < 1$ (to get a preimage in $R$.) Now $\varphi'(x) = 0$ at $x = 0, 1$; $\varphi(0) = -\epsilon$ and $\varphi(1) = -(\epsilon + \frac{1}{6})$. Note that $\varphi(-1) = -(\frac{5}{6} + \epsilon)$. Also $\varphi''(x) = 0$ at $x = 1/2$. An elementary calculus argument shows that we have no solutions for $\varphi(x) = 0$ in $(-1, 1)$ if $0 < \epsilon < 1/6$. Also, if $-\frac{1}{6} < \epsilon < 0$, $\varphi$ has two distinct zeros in $(-1, 1)$. Thus, for $|\epsilon|$ sufficiently small:
- If $\epsilon < 0$, two distinct preimages of $w = \epsilon$ in $R$ on the real axis.
- If $\epsilon > 0$, no preimages of $w = \epsilon$ in $R$ on the real axis.
Now, let $y^2 = 3 x\ (x + 1)$. We may assume $y \ne 0$, since that case has already been handled. Then $Re\ f = \epsilon$ becomes $\varphi(x) = 0$, where $\varphi(x) =
\frac{8}{3}x^3 + 2 x^2 - \frac{3}{2} x +
\epsilon$. By Claim 1, we also require $0 \le x < 1/4$. The zeros of $\varphi'(x)$ are $x = 1/4, -3/4$, while $\varphi''(x) = 0$ at $x = -1/4$. Notice that $\varphi(0) = \epsilon$, $\varphi(1) = \frac{19}{6} + \epsilon > 0$ for $|\epsilon|$ sufficiently small, and $\varphi(1/4) = -\frac{5}{24} + \epsilon < 0$ for $|\epsilon|$ sufficiently small. Thus, for $|\epsilon|$ sufficiently small we have:
- If $\epsilon < 0$, $\varphi(x)$ has no zeros in $[0, 1/4)$. So, no preimages of $w = \epsilon$ in $R$ off the real axis.
- If $\epsilon > 0$, $\varphi(x)$ has one zero in $[0, 1/4)$; that root is not zero. Since $y^2 = 3 x\ (x + 1) \ne 0$ for $x \in (0, 1/4)$, two distinct preimages of $w = \epsilon$ in $R$ off the real axis.
The claim follows by combining the two cases.
When does the partitioning set in the preceding corollary have empty interior?
Let $f$ be harmonic in a bounded open set $R \subset \mathbb{C}$. Suppose that $P = int (\overline{R}) \backslash R$ is empty or consists of a finite number of poles of $f$. Suppose also that $f$ has a $C^1$ extension to some open set $R_1$, where $R_1 \supset \overline{R} \backslash P$. Then $f(S) \cup f(\partial R \backslash P)$ has empty interior.
The following proof was suggested by D. Sarason. Let $F$ denote our $C^1$ extension of $f$ to $R_1$ and $S_F$ denote the critical set of $F$. Let $$\begin{aligned}
A &=& \partial R \backslash (S_F \cup P) \\
B &=& S \cup (S_F \cap (\partial R \backslash P))\end{aligned}$$ Then $S \cup (\partial R \backslash P) = A \cup B$. If $F(A) \cup F(B)$ is a countable union of nowhere dense sets, then the result follows from the Baire Category Theorem.
We claim that $F(B)$ is a countable union of nowhere dense sets. By construction, $B \subseteq S_F$; hence $F(B) \subseteq F(S_F)$ and it is enough to show that $F(S_F)$ is a countable union of nowhere dense sets. $F(S_F)$ has measure 0 by Sard’s Theorem. Since $S_F$ is closed in $R_1$, $S_F$ is a countable union of compact sets. Therefore $F(S_F)$ is a countable union of compact sets, each of which has measure 0, hence each nowhere dense.
We also claim that $F(A)$ is a countable union of nowhere dense sets. We note that $F$ is a local homeomorphism at each $z \in A$, so for each $z \in A$, there exists $\epsilon > 0$ such that $F|_{B(z, 2 \epsilon)}$ is a homeomorphism. In particular, $B(z, 2 \epsilon) \cap (S_F \cup P) = \varnothing$. Then $\mathcal{C} = \{B(z, \epsilon): z \in A\}$ is an open cover of $A$. By Lindelöf’s Theorem, $\mathcal{C}$ has a countable subcover, say $\{B(z_j, \epsilon_j): z_j \in A\}$. By construction, $K_j = \overline{B(z_j, \epsilon_j) \cap A}
\subseteq A$ and we see that $A = \cup K_j$. We note that $K_j \subseteq \partial R$ is compact and has empty interior. Since $F$ is a homeomorphism on $K_j$, $F(K_j)$ is nowhere dense. The claim follows by noting that $F(A) = \cup F(K_j)$.
Application to harmonic mappings in the entire complex plane {#sec:entire}
============================================================
If $f$ is harmonic in the entire complex plane, $C(f) = C(f, \infty)$. Also, we can write $f$ as $f = h + \overline{g}$ where $h$ and $g$ are entire (holomorphic) functions. This allows us to say more about the behavior of $f$ when $S$ has non-empty interior. In particular, we can apply Lemma \[lem:nullpartifsnotempty\] as follows:
\[lem:plasnotempty\] Let $f$ be harmonic on $\mathbb{C}$. If $int\ S \neq \varnothing$, then $f(\mathbb{C})$ is either a single point or a line. In either case, $S = \mathbb{C}$ and there exists a point $w_0 \in \mathbb{C}$ such that $Val(f, w_0) = \infty$.
Let $f = h + \overline{g}$, where $g$ and $h$ are entire (holomorphic) functions. The conclusions about $S$ and the existence of a point of infinite valence follow directly from Lemma \[lem:nullpartifsnotempty\].
Now we return to the proof of Lemma \[lem:nullpartifsnotempty\]. If the zeros of $g'$ are not isolated, then $f(\mathbb{C})$ is a single point. If the zeros of $g'$ are isolated, $g$ is a non-constant entire function and there exists a unimodular constant $\lambda$ such that $h' \equiv \lambda g'$. Choosing $\tau$ such that $\tau^2 = \lambda$ and choosing $z_0$ such that $f(z_0) = w_0$, we have $f = h + \overline{g} = \alpha + 2 \tau\ Re(\tau g)$ where $\alpha = w_0 - 2 \tau\ Re(\tau g(z_0))$.
Since $\tau$ is non-zero and $g$ is a non-constant entire function, $\varphi(z) = Re(\tau g(z))$ is a non-constant, real harmonic function in the entire plane. By Liouville’s Theorem, $\varphi(z)$ can be bounded neither above nor below. Noting that the range of $\varphi$ is connected, we see that the range of $\varphi$ is the entire real axis. Hence $f(\mathbb{C})$ is a line when the zeros of $g'$ are isolated.
\[lem:planullpart\] Let $f$ be harmonic on $\mathbb{C}$. Suppose that $int\ S \neq \varnothing$. Then $int(f(S) \cup C(f,\infty)) = \varnothing$. Moreover, if $w \in \mathbb{C} \backslash
(f(S) \cup C(f,\infty))$, then $Val(f, w) = 0$.
By Lemma \[lem:plasnotempty\], $S = \mathbb{C}$ and $f(\mathbb{C})$ is either a point or a line (each is a closed set with empty interior in $\mathbb{C}$.) Clearly, $C(f,\infty) \subseteq f(S) = f(\mathbb{C})$. Hence, $f(S) \cup C(f,\infty)$ has empty interior. Also, if $w \in \mathbb{C} \backslash
(f(S) \cup C(f,\infty))$, then $w \in \mathbb{C} \backslash f(\mathbb{C})$ and thus $Val(f,w) = 0$.
We also can say more about when the partitioning set $f(S) \cup C(f, \infty)$ has empty interior. We first recall a result of Wilmshurst [@W:thesis]:
\[thm:wfinval\] Let $f(z) = \overline{g(z)} + h(z)$ be a function harmonic in the (entire) complex plane. If $ \lim_{z \rightarrow \infty}
f(z) = \infty$, then $f$ has finitely many zeros.
Note that $ \lim_{z \rightarrow \infty}
f(z) = \infty$ implies that $Val(f,w)$ is finite for each $w$ (note that this does not imply that $f$ has finite valence.) We also note that if $ \lim_{z \rightarrow \infty}
f(z) = \infty$, then $C(f, \infty) = \varnothing$. Since $f(S)$ has empty interior by Sard’s Theorem, we have that $f(S) \cup C(f, \infty) = f(S)$ has empty interior. In particular,
\[cor:inflimpart\] If $f$ is a harmonic mapping on $\mathbb{C}$ such that $ \lim_{z \rightarrow \infty}
f(z) = \infty $, then $ f(S) $ partitions $ \mathbb{C} $ into non-empty regions of constant valence.
Corollary \[cor:inflimpart\] obviously applies if $f(z) = p(z) - \overline{q(z)}$, where $p$ and $q$ are polynomials in $z$ of different degree. D. Bshouty, W. Hengartner and M. Naghibi-Beidokhti in [@BHN:dilat] use approximation theory to show that there exist harmonic mappings on $\mathbb{C}$ which are not polynomials such that $ \lim_{z
\rightarrow \infty} f(z) = \infty$. An explicit example of such a function is due to D. Sarason; consider $f(z) = Re(e^{-z^2}) + z$. Calculations also show that $f$ is onto $\mathbb{C}$. While $Val(f, w)$ is finite for each $w$, $Val(f)$ is infinite.
We will show that $f(S) \cup C(f, \infty)$ has empty interior when $f$ is a harmonic polynomial, even when the Corollary \[cor:inflimpart\] does not apply. To do so, we need to state several results. We start with a result one can find in [@W:paper]:
\[thm:wconstarc\] Let a function $f$ be harmonic in some domain $D$ and have a sequence of distinct zeros $\{z_m\}$ converging to some point $z^*$ in $D$. Then $f(z) \equiv 0$ on some simple analytic arc containing $z^*$ as an interior point. Further, there are at most finitely many such arcs unless $f(z) \equiv 0$ in $D$.
We also note two forms of Bézout’s Theorem found in Wilmshurst [@W:paper]:
\[thm:bezout\] Let $A$ and $B$ be relatively prime polynomials in the real variables $x$ and $y$ with real coefficients, and let $deg\ A = n$ and $deg\ B = m$. Then the two algebraic curves $A(x, y) = 0$ and $B(x,y) = 0$ have at most $mn$ points in common.
\[thm:waltbezout\] Let $A$ and $B$ be polynomials in the real variables $x$ and $y$ with real coefficients. If $deg\ A = n$ and $deg\ B = m$, then either $A$ and $B$ have at most $mn$ common zeros or have infinitely many common zeros.
\[rem:algcurve\] Bézout’s Theorem also provides information about a real algebraic curve in the plane. Let $P(x, y)$ be a polynomial in the real variables $x$ and $y$ with real coefficients. There exist irreducible polynomials in $x$ and $y$ with real coefficients, say $A_1(x, y), \ldots, A_k(x, y)$, such that $P = A_1^{n_1} \ldots A_k^{n_k}$. Let $V = \{(x, y): P(x, y) = 0\}$. Then $V = \cup_{j = 1}^{k} V_j$, where $V_j = \{(x, y): A_j(x, y) = 0\}$. Let $D_j = \{(x, y): \partial A_j / \partial y = 0\}$. Since the $A_j$ are relatively prime, if $i \neq j$, then $V_i \cap V_j$ is finite by Theorem \[thm:bezout\]. If $\partial A_j / \partial y \equiv 0$, then $V_j$ consists of a finite number (possibly zero) of vertical lines. If $\partial A_j / \partial y$ is not identically zero, then $V_j \cap D_j$ is finite by Theorem \[thm:bezout\] since $A_j$ is irreducible. In that case, we can divide the plane into a finite number of vertical strips such that $\partial A_j / \partial y \neq 0$ for points in $V_j$ in the interior of a given vertical strip. By the implicit function theorem, using analytic continuation, we see that $V_j$ consists of a finite number (possibly zero) of non-intersecting curves in the interior of a given vertical strip[@S:primer p. 249-253] and possibly a finite number of vertical lines. If $V_j$ contains a closed loop, we see that $V_j$ must contain a simple loop. A closed loop in $V$ is either a closed loop in one of the $V_j$ or is formed by joining arcs in $V$ terminating at points in $V_i \cap V_j$ where $i \neq j$. Since there are finitely many points in the $V_i \cap V_j$, we see that if $V$ contains a closed loop, then $V$ contains a simple loop.
We also need a fact about harmonic polynomials from [@BC:brelotchoquetlem]:
\[M. Brelot and G. Choquet\] \[lem:brelotchoquet\] If a real harmonic polynomial $p$ in two variables has a nonconstant factor $q$, then the zeros of $q$ are not isolated.
\[lem:constpolyonloop\] Let $f$ be a harmonic polynomial. Let $\gamma$ be a closed loop in $S$. If $S$ has empty interior, then $f$ cannot be constant on $\gamma$.
Since $S$ has empty interior, we can view $S$ as a real algebraic curve. By Remark \[rem:algcurve\], if $S$ contains a closed loop $\gamma$, we can find a simple loop in $\gamma$. Without loss of generality, we may suppose that $\gamma$ is a Jordan curve. Then $\gamma$ encloses some bounded, connected region, say $R$, by the Jordan curve theorem.
Assume that $f$ is constant on $\gamma$. Let $u = Re\ f$ and $v = Im\ f$. Then $u$ and $v$ are real harmonic functions in $\mathbb{C}$. Since $f$ is constant on $\gamma = \partial R$, so are $u$ and $v$. Hence, $u$ must be constant in $R$ by the minimum and maximum principles for real harmonic functions. Similarly, $v$ is constant in $R$. Thus, $f$ is constant in $R$ and hence in $\mathbb{C}$. We have $S = \mathbb{C}$, a contradiction.
The preceding proof uses an argument similar to that used by Wilmshurst (pages 42-43 in [@W:thesis]) to show that the zeros of a harmonic polynomial are isolated under certain conditions.
\[cor:partpoly\] If $f$ is a harmonic polynomial, then $int (f(S) \cup C(f, \infty)) = \varnothing$.
The result is obvious if $f$ is constant, so we will assume that $f$ is a non-constant harmonic polynomial. Following Wilmshurst [@W:paper], finding all $z$ such that $f(z) - w = 0$ is equivalent to finding the common zeros of $Re(f-w)$ and $Im (f- w)$ in $\mathbb{R}^2$, identifying $\mathbb{C}$ with $\mathbb{R}^2$ in the obvious way.
Let $n = Max\{deg\,(Re\,f), deg\,(Im\,f)\}$. By Theorem \[thm:waltbezout\], $Re (f - w)$ and $Im (f - w)$ have either at most $n^2$ common zeros or have infinitely many common zeros. Hence $Val(f) \le n^2$ or there exists $w_0 \in \mathbb{C}$ such that $Val(f, w_0) = \infty$.
If $Val(f)$ is finite, the result follows from Corollary \[cor:goodpart\]. So we may suppose that there exists $w_0 \in \mathbb{C}$ such that $Val(f, w_0) = \infty$. If $S$ has non-empty interior, we are in the simple case where $f$ maps $\mathbb{C}$ to a point or a line by Lemma \[lem:plasnotempty\], and the result follows. So, suppose also that $S$ has empty interior. Since $S$ is closed in $\mathbb{C}$, we also have that $S$ is nowhere dense. Since $f$ is harmonic, $f(S)$ has empty interior by Sard’s Theorem. We want to show that $W_\infty = \{w: Val(f, w) = \infty\}$ is nowhere dense. If that is true, the result follows from Theorem \[thm:c1dense\] and Remark \[rem:emptyint\]. We will show that $W_\infty$ is finite.
Since $Val(f, w_0) = \infty$, there are infinitely many distinct values of $z$ such that $f(z) = w_0$. By Bézout’s Theorem (Theorem \[thm:bezout\]), $Re (f - w_0)$ and $Im (f - w_0)$ have a non-trivial common factor. Choose $z_0$ such that $f(z_0) = w_0$. By the Brelot-Choquet Lemma (Lemma \[lem:brelotchoquet\]), $z_0$ is not an isolated zero of $f(z) - w_0$. Hence, by Theorem \[thm:wconstarc\], there exists a simple analytic arc $\gamma$ such that $f\mid_\gamma\ \equiv\ w_0$ and such that $z_0$ is in the interior of $\gamma$. Since $f$ is not locally 1-1 at any point in the interior of $\gamma$, $int\ \gamma \subseteq S$. Further, we can apply Theorem \[thm:wconstarc\] to each of the endpoints of $\gamma$ to see that $\gamma \subseteq S$. If $\gamma$ contains a closed loop, then $S$ has non-empty interior by Lemma \[lem:constpolyonloop\], a contradiction. So we may suppose that $\gamma$ does not contain a closed loop. Thus, for each $w \in W_\infty$, we can find a simple curve $\gamma$ in $S$ such that $f|_\gamma \equiv w$.
Since $f$ is a harmonic polynomial, $J_f(z)$ can be viewed as a polynomial in $\mathbb{R}^2$ and Remark \[rem:algcurve\] applies to $S$. Moreover, if we extend $\gamma$ by repeated application of Theorem \[thm:wconstarc\], we see that $\gamma$ extends to an unbounded curve such that $f|_\gamma \equiv w_0$. Otherwise, if our extensions of $\gamma$ have a finite limit point $z_0$, by continuity, $f(z_0) = w_0$ and we can extend $\gamma$ through $z_0$, a contradiction. From Remark \[rem:algcurve\], we see that $\gamma$ can be extended to a simple, unbounded curve in $S$. From Remark \[rem:algcurve\], we see that $S$ contains at most finitely many simple, unbounded curves. Hence $W_\infty$ contains a finite number of points and the result follows.
Example \[ex:flatpoly\] is an example of a harmonic polynomial with $int\ S = \varnothing$ and a point with infinite valence. In this example, the preimages of this point are non-isolated and lie on a curve (the line $Re\ z = Im\ z$.)
The situation is not so nice when $f$ is not a harmonic polynomial. We will restrict our attention to the case $R = \mathbb{C}$. For example, if $f$ is an entire (holomorphic) transcendental function, $C(f, \infty)$ is the complex plane by Picard’s Theorem. The monograph of M. Balk [@Balk:book] concerns the theory of polyanalytic and polyentire functions; results similar to Picard’s Theorem are given for polyentire functions (see pages 107-109.)
Recall that $\varphi(z)$ is a polyanalytic function of order $n$ if it can be written as $\varphi(z) = \Sigma_{k = 0}^{n-1}\
a_k(z) \overline{z}^k$ where each $a_k(z)$ is a holomorphic function of $z$. If each $a_k(z)$ is entire, then $\varphi$ is polyentire. Let $h$ and $g$ be holomorphic in some region $R$ and let $f = h + \overline{g}$. Then the harmonic function $f$ can be thought of as a polyanalytic function (of countable order if $g$ is transcendental.) If $g(z) = \sum_{k=0}^m b_k z^k$ is a non-constant polynomial, we can think of $f$ as a polyanalytic function of finite order by putting $a_0(z) = h(z) + \overline{b_0}$ and $a_k(z) \equiv \overline{b_k}$ for $1 \le k \le deg\ g$.
Let $P(z, \overline{z})$ denote a polynomial in $z$ and $\overline{z}$. Note that $P$ can include terms of the form $z^n \overline{z}^m$. Consider the following result from [@Balk:book] about polyentire functions:
\[thm:polyentire\] Let $E(z)$ be an entire transcendental analytic function, and let $P(z, \overline{z})$ be an arbitrary polyanalytic but not analytic polynomial. Then the polyentire function $F(z) = E(z) + P(z, \overline{z})$ assumes in the complex plane $C$ every complex value $A$ (without any exceptions!) The set $M(F; A)$ of all A-points of the polyentire function $F(z)$ is for every $A$ unbounded and discrete in $C$.
If $f = h + \overline{g}$ is harmonic in $\mathbb{C}$, the preceding result tells us that $C(f, \infty) = \mathbb{C}$ if $h$ is transcendental and $g$ a non-constant polynomial. Picard’s Theorem takes care of the case when $h$ is transcendental and $g$ a constant. This gives us a necessary condition for $f(S) \cup C(f, \infty)$ to have empty interior:
\[cor:badpart\] Let $f = h + \overline{g}$ be harmonic in $\mathbb{C}$, where $h$ and $g$ are entire. If $f(S) \cup C(f, \infty)$ has empty interior, then one of the following holds: (i) $h$ and $g$ are both polynomials in $z$ or (ii) $h$ and $g$ are both entire transcendental functions in $z$.
Does the converse hold? Corollary \[cor:partpoly\] shows that the converse holds for harmonic polynomials. We have seen some examples where $h$ and $g$ are both transcendental and $f(S) \cup C(f, \infty)$ has empty interior (see Example \[ex:transharm\] and the comments after Corollary \[cor:inflimpart\].) Both examples are of the form $f(z) = z + Re\ h(z)$ where $h$ is an entire transcendental function.
Joining regions for $f$ harmonic
--------------------------------
Let $f$ be a light harmonic function in $\mathbb{C}$. Suppose that $R_1$ and $R_2$ are two distinct components of $\mathbb{C} \backslash
f^{-1}(f(S) \cup C(f, \infty))$. Suppose that that $R_1$ and $R_2$ share a common boundary arc. Let $f(R_1) = \Omega_1$ and $f(R_2) = \Omega_2$. If $\Omega_1$ and $\Omega_2$ are each simply connected, then $f$ is univalent in $R_1$ and in $R_2$ by Theorem \[thm:preimcover\]. Can we join $R_1$ and $R_2$ along the interior of the shared boundary arc to get a new region where $f$ is univalent?
If the regions we want to combine all lie in a convex domain, a partial answer is given in [@W:thesis]:
\[thm:univconvex\] If $g(z)$ is an analytic function in the convex domain $D$ and $|g'(z)| < 1$ in $D$ then $\overline{z} + g(z)$ is univalent on $D$.
Let $h(z)$ be holomorphic in some convex domain $D$ and suppose that $|h'(z)| < 1$ in $D$. It is easy to see that this result also holds for $f(z) = h(z) + \lambda \overline{z}$ where $\lambda$ is a unimodular constant (apply the theorem to $f_1(z) = \overline{\lambda} f(z)$.)
We can apply this result to each of the bounded components of $\mathbb{C} \backslash S$ in Example \[ex:cubictwos\] below to show that $f$ is univalent on each of these two bounded components. However, this result won’t help us in the case that a bounded component of $\mathbb{C} \backslash S$ is not convex; consider for example the critical set in Example \[ex:dumbbell\] below. Also, this result will not help us find univalent regions in the unbounded component of $\mathbb{C} \backslash S$ nor will it help us with more complicated harmonic functions.
What do the results in the preceding sections tell us about finding univalent regions? By identifying $\mathbb{C}$ with $\mathbb{R}^2$ in the obvious way, we may apply the preceding results.
\[lem:puncturepoints\] Let $f$ be a light harmonic function in $\mathbb{C}$. Let $R$ be a bounded component of $\mathbb{C} \backslash f^{-1}(f(S) \cup C(f, \infty))$. Choose a component $\Omega \subseteq
\mathbb{C} \backslash (f(S) \cup C(f, \infty))$ such that $f(R) = \Omega$. Suppose that $w_0$ is an isolated point in $\partial \Omega$. If $f^{-1}(w_0)\ \cap\ \partial R\ \cap\ S = \varnothing$, then $w_0$ has exactly $Val(f|_R, \Omega)$ distinct preimages in $\partial R$. These preimages are isolated points in $\partial R$.
Note that $\Omega$ exists by Theorem \[thm:preimonto\]. We first check that $\zeta \in f^{-1}(w_0) \cap \partial R$ is isolated in $\partial R$. Suppose not. We first note that $\zeta$ is isolated in $f^{-1}(w_0)$. Otherwise, by Theorem \[thm:wconstarc\], there exists a non-degenerate arc $\gamma$ such that $f$ is constant on $\gamma$. This contradicts $f$ being light; hence $\zeta$ is isolated in $f^{-1}(w_0)$. By Lemma \[lem:harmbdyim\], $f(\partial R) = \partial \Omega$. Hence if $\zeta$ is not isolated in $\partial R$, then there exists $\{\zeta_n\} \subseteq
\partial R \backslash \{\zeta\}$ such that $\zeta_n \rightarrow \zeta$. Since $\zeta$ is isolated in $f^{-1}(w_0)$, $f(\zeta_n) \neq w_0$ for $n$ sufficiently large. Since $\{f(\zeta_n)\} \subseteq \partial \Omega$ and $f(\zeta_n) \rightarrow w_0$ by continuity, $w_0$ is not isolated in $\partial \Omega$, a contradiction.
We need to check that the conditions in Theorem \[thm:harmbdypreim\] are satisfied. Since $R$ is bounded, $C(f|_R, \infty) = \varnothing$ and (i) holds. By assumption, (ii) holds. Since $R$ is bounded and $f$ is harmonic in $\mathbb{C}$, we can look at a sequence in $\Omega$ converging to $w_0$ and look at the corresponding sequence of preimages in $R$ to see that (iii) holds. Since the preimage points of $w_0$ are isolated in $\partial R$, (iv) also holds. The claim concerning the number of preimages follows from Theorem \[thm:harmbdypreim\].
\[cor:fillpuncture\] Let $f$ be a light harmonic function in $\mathbb{C}$. Let $\Omega$ be a component of $\ \mathbb{C} \backslash (f(S) \cup C(f, \infty))$. Suppose that $(int\ \overline{\Omega}) \backslash
\Omega$ consists of a finite number of points. Suppose that there exists a bounded component, say $R$, of $\ \mathbb{C} \backslash f^{-1}
(f(S) \cup C(f, \infty))$ such that $f(R) = \Omega$. Suppose that every point in $(int\ \overline{R}) \backslash R$ is mapped to $(int\ \overline{\Omega}) \backslash \Omega$. If $(int\ \overline{R}) \backslash R$ contains no point in $S$, then $f$ is $Val(f|_R, \Omega): 1$ in $int\ \overline{R}$. Moreover, if $int\ \overline{\Omega}$ is simply connected, then $f$ is univalent in $int\ \overline{R}$.
Since $R$ is bounded, we can apply Lemma \[lem:puncturepoints\] to $R$ to get that each of the isolated points in $\partial \Omega$ has $Val(f|_R, \Omega)$ preimages in $\partial R$, which are puncture points in $R$. Thus, each point in $int\ \overline{\Omega}$ has exactly $Val(f|_R, \Omega)$ distinct preimages in $int\ \overline{R}$, the first claim. Since $S \cap int\ \overline{R} = \varnothing$, we see that $int\ \overline{R}$ is a $Val(f|_R, \Omega)$-fold covering of $int\ \overline{\Omega}$ (see the proof of Theorem \[thm:preimcover\].) The second claim follows from Theorem \[thm:munch8th4.5\].
\[ex:plane\_isolcrit\] $f(z) = \frac{1}{3}\ z^3 - \frac{1}{2}\ \overline{z}^2$
We looked at this function restricted to $\{z: |z| < 1\}$ in Example \[ex:isolcrit\] above. We will now look at $f$ as a function in $\mathbb{C}$. Calculations show that $S = \{z: |z| = 1\} \cup \{0\}$ and that $C(f, \infty) = \varnothing$. The partitioning set for $f$ will be $f(S)$, which is graphed in Figure \[fig:isolcrit\_imaglabel\]. We can select a few points in each region of the partition and ask Mathematica to find the preimages. $Val(f, w) = 3$ in the unbounded component of $\mathbb{C} \backslash f(S)$. In each “point of the star” region, $Val(f, w) = 5$. $Val(f, w) = 7$ in the “center of the star” region (recall that the origin is in $f(S)$ and is not in this region.) We have numbered the cusps in $f(S)$ in Figure \[fig:isolcrit\_imaglabel\]. The critical points mapped to a given cusp are labeled with the same number in Figure \[fig:isolcrit\_preim\].
Recall that the origin is an isolated critical point of $f$. From the calculations in Example \[ex:isolcrit\], we see that the origin has six distinct preimages. From Figure \[fig:isolcrit\_preim\], we see that the preimages other than the origin lie in regions in the arms of the “star” (one in each such region.) From the calculation in Example \[ex:isolcrit\], we see that $f$ is 2:1 in the component with the origin as a puncture point. Hence $f$ must be univalent in each of the five other components which have $f^{-1}(0)$ as a puncture point. It’s not completely obvious from the figure that the preimage of the origin is an isolated point in each of the regions, because the Mathematica routines plot single points. However, in a region where $f^{-1}(0)$ is clearly a puncture point (for example, the arm in the lower right quadrant), this serves as an example of Corollary \[cor:fillpuncture\]. We can fill in the puncture point not in $S$ in each such component of the preimage to get a simply connected region where $f$ is univalent.
![Image of critical set for $f(z) =
\frac{1}{3}\ z^3 - \frac{1}{2}\ \overline{z}^2$, with labels. Note that $0 \in f(S)$.[]{data-label="fig:isolcrit_imaglabel"}](isolcrit_imag_lbl_m.eps)
![Preimage of $f(S)$ for $f(z) =
\frac{1}{3}\ z^3 - \frac{1}{2}\ \overline{z}^2$[]{data-label="fig:isolcrit_preim"}](isolcrit_preim.eps)
\[thm:jointwo\] Let $f$ be a light, harmonic function in $\mathbb{C}$. Let $R_1$ and $R_2$ be disjoint bounded components of $\ \mathbb{C} \backslash
f^{-1}(f(S) \cup C(f, \infty))$. Suppose that $R_1$ and $R_2$ each have at most a finite number of puncture points. Suppose that for each component $\Omega \subseteq
\mathbb{C} \backslash (f(S) \cup C(f, \infty))$, $\partial \Omega \cap int\ \overline{\Omega}$ consists of a finite number of points. Suppose that $f$ is univalent on $R_1$ and on $R_2$. Let $\gamma$ be a non-degenerate, simple arc in $\partial R_1 \cap \partial R_2$. Suppose that $f(int\ \gamma)$ is a simple arc. If $S\ \cap\ int\ \gamma = \varnothing$, then
1. $f(R_1)$ and $f(R_2)$ are distinct components of $\ \mathbb{C} \backslash (f(S) \cup C(f,\infty))$.
2. $f$ is a univalent on $R_1\ \cup\ R_2\ \cup\ int\ \gamma$.
This follows from Theorem \[thm:jointwoc1\] by associating $\mathbb{R}^2$ and $\mathbb{C}$ in the usual way.
\[rem:sharedarc\] Note that if $int\ \gamma$ contains a non-degenerate subarc in $S$, then there exists $z_0 \in int\ \gamma$ such that $f_{z_0} \sim z, \overline{z}$ (see the proof of Theorem \[thm:lyzval\].) Thus, $f(R_1) \cap f(R_2) \neq \varnothing$. By Theorem \[thm:preimonto\], $f(R_1) = f(R_2)$.
Extending these ideas to joining more than two adjacent regions is more difficult. Applying these results requires assumptions as to whether pairs of components (both in the domain and in the range) have boundaries intersecting in more than one simple arc. Even if we join two regions, it could be the case that the images of the two regions meet in more than one simple arc; joining along more than one of these arcs could cause trouble.
If we have components as in Theorem \[thm:jointwo\], we can fill in puncture points if the conditions in Corollary \[cor:fillpuncture\] are satisfied to get a simply connected region of univalence. The following theorem of M. Ortel and W. Smith [@OrtelSmith:univalence] will then help us join more than two univalent components, provided that the combined components in both the domain and range are simply connected.
\[thm:OSlcluniv\] If $\Omega_1$ and $\Omega_2$ are open, connected, and simply connected subsets of the complex plane, $f: \Omega_1 \rightarrow \Omega_2$ is continuous, surjective, and locally univalent, $N \in \{1,2,3, ...\}$ and $\# f^{-1}(w) \in \{1, N\}$ for all $w \in \Omega_2$, then $f$ is univalent.
For example, suppose that we have three disjoint simply connected components of $\mathbb{C} \backslash f^{-1}(f(S) \cup C(f, \infty))$ where $f$ is univalent and the image of each component is simply connected. Suppose also that the conditions of Theorem \[thm:jointwo\] hold pairwise. We thus have either two or three distinct image components. $f$ is univalent in the interior of each of the two boundary arcs used for joining regions. By construction, at least one of the image regions has preimages in only one of the original components. Thus, a point in the image of the combined image region must have either one or two preimages in the combined region. Univalence in the combined region follows from the theorem of Ortel and Smith, provided that we show that the combined image region is simply connected (see Examples \[ex:cubictwos\] and \[ex:dumbbell\].)
\[ex:cubictwos\] $f(z) = -\frac{1}{2} z^3 + \frac{3}{2} z -
\overline{z}$
Clearly, $C(f, \infty) = \varnothing$; thus the partitioning set is $f(S)$. The critical set of $f$ is graphed in Figure \[fig:cubictwos\_crit\]; it consists of two disjoint, closed curves. Figure \[fig:cubictwos\_imaglabel\] shows $f(S)$, with the cusps numbered and regions labeled in upper case letters. Using Mathematica to find preimages of a few points in each component of the partition, we see that $Val(f, w) = 3$ on the unbounded component, $Val(f, w) = 5$ on the tips of the star, and $Val(f, w) = 7$ in the “center” of the star.
![Critical set for $f(z) =
-\frac{1}{2} z^3 + \frac{3}{2} z - \overline{z}$[]{data-label="fig:cubictwos_crit"}](cubictwos_crit.eps)
Notice that each of the image regions is simply connected; hence we expect $f$ to be univalent on each of the corresponding preimage regions by Theorem \[thm:preimcover\]. This is consistent with the results for finding the preimages of a point in each component of $\mathbb{C} \backslash f(S)$ using Mathematica.
The preimages of the cusps on the critical set are labeled in Figure \[fig:cubictwos\_preim\] with the same numbers used in Figure \[fig:cubictwos\_imaglabel\]. We have labeled the preimage components corresponding to a given bounded component of $\mathbb{C} \backslash f(S)$ with the corresponding letter.
As expected from Theorem \[thm:jointwo\], we can join components sharing a common arc with interior off $S$ to get a larger region of univalence. From Figure \[fig:cubictwos\_preim\], it is apparent that we can continue joining such bounded components in the unbounded component of $\mathbb{C} \backslash S$ until we reach $S$. The combined regions in the image and in the domain appear to be simply connected, so univalence in the combined region should follow from the theorem of Ortel and Smith (Theorem \[thm:OSlcluniv\].)
![Image of critical set for $f(z) =
-\frac{1}{2} z^3 + \frac{3}{2} z - \overline{z}$, with labels[]{data-label="fig:cubictwos_imaglabel"}](cubictwos_imag_lbl_m.eps)
![Preimage of $f(S)$ for $f(z) =
-\frac{1}{2} z^3 + \frac{3}{2} z - \overline{z}$[]{data-label="fig:cubictwos_preim"}](cubictwos_preim.eps)
\[ex:dumbbell\] $f(z) = -\frac{1}{2} z^3 + \frac{9}{10}z -
\overline{z}$
$C(f, \infty) = \varnothing$; thus the partitioning set is $f(S)$. The critical set of $f$ is graphed in Figure \[fig:dumbbell\_crit\]; it looks like a dumbbell. Figure \[fig:dumbbell\_imaglabel\] shows $f(S)$, with the cusps numbered and two regions labeled in upper case letters. Using Mathematica to find preimages of a few points in each component of the partition, we see that $Val(f, w) = 3$ on the unbounded component and increases by two each time we cross an arc in $f(S)$ into a component containing the tangent to the arc. The components with maximum valence (7) are bisected by the imaginary axis.
![Critical set for $f(z) =
-\frac{1}{2} z^3 + \frac{9}{10}z - \overline{z}$[]{data-label="fig:dumbbell_crit"}](dumbbell_crit.eps)
Notice that each of the image regions appears to be simply connected; hence we expect $f$ to be univalent on each of the corresponding preimage regions by Theorem \[thm:preimcover\].
In Figure \[fig:dumbbell\_preim\], we label the regions of the partition of the domain inside the bounded component of $\mathbb{C} \backslash S$ which correspond to our two labeled image regions. Note that if we combine the three regions meeting at the point numbered $3$ (a preimage of the cusp numbered $3$ in Figure \[fig:dumbbell\_imaglabel\]), we run into trouble. Let’s call that point $z_0$. Note that $z_0$ is the only shared boundary point in $S$ for the three regions and is not included in the combined region. We note that $f(z_0)$ is a puncture point in the combined region, so the theorem of Ortel and Smith (Theorem \[thm:OSlcluniv\]) cannot be applied to claim that $f$ is univalent on the combined region in the domain.
In Figure \[fig:dumbbell\_preim\], we only labeled the preimages in $S$ of cusps.
![Image of critical set for $f(z) =
-\frac{1}{2} z^3 + \frac{9}{10}z - \overline{z}$, with labels[]{data-label="fig:dumbbell_imaglabel"}](dumbbell_imag_lbl_m.eps)
![Preimage of $f(S)$ for $f(z) =
-\frac{1}{2} z^3 + \frac{9}{10}z - \overline{z}$[]{data-label="fig:dumbbell_preim"}](dumbbell_preim.eps)
To make further progress with this, we need to have a better understanding of what the regions of the partitions look like (for example, is a component of the partition of the image ever an annulus?) An obvious starting point is to look at harmonic polynomials where $C(f, \infty) = \varnothing$. We make the following
**Conjecture:** *Let $f(z) = p(z) - \overline{z}$ with $p(z)$ being a polynomial in $z$ of degree two or higher. Let $\{R_j\}$ be a finite collection of bounded regions in the unbounded component of $\mathbb{C} \backslash S$ such that each $R_j$ is a component of $\mathbb{C} \backslash
f^{-1}(f(S))$. Let $R = int(\bigcup \overline{R_j})$. Suppose that we choose the $R_j$ such that $R$ is a connected region where $z \in \partial R$ implies one of the following holds: *(i)* $z \in S$ or *(ii)* every neighborhood of $z$ contains points mapped to the unbounded component of $\mathbb{C} \backslash f(S)$. Then $f$ is univalent on $R$. Moreover, $R$ is maximal in the sense that if we add another bounded component of $\mathbb{C} \backslash f^{-1}(f(S))$ to $R$, then $f$ will not be univalent on the combined region.*
Global valence and Lyzzaik’s local results {#sec:lyzzaik}
==========================================
The examples with $Val(f) < \infty$ where $f$ is defined in $\mathbb{C}$ in Sections \[sec:harm\] and \[sec:entire\] above share a common feature: the valence increases by an even number when one crosses an arc of $f(S)$ into the region containing the tangent line to that arc. Since $Val(f) < \infty$ in each such example, these are examples of light harmonic functions. We shall apply some results of Lyzzaik [@L:light] concerning the behavior of a light harmonic function defined in a simply connected domain to show that the change in valence illustrated in these examples is a property of light harmonic functions in $\mathbb{C}$.
First, we will consider the setting for Lyzzaik’s results. Let $W$ be a simply connected domain in $\mathbb{C}$ and $f$ a harmonic function in $W$. Then we can represent $f$ as $f = h + \overline{g}$ where $h$ and $g$ are holomorphic in $W$. Let $\psi(z) = (h' / g')(z)$ for $z \in W$; $\psi$ is the reciprocal of the dilatation of $f$. Lyzzaik denotes the critical set of $f$ by $J$; for consistency, we will denote it by $S$.
\[def:lyzdef2.1\] For a light harmonic mapping $f$ in $W$ every $z \in N = \{z \in S: |\psi(z)| \ne 1\}$ is called a non-folding critical point of $f$. Note that if $z \in N$, then $h'(z) = g'(z) = 0$.
\[lem:lyzlem2.2\] Every $z_0 \in N$ belongs to a neighborhood that contains no other point in $S$. $\psi$ is unimodular on $\Gamma_f = S \backslash N$ and $\Gamma_f$ consists of curves which are analytic except possibly for algebraic singularities.
Lyzzaik parametrizes a directed Jordan subarc $\gamma$ of $\Gamma_f$ by an analytic path $z(t)$ for $t \in I = [0, 1]$. He then defines a continuous, increasing function $\phi : I \rightarrow \mathbf{R}$ by requiring that $\psi(z(t)) =\ $exp$(i \phi(t))$. An easy calculation shows that
$$\frac{d}{dt} f(z(t)) =
2 (Re\ \omega(t)) \exp(i \phi(t) / 2)$$
where
$$\omega(t) = h'(z(t)) z'(t)
\exp(-i \phi(t) / 2)$$
for $z \in \gamma \subseteq S \backslash N$.
\[def:lyzdef2.2\] Let $f$ be a light harmonic mapping in $W$ and let $z_0 \in \Gamma_f$.
[()]{}
If $\psi'(z_0) \ne 0$, then $z_0$ is interior to a Jordan subarc $\gamma$ of $\Gamma_f$ which we assume given as above, with $z_0 = z(t_0)$, $0 < t_0 < 1$. Let $\omega(t)$ be as above. We call $z_0$ a critical point of $f$ of the
[()]{}
first kind if $Re\ \omega$ changes sign at $t_0$.
second kind if $z_0$ is not a critical point of the first kind and if $z_0$ is a zero of $h'$, or equivalently $g'$ (which yields $Re\ \omega(t_0)) = 0$.)
If $\psi'(z_0) = 0$, then we call $z_0$ a critical point of $f$ of the third kind.
Let $F_j$ where $j$= 1,2, or 3, denote the set of all critical points of $f$ of the $j$th kind, and let $F = \cup_{j=1}^3 F_j$. We call $F$ the set of folding critical points of $f$. Note that this classification is independent of $\gamma$ and its parametrization.
\[thm:lyzthm2.2\] Let $f$ be a light harmonic function in $W$. Then $F \cup N$ consists of isolated points.
\[rem:noaccpts\] We also claim that $F \cup N$ has no point of accumulation in $W$. Since $f$ is light, neither $F_3$ nor $N$ can have a point of a accumulation; otherwise $S$ has nonempty interior (in which case, the proof of Lemma \[lem:nullpartifsnotempty\] implies that $f$ is not light.) Further, as noted by Lyzzaik, $Re\ \omega$ can have only finitely many zeros on any Jordan arc in $S$ when $f$ is light. Hence, $F_1 \cup F_2$ has no point of accumulation.
We can now state one of Lyzzaik’s structure theorems for the local behavior of $f$ near the critical set. The notation $f_{z_0} \sim z^j, \overline{z}^k$ is explained in Section \[sec:lyzintro\] above.
\[thm:lyzthm5.1\] Let $f = \overline{g} + h$ be a light harmonic mapping in $W$, $z_0 \in \Gamma_f$, and $\ell \ge 0$ the order of $z_0$ as a zero of $h'$ or equivalently $g'$.
[()]{}
Suppose that $z_0 \in \Gamma_f \backslash
(F_1 \cup F_3)$. Then $f$ satisfies $f_{z_0} \sim z^{\ell+1},
\overline{z}^{\ell+1}$ if $\ell$ is even (including zero), and $f_{z_0} \sim z^{\ell + 2}, \overline{z}^\ell$ or $f_{z_0} \sim z^\ell, \overline{z}^{\ell + 2}$ if $\ell$ is odd.
Suppose that $z_0 \in F_1$. Then $f$ satisfies $f_{z_0} \sim z^{\ell+1},\overline{z}^{\ell+3}$ or $f_{z_0} \sim z^{\ell+3},\overline{z}^{\ell+1}$ if $\ell$ is even, and $f_{z_0} \sim z^{\ell + 2},
\overline{z}^{\ell + 2}$ if $\ell$ is odd.
Lyzzaik defines a convex arc to be a directed simple arc that has the slope of its tangent continuously increasing. An arc is locally convex at $z_0$ if $z_0$ belongs to some open subarc which is convex. Let $\gamma \subseteq \Gamma_f$ and $z_0 \in int\ \gamma$. Lyzzaik examines arg $\frac{d}{dt} f(z(t))$ at $z_0 = z(t_0)$ and shows that $f(\gamma)$ is locally convex at $z_0 \in \Gamma_f \backslash (F_1 \cup F_3)$. Thus we can speak of the region which contains the tangent line to $f(\gamma)$ at $f(z_0)$ if $z_0 \notin F$. (Another approach is to note that if $\gamma \subseteq \Gamma_f$, then $f(\gamma)$ cannot be piecewise constant; otherwise, a connected subset of $\gamma$ is mapped to a point and $f$ is not light. One can then apply a result of P. Duren and D. Khavinson [@DK:concave] to show that $f(\gamma)$ is concave, provided that $f$ is univalent in a region with $\gamma$ in its boundary. However, this approach requires us to show that we can find such a region of univalence.)
Let $f$ be a light harmonic function in $\mathbb{C}$. Let $\gamma$ be a simple analytic arc in the critical set such that $\beta = f(\gamma)$ is convex (and contains no points of inflection) and such that $\gamma\ \cap\ (F \cup N)
= \varnothing$. An arc will be called non-degenerate if it is a non-empty arc that does not consist of a single point. Here, $\gamma$ is non-degenerate. Then, given $z_0 \in\ int\ \gamma$, there is a neighborhood $U$ of $z_0$ such that $\gamma$ partitions $U$ into a sense-preserving region $U^+$ and a sense-reversing region $U^-$. If we choose $z_0 \in int\ \gamma$, then $\ell = 0$ in Theorem \[thm:lyzthm5.1\] (since $g'(z_0) \neq 0$) and $f_{z_0} \sim z, \overline{z}$. By Theorem \[thm:lyzthm5.1\] and its proof, we can choose $U$ and some neighborhood of $f(z_0)$, say $B$, such that $\beta$ partitions $B$ into two regions $V^+$ and $V^-$ and such that $f$ maps $U^+$ 1-1 onto $V^+$ and maps $U^-$ 1-1 onto $V^+$. Here, $V^+$ is chosen such that the tangent to $\beta$ at $f(z_0)$ lies in $V^+$. We will use these ideas to prove:
\[thm:lyzval\] Let $f$ be a finite valence harmonic mapping on $\mathbb{C}$. Let $\Omega_+$ and $\Omega_-$ be distinct components of $\mathbb{C}\backslash (f(S) \cup C(f,\infty))$ that share a common non-degenerate boundary arc $\beta_0$. Suppose that $int\ \beta_0$ does not lie completely in $C(f, \infty)$. Then there is a non-degenerate subarc $\beta$ of $\beta_0$ that lies outside of $C(f,\infty)$ and such that $f^{-1}(\beta) \cap
(F \cup N) = \varnothing$. Choose $w_0 \in\ int\ \beta$ and choose $\Omega_+$ to be the region containing the tangent to $\beta$ at $w_0$. Suppose that $w_0$ has $N_0 \ge 0$ distinct preimages off $S$ and $N_1 > 0$ distinct preimages in $S$. Then $Val(f, \Omega_-) = N_0$ and $Val(f, \Omega_+) = N_0 + 2 N_1$.
Since $f$ has finite valence, $f$ is light. We wish to choose a non-degenerate, simple analytic arc $\beta \subseteq \beta_0$ such that $\beta \cap C(f,\infty) = \varnothing$, with $\beta$ in the boundary between two regions of constant valence, and such that $\beta \cap f(F \cup N) = \varnothing$. The two regions are non-empty by Corollary \[cor:goodpart\], since $Val(f) < \infty$. By our choice of $\beta$, a structure theorem of Lyzzaik for light harmonic functions (Theorem \[thm:lyzthm5.1\]) should help us count the distinct preimages of points in some sufficiently small neighborhood $B$ of $w_0$.
1. Existence of $\beta$: Since $int\ \beta_0$ does not lie completely in $C(f,\infty)$, we can find a bounded, closed, non-degenerate subarc $\tilde{\beta}$ such that $\exists\ w \in int\ \tilde{\beta}$ with $w \notin C(f,\infty)$. Since $C(f, \infty)$ is closed, there exists a neighborhood of $w$ that is disjoint from $C(f, \infty)$. Take $\beta_1$ to be a closed, non-degenerate subarc of $\tilde{\beta}$ contained in this neighborhood and to have $w$ in its interior.
If $K$ is an arbitrary compact set, then $(F \cup N) \cap K$ must be finite by Remark \[rem:noaccpts\]. Since $\beta_1 \cap C(f, \infty) = \varnothing$, the closed set $f^{-1}(\beta_1)$ is also bounded. Thus, $f^{-1}(\beta_1)$ is compact and $\beta_1$ contains at most a finite number of points with preimages in $F \cup N$. So, we may find a bounded subarc $\beta$ of $\beta_0$ that neither intersects $C(f,\infty)$ nor contains points in $f(F \cup N)$. Moreover, we may assume that this arc is convex and without any points of inflection. Since $f$ has finite valence, $f(S) \cup C(f,\infty)$ has empty interior. By construction, $\beta$ will be the local boundary between $\Omega_+$ and $\Omega_-$.
2. Preimages of $w_0$: Choose $w_0 \in\ int\ \beta$. Let $f^{-1}(w_0) \cap S = \{z_1, z_2, ... , z_{N_1}\}$. In a sufficiently small neighborhood of $z_j$, $S$ consists of a single non-degenerate arc since $z_j \notin N \cup F_3$ by (1). A non-degenerate subarc of this arc is mapped to a subarc of $\beta$. This follows from our assumption that $\beta_0$ locally separates components of the partition; *i.e.*, if two distinct curves in $f(S)$ cross at $w_0$, then $\beta_0$ doesn’t separate $\Omega_+$ and $\Omega_-$ in a sufficiently small neighborhood of $w_0$. Since $w_0 \in f(S)$, we have $N_1 > 0$. Let $f^{-1}(w_0) \cap (\mathbb{C}\backslash S)
= \{\zeta_1, \zeta_2, ... , \zeta_{N_0}\}$, where $N_0 \ge 0$.
3. Construction of neighborhood $B$ of $w_0$: Consider a preimage $\zeta_j$ of $w_0$ off of $S$. By the inverse function theorem, there exist $Q_j$, an open neighborhood of $\zeta_j$, and $V_j$, an open neighborhood of $w_0$, such that $f: Q_j \rightarrow V_j$ is 1-1, onto. Without loss of generality, we may assume $Q_j \subseteq
\mathbb{C} \backslash f^{-1}(f(S) \cup C(f,\infty))$. Now consider a preimage $z_j$ in $S$. By construction, $z_j \notin N \cup F_3$. Let $\gamma$ be a simple arc in $S$ such that $z_j \in\ int\ \gamma$ and $f(\gamma) \subseteq \beta$. By our choice of $\beta$, $\gamma \cap (F \cup N) = \varnothing$. Hence, $g'(z_j) \neq 0$. We may apply our interpretation above of Theorem \[thm:lyzthm5.1\] to find an open neighborhood $U_j$ of $z_j$ and an open neighborhood $W_j$ of $w_0$ such that
- $\gamma$ partitions $U_j$ into a sense-preserving region ${U_j}^+$ and a sense-reversing region ${U_j}^-$.
- ${U_j}^+$ and ${U_j}^-$ are each mapped 1-1 onto $W_j \cap\ \Omega_+$.
Let $B_0 = \cap_{j=1}^{N_0} V_j$ and let $B_1 = \cap_{j=1}^{N_1} W_j$. Clearly, $B_1 \neq \varnothing$. If $N_0 > 0$, $B_0 \neq\ \varnothing$, so let $B = B_0 \cap B_1$. Otherwise, $B_0$ is empty, so let $B = B_1$. By construction, $B$ is open. Finally, let $T = f^{-1}(B) \cap ((\cup_{j=1}^{N_0} Q_j)
\cup (\cup_{j=1}^{N_1} U_j))$. Then every preimage of $w_0$ is in $int\ T$. We may choose the $Q_j$ and the $U_j$ such that these sets are all pairwise disjoint.
4. Valence in $\Omega_+$: By construction, each $w \in B \cap \Omega_+$ has exactly $N_0 + 2 N_1$ distinct preimages in $T$ and thus has at least $N_0 + 2 N_1$ distinct preimages in $\mathbb{C}$. We claim that $\exists\ w_+ \in B \cap \Omega_+$ such that $Val(f, w_+) = N_0 + 2 N_1$. Suppose not. Then $\exists\ \{w_n\} \subset B \cap \Omega_+$ such that $w_n \rightarrow w_0$ and such that each $w_n$ has at least $N_0 + 2 N_1 + 1$ distinct preimages in $\mathbb{C}$. In particular, each $w_n$ has a preimage $\xi_n \notin T$. Since $w_n \rightarrow w_0$ and $w_0 \notin C(f,\infty)$, $\{\xi_n\}$ is bounded, so has a convergent subsequence, which we will also denote $\{\xi_n\}$. Let $\xi_n \rightarrow \xi_0$. Since $f(\xi_0) = w_0$, $\xi_0 \in int\ T$. Thus, $\xi_n \in T$ for $n$ sufficiently large, a contradiction. Thus, $\exists\ w_+ \in B \cap \Omega_+$ such that $Val(f, w_+) = N_0 + 2 N_1$. Since, by Theorem \[thm:harmpart\], the valence of $f$ is constant on the region $\Omega_+$, $Val(f, \Omega_+) = Val(f, w_+) = N_0 + 2 N_1$.
5. Valence in $\Omega_-$: By construction, each $w \in B \cap \Omega_-$ has exactly $N_0$ distinct preimages in $T$ (one preimage in each $Q_j$) and thus has at least $N_0$ distinct preimages in $\mathbb{C}$. We claim that $\exists\ w_- \in B \cap \Omega_-$ such that $Val(f, w_-) = N_0$. Suppose not. Then $\exists\ \{w_n\} \subset B \cap \Omega_-$ such that $w_n \rightarrow w_0$ and such that each $w_n$ has at least $N_0 + 1$ preimages in $\mathbb{C}$. In particular, each $w_n$ has a preimage $\xi_n \notin f^{-1}(B) \cap (\cup_{j=1}^{N_0} Q_j)$. Since $w_0 \notin C(f,\infty)$, $\{\xi_n\}$ is bounded, so has a convergent subsequence, which we will also denote $\{\xi_n\}$. Let $\xi_n \rightarrow \xi_0$. Since $f(\xi_0) = w_0$, $\xi_0 \in int\ T$. Thus, for $n$ sufficiently large, $\xi_n \in T$. Since $f$ maps $\cup_{j=1}^{N_1} U_j$ into $\beta \cup \Omega_+$, $\xi_n \in f^{-1}(B) \cap (\cup_{j=1}^{N_0} Q_j)$ for sufficiently large $n$, a contradiction. Thus, $\exists\ w_- \in B \cap \Omega_-$ such that $Val(f, w_-) = N_0$. Since the valence of $f$ is constant on the region $\Omega_-$ (Theorem \[thm:harmpart\] again), $Val(f, \Omega_-) = Val(f, w_-) = N_0$.
The proof above can also be extended to the case where $f$ is a finite-valence, light harmonic function in a simply connected, open set. In that case, $C(f)$ replaces $C(f, \infty)$ and $S$ is defined only for points in our open set.
Consider Example \[ex:isolcrit\] above. We can select a few points in each region of the partition and ask Mathematica to find the preimages in $B(0,1)$. $Val(f, w) = 0$ in the unbounded component of $\mathbb{C} \backslash f(S)$. In each “point of the star” region, $Val(f, w) = 1$. $Val(f, w) = 2$ in the “center of the star” region (recall that the origin is in $f(S)$ and is not in this region.) The origin has one preimage in $B(0,1)$. This is not a counterexample to this extended version of Theorem \[thm:lyzval\], because the “star” is $C(f)$, not $f(S)$. Effectively, when we restrict $f$ to our region, there are no points with $|z| > 1$ for $f$ to “fold” over $|z| = 1$.
[NAM99]{} Y. Abu-Muhanna and A. Lyzzaik, *A geometric criterion for decomposition and multivalence*, Math. Proc. Cambridge Phil. Soc. **103** (1988), 487–495. MR 89e:30010. Mark Benevich Balk, *Polyanalytic Functions*, Mathematical research, volume 63, Akademie Verlag GmbH (1991). MR 93k:30076. Marcel Brelot and Gustave Choquet, *Polynômes harmoniques et polyharmoniques*, Second colloque sur les équations aux dérivées partielles, Bruxelles, 1954, pp. 45–66. Georges Thone, Liège; Masson $\&$ Cie (1955). MR 16,1108e. Daoud Bshouty, Walter Hengartner, and M. Naghibi-Beidokhti, *p-valent harmonic mappings with finite Blaschke dilatations*, XII-th Conference on Analytic Functions (Lublin, 1998), Ann. Univ. Mariae Curie-Sklodowska Sect. A, **53** (1999), 9–26. MR 2001j:30016. Daoud Bshouty, Walter Hengartner, and Tiferet Suez, *The exact bound on the number of zeros of harmonic polynomials*, J. Anal. Math. **67** (1995), 207–218. MR 97f:30025. E. F. Collingwood and A. J. Lohwater, *The Theory of Cluster Sets*, Cambridge University Press (1966). MR 38\#325. Peter Duren and Dmitry Khavinson, *Boundary correspondence and dilatation of harmonic mappings*, Complex Variables Theory Appl., **33** (1997), 105–111. MR 98m:30039. Dmitry Khavinson and Grzegorz Światek, *On the number of zeros of certain harmonic polynomials*, Proc. Amer. Math. Soc. **131** (2003), 409–414. Hans Lewy, *On the non-vanishing of the Jacobian in certain one-to-one mappings*, Bull. Amer. Math. Soc. **42** (1936), 689–692. Abdallah Lyzzaik, *Local properties of light harmonic mappings*, Canad. J. Math. **44** (1992), 135–153. MR 93e:30048. James R. Munkres, *Topology: A First Course*, Prentice-Hall, Inc. (1975). MR 57\#4063. Genevra Chasanov Neumann, Valence of harmonic functions, Ph.D. dissertation, University of California, Berkeley. 2003. M. Ortel and W. Smith, *A covering theorem for continuous locally univalent maps of the plane*, Bull. London Math. Soc. **18** (1986), 359–363. MR 88b:30013. Kennan T. Smith, *Primer of Modern Analysis*, Bogden & Quigley, Inc. (1971). MR 84m:26002. S. Stöilow, *Leçons sur les principes topologiques de la theórie des fonctions analytiques*, deuxième edition, Gauthier-Villars (1956). MR 18,568b. T. J. Suffridge and J. W. Thompson, *Local behavior of harmonic mappings*, Complex Variables Theory Appl., **41** (2000), 63–80. MR 2001a:30019. Alan Stephen Wilmshurst, *Complex harmonic mappings and the valence of harmonic polynomials*, D.Phil. thesis, University of York, England. 1994. A. S. Wilmshurst, *The valence of harmonic polynomials*, Proc. Amer. Math. Soc. **126** (1998), 2077–2081. MR 98h:30029.
[^1]: We will refer to [@W:paper] when a result appears in both [@W:thesis] and [@W:paper].
[^2]: Mathematica is a registered trademark of Wolfram Research, Inc.
[^3]: These routines are enhanced versions of the Mathematica routines in [@gcn:thesis]. For the figures in this paper, $z \in f^{-1}(w)$ if $|Re\,f(z) - Re\,w| +
|Im\,f(z) - Im\,w|
< 10^{-10}$.
[^4]: Lyzzaik defines a convex arc to be a directed simple arc where the slope of the tangent is continuously increasing.
|
---
author:
- 'R. Tylenda'
- 'T. Kamiński'
date: 'Received; accepted'
title: |
Light echo of V838 Monocerotis:\
properties of the echoing medium [^1]
---
Introduction \[intro\]
======================
V838 Monocerotis was discovered as an eruptive star at the beginning of January 2002. The main eruption started at the beginning of February 2002, however, and lasted about two months [see e.g. @muna02; @kimes02; @crause03]. The main characteristic of the eruption, which is different in V838 Mon from other stellar eruptions (e.g. classical novae), was that the object evolved to progressively lower effective temperatures and declined as a very late M-type supergiant [e.g. @tyl05].
@ts06 showed that the eruption of V838 Mon cannot be explained by a thermonuclear runaway similar to classical novae, nor by a late He-shell flash. They proposed, following @st03, that the event resulted from a merger of two stars. This idea obtained strong support from an analysis of archive observations of the progenitor of V1309 Sco. The eruption of this object was observed in 2008 and was of the same type as that of V838 Mon [@mason10]. @thk11 showed that the event resulted from a merger of a contact binary.
The eruption of V838 Mon was accompanied by a spectacular light echo. A series of fantastic images of the light echo have been obtained by Hubble Space Telescope [HST, see e.g. @bond03; @bond07]. Analyses of the light echo’s structure and properties allowed astrophysicists to study the dusty medium around V838 Mon [e.g. @tyl04] and to determine the distance to the object [@sparks].
V838 Mon has a spectroscopic companion, a B-type star discovered by @mdh02 in September–October 2002, when the eruption remnant became so cool that it almost disappeared from the optical. The spectrum of the companion, classified as B3V by @mdh02, was seen undisturbed until the fall of 2006, when an eclipse-like event was observed [e.g. @muna07]. The companion was eclipsed for about two months. In 2004/2005, emission lines of [\[\]]{} appeared in the spectrum of V838 Mon. The lines strengthened with time and reached a maximum at the time of the eclipse-like event. @tks09 showed that the profiles and evolution of the [\[\]]{} emission lines, as well as the eclipse-like event resulted from interactions of the B-type star radiation with the matter ejected by V838 Mon in the 2002 eruption. After reappearance from the eclipse-like event, the companion began to fade again and in 2008 traces of the B-type spectrum, as well as of the [\[\]]{} emission lines, disappeared. Apparently the companion became completely embedded in the dusty ejecta of V838 Mon [@tks11].
@afbond found a sparse, young cluster, containing three B-type main-sequence stars, in the field of V838 Mon. The cluster is at a distance of $\sim$6.2 kpc, which is very close to the value of $\sim$6.1 kpc derived by @sparks for V838 Mon from their analysis of the polarization of the light echo. The reddening of the cluster and that of V838 Mon and its companion are also very similar ($E_{B-V} = 0.84$ versus 0.9, respectively). It seems very likely that V838 Mon and its B-type companion are also members of the cluster.
However, there is a problem with the observed brightness of the B-type companion of V838 Mon when compared to the magnitudes of the B-type stars of the cluster. @muna05 derived $V$ = 16.05 $\pm$ 0.05 for the companion of V838 Mon. Using the compilation of photometric data of V. Goranskij[^2], one derives $V$ =16.19 $\pm$ 0.03 from the data obtained in September–November 2002. The B-type companion is therefore fainter by 1.2–1.4 mag when compared to star 9 (classified as B3V) in the cluster of @afbond. @afbond interpreted this as evidence that the companion is partly submerged in the dusty ejecta of V838 Mon. However, if a dusty medium results from a stellar eruption, it is very unlikely that it is homogeneous. Therefore, if the ejecta flows in front of a star, one expects to observe a high variability of the stellar light. This was not observed in the case of the B-type companion of V838 Mon until the fall of 2006. As can be seen from the light curve of V838 Mon between fall 2002 and fall 2006 (see e.g. the data of Goranskij$^1$), the $U$ brightness (dominated by the B-type companion in this epoch) remained constant within $\pm$0.1 mag. The light curve exhibits the same behaviour in the $B$ band after the 2002 eruption [see @muna05]. As discussed above, interactions of the V838 Mon ejecta with the companion started at the earliest in the fall of 2006 and then indeed resulted in a high variability of the observed light of the companion, eventually leading to the complete disappearance of the star.
The problem of the observed brightness of the B-type companion remains unresolved. A possible explanation is that V838 Mon and its companion suffer from additional, local extinction of $\Delta A_{V} \simeq 1.3$, compared to the three B-type stars of the cluster. A standard extinction of this value would imply additional reddening of $\Delta E_{B-V} \simeq 0.4$. This is significantly higher than the observed difference between the reddening of the V838 Mon and its companion, and that of the cluster ($\Delta E_{B-V}\la 0.1$, see above). This implies that the local extinction, if present, would have to be almost grey in the optical, i.e. $R = A_{V}/E_{B-V} \ga 10$, which would additionally imply local dust dominated by large grains ($a \ga 0.3~\mu$m).
This idea seems to be conceivable. Young stellar clusters are usually known to contain a significant amount of diffuse matter consisting of remnants of a more massive complex, from which the cluster was formed. For V838 Mon there is observational evidence that diffuse matter is indeed present in the near vicinity of the object. One piece of evidence is the light echo. Radio observations in the CO rotational lines also show the presence of molecular matter, most probably related to the dusty medium seen in the light echo [@kmt; @ktd]. The light echo offers possibilities to study parameters and properties of the scattering matter. This is the aim of the present study.
As we show in detail below, an analysis of the brightness of the echo at different wavelengths and its evolution with time allows us to estimate several parameters of the scattering matter. This includes the optical thickness of the matter, as well as dependeces of the scattering coefficient on the wavelength and the scattering angle. In this way we can verify whether the local extinction can be responsible for the relative faintness of the B-type companion of V838 Mon when compared to the other B-type stars of the cluster. The results of the echo study also allow us to derive constraints on the nature of the dust grains and on the mass of the scattering material. The latter is important for discussing the nature of the matter, i.e. whether it is of interstellar origin or rather, as suggested by some authors [e.g. @bond03; @bond07], that it resulted from past mass loss activity of the progenitor of V838 Mon. This can have consequences for the nature of the progenitor itself and the mechanism of the 2002 eruption.
Observational material and its reduction\[obs\_sect\]
=====================================================
We have used the archival images of V838 Mon’s light echo obtained with HST in 2002 on April 30, May 20, September 2, October 28, and December 17 (HST proposals 9587, 9588, and 9694). The first image was taken in the F435W filter only. The other ones were obtained in the F435W, F606W, and F814W filters. The observations were performed with the Advanced Camera for Surveys (ACS) combined with the detector of the Wide Field Channel (WCS). Some of the images were taken in the polarimetric mode, the rest in the standard (non-polarimetric) mode. The observations have been described in more detail in [@sparks]. The pipeline-reduced data, which we extracted from the HST archive, were further processed using the [*multidrizzle*]{}[^3] package (version 3.3.8). The non-polarimetric observations for a given date and filter were combined in [*multidrizzle*]{} to produce images in units of counts per second. The polarimetric observations were obtained with three retarder angles (0, 60, 120). We first combined frames with the same retarder angle for a given filter and date. Then the polarimetric images were scaled using calibration corrections taken from [*ACS Data Handbook*]{}[^4]. Finally the data were combined into the Stokes $I$ (total intensity) image in each filter.
The following reduction was performed using Starlink[^5] packages, mainly with KAPPA and GAIA. We measured the background sky level and its variations in each image by performing statistics on a large number of pixels free of stellar and echo emission and instrumental defects. After correcting images for the sky level, those measurements were used to blank all pixels below 1.5$\sigma$ of sky variations and above the maximum echo brightness for a given date. From the resulting images, we manually removed the remaining parts of stellar diffraction patterns and cosmic-ray hits. Finally, we measured the light echo region.
Measurements \[resobs\_sect\]
=============================
The results of the measurements are given in Table \[measur\_tab\]. The upper, middle, and bottom parts of the table present the results obtained with the F435W, F606W, and F814W filters, respectively. The dates of the observations are given in column (1), while column (2) notes the observing mode used to obtain the images. Total count rates from the echo are given in column (3). Column (4) gives the total number of pixels, which registered a measurable signal from the echo. A total flux from the light echo can be calculated by multiplying the total count rate number by the [*photflam*]{} conversion factor [@sirianni], which is equal to $3.08 \times 10^{-18}$, $7.725 \times 10^{-19}$, and $6.94 \times 10^{-19}$ Wm$^{-2}$$\mu$m$^{-1}$ per count s$^{-1}$ for the F435W, F606W, and F814W filters, respectively. The resulted fluxes are presented in column (5). The ACS pixel scale is $0\farcs05 \times 0\farcs05$, which allows one to easily calculate a mean surface brightness, $S_{\rm B}$, from the mean count rate summed over 400 pixels (corresponding to 1 arcsec$^2$) and the [*photflam*]{}. The results are listed in column (6).
The data in columns (3)–(6) refer only to the part of the light echo surface that was not contaminated by field stars and V838 Mon itself. The procedure of removing diffraction patterns of field stars introduced uncertainties in the measurements. This is particularly the case for the total flux, which is expected to be systematically underestimated in column (5) of Table \[measur\_tab\]. The effect is difficult to estimate, as each time different parts of the echo were contaminated by stars. The mean surface brightness of the echo is expected to be less vulnerable to the procedure of cutting out the stellar patterns.
We have attempted to correct the measured total flux for this effect. From the radius of the outer rim of the echo measured in @tss05, we calculated the total number of pixels expected inside the echo. Then we defined a covering factor, $f_{\rm c}$, as a ratio of the number of pixels that detected the echo (i.e. counted in column 4 of Table \[measur\_tab\]), to the total number expected from the size of the echo. This factor is generally between 0.7–1.0. Assuming that $f_{\rm c}$ is $<$1.0 because of the field stars’ patterns and that the mean surface brightness is statistically the same over the echo image, the corrected echo flux can be calculated as the measured flux (listed in column 5) devided by $f_{\rm c}$. The resulting value should, however, be treated as an upper limit to the echo flux. The missing pixels do not result from the field stars only, but also from the fact that there were regions of the echo, not contaminated by stars, where the echo was physically too faint to be measured. Therefore, as a finally corrected flux, we took a mean value from the flux estimated in the above way and the measured flux (column 5). For all except the observations from December 17, both fluxes were taken with the same weight when averaging. The echo images obtained on December 17 show empty regions significantly more extended than on earlier dates, so we took the measured flux with a double weight when calculating the finally corrected flux for this date. The echo fluxes corrected with the above procedure and converted to the ST magnitudes [@sirianni] are given in column (7) of Table \[measur\_tab\]. The corrections to the echo total flux were largest for the F435W filter, but even in this case they do not exceed 10% (0.10 mag.).
------ ------------------------------------------------------------------------------------------------------------------------ ---------------- -------- ------------------------ ------------------------------------- ----------- -------------------------------------
date obs.mode total pixels flux surf.brightness STmag surf.brightness($f_{\rm c}=0.5$)
2002 count s$^{-1}$ number Wm$^{-2}$$\mu$m$^{-1}$ Wm$^{-2}$$\mu$m$^{-1}$arcsec$^{-2}$ corrected Wm$^{-2}$$\mu$m$^{-1}$arcsec$^{-2}$
(1) (2) (3) (4) (5) (6) (7) (8)
**[F435W]{}\
Apr.30 & polarimetric & 9.59E4 & 4.20E5 & 2.95E–13$\pm$7.3% & 2.81E–16$\pm$10.7% & 12.72$\pm$0.07 & 4.14E–16$\pm$7.5%\
May 20 & polarimetric & 7.55E4 & 4.80E5 & 2.33E–13$\pm$8.6% & 1.94E–16$\pm$13.0% & 12.92$\pm$0.08 & 2.65E–16$\pm$7.4%\
Sept.2 & standard & 3.58E4 & 9.44E5 & 1.10E–13$\pm$7.8% & 4.67E–17$\pm$12.0% & 13.69$\pm$0.08 & 6.36E–17$\pm$7.5%\
Oct.28 & standard & 2.90E4 & 1.21E6 & 8.92E–14$\pm$7.3% & 2.96E–17$\pm$10.3% & 13.94$\pm$0.07 & 4.16E–17$\pm$7.5%\
Dec.17 & standard & 2.48E4 & 1.27E6 & 7.64E–14$\pm$11.2% & 2.41E–17$\pm$17.5% & 14.10$\pm$0.11 & 3.10E–17$\pm$7.4%\
&\
& **[F606W]{}\
May 20 & polarimetric & 1.02E5 & 6.28E5 & 7.90E–13$\pm$5.0% & 5.82E–16$\pm$7.5% & 11.66$\pm$0.05 & 9.46E–16$\pm$5.8%\
Sept.2 & polarimetric & 5.19E4 & 1.16E6 & 4.01E–13$\pm$5.0% & 1.41E–16$\pm$7.3% & 12.39$\pm$0.06 & 2.25E–16$\pm$5.8%\
Oct.28 & standard & 3.94E4 & 1.36E6 & 3.05E–13$\pm$5.0% & 8.93E–17$\pm$7.0% & 12.68$\pm$0.05 & 1.44E–16$\pm$5.8%\
Dec.17 & polarimetric & 3.52E4 & 1.45E6 & 2.72E–13$\pm$5.0% & 7.48E–17$\pm$7.2% & 12.77$\pm$0.05 & 1.09E–16$\pm$5.7%\
&\
& **[F814W]{}\
May 20 & standard & 1.29E5 & 5.07E5 & 8.98E–13$\pm$4.4% & 7.08E–16$\pm$8.9% & 11.49$\pm$0.05 & 1.16E–15$\pm$5.6%\
Sept.2 & standard & 7.04E4 & 1.07E6 & 4.89E–13$\pm$4.3% & 1.83E–16$\pm$6.8% & 12.14$\pm$0.05 & 2.87E–16$\pm$5.3%\
Oct.28 & standard & 5.96E4 & 1.36E6 & 4.14E–13$\pm$4.3% & 1.22E–16$\pm$5.8% & 12.34$\pm$0.04 & 1.97E–16$\pm$5.2%\
Dec.17 & standard & 5.40E4 & 1.51E6 & 3.75E–13$\pm$4.5% & 9.95E–17$\pm$8.4% & 12.43$\pm$0.05 & 1.50E–16$\pm$5.1%\
******
------ ------------------------------------------------------------------------------------------------------------------------ ---------------- -------- ------------------------ ------------------------------------- ----------- -------------------------------------
As noted above, the covering factor, $f_{\rm c}$, is generally $<$1.0, but it varies from image to image. It is usually lowest for the F435W filter and shows a tendency to decrease with the time of observations. As a result the surface brightness listed in column (6) of Table \[measur\_tab\] is not entirely comparable between different filters and different dates. This may create uncertainties and ambiguities when comparing the results of the echo simulations to the observational measurements. Therefore we also derived another mean surface brightness, which was calculated from the brightest pixels in each image, counted until they fill $f_{\rm c} = 0.5$. The resulting values, named $S_{\rm B\,0.5}$ below, are listed in column (8) of Table \[measur\_tab\].
Table \[measur\_tab\] lists uncertainties of the obtained values in addition to the results of measurements. The errors of the total flux and the surface brightness (columns 5, 6, and 8) are expressed in percent of the values, while those of STmag (column 7) are given in magnitudes. When evaluating the errors we took into account count statistics, uncertainities due to subtracting the sky background, uncertainties of the echo dimension, and calibration errors. The first source of errors, which we evaluated from the standard deviation of the mean sky background, was negligible compared to the other ones and resulted in errors $\la$0.1% in the flux and surface brightness, and $\la$0.001 mag in STmag. As noted in Sect. \[obs\_sect\], before measuring the echo, we clipped the pixels with signal below 1.5$\sigma$ of the sky variations. This value is of course somewhat arbitrary. To estimate the uncertainties caused by this procedure we repeated measurments by allowing for 1$\sigma$ variations of the clipping level. The surface brightness, $S_{\rm B}$, was most vulnerable to this uncertainty, which was 4–7% in the F606W and F814W filters, and 8–16% in the F435W filter. The total flux was uncertain to 0.3–1.3% in the F606W and F814W filters and 2–8% in the F435W filter. $S_{\rm B\,0.5}$ is insensitive to the clipping level. Errors in determining the echo radius derived in @tss05 propagate through the correction procedure (described above) when deriving STmag, resulting in a $\sim$0.05 mag uncertainty. They also affect $S_{\rm B\,0.5}$ through the evaluation of the total number of pixels inside the echo and result in 2.5–3.5% uncertainty. The deriviation of $S_{\rm B}$ does not involve the echo radius. To evaluate uncertainties of the image flux calibration, we selected ten field stars located outside the light echoes and measured their corresponding counts in all images. Calculating mean values and standard deviations, we found that the calibration is uncertain to 7%, 5%, and 4% in the F435W, F606W, and F814W filters, respectively. Propagation of all these errors through the measuring procedure results in the final errors listed in Table \[measur\_tab\].
The model \[model\_sect\]
=========================
Basic assumptions and formulae of the light echo geometry and its modelling can be found in @tyl04 [see also @sparks]. Briefly, we assume a single-scattering approximation and that the echo structure is described by the paraboloid equation, i.e. $$r = z + ct,
\label{parabol}$$ where $r$ is a radial distance of the scattering point from the source of radiation, $z$ is a projection of $r$ on the line of sight of the source, and $t$ is a time delay between the moment of observations of the scattered radiation and that of the radiation directly recorded from the source.
![Cross section, $y=0$, of the geometrical model adopted in our light-echo modelling. The source of the light is at $x=0$, $y=0$, $z=0$ (black point in the figure). The observer is at $x=0$, $y=0$, $z=\infty$. $x$ and $z$ are in units of $ct$. Dust is uniformely distributed in the space satisfying $z \le z_0$ and $r = \sqrt{x^2 + y^2 + z^2} \ge r_0$. Dashed curves: light-echo paraboloids, Eq. (\[parabol\]), for two epochs differing by a factor of 2. If the source produced a flash of light, which is observed between these two epochs, dust scattering the light at present ($ct=0$) towards the observer is situated between these two paraboloids. []{data-label="scatch_fig"}](scatch.eps)
The geometry assumed in our light-echo simulations is shown in Fig. \[scatch\_fig\]. The echoing medium surrounding the flaring object is uniform and has a semi-infinite geometry. Its boundary is in a form of a plane perpendicular to the line of sight, situated in front of the flaring object at a certain distance, $z_0$, from the object. The medium extends well beyond the object so that the light echo paraboloid never reaches its possible boundary behind the object during the time span covered in our simulations. It was shown in @tyl04 that V838 Mon is situated in a cavity in the surrounding dusty medium. We assume, for simplicity, that this dust-free hole is spherical in respect to the object and has a radius $r_0$.
In the above approximation, the echoing medium is parametrized, apart from $z_0$ and $r_0$, with the optical thickness of the medium along the line of sight of the central object, $\tau_0$. The thickness devided by $z_0 - r_0$ gives the extinction coefficient, $Q_{\rm ext} = Q_{\rm abs} + Q_{\rm sca}$, where $Q_{\rm abs}$ and $Q_{\rm sca}$ are the absorption and scattering coefficients, respectively. The scattering on dust grains is usually anisotropic, so in order to model it properly, we adopted a standard anisotropy phase function, i.e. $$f(\theta) = \frac{1 - g^2}{(1 + g^2 -2g\cos\theta)^{3/2}},$$ where $\theta$ is a scattering angle and $g \equiv \langle \cos \theta \rangle$ is an anisotropy factor. Generally, the anisotropy factor is determined by properties of dust grains [see e.g. @draine]. In our modelling, $\langle \cos \theta \rangle$ is treated as a free parameter to be determined from fitting the results of simulations to the observational measurements. Extinction of the light travelling through the echoing medium is taken into account in the modelling.
The observed light curve of V838 Mon was obtained in the Johnson-Cousins filters. The HST images of the light echo were obtained with the HST filters, however, which differ from the Johnson-Cousins system. For the purpose of our modelling we converted the photometric measurements of V838 Mon in the $B$, $V$, $R_c$, and $I_c$, taken from the same references as in @tyl05, into the light curves in F435W, F606W, and F814W using transformation formulae of @sirianni. These light curves were then used to model the evolution of the light echo images.
As discussed in Sect. \[obs\_sect\], the observed echo image was contaminated by field stars, in particular by the diffraction pattern of V838 Mon itself. The latter had a typical dimension of $\sim$3 arcsec in radius (note that the diffraction cross was much more extended), and was removed in the reduction precedure of all images before measuring the echo. This effect can be easily taken into account in the model simulations and we removed the central circular region of the above radius from calculating the model echo flux and surface brightness.
![Observed expansion of the outer rim of the light echo, as measured in @tss05 (symbols), compared to our modelling (lines) assuming a distance to V838 Mon of 6.1 kpc. The curves were obtained taking $z_0$ = 2.1 pc (middle curve) $\pm$ 10% (outer curves). The outer rim in the simulations was defined as the radius at which the echo surface brightness drops to 10% of its mean value. The light curve of V838 Mon, used in the simulations, is the same as that used to model the light echo in the F435W filter, see upper panel of Fig. \[mag\_fig\].[]{data-label="radius_fig"}](radius.eps)
Following @sparks and @afbond, we assume that V838 Mon is at a distance of 6.1 kpc. For a fixed distance, the expansion of the echo outer rim is uniquely determined by $z_0$. The observed expansion of the light echo outer rim, as measured in @tss05 and compared to the results of our modelling in Fig. \[radius\_fig\], results in $z_0 \simeq 2.1$ pc. Following @tyl04 and the adopted distance, we assume $r_0 = 0.1$ pc. Note that the latter value is not crucial for the results of our analysis. The albedo of dust grains, i.e. $Q_{\rm sca}/Q_{\rm ext}$, was taken from the interstellar extinction curve of @wd01 with $R_V = 3.1$.
Results of model fitting to observations \[res\_sect\]
======================================================
The results of our modelling with the fixed values of the distance, $z_0$, $r_0$, and albedo, as described in Sect. \[model\_sect\], depend on two parameters, i.e. optical thickness of the echoing matter along the line of sight in front of V838 Mon, $\tau_0$, and the scattering anisotropy factor, $\langle \cos \theta \rangle$. This is illustrated in Fig. \[taucos\_fig\]. Note that each of the two parameters affects the model results in a different way. As can be seen from the upper panel of Fig. \[taucos\_fig\], $\tau_0$ does not change the shape of the echo flux evolution but it moves the curve vertically in the diagram. On the other hand, $\langle \cos \theta \rangle$ determines the rate of fading of the light echo with time (see the bottom panel of Fig. \[taucos\_fig\]). Therefore fitting the model results to the observed evolution of the light echo in different filters can constrain the values of these two parameters and their dependence on the wavelength.
![Evolution of the model echo total flux (in the magnitude scale) as a function of the optical thickness, $\tau_0$, along the line of sight of the central source (upper panel) and the anisotropy factor, $\langle \cos \theta \rangle$, (bottom panel). Dashed curve in both panels: the light curve of the central source (the same as in the upper panel of Fig. \[mag\_fig\]). Full curves in the upper panel: the light echo flux evolution with $\langle \cos \theta \rangle$ fixed at 0.6 but for $\tau_0$ varying from 0.05 (bottom curve), through 0.1, 0.2, 0.3, and 0.5 (uppermost curve). Full curves in the bottom panel: the light echo flux evolution with $\tau_0$ fixed at 0.2 but for $\langle \cos \theta \rangle$ varying from 0.0 (bottom curve), through 0.2, 0.4, 0.6, and 0.8 (uppermost curve)[]{data-label="taucos_fig"}](echo_mag_tau.eps "fig:") ![Evolution of the model echo total flux (in the magnitude scale) as a function of the optical thickness, $\tau_0$, along the line of sight of the central source (upper panel) and the anisotropy factor, $\langle \cos \theta \rangle$, (bottom panel). Dashed curve in both panels: the light curve of the central source (the same as in the upper panel of Fig. \[mag\_fig\]). Full curves in the upper panel: the light echo flux evolution with $\langle \cos \theta \rangle$ fixed at 0.6 but for $\tau_0$ varying from 0.05 (bottom curve), through 0.1, 0.2, 0.3, and 0.5 (uppermost curve). Full curves in the bottom panel: the light echo flux evolution with $\tau_0$ fixed at 0.2 but for $\langle \cos \theta \rangle$ varying from 0.0 (bottom curve), through 0.2, 0.4, 0.6, and 0.8 (uppermost curve)[]{data-label="taucos_fig"}](echo_mag_cos.eps "fig:")
Figures \[mag\_fig\] and \[sb\_fig\] compare the results of our models of the light echo (curves) fitted to the observational measurements (symbols with error bars) collected in Table \[measur\_tab\]. The fitting was made with the $\chi^2$ method. Figure \[mag\_fig\] compares the total brightness of the light echo in the ST magnitude scale (data from column 7 in Table \[measur\_tab\]). Here, we also plotted the observed light curves of V838 Mon (dashed curves) that were used to model the echo. Figure \[sb\_fig\] presents the observed and model evolution of the mean surface brightness of the echo. Open symbols and full curves refer to the measurments and modelling of $S_{\rm B}$ (data from column 6 in Table \[measur\_tab\]). Filled symbols and dashed curves shows the same but for $S_{\rm B\,0.5}$ (data from column 8 in Table \[measur\_tab\]). The parameters of all models displayed in Figs. \[mag\_fig\] and \[sb\_fig\] are given in Table \[res\_tab\]. The errors of the parameters were obtained form the 90% confidence level of the $\chi^2$ fitting.
![Evolution of the light-echo total-flux in the ST magnitude scale as observed in the F435W (upper panel), F606W (middle panel), and F814W (bottom panel) filters. Dashed curve: observed light curve of V838 Mon. Asterisks: observed light-echo magnitudes (column 7 in Table \[measur\_tab\]). Full curve: best fit of the modelled evolution to the observational points. Parameters of the fit can be found in Table \[res\_tab\] upper rows. []{data-label="mag_fig"}](B_mag.eps "fig:") ![Evolution of the light-echo total-flux in the ST magnitude scale as observed in the F435W (upper panel), F606W (middle panel), and F814W (bottom panel) filters. Dashed curve: observed light curve of V838 Mon. Asterisks: observed light-echo magnitudes (column 7 in Table \[measur\_tab\]). Full curve: best fit of the modelled evolution to the observational points. Parameters of the fit can be found in Table \[res\_tab\] upper rows. []{data-label="mag_fig"}](V_mag.eps "fig:") ![Evolution of the light-echo total-flux in the ST magnitude scale as observed in the F435W (upper panel), F606W (middle panel), and F814W (bottom panel) filters. Dashed curve: observed light curve of V838 Mon. Asterisks: observed light-echo magnitudes (column 7 in Table \[measur\_tab\]). Full curve: best fit of the modelled evolution to the observational points. Parameters of the fit can be found in Table \[res\_tab\] upper rows. []{data-label="mag_fig"}](I_mag.eps "fig:")
![Evolution of the light-echo surface-brightness in the F435W (upper panel), F606W (middle panel), and F814W (bottom panel) filters. Open symbols and full curve: observed light-echo surface-brightness $S_{\rm B}$ (column 6 in Table \[measur\_tab\]) and its best-fitting model. Filled symbols and dashed curve: the same but for $S_{\rm B\,0.5}$ (column 8 in Table \[measur\_tab\]). Parameters of the fit can be found in Table \[res\_tab\] middle and bottom rows.[]{data-label="sb_fig"}](B_sb.eps "fig:") ![Evolution of the light-echo surface-brightness in the F435W (upper panel), F606W (middle panel), and F814W (bottom panel) filters. Open symbols and full curve: observed light-echo surface-brightness $S_{\rm B}$ (column 6 in Table \[measur\_tab\]) and its best-fitting model. Filled symbols and dashed curve: the same but for $S_{\rm B\,0.5}$ (column 8 in Table \[measur\_tab\]). Parameters of the fit can be found in Table \[res\_tab\] middle and bottom rows.[]{data-label="sb_fig"}](V_sb.eps "fig:") ![Evolution of the light-echo surface-brightness in the F435W (upper panel), F606W (middle panel), and F814W (bottom panel) filters. Open symbols and full curve: observed light-echo surface-brightness $S_{\rm B}$ (column 6 in Table \[measur\_tab\]) and its best-fitting model. Filled symbols and dashed curve: the same but for $S_{\rm B\,0.5}$ (column 8 in Table \[measur\_tab\]). Parameters of the fit can be found in Table \[res\_tab\] middle and bottom rows.[]{data-label="sb_fig"}](I_sb.eps "fig:")
F435W F606W F814W
------------ --------------- --------------- ---------------
: Values of $\tau_0$ and $\langle \cos \theta \rangle$ derived from the model fitting to the observations.[]{data-label="res_tab"}
\
total flux (STmag)\
------------------------------- ------------------ ------------------ ------------------
$\tau_0$ $0.204 \pm .016$ $0.151 \pm .003$ $0.087 \pm .003$
$\langle \cos \theta \rangle$ $0.67 \pm .06$ $0.62 \pm .02$ $0.59 \pm .03$
------------------------------- ------------------ ------------------ ------------------
: Values of $\tau_0$ and $\langle \cos \theta \rangle$ derived from the model fitting to the observations.[]{data-label="res_tab"}
\
\
surface brightness $S_{\rm B}$\
------------------------------- ------------------ ------------------ ------------------
$\tau_0$ $0.218 \pm .028$ $0.159 \pm .009$ $0.087 \pm .006$
$\langle \cos \theta \rangle$ $0.64 \pm .10$ $0.57 \pm .06$ $0.54 \pm .06$
------------------------------- ------------------ ------------------ ------------------
: Values of $\tau_0$ and $\langle \cos \theta \rangle$ derived from the model fitting to the observations.[]{data-label="res_tab"}
\
\
surface brightness $S_{\rm B\,0.5}$\
------------------------------- ------------------ ------------------ ------------------
$\tau_0$ $0.306 \pm .020$ $0.249 \pm .010$ $0.137 \pm .004$
$\langle \cos \theta \rangle$ $0.64 \pm .05$ $0.60 \pm .04$ $0.56 \pm .03$
------------------------------- ------------------ ------------------ ------------------
: Values of $\tau_0$ and $\langle \cos \theta \rangle$ derived from the model fitting to the observations.[]{data-label="res_tab"}
Analysis and discussion \[discuss\_sect\]
=========================================
![Dependence of the scattering coefficient, $Q_{\rm sca}$, on the wavelength as derived from modelling of the echo. The values are normalized to $Q_{\rm sca}$ at the effective wavelength of the F814W filter. Asterisks: results of the models fitting the total flux, STmag, (upper rows in Table \[res\_tab\]). Open circles: results of the models fitting the mean surface brightness, $S_{\rm b}$, (middle rows in Table \[res\_tab\]). Filled circles: results of the models fitting the mean surface brightness derived from the brightest pixels filling $f_{\rm c} = 0.5$, $S_{\rm B\,0.5}$, (bottom rows in Table \[res\_tab\]). The horizontal error bars represent the widths of the photometric bands. Open and filled symbols are shifted by $\pm 0.05\,\mu$m in the abscissa axis for clarity. Curves: relations expected from modelling the interstellar extinction curve of @wd01 for $R_{V} = 3.1$ (dotted) and $R_{V} = 5.5$ (dashed).[]{data-label="qsca_fig"}](q_sca.eps)
![Dependence of the anisotropy factor, $\langle \cos \theta \rangle$, on the wavelength, as derived from our modelling of the echo. The symbols and curves have the same meaning as in Fig. \[qsca\_fig\].[]{data-label="cos_fig"}](cos.eps)
As can be seen from Table \[res\_tab\], the optical thickness of the dusty matter in front of V838 Mon, $\tau_0$, as determined from modelling the surface brightness, $S_{\rm B}$, (middle rows in the table) is systematically higher than the values derived from fitting the total echo flux (upper rows in the table). The difference is within the error bars, however. Modelling of the surface brightness of the bright regions covering 50% of the echo surface, $S_{\rm B\,0.5}$, (bottom rows in the table) gives significantly higher values of $\tau_0$ than in the two previous cases. This is not surprising, as brighter regions mean more effective scattering and thus a higher extinction coefficient. The values listed in the upper and middle rows of Table \[res\_tab\] can be considered as typical for the echoing medium of V838 Mon. Those in the bottom rows, when compared to the other ones, illustrate how inhomogeneous the medium is.
The results in Table \[res\_tab\] allow us to study properties of dust grains in the echoing medium. Figures \[qsca\_fig\] and \[cos\_fig\] show values of the scattering coefficient and the anisotropy factor, respectively, as functions of wavelength and compare them to the analogous values obtained by @wd01 from modelling the interstellar extinction curve with $R_{V} = 3.1$ and 5.5.
As can be seen from Fig. \[qsca\_fig\], our modelling of different observational quantities (total flux, surface brightness) results in a similar wavelength dependence of the scattering coefficient. We can also conclude that this relation is not significantly different from that expected from the standard composition and size distribution of dust grains used to model the interstellar extinction curve of @wd01 with $R_{V} = 3.1$. This suggests that the dusty matter responsible for the observed echo is not significantly different from the typical interstellar dusty medium.
The scattering anisotropy factor, $\langle \cos \theta \rangle$, as displayed in Fig. \[cos\_fig\], is more sensitive to the observational uncertainties than the scattering coefficient. There are systematic differences in the derived $\langle \cos \theta \rangle$ values, depending on which quantity was analysed, but the differences are within the error bars. The value of $\langle \cos \theta \rangle$ is determined from the rate of the echo fading with time. This rate, if derived from the measurements of all pixels detecting emission from the echo, is likely to be distorted because the portion of the echo surface covered by these pixels is different for different dates of the observations. Therefore we expect that the results of fitting the mean surface brightness, $S_{\rm B\,0.5}$, (bottom row in Table \[res\_tab\] and filled circles in Fig. \[cos\_fig\]) are more reliable than the other two.
Fig. \[cos\_fig\] shows that our determinations of $\langle \cos \theta \rangle$ gave values systematically greater than those expected from the interestellar extinction curve with $R_{V} = 3.1$. They are better reproduced by the extinction curve with $R_{V} = 5.5$, but even in this case the points mostly lie above the curve. Greater $\langle \cos \theta \rangle$ is expected for bigger dust grains, but this usually implies that the scattering coefficient is less dependent on the wavelength, which would be in conflict with the points in Fig. \[qsca\_fig\]. However, the model extinction curve and $\langle \cos \theta \rangle$ depend not only on the grain size, but also on the chemical composition and structure of the grains. It is thus possible that playing with different parameters of the dust grains would allow one to better reproduce the dust properties we obtained from the analysis of the light echo of V838 Mon. This is out of the scope of the present study, however.
We can estimate from Table \[res\_tab\] (upper and middle rows) that the additional extinction of V838 Mon due to local dust responsible for the light echo is $A_{V} \simeq 1.2\,\tau_0({\rm F606W}) \simeq 0.18$ ($E_{B-V} \simeq
0.05$). Of course, this is the most likely value derived from the mean properties of the local dusty medium seen in the echo. This medium is inhomogeneous, however, as is shown by the uneven distribution of the surface brightness over the echo images. Therefore the real extinction along the line of sight of V838 Mon can be different from the above estimate. Nevertheless, the results obtained from the brighter half part of the echo surface, i.e. from fitting $S_{\rm B\,0.5}$ (bottom rows in Table \[res\_tab\]), indicate that it is rather improbable that $A_{V} > 0.3$. With the observed extinction curve (Fig. \[qsca\_fig\]) this would also imply $E_{B-V} \ga 0.10$ from the local dust, which would be too large compared to the observed difference in $E_{B-V}$ between V838 Mon and the three B-type main-sequence stars observed by @afbond (see Sect. \[intro\]).
Using the standard interstellar-medium relation, $A_{V}/N_{\rm H} \simeq 5.3\times10^{-22}$ cm$^2$ [@bohlin], where $N_{\rm H}$ is the column density of hydrogen (atomic plus molecular), and taking $A_{V} \simeq 0.18$, we obtain a surface density of the echoing matter in front of V838 Mon to be $\sim 8 \times 10^{-4}$ gcm$^{-2}$. In 2004 the angular radius of the light echo reached $\sim$1 arcmin [@tss05]. With a distance of 6.1 kpc this allows us to estimate the mass of the diffuse matter of the light echo to be $\sim$35 M$_\odot$. This is a lower limit to the mass of the circumstellar matter of V838 Mon, since, first, in later epochs the echo expanded to even larger radii, second, the estimate is based on the optical thickness [*in front of*]{} V838 Mon only, while the matter almost certainly extends beyond the object. The above estimate can be compared to 90–100 M$_\odot$ obtained by @ktd for the molecular complex seen on the CO rotational lines in the vicinity of V838 Mon and a hundred M$_\odot$ estimated by @baner06 for the mass of the echoing matter in the infrared. All these estimates clearly show that the diffuse matter in the vicinity of V828 Mon, partly seen in the light echo, could not have resulted from a past mass loss activity of V838 Mon, as advocated, e.g. in @bond03 and @bond07. It is much more probable that we see remnants of an interstellar complex, from which V838 Mon, its companion, and perhaps the other members of the observed cluster were formed.
It is worth noting that @kervella recently found that the mass of the dusty matter echoing pulsating radiation from the Cepheid RS Pup is $\sim$290 M$_\odot$. It seems therefore that, in general, a stellar light echo to be well resolved in imaging observations requires a flaring star to be associated with a dusty medium of interstellar origin rather than formed from a mass loss activity of the star.
On the basis of our results, we can quite safely conclude that the optical thickness of the dusty matter seen in the light echo is too low to account for the observed dimming of the B-type companion by $\sim$1.3 mag compared to the other B-type stars observed by @afbond in the vicinity of V838 Mon (see Sect. \[intro\]). Other explanations of this discrepancy have to be considered, e.g. mis-classification of the B-type companion, an evolutionary status of the companion different from that of the three B-type stars of the cluster, or a mistaken distance to V838 Mon and its companion.
The classification of the companion as a B3 main-sequence star comes from @mdh02 [see also @mnv07]. This classification was essentially confirmed by @afbond, but the B-type component in their spectrum, similarly as in the spectrum of @kst09, was then significantly contaminated by V838 Mon. The observed $V$ brightness of the companion would be in accord with those of the three B-type stars of @afbond if it were of a $\sim$B7V spectral type. The intrinsic colours of such a star would be $(B-V)_0 = -0.13$ and $(U-B)_0
= -0.43$ [@sk82]. From Goranskij’s$^1$ compilation of the photometric data we derived $V = 16.19 \pm 0.03$, $B - V = 0.55 \pm 0.05$, and $U - B = -0.15 \pm 0.07$ for the B-type companion. The observed $B-V$ colour can be reconciled with that of the above standard if $E_{B-V} \simeq 0.69$. But then the reddened $U - B$ colour of the standard star is $+0.08$, i.e. it is significantly redder than the observed one. Using the photometric measurements of the B-type companion of @mnv07, we obtain $E_{B-V} \simeq 0.82$ and $U-B = +0.17$ for the reddened standard, which is to be compared to the observed one of $-0.06$. We thus conclude that B7V cannot be reconciled with the observed colours of the B-type companion. Note that the best fit of a standard star to the colours derived from the Goranskij data can be obtained taking the spectral type of B2.5V and $E_{B-V} \simeq 0.80$, which is not significantly different from the B3V spectral type obtained by @mdh02.
The second possibility is that the B-type companion of V838 Mon is intrinsically fainter because it is in a somewhat different evolutionary stage than the three B-type stars of @afbond.[^6] All existing observational data indicate that all four stars are on the main sequence (MS) or not far from it. The only reasonable situation to be discussed in this context is that the B-type companion of V838 Mon has a similar effective temperature but is less luminous (by $\sim$1.3 mag.) than star 9 (B3V) of @afbond because it is less massive but is at its hottest stage, namely near the zero age main sequence (ZAMS), while star 9 is more evolved on the MS. Taking the calibrations of the absolute magnitude versus spectral type and luminosity class from @sk82, one easily concludes that to account for a $\sim$1.3 magnitude difference between two B-type stars of a similar spectral type with one of them being on ZAMS, the second star has to be of luminosity class IV, i.e. slightly off MS. This would probably not be in a significant conflict with the spectroscopically determined class V, given the quality of the data of @afbond. However, the differences in magnitude between star 9, star 8 (B4V), and star 7 (B6V) imply that the latter two stars have also to be of luminosity class IV. To have a situation in which three stars of different masses are about to leave or just left MS at the same time, one has to conclude that the three stars have different ages, e.g. star 7 would have to be about twice as old as star 9. This disproves the idea that these three stars form a cluster. Futhermore, a case in which three B-type stars formed at different times happen to lie in a close vicinity in the sky and mimic an isochronic sequence of a cluster in the colour-magnitude diagram is extremely improbable. So is the whole scenario discussed in this paragraph.
If two MS stars of the same spectral type suffer from a similar extinction but are significantly different in the observed brightness then the only reasonable conclusion is that the stars are at different distances. In this way we return to the study of @muna07, where the authors concluded, primarily on the basis of the spectroscopic distance of the B-type companion, that V838 Mon is at a distance of $\sim$10 kpc. In this case V838 Mon and its companion would have nothing to do with the cluster of @afbond. This possibility creates a problem with the observed linear polarization of the light echo, however, as analysed in @sparks. To reconcile the latter with the distance of 10 kpc, one has to postpone that maximum polarization occurs at a scattering angle of $\sim$60 and not at 90, as assumed in @sparks. Theoretically this cannot be excluded [e.g. Chapter 4.2 in @krug] but requires rather unusual dust grains, e.g. metallic particles.
We performed light echo simulations, similar to those described in Sect. \[res\_sect\], but assuming a distance of 10 kpc. They gave results not significantly different from those obtained in Sect. \[res\_sect\], except that $\langle \cos \theta \rangle$ was systematically higher by $\sim$20% than the values in Table \[res\_tab\].
We can conclude that if the distance of V838 Mon were $\sim$10 kpc, dust in the echoing medium would have to be quite peculiar, i.e. giving an extinction curve close to the standard one but scattering strongly in forward directions and giving maximum polarization at scattering angles significantly lower than 90. With the present state of our knowledge of interstellar dust grains, we consider the above result as an argument against a distance as large as $\sim$10 kpc.
In summary, we cannot present any strightforward scenario that could explain the $\sim$1.3 magnitude difference between the B-type companion of V838 Mon and the B3V star of @afbond and which would be in accord with the observed properties of dust in the echoing matter of V838 Mon. Perhaps this could be a dust cloud giving a grey extinction, lying well in front of V838 Mon (not to be seen in the light echo), and not intervening with the lines of sight of the three B-type stars of @afbond. A detailed spectroscopic study, particularly of the interstellar line profiles (NaI, KI), as well as multi-colour photometric measurements of a large sample of stars in the field of V838 Mon would perhaps enable one to be conclusive in this subject.
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[^1]: Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555.
[^2]: http://jet.sao.ru/jet/$\sim$goray/v838mon.htm
[^3]: http://stsdas.stsci.edu/multidrizzle
[^4]: http://www.stsci.edu/hst/acs/documents/handbooks/currentDHB
[^5]: http://starlink.jach.hawaii.edu/starlink
[^6]: As sugested by V. Goranskij during the STScI workshop “Intermediate-Luminosity Red Transients”, June 2011.
|
---
abstract: 'In this paper we prove the dry version of the Ten Martini problem: Cantor spectrum with all gaps open, for the extended Harper’s model in the non self-dual region for Diophantine frequencies.'
address: 'Department of Mathematics, University of California, Irvine, CA, 92697-3875, United States of America'
author:
- Rui Han
title: 'Dry Ten Martini problem for the non-self-dual extended Harper’s model'
---
Introduction
============
The study of independent electrons on a two-dimensional lattice exposed to a perpendicular magnetic field and periodic potentials can be reduced via an appropriate choice of gauge field to the study of discrete one-dimensional quasiperiodic Jacobi matrices. The most extensively studied case is the almost Mathieu operator (AMO) acting on $l^2({{\mathbb Z}})$ defined by $$\begin{aligned}
(H_{\lambda, \alpha, \theta}u)_n=u_{n+1}+u_{n-1}+2\lambda \cos{2\pi (\theta+n\alpha)} u_n.\end{aligned}$$ This is a one-dimensional tight-binding model with anisotropic nearest neighbor couplings in general. A more general model, called the extended Harper’s model (EHM), is the operator acting on $l^2({{\mathbb Z}})$ defined by: $$\begin{aligned}
(H_{\lambda, \alpha,\theta}u)_n=c(\theta+n\alpha)u_{n+1}+\tilde{c}(\theta+(n-1)\alpha)u_{n-1}+2\cos{2\pi(\theta+n\alpha)}u_{n}.\end{aligned}$$ where $c(\theta)=\lambda_1 e^{-2\pi i(\theta+\frac{\alpha}{2})}+\lambda_2+\lambda_3 e^{2\pi i(\theta+\frac{\alpha}{2})}$ and $\tilde{c}(\theta)=\lambda_1 e^{2\pi i (\theta+\frac{\alpha}{2})}+\lambda_2+\lambda_3 e^{-2\pi i (\theta+\frac{\alpha}{2})}$. It is obtained when both the nearest neighbor coupling (expressed through $\lambda_2$) and the next-nearest couplings (expressed through $\lambda_1$ and $\lambda_3$) are included. This model includes AMO as a special case (when $\lambda_1=\lambda_3=0$).
For the AMO, it was proved in [@AJ1] that the spectrum is a Cantor set for any $\alpha\in {{\mathbb R}}\setminus {{\mathbb Q}}$ and $\lambda\neq 0$. This is the [*Ten Martini Problem*]{} dubbed by Barry Simon, after an offer of Mark Kac. A much more difficult problem, known as the dry version of the Ten Martini Problem, is to prove that the spectrum is not only a Cantor set, but that all gaps predicted by the Gap-Labelling theorem [@BLT], [@JM] are open. The first result was obtained for Liouvillean $\alpha$ [@CEY], and later it was proved for a set of $(\lambda, \alpha)$ of positive Lebesgue measure [@Puig]. The most recent result is [@AJ2], in which they were able to deal with all Diophantine frequencies and $\lambda\neq 1$. A solution for all irrational frequencies and $\lambda\neq 1$ was also recently announced in [@AYZ].
Recently, there have been several important advances on the spectral theory of the EHM: purely point spectrum for Diophantine $\alpha$ and a.e.$\theta$ in the positive Lyapunov exponent region [@JKS]; the exact formula for Lyapunov exponent for all coupling constants [@Jm]; the spectral decomposition for a.e.$\alpha$ [@AJM]. However the results that study the spectrum as a set have not been obtained for the EHM.
For EHM, depending on the values of the parameters $\lambda_1, \lambda_2, \lambda_3$, we could divide the parameter space into three regions as shown in the picture below:
(-10,-1) – (-3,-1) node\[below\] [$\lambda_2$]{}; (-10,-1) – (-10,6) node\[right\] [$\lambda_1+\lambda_3$]{}; plot \[smooth\] coordinates [ (- 7, 2) (-4, 5) ]{}; plot \[smooth\] coordinates [ (- 7, 2) (-7,-1) ]{}; plot \[smooth\] coordinates [ (-10, 2) (-7, 2) ]{};
(- 4, 5) node \[above\] [$\lambda_1+\lambda_3=\lambda_2$]{}; (- 7, -1) node \[below\] [$1$]{}; (- 10, 2) node \[left\] [$1$]{}; (-9.2, 0.5) node \[color=blue\]\[right\] [region I]{}; (- 5, 1) node \[color=blue\]\[right\] [region II]{}; (-9.2, 3.6) node \[color=blue\]\[right\] [region III]{};
(- 7, 0.5) node \[color=red\]\[right\] [$L_{II}$]{}; (-8.5, 2) node \[color=red\]\[above\] [$L_{I}$]{}; (-5.5, 3.7) node \[color=red\]\[left\] [$L_{III}$]{};
$$\begin{aligned}
®ion\ I : 0<\max{(\lambda_1+\lambda_3,\lambda_2)}<1,\\
®ion\ II : 0<\max{(\lambda_1+\lambda_3, 1)} < \lambda_2,\\
®ion\ III : 0<\max{(1, \lambda_2)} < \lambda_1+\lambda_3.\end{aligned}$$
According to the action of the duality transformation $\sigma: \lambda=(\lambda_1, \lambda_2, \lambda_3)\rightarrow \hat{\lambda}=(\frac{\lambda_3}{\lambda_2}, \frac{1}{\lambda_2}, \frac{\lambda_1}{\lambda_2})$, region I and region II are dual to each other and region III is a self-dual region. Region I is the positive Lyapunov exponent region, which is a natural extension of the segment $\{\lambda_1+\lambda_3=0, 0<\lambda_2<1\}$ corresponding to the case $\lambda>1$ in the AMO. Region II is the subcritical region, which is an extension of the segment $\{\lambda_1+\lambda_3=0, 1<\lambda_2\}$ corresponding to the case $\lambda<1$ in the AMO.
In this paper we prove the dry version of the Ten Martini Problem in region I and region II under the Diophantine condition.
Let ${p_n}/{q_n}$ be the continued fraction appoximants of $\alpha\in {{{\mathbb R}}}\setminus {{{\mathbb Q}}}$. Let $$\begin{aligned}
\beta(\alpha)=\limsup_{n\rightarrow\infty} \frac{\ln{q_{n+1}}}{q_n}.\end{aligned}$$ If $\beta(\alpha)=0$, we say $\alpha$ satisfies the Diophantine condition, denoted by $\alpha\in \mathrm{DC}$. It is easily seen that such $\alpha$ form a full measure subset of ${{\mathbb T}}$.
It is known that when $E$ is in the closure of a spectral gap, the integrated density of states (IDS) $N(E)\in \alpha {{\mathbb Z}}+{{\mathbb Z}}$ (refer to ($\ref{IDS}$) for the definition of IDS) [@BLT], [@JM]. Here we prove the inverse is true.
\[dry\] If $\alpha\in \mathrm{DC}$ and $\lambda$ belongs to region I or region II, all possible spectral gaps are open.
We note the Dry Ten Martini problem has not yet been solved for the self-dual AMO. In the self-dual region III, Cantor spectrum is known in the isotropic case (when $\lambda_1=\lambda_3$), see Fact $2.1$ in [@AJM]. In fact one could prove the operator has zero Lebesgue measure spectrum for all frequencies.
In region I and II, for Liouvillean $\alpha$ (where $\beta(\alpha)$ is large), it is not clear whether even the Cantor spectrum holds. The proof may require a non-trivial adjustment of the proof for AMO in [@CEY].
We first establish almost localization (see section 3.1) in region I, then a quantitative version of Aubry duality to obtain almost reducibility (see section 3.2) in region II which enables us to deal with all energies whose rotation numbers are $\alpha$-rational.
Thus the strategy follows that of [@AJ2], but we need to extend the almost localization and quantitative duality, as well as the final argument to our Jacobi setting, which is non-trivial on a technical level. At the same time unlike [@AJ2], we only deal with a short-range dual operator, leading to a significant streamlining of some arguments of [@AJ2].
We organize the paper as follows: in section 2 we present some preliminaries, in section 3 we state our main results about almost localization and almost reducibility, relying on which we provide a proof of Theorem $\ref{dry}$. In section 4 and 5 we prove the main results that we present in section 3.
preliminaries
=============
Cocycles
--------
Let $\alpha\in {{\mathbb R}}\setminus {{\mathbb Q}}$ and $A\in C^0({{\mathbb T}}, M_2({{\mathbb C}}))$ measurable with $\log{\|A(x)\|}\in L^1({{\mathbb T}})$. The quasi-periodic $cocycle$ $(\alpha, A)$ is the dynamical system on ${{\mathbb T}}\times {{\mathbb C}}^2$ defined by $(\alpha, A)(x, v)=(x+\alpha, A(x)v)$. The [*Lyapunov exponent*]{} is defined by $$\begin{aligned}
L(\alpha, A)=\lim_{n\rightarrow\infty}\frac{1}{n}\int_{{{\mathbb T}}}\log{\|A_n(x)\|}\mathrm{d}x=\inf_{n}\frac{1}{n}\int_{{{\mathbb T}}}\log{\|A_n(x)\|}\mathrm{d}x.\end{aligned}$$ where $$\begin{aligned}
\begin{cases}
A_n(x)= A(x+(n-1)\alpha)\cdots A(x) \ \ \mathrm{for}\ n\geq 0,\\
A_n(x)=A^{-1}(x+n\alpha)\cdots A^{-1}(x-\alpha) \ \ \mathrm{for}\ n<0.
\end{cases}\end{aligned}$$
\[uniformupp\](e.g.[@AJ2]) Let $(\alpha, A)$ be a continous cocycle, then for any $\delta>0$ there exists $C_{\delta}>0$ such that for any $n\in {{\mathbb N}}$ and $\theta\in {{\mathbb T}}$ we have $$\begin{aligned}
\|A_n(\theta)\|\leq C_{\delta} e^{(L(\alpha, A)+\delta)n}.\end{aligned}$$
We say that $(\alpha, A)$ is [*uniformly hyperbolic*]{} if there exists continuous splitting ${{\mathbb C}}^2=E^s(x)\bigoplus E^u(x)$, $x\in {{\mathbb T}}$ such that for some constant $C, \eta>0$ and all $n\geq 0$, $\|A_n(x) v\|\leq Ce^{-\eta n}\|v\|$ for $v\in E^s(x)$ and $\|A_{-n}(x) v\|\leq C e^{-\eta n}\|v\|$ for $v\in E^{u}(x)$.
Given two complex cocycles $(\alpha, A^{(1)})$ and $(\alpha, A^{(2)})$, we say they are [*complex conjugate*]{} to each other if there is $M\in C^0({{\mathbb T}}, SL(2,{{\mathbb C}}))$ such that $$\begin{aligned}
M^{-1}(x+\alpha)A^{(1)}(x)M(x)=A^{(2)}(x).\end{aligned}$$ We assume now that $A$ is a real cocycle, $A\in C^0({{\mathbb T}}, SL(2,{{\mathbb R}}))$. The notation of [*real conjugacy*]{} (between real cocycles) is the same as before, except that we look for $M\in C^0({{\mathbb T}}, PSL(2,{{\mathbb R}}))$. A reason why we look for $M\in C^0({{\mathbb T}}, PSL(2,{{\mathbb R}}))$ instead of $M\in C^0({{\mathbb T}}, SL(2,{{\mathbb R}}))$ is given by the following well-known result.
\[uhconjugate\] Let $(\alpha, A)$ be uniformly hyperbolic, assume $\alpha\in \mathrm{DC}$ and $A$ analytic, then there exists $M\in C^{\omega}({{\mathbb T}}, PSL(2,{{\mathbb R}}))$ [^1] such that $M^{-1}(x+\alpha)A(x)M(x)$ is constant.
We say $(\alpha, A)$ is (analytically) [*reducible*]{} if it is real conjugate to a constant cocycle by an analytic conjugacy.
Let $$\begin{aligned}
R_{\theta}=
\left(
\begin{matrix}
\cos{2\pi \theta} \ \ &-\sin{2\pi \theta}\\
\sin{2\pi \theta} \ \ &\cos{2\pi \theta}
\end{matrix}
\right).\end{aligned}$$ Any $A\in C^0({{\mathbb T}}, PSL(2,{{\mathbb R}}))$ is homotopic to $x\rightarrow R_{\frac{k}{2}x}$ for some $k\in {{\mathbb Z}}$ called the ${\it degree}$ of $A$, denoted by $\deg{A}=k$.
Assume now that $A\in C^0({{\mathbb T}}, SL(2,{{\mathbb R}}))$ is homotopic to identity. Then there exists $\phi:{{\mathbb R}}/{{\mathbb Z}}\times {{\mathbb R}}/{{\mathbb Z}}\to {{\mathbb R}}$ and $v:{{\mathbb R}}/{{\mathbb Z}}\times {{\mathbb R}}/{{\mathbb Z}}\rightarrow {{\mathbb R}}^+$ such that $$\begin{aligned}
A(x)
\left(
\begin{matrix}
\cos 2 \pi y \\
\sin 2 \pi y
\end{matrix}
\right)
=v(x,y)
\left(
\begin{matrix}
\cos 2 \pi (y+\phi(x,y)) \\
\sin 2 \pi (y+\phi(x,y))
\end{matrix}
\right).\end{aligned}$$ The function $\phi$ is called a lift of $A$. Let $\mu$ be any probability on ${{\mathbb R}}/{{\mathbb Z}}\times {{\mathbb R}}/{{\mathbb Z}}$ which is invariant under the continuous map $T:(x,y) \mapsto (x+\alpha,y+\phi(x,y))$, projecting over Lebesgue measure on the first coordinate. Then the number $$\begin{aligned}
\rho(\alpha,A)=\int \phi\ d\mu {\operatorname{mod}}{{\mathbb Z}}\end{aligned}$$ is independent of the choices of $\phi$ and $\mu$, and is called the [*fibered rotation number*]{} of $(\alpha,A)$.
It can be proved directly by the definition that $$\begin{aligned}
\label{rho0}
|\rho(\alpha,A)-\theta|<C\|A-R_{\theta}\|_0.\end{aligned}$$
If $(\alpha,A^{(1)})$ and $(\alpha,A^{(2)})$ are real conjugate, $M^{-1}(x+\alpha)A^{(2)}(x)M(x)=A^{(1)}(x)$, and $M:{{\mathbb R}}/{{\mathbb Z}}\to PSL(2,{{\mathbb R}})$ has degree $k$ then $$\begin{aligned}
\label{rhoconju}
\rho(\alpha,A^{(1)})=\rho(\alpha,A^{(2)})-k\alpha/2.\end{aligned}$$
For uniformly hyperbolic cocycles there is the following well-known result.
\[uhrho\] Let $(\alpha,A)$ be a uniformly hyperbolic cocycle, with $\alpha \in {{\mathbb R}}\setminus {{\mathbb Q}}$. Then $2 \rho(\alpha,A) \in \alpha {{\mathbb Z}}+{{\mathbb Z}}$.
Extended Harper’s model
-----------------------
We consider the extended Harper’s model $\{H_{\lambda,\theta}\}_{\theta \in {{\mathbb T}}}$. The formal solution to $H_{\lambda, \theta}u=Eu$ can be reconstructed via the following equation $$\begin{aligned}
\left(
\begin{matrix}
u_{n+1}\\
u_n
\end{matrix}
\right)
=
A_{\lambda, E}(\theta+n\alpha)
\left(
\begin{matrix}
u_n\\
u_{n-1}
\end{matrix}
\right).\end{aligned}$$ where $A_{\lambda, E}(\theta)=
\frac{1}{c(\theta)}
\left(
\begin{matrix}
E-2\cos{2\pi \theta}\ \ &-\tilde{c}(\theta-\alpha)\\
c(\theta)\ \ &0
\end{matrix}
\right)$. Notice that since $A_{\lambda, E}(\theta)\notin SL(2,{{\mathbb R}})$, we introduce the following matrix (see Lemma $\ref{conjugate}$) $$\begin{aligned}
\tilde{A}_{\lambda, E}(\theta)=
\frac{1}{\sqrt{|c|(\theta)|c|(\theta-\alpha)}}
\left(
\begin{matrix}
E-2\cos{2\pi \theta}\ \ &-|c|(\theta-\alpha)\\
|c|(\theta)\ \ &0
\end{matrix}
\right)=
Q_{\lambda}(\theta+\alpha)A_{\lambda, E}(\theta)Q_{\lambda}^{-1}(\theta),\end{aligned}$$ where $|c|(\theta)=\sqrt{c(\theta)\tilde{c}(\theta)}$ (which is not the same as $|c(\theta)|=\sqrt{c(\theta)\overline{c(\theta)}}$ when $\theta\notin {{\mathbb T}}$) and $Q_{\lambda}(\theta)$ is analytic on $|\mathrm{Im}\theta|\leq \frac{\epsilon_1}{2\pi}$.
The spectrum of $H_{\lambda, \theta}$ denoted by $\Sigma_{\lambda}$, does not depend on $\theta$ [@as], and it is the set of $E$ such that $(\alpha, \tilde{A}_{\lambda, E})$ is not uniformly hyperbolic.
The Lyapunov exponent is defined by $L_{\lambda}(E)=L(\alpha,A_{\lambda, E})=L(\alpha, \tilde{A}_{\lambda, E})$.
For a matrix-valued function $M(\theta)$, let $M_{\epsilon}(\theta)=M(\theta+i\epsilon)$ be the phase-complexified matrix.
In [@A3], Avila divides all the energies in the spectrum into three catagories: super-critical, namely the energy with positive Lyapunov exponent; subcritical, namely the energy whose Lyapunov exponent of the phase-complexified cocycle is identically equal to zero in a neighborhood of $\epsilon=0$; critical, otherwise.
The following theorem is shown in [@Jm] (see also the appendix):
\[LE\] Extended Harper’s model is [*super-critical*]{} in region I and [*sub-critical*]{} in region II. Indeed
- when $\lambda$ belongs to region II, $L_{\lambda}(E)=L(\alpha, A_{\lambda, E,\epsilon})=L(\alpha, \tilde{A}_{\lambda, E,\epsilon})=0$ on $|\epsilon|\leq \frac{1}{2\pi}\epsilon_1(\lambda)$,
- when $\lambda$ belongs to region II, we have $\hat{\lambda}=(\frac{\lambda_3}{\lambda_2}, \frac{1}{\lambda_2}, \frac{\lambda_1}{\lambda_2})$ belongs to region I and $$\label{LE1}
L_{\hat{\lambda}}(E)= \epsilon_1(\lambda),$$ where $$\label{epsilon1}
\epsilon_1(\lambda)=\ln{\frac{\lambda_2+\sqrt{\lambda_2^2-4\lambda_1\lambda_3}}{\max{(\lambda_1+\lambda_3, 1)}+\sqrt{\max{(\lambda_1+\lambda_3,1)}^2-4\lambda_1\lambda_3}}}>0.$$
Fix a $\theta$ and $f\in l^2({{\mathbb Z}})$. Let $\mu_{\lambda, \theta}^f$ be the [*spectral measure*]{} of $H_{\lambda ,\theta}$ corresponding to $f$, $$\begin{aligned}
\langle (H_{\lambda, \theta}-z)^{-1}f, f \rangle=\int_{{{\mathbb R}}}\frac{1}{E-z}\mathrm{d}\mu_{\lambda, \theta}^f (E).\end{aligned}$$ for $z$ in the resolvent set ${{\mathbb C}}\setminus \Sigma_{\lambda}$.
The [*integrated density of states*]{} ($\mathrm{IDS}$) is the function $N_{\lambda}: {{\mathbb R}}\to [0,1]$ defined by $$\begin{aligned}
\label{IDS}
N_{\lambda}(E)=\int_{{{\mathbb T}}}\mu_{\lambda, \theta}^f(-\infty, E]\mathrm{d}\theta,\end{aligned}$$ where $f\in l^2({{\mathbb Z}})$ is such that $\|f\|_{l^2({{\mathbb Z}})}=1$. It is a continuous non-decreasing surjective funtion.
Notice that $\tilde{A}_{\lambda, E}(\theta)\in SL(2,{{\mathbb R}})$ is homotopic to identity in $C^{0}({{\mathbb T}}, SL(2,{{\mathbb R}}))$, in fact just consider $$\begin{aligned}
H_t(\lambda, E, \theta)=\frac{1}{\sqrt{|c|(\theta)|c|(\theta-t\alpha)}}
\left(
\begin{matrix}
t(E-v(\theta))\ \ &-|c|(\theta-t\alpha)\\
|c|(\theta) \ \ &0
\end{matrix}
\right).\end{aligned}$$ which establishes a homotopy of $\tilde{A}_{\lambda, E}(\theta)$ to $R_{\frac{1}{4}}$ and hence to the identity. Therefore we can define the rotation number $\rho(\alpha, \tilde{A}_{\lambda, E})$. Let $\rho_{\lambda}(E)=\rho(\alpha, {\tilde{A}_{\lambda, E}})$. Notice that $\rho_{\lambda}(E)$ is associated to the operator $$\begin{aligned}
(\tilde{H}_{\lambda, \theta}u)_n=|c|(\theta+n\alpha)u_{n+1}+|c|(\theta+(n-1)\alpha)u_{n-1}+2\cos{2\pi (\theta+n\alpha)} u_n.\end{aligned}$$ It is easily seen that for each $\theta$, $\tilde{H}_{\lambda ,\theta}$ and $H_{\lambda ,\theta}$ differ by a unitary operator, thus they share the same spectrum and integrated density of states, $\tilde{N}_{\lambda}(E)=N_{\lambda}(E)$. The relation between the integrated density of states and rotation number of $\tilde{H}_{\lambda, \theta}$ yields the following $$\label{IDSROT}
N_{\lambda}(E)=\tilde{N}_{\lambda}(E)=1-2\rho_{\lambda}(E).$$
The dual model
--------------
It turns out the spectrum $\Sigma_{\lambda}$ of $H_{\lambda, \theta}$ is related to the spectrum $\Sigma_{\hat{\lambda}}$ of $H_{\hat{\lambda}, \theta}$ in the following way $$\begin{aligned}
\Sigma_{\lambda}=\lambda_2 \Sigma_{\hat{\lambda}}\end{aligned}$$ by Aubry duality. This map $\sigma:\lambda\to \hat{\lambda}$ establishes the duality between region I and region II. The $\mathrm{IDS}$ $N_{\lambda}(E)$ of $H_{\lambda, \theta}$ coincide with the $\mathrm{IDS}$ $N_{\hat{\lambda}}({E}/{\lambda_2})$ of $H_{\hat{\lambda},\theta}$. Since $\Sigma_{\lambda}=\lambda_2 \Sigma_{\hat{\lambda}}$, we have the following
[@ber], [@sim] For any $\lambda, \theta$, there exists a dense set of $E\in \Sigma_{\lambda}$ such that there exists a non-zero solution of $H_{\hat{\lambda}, \theta}u=\frac{E}{\lambda_2}u$ with $|u_k|\leq 1+|k|$.
Bounded eigenfunction for every energy
--------------------------------------
The next result from [@AJ2] allows us to pass from a statement of every $\theta$ to every $E$.
\[Etheta\][@AJ2] If $E\in \Sigma_{\lambda}$ then there exists $\theta(E) \in {{\mathbb T}}$ and a bounded solution of $H_{\hat{\lambda},\alpha,\theta}u=\frac{E}{\lambda_2}u$ with $u_0=1$ and $|u_k|\leq 1$.
Localization and reducibility
-----------------------------
\[reducible\] Given $\alpha$ irrational, $\theta\in {{\mathbb R}}$ and $\lambda$ in region II, fix $E\in \Sigma_{\lambda}$, and suppose $H_{\hat{\lambda}, \theta}u=\frac{E}{\lambda_2}u$ has a non-zero exponentially decaying eigenfunction $u={\{u_k\}}_{k\in{{\mathbb Z}}}$, $|u_k|\leq e^{-c |k|}$ for $k$ large enough. Then the following hold:
- \(A) If $2\theta\notin \alpha{{\mathbb Z}}+{{\mathbb Z}}$, then there exists $M: {{{\mathbb R}}}/{{{\mathbb Z}}}\rightarrow\ SL(2, {{\mathbb R}})$ analytic, such that $${M^{-1}(x+\alpha)}\tilde{A}_{\lambda, E}(x)M(x)=R_{\pm\theta}.$$ In this case $\rho (\alpha, \tilde{A}_{\lambda, E})=\pm\theta+\frac{m}{2}\alpha$ $\mathrm{mod} {{\mathbb Z}}$, where $m=\deg{M}$ (here since $M\in SL(2,{{\mathbb R}})$, we have that $m$ is an even number) and $2\rho (\alpha, \tilde{A}_{\lambda,E})\notin \alpha{{\mathbb Z}}+{{\mathbb Z}}$.
- \(B) If $2\theta \in \alpha{{\mathbb Z}}+{{\mathbb Z}}$ and $\alpha\in\DC$, then there exists $M: {{{\mathbb R}}}/{{{\mathbb Z}}}\rightarrow\ PSL(2, {{\mathbb R}})$ analytic, such that $${M^{-1}(x+\alpha)}\tilde{A}_{\lambda, E}(x)M(x)=\left(\begin{matrix}\pm 1 &a\\ 0 &\pm 1\end{matrix}\right)$$ with $a\neq 0$. In this case $\rho(\alpha, \tilde{A}_{\lambda, E})=\frac{m}{2}\alpha$ $\mathrm{mod}{{\mathbb Z}}$, where $m=\deg{M}$, i.e. $2\rho(\alpha, \tilde{A}_{\lambda, E})\in \alpha{{\mathbb Z}}+{{\mathbb Z}}$.
Let $u(x)=\sum_{k\in{{\mathbb Z}}} \hat{u}_k e^{2\pi i k x}$, $U(x)=\left(\begin{matrix} e^{2\pi i \theta}u(x)\\ u(x-\alpha)\end{matrix}\right)$. Then $$\begin{aligned}
A_{\lambda , E}(x)U(x)=e^{2\pi i \theta}U(x+\alpha),\end{aligned}$$ $$\begin{aligned}
\tilde{A}_{\lambda , E}(x)\tilde{U}(x)=e^{2\pi i \theta}\tilde{U}(x+\alpha).\end{aligned}$$ Notice $\tilde{U}(x)=Q_{\lambda}(x)U(x)$ is analytic in $|\mathrm{Im}x|<\frac{\tilde{c}}{2\pi}$, where $\tilde{c}=\min{(\epsilon_1,c)}$, $\epsilon_1$ as in $\ref{epsilon1}$ and $Q_{\lambda}$ as in $\ref{conjugate}$. Define $\overline{\tilde{U}(x)}$ to be the complex conjugate of $\tilde{U}(x)$ on ${{\mathbb T}}$ and its analytic extension to $|\mathrm{Im}x|<\frac{\tilde{c}}{2\pi}$. Let $M(x)$ be the matrix with columns $\tilde{U}(x)$ and $\overline{\tilde{U}(x)}$. Then, $$\begin{aligned}
\tilde{A}_{\lambda, E}(x)M(x)=M(x+\alpha)\left(\begin{matrix}e^{2\pi i\theta} &0\\ 0 &e^{-2\pi i\theta}\end{matrix}\right)\ \ \mathrm{on}\ {{\mathbb T}}.\end{aligned}$$ Then since $\det{M(x+\alpha)}=\det{M(x)}$, we know $\det{M(x)}$ is a constant on ${{\mathbb T}}$.
Case 1. If $\det{M(x)}\neq 0$, then let $M(x)=\tilde{M}(x)\left(\begin{matrix}1 &1\\ i &-i\end{matrix}\right)$.
$$\begin{aligned}
{\tilde{M}^{-1}(x+\alpha)}\tilde{A}_{\lambda, E}(x)\tilde{M}(x)=R_{\theta}=
\left(
\begin{matrix}
\cos{2\pi \theta}\ \ &-\sin{2\pi \theta}\\
\sin{2\pi \theta}\ \ &\cos{2\pi \theta}
\end{matrix}
\right).\end{aligned}$$
Case 2. If $\det{M(x)}=0$, then if we denote $\tilde{U}(x)=\left(\begin{matrix}u_1(x) \\ u_2(x)\end{matrix}\right)$, then $\det{M(x)}=0$ means there exists $\eta(x)$ such that $u_1(x)=\eta (x)\overline{u_1(x)}$ and $u_2(x)=\eta (x)\overline{u_2(x)}$. This implies that $\eta(x)\in {{\mathbb C}}^{\omega}({{\mathbb T}}, {{\mathbb C}})$, and $|\eta(x)|=1$ on ${{\mathbb T}}$. Therefore there exists $\phi(x)\in {{\mathbb C}}^{\omega}({{\mathbb R}}/{2{{\mathbb Z}}}, {{\mathbb C}})$ such that $\phi^2(x)=\eta(x)$ and $|\phi(x)|=1$. It is easy to see $\overline{\phi(x)}u_1(x)=\phi(x)\overline{u_1(x)}$ and $\overline{\phi(x)}u_2(x)=\phi(x)\overline{u_2(x)}$. Then we define $W(x)=\left(\begin{matrix}\overline{\phi(x)}u_1(x)\\ \overline{\phi(x)}u_2(x)\end{matrix}\right)$, it is a real vector on ${{{\mathbb R}}}/{2{{\mathbb Z}}}$ with $W(x+1)=\pm W(x)$, and $\tilde{U}(x)=\phi(x) W(x)$. Now let us define $\tilde{M}(x)$ to be the matrix with columns $W(x)$ and $\frac{1}{{\|W(x)\|}^{-2}}R_{\frac{1}{4}}W(x)$, then $\det{\tilde{M}(x)}=1$ and $\tilde{M}(x)\in PSL(2, {{\mathbb R}})$. Since $$\tilde{A}_{\lambda, E}(x)W(x)=\frac{e^{2\pi i\theta}\phi(x+\alpha)}{\phi(x)}W(x+\alpha).$$ We have $$\begin{aligned}
\tilde{A}_{\lambda, E}(x)\tilde{M}(x)=\tilde{M}(x+\alpha)\left(\begin{matrix} d(x) &\tau(x)\\ 0 &{d(x)}^{-1}\end{matrix}\right)\end{aligned}$$ where $d(x)=\frac{e^{2\pi i\theta}\phi(x+\alpha)}{\phi(x)}$, $|d(x)|=1$ and $d(x)$ being real number, therefore $d(x)=\pm 1$. Also $\tau(x)\in {{\mathbb C}}^{\omega}({{{\mathbb R}}}/{2{{\mathbb Z}}}, {{\mathbb C}})$. But in fact ${\tilde{M}^{-1}(x+\alpha)}\tilde{A}_{\lambda, E}(x)\tilde{M}(x)$ is well-defined on ${{\mathbb T}}$. Therefore $\tau(x)\in {{\mathbb C}}^{\omega}({{\mathbb T}}, {{\mathbb C}})$. Now since we assumed $\alpha\in\DC$, we can further reduce $\tau(x)$ to the constant $\tau=\int_{{{\mathbb T}}}\tau(x) \mathrm{d}x$. In fact there exists $\psi(x)\in {{\mathbb C}}^{\omega}({{\mathbb T}}, {{\mathbb C}})$ such that $-\psi(x+\alpha)+\psi(x)+\tau(x)=\int_{{{\mathbb T}}}\tau(x) \mathrm{d}x$. This implies
$$\begin{aligned}
\left(\begin{matrix} 1 &-\psi(x+\alpha)\\ 0 &1\end{matrix}\right)\tilde{M}^{-1}(x+\alpha)\tilde{A}_{\lambda, E}(x)\tilde{M}(x)\left(\begin{matrix} 1 &\psi(x)\\ 0 &1\end{matrix}\right)=\left(\begin{matrix} \pm1 &\tau\\ 0 &\pm1\end{matrix}\right).\end{aligned}$$
In fact if $\det{M(x)}=0$, then $\frac{e^{2\pi i\theta}\phi(x+\alpha)}{\phi(x)}=\pm 1$, which implies that $2\theta\in \alpha{{\mathbb Z}}+{{\mathbb Z}}$. Therefore if $2\theta\notin \alpha{{\mathbb Z}}+{{\mathbb Z}}$, we must be in case (A). If on the other hand, $2\theta\in \alpha{{\mathbb Z}}+{{\mathbb Z}}$, $2\theta=k\alpha+n$, suppose $\tilde{M}^{-1}(x+\alpha)\tilde{A}_{\lambda, E}(x)\tilde{M}(x)=R_{\theta}$, then $R_{-\frac{k}{2}(x+\alpha)}\tilde{M}^{-1}(x+\alpha)\tilde{A}_{\lambda, E}(x)\tilde{M}(x)R_{\frac{k}{2}x}=R_{\frac{n}{2}}=\pm I$ leading to a contradiction. Therefore if $2\theta\in \alpha{{\mathbb Z}}+{{\mathbb Z}}$, we must be in case (B). $\hfill{} \Box$
Continued fractions
-------------------
Let $\{q_n\}$ be the denominators of the continued fraction approximants of $\alpha$. We recall the following properties: $$\begin{aligned}
\|q_n\alpha\|_{{{\mathbb R}}/{{\mathbb Z}}}=\inf_{1\leq |k|\leq q_{n+1}-1} \|k\alpha\|_{{{\mathbb R}}/{{\mathbb Z}}},\end{aligned}$$ $$\begin{aligned}
\frac{1}{2q_{n+1}}\leq \|q_n\alpha\|_{{{\mathbb R}}/{{\mathbb Z}}}\leq \frac{1}{q_{n+1}}.\end{aligned}$$
Recall that the Diophantine condition of $\alpha$ is $\beta(\alpha)=\limsup_{n\rightarrow\infty} \frac{\ln{q_{n+1}}}{q_n}=0$. Thus for any $\xi>0$, there exists $C_{\xi}>0$ such that $$\|k\alpha\|_{{{\mathbb R}}/{{\mathbb Z}}}\geq C_{\xi}e^{-\xi |k|}\ \ \mathrm{for}\ \mathrm{any}\ k\neq 0.$$
\[smallest\][@AJ1] Let $\alpha\in {{\mathbb R}}\backslash{{\mathbb Q}}$, $x\in{{\mathbb R}}$ and $0\leq l_0\leq q_n-1$ be such that $|\sin\pi(x+l_0\alpha)|=\inf_{0\leq l\leq q_n-1}|\sin\pi(x+l\alpha)|$, then for some absolute constant $C_1>0$, $$-C_1\ln q_n\leq \sum_{0\leq l\leq q_n-1, l\neq l_0} \ln|\sin\pi(x+l\alpha)|+(q_n-1)\ln 2\leq C_1\ln q_n$$
\[polynomialestimate\][@AJ2] Let $1\leq r\leq [q_{n+1}/q_n]$. If $p(x)$ has essential degree at most $k=rq_n-1$ and $x_0\in{{\mathbb R}}/{{{\mathbb Z}}}$, then for some absolute constant $C_2$, $$\begin{aligned}
\|p(x)\|_0\leq C_2 q_{n+1}^{C_2 r}\sup_{0\leq j\leq k}|p(x_0+j\alpha)|.\end{aligned}$$
Main estimates and proof of Theorem $\ref{dry}$
===============================================
Almost localization for every $\theta$
--------------------------------------
Let $\alpha\in {{\mathbb R}}\setminus{{\mathbb Q}}$, $\theta\in{{\mathbb R}}$, $\epsilon_0>0$. We say that $k$ is an $\epsilon_0-$resonance of $\theta$ if $\|2\theta-k\alpha\|\leq e^{-\epsilon_0 |k|}$ and $\|2\theta-k\alpha\|=\min_{|l|\leq |k|}\|2\theta-l\alpha\|$.
Let $0=|n_0|<|n_1|<...$ be the $\epsilon_0-$resonances of $\theta$. If this sequence is infinite, we say $\theta$ is $\epsilon_0-$resonant, otherwise we say it is $\epsilon_0-$non-resonant.
We say the extended Harper’s model $\{H_{\lambda,\alpha,\theta}\}_{\theta}$ exhibits almost localization if there exists $C_0, C_3, \epsilon_0, \tilde{\epsilon}_0>0$, such that for every solution $\phi$ to $H_{\lambda,\alpha,\theta}\phi=E\phi$ satisfying $\phi(0)=1$ and $|\phi(m)|\leq 1+|m|$, and for every $C_0 (1+|n_j|)<|k|<C_0^{-1}|n_{j+1}|$, we have $|\phi(k)|\leq C_3 e^{-\tilde{\epsilon}_0 |k|}$ (where $n_j$ are the $\epsilon_0-$resonances of $\theta$).
\[al\] If $\lambda$ belongs to region II, $\{H_{\hat{\lambda},\alpha,\theta}\}_{\theta}$ is almost localized for every $\alpha\in \mathrm{DC}$.
It is clear from Theorem $\ref{al}$ that almost localization implies localization for non-resonant $\theta$.
We will actually prove the following explicit lemma:
\[alexplicit\] Let $\lambda$ be in region II. Let $C_4$ be the absolute constant in Lemma $\ref{uniform}$, $\epsilon_1=\epsilon_1(\lambda)$ be as in ($\ref{epsilon1}$), then for any $0<\epsilon_0<\frac{\epsilon_1}{100C_4}$, there exists constant $C_3>0$, which depends on $\lambda, \alpha$ and $\epsilon_0$, so that for every solution $u$ of $H_{\hat{\lambda}, \alpha, \theta}u=Eu$ satisfying $u(0)=1$ and $|u_k|\leq 1+|k|$, if $3(|n_j|+1)<|k|<\frac{1}{3}|n_{j+1}|$, then $|u_k|\leq C_3 e^{-\frac{\epsilon_1}{5} |k|}$, where $\{n_j\}$ are the $\epsilon_0$-resonances of $\theta$.
The proof of Lemma \[alexplicit\] (and thus of Theorem $\ref{al})$ is given in Section 4.
Almost reducibility
-------------------
Let $\lambda$ be in region II. For every $E\in \Sigma_{\lambda}$, let $\theta(E)\in {{\mathbb T}}$ be given in Theorem $\ref{Etheta}$. Let $0<\epsilon_0<\frac{\epsilon_1}{100C_4}$ and $\{n_j\}$ be the set of $\epsilon_0-$ resonances of $\theta(E)$. Then for some positive constants $N_0$, $C$ and $c$, independent of $E$ and $\theta$, we have the following theorem:
\[ar\] For any fixed $j$, with $N_0<n=|n_j|+1<\infty$, let $N=|n_{j+1}|$, $L^{-1}=\|2\theta-n_j\alpha\|$. Then there exists $W:{{\mathbb T}}\rightarrow SL(2,{{\mathbb R}})$ analytic such that $|\deg{W}|\leq Cn$, $\|W\|_0\leq CL^C$ and $\|W^{-1}(x+\alpha)\tilde{A}_{\lambda, E}(x)W(x)-R_{\mp \theta}\|\leq Ce^{-cN}.$
Notice that this theorem requires $n>N_0$, which is not always ensured when $\theta(E)$ is non-resonant, however in that case we have localization for $H_{\hat{\lambda},\alpha, \theta}$ instead of almost localization. We will prove Theorem $\ref{ar}$ in Section 5.
Spectral consequences of Almost reducibility
--------------------------------------------
Let $\epsilon_1=\epsilon_1(\lambda)$ and $C_4$ be as in Lemma \[alexplicit\].
\[rhotheta\] Assume $\alpha\in \mathrm{DC}$. For $\lambda$ in region II, fix $E\in\Sigma_{\lambda}$. Assume $\theta(E) \in {{\mathbb T}}$ is such that $H_{\hat{\lambda},\alpha,\theta} u=\frac{E}{\lambda_2} u$ has solution satisfying $u_0=1$ and $|u_k|\leq 1$. Let $C$ be the constant in Theorem $\ref{ar}$. Then $\theta(E)$ and $\rho(\alpha,\tilde{A}_{\lambda,E})$ have the following relation:
- \(A) If $\theta$ is $\epsilon_0$-non-resonant for some $\frac{\epsilon_1}{100C_4}>\epsilon_0>0$, then $2\theta\in {{\mathbb Z}}\alpha+{{\mathbb Z}}$ if and only if $2\rho(\alpha, \tilde{A}_{\lambda, E})\in {{\mathbb Z}}\alpha+{{\mathbb Z}}$.
- \(B) If $\theta$ is $\epsilon_0$-resonant for some $\frac{\epsilon_1}{100C_4}>\epsilon_0>0$, then $\rho(\alpha,\tilde{A}_{\lambda, E})$ is $\frac{\epsilon_0}{C+2}$-resonant.
(A): When $\theta$ is $\epsilon_0$-non-resonant for some $\frac{\epsilon_1}{100 C_4}>\epsilon_0>0$, Theorem $\ref{al}$ implies $H_{\hat{\lambda},\alpha, \theta}$ has exponentially decaying eigenfunction. Then applying Theorem $\ref{reducible}$ we get $2\theta\in{{\mathbb Z}}\alpha+{{\mathbb Z}}$ if and only if $2\rho(\alpha, \tilde{A}_{\lambda, E})\in {{\mathbb Z}}\alpha+{{\mathbb Z}}$.
(B): Assume $\theta$ is $\epsilon_0$-resonant for some $\frac{\epsilon_1}{100 C_4}>\epsilon_0>0$. Fix any $\xi<\frac{\epsilon_0}{2C+2}$, then there exists $C_{\xi}>0$ such that for any $k\neq 0$ we have $\|k\alpha\|\geq C_{\xi}e^{-\xi |k|}$. Now take an $\epsilon_0$-resonance $n_j$ of $\theta$ such that $n=|n_j|>\max{(\frac{-\ln{C_{\xi}/2}}{\epsilon_0-(2C+2)\xi}, N_0)}$. Then there exists $|m|\leq Cn$ such that $2\rho(\alpha, \tilde{A}_{\lambda, E})-m\alpha=-2\theta$. Then $$\|2\rho(\alpha,\tilde{A}_{\lambda, E})-(m-n_j)\alpha\|=\|2\theta-n_j\alpha\|<e^{-\epsilon_0 n}\leq e^{-\frac{\epsilon_0}{C+2}|m-n_j|}.$$ Take any $|l|\leq |m-n_j|$, $l\neq m-n_j$. Then $$\|(l-(m-n_j))\alpha\|\geq C_{\xi} e^{-2\xi |m-n_j|}> 2e^{-\epsilon_0 n}>2\|2\rho(\alpha,\tilde{A}_E)-(m-l_0)\alpha\|.$$ Thus $\|2\rho(\alpha,\tilde{A}_E)-l\alpha\|>\|2\rho(\alpha,\tilde{A}_E)-(m-n_j)\alpha\|$ for any $|l|\leq |m-n_j|$, $l\neq m-n_j$. This by definition means $\rho(\alpha,\tilde{A}_{\lambda, E})$ is $\frac{\epsilon_0}{C+2}$-resonant.
$\hfill{} \Box$
Now based on Theorem $\ref{rhotheta}$, we can complete the proof of the dry version of Ten Martini Problem for extended Harper’s model in regions I and II.
[**Proof of Theorem \[dry\]**]{}
It is enough to consider $\lambda$ in region II. Let $E\in \Sigma_{\lambda}$ be such that $N_{\lambda}(E)\in {{\mathbb Z}}\alpha+{{\mathbb Z}}$. We are going to show $E$ belongs to the boundary of a component of ${{\mathbb R}}\setminus \Sigma_{\lambda}$. Now by $(\ref{IDSROT})$ we have $2\rho(\alpha,\tilde{A}_{\lambda, E})\in \alpha{{\mathbb Z}}+{{\mathbb Z}}$, thus by Theorem $\ref{rhotheta}$, $2\theta(E) \in \alpha{{\mathbb Z}}+{{\mathbb Z}}$. By Theorem $\ref{reducible}$, this means there exist $M(x)\in C^{\omega}_h ({{\mathbb T}},PSL(2,{{\mathbb R}}))$ such that $M^{-1}(x+\alpha)\tilde{A}_{\lambda, E}(x)M(x)=
\left(\begin{matrix}
\pm 1\ \ &a\\
0\ \ &\pm 1
\end{matrix}
\right).$ Without loss of generality, we assume $M^{-1}(x+\alpha)\tilde{A}_{\lambda, E}(x)M(x)=
\left(\begin{matrix}
1\ \ &a\\
0\ \ & 1
\end{matrix}
\right).$ Let $\tilde{M}(x)=\frac{M(x)}{\sqrt{|c|(x-\alpha)}}$, then $$\begin{aligned}
\tilde{M}^{-1}(x+\alpha)
\left(
\begin{matrix}
\frac{E-v(x)}{|c|(x)}\ \ &-\frac{|c|(x-\alpha)}{|c|(x)}\\
1\ \ &0
\end{matrix}
\right)
\tilde{M}(x)=
\left(\begin{matrix}
1\ \ &a\\
0\ \ & 1
\end{matrix}
\right).\end{aligned}$$ Now let $\tilde{M}(x)=
\left(
\begin{matrix}
M_{11}(x)\ \ &M_{12}(x)\\
M_{21}(x)\ \ &M_{22}(x)
\end{matrix}
\right).$ Then $M_{21}(x)=M_{11}(x-\alpha)$ and $M_{22}(x)=M_{12}(x-\alpha)-aM_{11}(x-\alpha)$ amd $$\begin{aligned}
&\tilde{M}^{-1}(x+\alpha)
\left(
\begin{matrix}
\frac{E+\epsilon-v(x)}{|c|(x)}\ \ &-\frac{|c|(x-\alpha)}{|c|(x)}\\
1\ \ &0
\end{matrix}
\right)
\tilde{M}(x)\\
=&
\left(
\begin{matrix}
1\ \ &a\\
0\ \ &1
\end{matrix}
\right)
+
\epsilon
\left(
\begin{matrix}
M_{11}(x)M_{12}(x)-aM_{11}^2(x)\ \ &M_{12}^2(x)-aM_{11}(x)M_{12}(x)\\
-M^2_{11}(x)\ \ &-M_{11}(x)M_{12}(x)
\end{matrix}
\right).\\
\triangleq &M_0+ \epsilon M_1(x).\end{aligned}$$ Now we look for $Z_{\epsilon}(x)$ of the form $e^{\epsilon Y(x)}$ such that $$\begin{aligned}
Z^{-1}_{\epsilon}(x+\alpha) (M_0+\epsilon M_1(x)) Z_{\epsilon}(x)=M_0+\epsilon [M_1]+O(\epsilon^2).\end{aligned}$$ We then just need to solve the equation: $$\begin{aligned}
(I-\epsilon Y(x+\alpha)+O(\epsilon^2))(M_0+\epsilon M_1(x)) (I+\epsilon Y(x)+O(\epsilon^2))=M_0+\epsilon [M_1]+O(\epsilon^2).\end{aligned}$$ It is sufficient to solve the coholomogical equation: $$\begin{aligned}
Y(x+\alpha)M_0-M_0 Y(x)=M_1(x)-[M_1],\end{aligned}$$ which is guaranteed by the Diophantine condition on $\alpha$. Thus $$\begin{aligned}
&(M(x+\alpha)Z_{\epsilon}(x+\alpha))^{-1}
\tilde{A}_{\lambda, E}(x)
(M(x)Z_{\epsilon}(x))
\\
=&
\left(
\begin{matrix}
1+\epsilon [M_{11}M_{12}]-a\epsilon [M_{11}^2]\ \ &a+\epsilon [M_{12}^2]-a\epsilon [M_{11}M_{12}]\\
-\epsilon [M_{11}^2] \ \ &1-\epsilon [M_{11}M_{12}]
\end{matrix}
\right)+O(\epsilon^2)\\
\triangleq &M_{\epsilon}+O(\epsilon^2).\end{aligned}$$ Notice that $\tilde{A}_{\lambda,E}$ is uniformly hyperbolic iff $\mathrm{Trace}(M_{\epsilon})>2$ which is fulfilled when $-a\epsilon [M_{11}^2]>0$. Thus for $\epsilon$ small, satisfying $-a\epsilon [M_{11}^2]>0$, $E+\epsilon\notin \Sigma_{\lambda}$, which means this spectral gap is open. $\hfill{} \Box$
Almost localization in region I
===============================
In this section we will prove Lemma $\ref{alexplicit}$. For fixed $\lambda$ in region II and $E$, let $D_{\hat{\lambda}, E} (\theta)=c_{\hat{\lambda}}(\theta)A_{\hat{\lambda},E}(\theta)$, where $c_{\hat{\lambda}}(\theta)=\frac{\lambda_3}{\lambda_2}e^{-2\pi i (\theta+\frac{\alpha}{2})}+\frac{1}{\lambda_2}+\frac{\lambda_1}{\lambda_2}e^{2\pi i (\theta+\frac{\alpha}{2})}$. Regarding the Lyapunov exponent, we recall the following result in [@Jm], $$\begin{aligned}
L(\alpha, A_{\hat{\lambda}, E})=L(\alpha, D_{\hat{\lambda},E})-\int_{{{\mathbb T}}}\ln{|c_{\hat{\lambda}}(\theta)|}\mathrm{d}\theta \triangleq \tilde{L}-\int \ln{|c_{\hat{\lambda}}|}>0,\end{aligned}$$ where $\tilde{L}=\ln{\frac{\lambda_2+\sqrt{\lambda_2^2-4\lambda_1\lambda_3}}{2\lambda_2}}$ and $\int \ln{|c_{\hat{\lambda}}|}=\ln{\frac{\max{(\lambda_1+\lambda_3,1)}+\sqrt{\max{(\lambda_1+\lambda_3,1)}^2-4\lambda_1\lambda_3}}{2\lambda_2}}$.
[**Proof of of Lemma $\ref{alexplicit}$**]{} Suppose $u$ is a solution satisfying the condition of Lemma $\ref{alexplicit}$. For an interval $I=[x_1, x_2]$, let $\Gamma_I$ be the coupling operator between $I$ and ${{\mathbb Z}}\setminus I$: $$\begin{aligned}
\Gamma_I(i,j)=
\left\lbrace
\begin{matrix}
&\tilde{c}(\theta+(x_1-1)\alpha),\ \ (i,j)=(x_1, x_1-1)\\
&c(\theta+(x_1-1)\alpha),\ \ (i,j)=(x_1-1, x_1)\\
&\tilde{c}(\theta+x_2\alpha),\ \ \ \ \ \ \ \ \ (i,j)=(x_2+1, x_2)\\
&c(\theta+x_2\alpha),\ \ \ \ \ \ \ \ \ (i,j)=(x_2, x_2+1)\\
&0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{otherwise}.
\end{matrix}
\right.\end{aligned}$$ Let $H_I=R_IH_{\hat{\lambda}, \theta} R_I^*$ be the restricted operator of $H_{\hat{\lambda} ,\theta}$ to $I$. Then for $x\in I$, we have $(H_I+\Gamma_I-E)u(x)=0$. Thus $u(x)=G_I\Gamma_Iu(x)$, where $G_I=(E-H_I)^{-1}$. By matrix multiplication:
$$\begin{aligned}
u(x)&=\sum_{y\in I,(y,z)\in \Gamma_I}G_I(x,y)\Gamma_I (y,z)u(z)\\
&=\tilde{c}(\theta+(x_1-1)\alpha) G_I(x,x_1)u (x_1-1)+ c(\theta+x_2\alpha) G_I(x,x_2)u(x_2+1).\end{aligned}$$
Let us denote $P_k(\theta)=\det{(E-H_{[0,k-1]}(\theta))}$. Then the $k-$step matrix $D_{\hat{\lambda}, E, k}(\theta)$ satisfies: $$\begin{aligned}
D_{\hat{\lambda}, E, k}(\theta)=
\left(
\begin{matrix}
P_k(\theta)\ \ &-\tilde{c}(\theta-\alpha)P_{k-1}(\theta+\alpha)\\
c(\theta+(k-1)\alpha)P_{k-1}(\theta)\ \ &-\tilde{c}(\theta-\alpha)c(\theta+(k-1)\alpha)P_{k-2}(\theta+\alpha)
\end{matrix}
\right).\end{aligned}$$ This relation between $P_k(\theta)$ and $D_{\hat{\lambda}, E, k}(\theta)$ gives a general upper bound of $P_k(\theta)$ in terms of $\tilde{L}$. Indeed by Lemma $\ref{uniformupp}$, for any $\epsilon>0$ there exists $C(\epsilon)>0$ so that $$\begin{aligned}
|P_n(\theta)|\leq C(\epsilon) e^{(\tilde{L}+\epsilon)n}\ \ \mathrm{for}\ \mathrm{any}\ n\in {{\mathbb N}}.\end{aligned}$$
By Cramer’s rule: $$\begin{aligned}
&|G_I(x_1,y)|=\prod_{j=x_1}^{y-1}|c(\theta+j\alpha)||\frac{\det{(E-H_{[y+1,x_2]}(\theta))}}{\det{(E-H_I(\theta))}}|=\prod_{j=x_1}^{y-1}|c(\theta+j\alpha)||\frac{P_{x_2-y}(\theta+(y+1)\alpha)}{P_k(\theta+x_1\alpha)}|,\\
&|G_I(y,x_2)|=\prod_{j=y+1}^{x_2}|c(\theta+j\alpha)||\frac{\det{(E-H_{[x_1, y-1]}(\theta))}}{\det{(E-H_I(\theta))}}|=\prod_{j=y+1}^{x_2}|c(\theta+j\alpha)||\frac{P_{y-x_1}(\theta+x_1\alpha)}{P_k(\theta+x_1\alpha)}|.\end{aligned}$$
Notice that $P_k(\theta)$ is an even function about $\theta+\frac{k-1}{2}\alpha$, it can be written as a polynomial of degree $k$ in $\cos{2\pi(\theta+\frac{k-1}{2}\alpha)}$. Let $P_k(\theta)=Q_k(\cos{2\pi (\theta+\frac{k-1}{2}\alpha)})$. Let $M_{k,r}=\{\theta\in {{\mathbb T}},\ |Q_k(\cos{2\pi \theta})|\leq e^{(k+1)r}\}$.
Fix $m>0$. A point $y\in{{\mathbb Z}}$ is called $(k,m)-$regular if there exists an interval $[x_1, x_2]$ containing $y$, where $x_2=x_1+k-1$ such that $$|G_I(y, x_i)|\leq e^{-m|y-x_i|} \ \mathrm{and}\ \mathrm{dist}(y, x_i)\geq \frac{1}{3} k\ \mathrm{for}\ i=1,2,$$ otherwise $y$ is called $(k,m)-$singular.
Suppose $y\in{{\mathbb Z}}$ is $(k,\tilde{L}-\int \ln{|c_{\hat{\lambda}}|}-\rho)-$singular. Then for any $\epsilon>0$ and any $x\in{{\mathbb Z}}$ satisfying $y-\frac{2}{3}k\leq x\leq y-\frac{1}{3} k$, we have $\theta+(x+\frac{1}{2}(k-1))\alpha$ belongs to $M_{k, \tilde{L}-\frac{1}{3} \rho+\epsilon}$ for $k>k(\lambda,\epsilon,\rho)$.
Suppose there exists $\epsilon>0$ and $x_1$: $y-(1-\delta)k\leq x_1\leq y-\delta k$, such that $\theta+(x_1+\frac{1}{2}(k-1))\alpha$ does not belong to $M_{k,\tilde{L}-\frac{1}{3} \rho+\epsilon}$, that is $|P_k(\theta+x_1\alpha)|>e^{(k+1)(\tilde{L}-\rho\delta+\epsilon)}$, $$\begin{aligned}
|G_I(x_1, y)| & \leq \prod_{j=x_1}^{y-1}|c_{\hat{\lambda}}(\theta+j\alpha)|e^{(k-|x_1-y|)(\tilde{L}+\epsilon)} e^{-(k+1)(\tilde{L}-\frac{1}{3}\rho+\epsilon)}\\
&<e^{-(\tilde{L}-\int \ln{|c_{\hat{\lambda}}|}-\rho)|y-x_1|}\ \ \mathrm{for}\ k>k(\lambda,\epsilon,\rho).\end{aligned}$$ Similarly $$|G_I(x_2,y)|\leq e^{-(\tilde{L}-\int \ln{|c_{\hat{\lambda}}|}-\rho)|y-x_2|}.$$
(-7,0) – (7,0) node \[right\] [$x$]{} ;
(4.2,0) node \[below\] [$y$]{}; (2.0,0) node \[below\] [$y-(\frac{1}{2}-\delta)k$]{}; (6.4,0) node \[below\] [$y+(\frac{1}{2}-\delta)k$]{};
(-6.0,0) node \[below\] [$y-(1-\delta)k$]{}; (-1.6,0) node \[below\] [$y-\delta k$]{};
(2.0,0.2) – (6.4,0.2)node\[pos=0.5,above\][$x+\frac{1}{2}(k-1)\alpha$]{}; (-6,0.2) – (-1.6,0.2)node\[pos=0.5,above\][$x$]{};
$\hfill{} \Box$
We say that the set $\{\theta_1, ..., \theta_{k+1}\}$ is $\gamma-$uniform if $$\max_{x\in [-1, 1]}\max_{i=1,...,k+1}\prod_{j=1, j\neq i}^{k+1}\frac{|x-\cos{2\pi\theta_j}|}{|\cos{2\pi\theta_i}-\cos{2\pi\theta_j}|}<e^{k\gamma}$$
\[gamma1\] Let $\gamma_1<\gamma$. If $\theta_1, ..., \theta_{k+1}\in M_{k, \tilde{L}-\gamma}$, then $\{\theta_1, ..., \theta_{k+1}\}$ is not $\gamma_1-$uniform for $k>k(\gamma, \gamma_1)$.
Otherwise, using Lagrange interpolation form we can get $|Q_k(x)|<e^{k\tilde{L}}$ for all $x\in [-1,1]$. This implies $|P_k(x)|<e^{k\tilde{L}}$ for all $x$. But by Herman’s subharmonic function argument, $\int_{{{\mathbb R}}/{{\mathbb Z}}}\ln|P_k(x)|\mathrm{d}x \geq k\tilde{L}$. This is impossible.
$\hfill{} \Box$ Now take $\xi$ and $\epsilon_0$ such that $0<1000\xi<\epsilon_0$. Then for $|n_{j+1}|>N(\xi)$ we have $$\begin{aligned}
2e^{-4\xi |n_{j+1}|} \leq C_{\xi} e^{-2\xi|n_{j+1}|}\leq \|(n_{j+1}-n_j)\alpha\|= \|n_{j+1}\alpha-2\theta+2\theta-n_j\alpha\|\leq 2\|2\theta-n_j\alpha\|\leq 2e^{-\epsilon_0 |n_j|},\end{aligned}$$ which yields that $$\label{250}
|n_{j+1}|>\frac{\epsilon_0}{4\xi} |n_j|>250|n_j|.$$
Without loss of generality, assume $3(|n_j|+1)<y<\frac{|n_{j+1}|}{3}$ and $y>N(\xi)$. Select $n$ such that $q_n\leq \frac{y}{8}< q_{n+1}$ and let $s$ be the largest positive integer satisfying $sq_n\leq \frac{y}{8}$. Set $I_1, I_2\subset {{\mathbb Z}}$ as follows $$\begin{aligned}
I_1=[1-2sq_n, 0]\ and\ I_2=[y-2sq_n+1, y+2sq_n],\ &\mathrm{if}\ n_j<0 \\
I_1=[0, 2sq_n-1]\ and\ I_2=[y-2sq_n+1, y+2sq_n],\ &\mathrm{if}\ n_j\geq 0\end{aligned}$$
\[uniform\] Let $\theta_j=\theta+j\alpha$, then set $\{\theta_{j}\}_{j\in I_1\cup I_2}$ is $C_4\epsilon_0+C_4\xi-$uniform for some absolute constant $C_4$ and $y>y(\alpha, \epsilon_0, \xi)$.
Without loss of generality, we assume $n_j>0$. Take $x=\cos{2\pi a}$. Now it suffices to estimate $$\begin{aligned}
\sum_{j\in I_1\cup I_2,\ j\neq i}\left( \ln{|\cos{2\pi a}-\cos{2\pi \theta_j}|}-\ln{|\cos{2\pi \theta_i}-\cos{2\pi \theta_j}|}\right) \triangleq \sum_1-\sum_2.\end{aligned}$$ Lemma $\ref{smallest}$ reduces this problem to estimating the minimal terms.
First we estimate $\sum_1$: $$\begin{aligned}
\sum_{1}
&=\sum_{j\in I_1\cup I_2, j\neq i} \ln|\cos{2\pi a}-\cos{2\pi\theta_{j}}|\\
&=\sum_{j\in I_1\cup I_2, j\neq i} \ln|\sin{\pi(a+\theta_{j})}|+\sum_{j\in I_1\cup I_2, j\neq i} \ln|\sin{\pi(a-\theta_{j})}|+(6sq_n-1)\ln 2\\
&\triangleq\sum_{1,+}+\sum_{1,-}+(6sq_n-1)\ln 2.\\\end{aligned}$$
We cut $\sum_{1,+}$ or $\sum_{1,-}$ into $6s$ sums and then apply Lemma $\ref{smallest}$, we get that for some absolute constant $C_1$: $$\begin{aligned}
\sum_{1}\leq -6sq_n\ln 2+C_1 s\ln q_n.\end{aligned}$$
Next, we estimate $\sum_2$. $$\begin{aligned}
\sum_2
&=\sum_{j\in I_1\cup I_2, j\neq i}\ln |\cos 2\pi\theta_j-\cos 2\pi\theta_i|\\
&=\sum_{j\in I_1\cup I_2, j\neq i} \ln|\sin{\pi(2\theta+(i+j)\alpha)}|+\sum_{j\in I_1\cup I_2, j\neq i} \ln|\sin{\pi(i-j)\alpha}|+(6sq_n-1)\ln 2\\
&\triangleq \sum_{2,+} + \sum_{2,-} +(6sq_n-1)\ln 2.\end{aligned}$$ We need to carefully estimate the minimal terms. For $\sum_{2,+}$, we use the property of resonant set; and for $\sum_{2,-}$, we use the Diophantine condition on $\alpha$.
For any $0<|j|< q_{n+1}$ , we have $\|j\alpha\|\geq \|q_n\alpha\|\geq C_{\xi}e^{-\xi q_n}$. Therefore $$\max(\ln|\sin{x}|, \ln|\sin(x+\pi j\alpha)|)\geq -2\xi q_n\ \ \mathrm{for}\ y>y(\alpha, \xi).$$ This means in any interval of length $sq_n$, there can be at most one term which is less than $-2\xi q_n$. Then there can be at most $6$ such terms in total.
For the part $\sum_{2,-}$, since $\|(i-j)\alpha\|\geq C_{\xi}e^{-\xi |i-j|}\geq e^{-20\xi sq_n}$, these 6 smallest terms must be bounded by $-20\xi sq_n$ from below. Hence $\sum_{2,-}\geq -6sq_n\ln 2-C \xi sq_n-Cs\ln{q_n}$ for $y>y(\xi)$ and some absolute constant $C$.
For the part $\sum_{2,+}$, notice $|i+j|\leq 2y+4sq_n<3y<|n_{j+1}|$ and $i+j>0>-n_j$. Suppose $\|2\theta+k_0\alpha\|=\min_{j\in I_1\cup I_2}\|2\theta+(i+j)\alpha\|\leq e^{-100\epsilon_0 sq_n}<e^{-\epsilon_0 |k_0|}$. Then for any $|k|\leq |k_0| \leq 40 sq_n$ (including $|n_j|$), $$\begin{aligned}
\|2\theta-k\alpha\|\geq \|(k+k_0)\alpha\|-\|2\theta+k_0\alpha\| >\|2\theta+k_0\alpha\|\ \ \mathrm{for}\ y>y(\alpha, \epsilon_0,\xi).\end{aligned}$$ This means $-k_0$ must be a $\epsilon_0-$resonance, therefore $|k_0|\leq |n_{j-1}|$. Then $$\|2\theta-n_j\alpha\|\geq \|(n_j+k_0)\alpha\|-\|2\theta+k_0\alpha\|\geq C_{\xi}e^{-12\xi sq_n}-e^{-100 \epsilon_0 sq_n}>e^{-100 \epsilon_0 sq_n}\geq \|2\theta+k_0\alpha\|$$ leads to a contradiction. Thus the smallest terms must be greater than $-100\epsilon_0 sq_n$. We can bound $\sum_{2,+}$ by $-6sq_n\ln 2-600\epsilon_0 sq_n-12\xi sq_n-Cs\ln{q_n}$ from below. Therefore $\sum_{2}\geq -6sq_n\ln 2-C\epsilon_0 sq_n-C \xi sq_n-Cs\ln{q_n}$. Thus the set $\{\theta_j\}_{j\in I_1\cup I_2}$ is $C_4\epsilon_0+C_4\xi-$uniform for $y>y(\alpha, \epsilon_0, \xi)$ and some absolute constant $C_4$.
$\hfill{} \Box$
Now let $C_4$ be the absolute constant in Lemma $\ref{uniform}$. Choose $0<1000\xi<\epsilon_0<\frac{\epsilon_1}{100C_4}$. Combining Lemma $\ref{gamma1}$ and Lemma $\ref{uniform}$, we know that when $y>y(\alpha, \epsilon_0, \xi)$, $\{\theta_j\}_{j\in I_1\cup I_2}$ can not be inside the set $M_{6sq_n-1, \tilde{L}-2C_4 \epsilon_0}$ at the same time. Therefore $0$ and $y$ can not be $(6sq_n-1, \tilde{L}-\int \ln{|c_{\hat{\lambda}}|}-9C_4 \epsilon_0)$ at the same time. However $0$ is $(6sq_n-1, \tilde{L}-\int \ln{|c_{\hat{\lambda}}|}-9C_4 \epsilon_0)-$singular given $n$ large enough. Therefore $$\{\theta_j\}_{j\in I_1}\subset M_{6sq_n-1, \tilde{L}-2C_4 \epsilon_0}.$$ Thus $y$ must be $(6sq_n-1,\tilde{L}-\int \ln{|c_{\hat{\lambda}}|}-9C_4 \epsilon_0)-$regular. This implies $$\begin{aligned}
|u(y)|\leq e^{-(\tilde{L}-\int \ln{|c_{\hat{\lambda}}|}-9C_4 \epsilon_0)\frac{1}{4}|y|}<e^{-\frac{\epsilon_1}{5} |y|}\ \ \mathrm{for}\ |y|\geq y(\lambda, \alpha, \epsilon_0, \xi).\end{aligned}$$ Thus there exists $C_3=C_{\lambda, \alpha, \epsilon_0, \xi}$ such that $|u(y)|\leq C_3 e^{-\frac{\epsilon_1}{5}|y|}$ for any $3|n_j|\leq |y|\leq \frac{1}{3}|n_{j+1}|$ and $j\in {{\mathbb N}}$.
Almost reducibility in region II
================================
[**Proof of Theorem $\ref{ar}$**]{} For any $E\in \Sigma_{\lambda}$, take $\theta(E)$ and $\{u_k\}$ as in Theorem $\ref{Etheta}$. Let $\epsilon_1$ be as in (\[epsilon1\]), $C_4$ be the absolute constant from Lemma $\ref{uniform}$, and $C_2$ be the absolute constant from Lemma $\ref{polynomialestimate}$. Fix $\max{(32C_2 \xi, 1000\xi)}<\epsilon_0<\min{(\frac{\epsilon_1}{200}, \frac{\epsilon_1}{100C_4})}$. By Lemma $\ref{alexplicit}$, there exists $C$ depending on $\lambda$ and $\alpha$ such that for any $3|n_j|<|k|<\frac{1}{3}|n_{j+1}|$, we have $|u_k|\leq C e^{-\frac{\epsilon_1}{5} |k|}$.
For any $n$, $9|n_j|<n<\frac{1}{9}|n_{j+1}|$, of the form $$\label{nform}
n=rq_m-1<q_{m+1}.\ \footnote{The existence of such $n$ comes from ($\ref{250}$).}$$ Let $u(x)=u^{I}(x)=\sum_{k\in I} u_k e^{2\pi i kx}$ with $I=[-[\frac{n}{2}], [\frac{n}{2}]]=[x_1,x_2]$. Define $$\begin{aligned}
U(x)=
\left(
\begin{matrix}
e^{2\pi i\theta}u(x)\\
u(x-\alpha)
\end{matrix}
\right).\end{aligned}$$ Let $A(\theta)=A_{\lambda, E}(\theta)$. By direct computation: $$\begin{aligned}
A(x)U(x)=
e^{2\pi i\theta}U(x+\alpha)+
\left(
\begin{matrix}
g(x)\\
0
\end{matrix}
\right)
\triangleq
e^{2\pi i \theta} U(x+\alpha)+G(x).\end{aligned}$$ The Fourier coefficients of $g(x)$ are possibly nonzero only at four points $x_1$, $x_2$, $x_1-1$ and $x_2+1$. Since $|u_k|\leq C_1 e^{-\frac{\epsilon_1}{5} |k|}$ when $3|n_j|<|k|<\frac{1}{3}|n_{j+1}|$, we know that ${\|G(x)\|}_{\frac{\epsilon_1}{20\pi}}\leq C_1 e^{- \frac{\epsilon_1}{20} n}$.
Combining Lemma $\ref{LEregion2}$ and $\ref{uniformupp}$, we have exponential control of the growth of the transfer matrix, for any $\delta>0$ there exists $C_{\delta}>0$ such that $$\begin{aligned}
\|\tilde{A}_k(x)\|_{\frac{\epsilon_1}{2\pi}}\leq C_{\delta}e^{\delta |k|},\ \ \mathrm{for}\ \mathrm{any}\ k.\end{aligned}$$ With some effort we are able to get the following significantly improved upper bound:
\[polycontrol\] For some $C>0$ depending on $\lambda$ and $\alpha$, $$\begin{aligned}
\|\tilde{A}_k(x)\|_{{{\mathbb T}}}\leq C{(1+|k|)}^{C}.\end{aligned}$$
Let $\tilde{U}(x)=Q(x)U(x)$, $\tilde{G}(x)=Q(x+\alpha)G(x)$, where $Q=Q_{\lambda}$ is given in $(\ref{conjugate})$. Since $$\max{(\|Q(x)\|_{\frac{\epsilon_1}{20\pi}}, \|Q^{-1}(x)\|_{\frac{\epsilon_1}{20\pi}})} \leq C,$$ we have $$\begin{aligned}
\tilde{A}(x)\tilde{U}(x)=e^{2\pi i \theta} \tilde{U}(x+\alpha)+\tilde{G}(x),\end{aligned}$$ where $\|\tilde{G}(x)\|_{\frac{\epsilon_1}{20\pi}}\leq C e^{-\frac{\epsilon_1}{20}n}$.
\[Ulower\] Let $C_2$ be the constant from Lemma $\ref{polynomialestimate}$, then for any $\delta$, $2C_2\xi<\delta<\frac{\epsilon_0}{16}$, we have $$\begin{aligned}
\inf_{|\mathrm{Im}{(x)}|\leq\frac{\epsilon_1}{20\pi}}\|\tilde{U}(x)\|\geq e^{-2\delta n},\end{aligned}$$ for $n>n(\alpha, \delta)$.
We will prove the statement by contradiction. Suppose for some $x_0\in \{|\mathrm{Im}{(x)}|\leq \frac{\epsilon_1}{20\pi}\}$ we have $\|\tilde{U}(x_0)\|< e^{-2\delta n}$. Notice that for any $l\in{{\mathbb N}}$, $$\begin{aligned}
e^{2\pi i l\theta}\tilde{U}(x_0+l\alpha)=\tilde{A}_l(x_0) \tilde{U}(x_0)-\sum_{m=1}^l e^{2\pi i (m-1)\theta}\tilde{A}_{l-m}(x_0+m\alpha) \tilde{G}(x_0+(m-1)\alpha).\end{aligned}$$ This implies for $n>n(\delta)$ large enough and for any $0\leq l\leq n$, $\|\tilde{U}(x_0+l\alpha)\| \leq e^{-\delta n}$, thus $\|u(x_0+l\alpha)\|\leq C_{\delta} e^{-\delta n}$. By Lemma $\ref{polynomialestimate}$, $\|u(x+i\mathrm{Im}(x_0))\|_{{{\mathbb T}}}\leq C_2 C_{\delta} e^{C_2\xi n}e^{-\delta n}\leq e^{-\frac{\delta}{2} n}$. This contradicts with $\int_{{{\mathbb T}}} u(x+i\mathrm{Im}(x_0)) \mathrm{d}x=u_0=1$.
$\hfill{} \Box$
\[columnmatrix\][@A2] Let $V$ : ${{\mathbb T}}\rightarrow {{\mathbb C}}^2$ be analytic in $|\mathrm{Im}(x)|<\eta$. Assume that $\delta_1<\|V(x)\|<\delta_2^{-1}$ holds on $|\mathrm{Im}(x)|<\eta$. Then there exists $M$ : ${{\mathbb T}}\rightarrow SL(2, {{\mathbb C}})$ analytic on $|\mathrm{Im}(x)|<\eta$ with first column $V$ and $\|M\|_\eta\leq C\delta_1^{-2}\delta_2^{-1}(1-\ln(\delta_1\delta_2))$.
Applying Lemma $\ref{columnmatrix}$, let $M(x)$ be the matrix with first column $\tilde{U}(x)$. Then $e^{-2\delta n}\leq \|\tilde{U}(x)\|_{\frac{\delta}{\pi}}\leq e^{\delta n}$ and hence $\|M(x)\|_{\frac{\delta}{\pi}}\leq Ce^{6\delta n}$. Therefore $$\begin{aligned}
M^{-1}(x+\alpha)\tilde{A}(x)M(x)=
\left(
\begin{matrix}
e^{2\pi i \theta} &0\\
0 &e^{-2\pi i\theta}
\end{matrix}
\right)
+
\left(
\begin{matrix}
\beta_1(x) &b(x)\\
\beta_3(x) &\beta_4(x)
\end{matrix}
\right)\end{aligned}$$ where $\|\beta_1(x)\|_{\frac{\delta}{\pi}},\ \|\beta_3(x)\|_{\frac{\delta}{\pi}},\ \|\beta_4(x)\|_{\frac{\delta}{\pi}}\leq C e^{-\frac{\epsilon_1}{40}n}$, and $\|b(x)\|_{\frac{\delta}{\pi}}\leq C e^{13\delta n}$. Let $$\begin{aligned}
\Phi(x)=M(x)
\left(
\begin{matrix}
e^{\frac{\epsilon_1}{160}n} &0\\
0 &e^{-\frac{\epsilon_1}{160}n}
\end{matrix}
\right).\end{aligned}$$ Then we would have:
$$\begin{aligned}
{\Phi(x+\alpha)}^{-1}\tilde{A}(x)\Phi(x)=
\left(
\begin{matrix}
e^{2\pi i\theta} &0\\
0 &e^{-2\pi i\theta}
\end{matrix}
\right)+
H(x),\end{aligned}$$
where $\|H(x)\|_{\frac{\delta}{\pi}}\leq Ce^{-\frac{\epsilon_1}{160}n}$, and $\|\Phi(x)\|_{\frac{\delta}{\pi}}\leq Ce^{\frac{\epsilon_1}{80}n}$. Thus $$\begin{aligned}
\sup_{0\leq s\leq e^{\frac{\epsilon_1}{320}n}}\|\tilde{A}_s(x)\|_{{{\mathbb T}}}\leq e^{\frac{\epsilon_1}{20}n}\end{aligned}$$ for $n\geq n(\lambda, \alpha)$ satisfying ($\ref{nform}$). For $s$ large, there always exists $9|n_j|<n<\frac{1}{9}|n_{j+1}|$ satisfying $(\ref{nform})$ such that $cn \leq\frac{320}{\epsilon_1}\ln{s}\leq n$ with some absolute constant $c$. Thus there exists $C$ depending on $\lambda$ and $\alpha$ such that $\|\tilde{A}_k(x)\|_{{{\mathbb T}}}\leq C(1+|k|)^{C}$. $\hfill{} \Box$
Now we come back to the proof of Theorem $\ref{ar}$. Fix some $n=|n_j|$, and $N=|n_{j+1}|$. Let $u(x)=u^{I_2}(x)$ with $I_2=[-[\frac{N}{9}], [\frac{N}{9}]]$ and $U(x)=\left(\begin{matrix}e^{2\pi i\theta}u(x)\\ u(x-\alpha)\end{matrix}\right)$. Then $$\begin{aligned}
A(x)U(x)=e^{2\pi i \theta}U(x+\alpha)+G(x) \ \ with\ \ {\|G(x)\|}_{\frac{\epsilon_1}{20\pi}}\leq C e^{-\frac{\epsilon_1}{90}N}.\end{aligned}$$ Define $U_0(x)=e^{\pi i n_j x}U(x)$. Notice that if $n_j$ is even, then $U_0(x)$ is well-defined on ${{\mathbb T}}$, otherwise $U_0(x+1)=-U_0(x)$. $$\begin{aligned}
\tilde{A}(x)\tilde{U}_0(x)&=e^{2\pi i \tilde{\theta}} \tilde{U}_0(x+\alpha)+H(x),\end{aligned}$$ where $\tilde{\theta}=\theta-\frac{n_j}{2}\alpha$, $\tilde{U}_0(x)=Q(x)U_0(x)$ and $\|H(x)\|_{\frac{\epsilon_1}{20\pi}}\leq Ce^{-\frac{\epsilon_1}{100}N}$. Consider the matrix $W(x)$ with $\tilde{U}_0(x)$ and $\overline{\tilde{U}_0(x)}$ being its two columns. Then $$\begin{aligned}
\tilde{A}(x)W(x)=W(x+\alpha)
\left(
\begin{matrix}
e^{2\pi i\tilde{\theta}} &0\\
0 &e^{-2\pi i \tilde{\theta}}
\end{matrix}
\right)
+\tilde{H}(x).\end{aligned}$$
\[lowerdet\]Let $L^{-1}=\|2\theta-n_j\alpha\|$. Then for $n>N_0(\lambda, \alpha)$ we have $$|\det{W(x)}|
\geq L^{-4C}\ \ \mathrm{for}\ \mathrm{any}\ x\in {{\mathbb T}},$$ where $C$ is the constant appeared in Theorem $\ref{polycontrol}$.
First, we fix $\xi_1<\frac{\epsilon_0}{1600}$ so that $\|k\alpha\|\geq C_{\xi_1}e^{-\xi_1 |k|}$ for any $k\neq 0$. We have the following estimate about $L$:
$e^{\epsilon_0 n}\leq L\leq e^{4\xi_1 N}$. $$\begin{aligned}
e^{-2\xi_1 N}\leq \|(n_{j+1}-n_j)\alpha\|\leq 2\|n_j\alpha-2\theta\|=2L^{-1}\leq 2 e^{-\epsilon_0 n}\ \ \mathrm{for}\ n\geq N(\xi_1).\end{aligned}$$
Now we prove by contradiction. Suppose there exists $\kappa$ and $x_0\in {{\mathbb T}}$ such that $\|\tilde{U}_0(x_0)-\kappa \overline{\tilde{U}_0(x_0)}\|<L^{-4C}$. Then $$\begin{aligned}
&\|\tilde{U}_0(x_0+l\alpha)e^{2\pi i l\tilde{\theta}}-\kappa \overline{\tilde{U}_0(x_0+l\alpha)}e^{-2\pi il\tilde{\theta}}\|\\
\leq &\|\sum_{m=0}^{l-1}\tilde{A}_{l-m}(x_0+m\alpha)H(x_0+m\alpha)-\kappa \sum_{m=0}^{l-1}\tilde{A}_{l-m}(x_0+m\alpha)\overline{H(x_0+m\alpha)}\|+\|A_l(x_0)\| L^{-4C} \\
\leq &C L^{2C} e^{-\frac{\epsilon_1}{100}N}+CL^{-2C}<L^{-C}.\end{aligned}$$ for $0\leq |l|\leq L^2$. If we take $j=\frac{L}{4}$, then $$\begin{aligned}
\|\tilde{U}_0(x_0+\frac{L}{4}\alpha)+\kappa \overline{\tilde{U}_0(x_0+\frac{L}{4}\alpha)}\|<L^{-1}.\end{aligned}$$ Next since $\|U_0(x)\|_{{{\mathbb T}}}\leq n$, we have $\|\tilde{U}_0(x)\|_{{{\mathbb T}}}\leq C n$. Thus $$\begin{aligned}
\|\tilde{U}_0(x_0+l\alpha)-\kappa \overline{\tilde{U}_0(x_0+l\alpha)}\|<L^{-\frac{1}{3}}\ \ \mathrm{for}\ 0\leq |l|\leq L^{\frac{1}{2}}.\end{aligned}$$ For any analytic function $f(x)=\sum_{k\in {{\mathbb Z}}} \hat{f}_k e^{2\pi i kx}$, define $f_{[-m,m]}(x)=\sum_{|k|\leq m} \hat{f}_k e^{2\pi i k x}$. For any column vector $V(x)=\left(\begin{matrix} v^{(1)}(x)\\ v^{(2)}(x)\end{matrix}\right)$, let $V_{[-m,m]}(x)=\left(\begin{matrix} v^{(1)}_{[-m,m]}(x)\\ v^{(2)}_{[-m,m]}(x)\end{matrix}\right)$. Now let us define $\tilde{U}_0^{[9n]}(x)=Q(x)e^{\pi i n_j x}U_{[-9n, 9n]}(x)$. Then $$\begin{aligned}
\|\tilde{U}_0^{[9n]}(x)-\tilde{U}_0(x)\|_{{{\mathbb T}}}\leq C e^{-\frac{9}{5}\epsilon_1 n}.\end{aligned}$$ Consider $[e^{-\pi i n_jx}\tilde{U}_0^{[9n]}(x)]_{[-18n, 18n]}(x)e^{\pi i n_jx}$. This function differs from a polynomial with essential degree $36n$ only by a multiple of $e^{\pi i n_jx}$. Notice that $Q(x)$ is analytic in $\{x:|\mathrm{Im}(x)|\leq \frac{\epsilon_1}{4\pi}\}$, thus $|\hat{Q}(k)|\leq C e^{-\frac{\epsilon_1}{2} |k|}$. Then $$\begin{aligned}
|\widehat{e^{-\pi in_jx}\tilde{U}_0^{[9n]}}(k)|\leq \sum_{|m|\leq 9n}|\hat{Q}(k-m)\hat{U}(m)|\leq C n e^{-\frac{\epsilon_1}{2} (|k|-9n)}\ \ \mathrm{for}\ |k|\geq 18n.\end{aligned}$$ Thus $$\begin{aligned}
\|e^{-\pi i n_jx}\tilde{U}_0^{[9n]}(x)-[e^{-\pi i n_jx}\tilde{U}_0^{[9n]}]_{[-18n, 18n]}(x)\|_{{{\mathbb T}}}\leq e^{-4\epsilon_1 n},\end{aligned}$$ $$\begin{aligned}
\|\tilde{U}_0(x)-[e^{-\pi i n_jx}\tilde{U}_0^{[9n]}]_{[-18n, 18n]}(x)e^{\pi i n_jx}\|_{{{\mathbb T}}} \leq e^{-4\epsilon_1 n}.\end{aligned}$$ Hence $$\begin{aligned}
&\|[e^{-\pi i n_jx}\tilde{U}_0^{[9n]}]_{[-18n, 18n]}(x_0+l\alpha)e^{2\pi i n_j(x_0+l\alpha)}-
\kappa\overline{[e^{-\pi i n_jx}\tilde{U}_0^{[9n]}]_{[-18n, 18n]}(x_0+l\alpha)}\|_{{{\mathbb T}}}\\
< &2L^{-\frac{1}{3}}+e^{-4\epsilon_1 n},\end{aligned}$$ for $|l|\leq L^{\frac{1}{2}}$. Notice that $$\begin{aligned}
[e^{-\pi i n_jx}\tilde{U}_0^{[9n]}]_{[-18n, 18n]}(x)e^{2\pi i n_jx}-\kappa\overline{[e^{-\pi i n_jx}\tilde{U}_0^{[9n]}]_{[-18n, 18n]}(x)}\end{aligned}$$ is a polynomial whose essential degree is at most $37n$. Thus by Lemma $\ref{polynomialestimate}$, we would have $$\begin{aligned}
\|[e^{-\pi i n_jx}\tilde{U}_0^{[9n]}]_{[-18n, 18n]}(x)e^{\pi i n_jx}-\kappa\overline{[e^{-\pi i n_jx}\tilde{U}_0^{[9n]}]_{[-18n, 18n]}(x)e^{\pi i n_jx}}\|_{{{\mathbb T}}}<L^{-\frac{1}{4}}+e^{-2\epsilon_1 n}.\end{aligned}$$ Hence $\|\tilde{U}_0(x)-\kappa \overline{\tilde{U}_0(x)}\|_{{{\mathbb T}}}<L^{-\frac{1}{4}}+2e^{-2\epsilon_1 n}$. But combining with $(9.1)$ we would get $\|\tilde{U}_0(x_0+\frac{L}{4}\alpha)\|<2L^{-\frac{1}{4}}+2e^{-2\epsilon_1 n}$, but this contradicts with $\inf_{x\in {{\mathbb T}}}\|\tilde{U}_0(x)\|>e^{-2\delta n}$ since $\delta<\frac{\epsilon_0}{16}$.$\hfill{} \Box$
Now for $n>N_0(\lambda,\alpha)$, take $S(x)=\mathrm{Re}\tilde{U}_0(x)$ and $T(x)=\mathrm{Im}\tilde{U}_0(x)$. Let $W_1(x)$ be the matrix with columns $S(x)$ and $T(x)$. Notice that $\det{W}_1(x)$ is well-defined on ${{\mathbb T}}$ and $\det{W}_1 (x)\neq 0$ on ${{\mathbb T}}$, hence without loss of generality we could assume $\det{W_1}(x)>0$ on ${{\mathbb T}}$, otherwise we simply take $W_1(x)$ to be the matrix with columns $S(x)$ and $-T(x)$. Then $$\begin{aligned}
\|\tilde{A}(x) W_1(x)-W_1(x+\alpha)R_{-\tilde{\theta}}\|_{{{\mathbb T}}}\leq Ce^{-\frac{\epsilon_1}{45}N}.\end{aligned}$$ By taking determinant, we get $$\begin{aligned}
\det{W_1}(x)=\det{W_1}(x+\alpha)+O(e^{-\frac{\epsilon_1}{50}N})\ \ \mathrm{on}\ {{\mathbb T}}.\end{aligned}$$ Since $\det{W_1}(x)$ is analytic on $|\mathrm{Im}x|\leq \frac{\epsilon_1}{20\pi}$, by considering the Fourier coefficients we could get $$\begin{aligned}
\det{W_1}(x)=w_0+O(e^{-\frac{\epsilon_1}{100}N})\ \ \mathrm{on}\ {{\mathbb T}},\end{aligned}$$ where $w_0\geq L^{-5C}$. Thus $\det{W_1}(x)$ is almost a positive constant.
Define $W_2(x)={\det{W_1(x)}}^{-\frac{1}{2}}W_1(x)$. Then $W_2(x)\in C^{\omega}({{\mathbb T}})$ and $\det{W_2(x)}=1$. We have $$\begin{aligned}
W_2^{-1}(x+\alpha)\tilde{A}(x)W_2(x)=\frac{{\det{W_1(x+\alpha)}}^{\frac{1}{2}}}{{\det{W_1(x)}}^{\frac{1}{2}}}R_{-\tilde{\theta}}+O(e^{-\frac{\epsilon_1}{100}N})\ \ \mathrm{on}\ {{\mathbb T}},\end{aligned}$$ $$\begin{aligned}
{W_2^{-1}(x+\alpha)} \tilde{A}(x) W_2(x)=R_{-\tilde{\theta}} + O(e^{-\frac{\epsilon_1}{200}N})\ \ \mathrm{on}\ {{\mathbb T}}.\end{aligned}$$ Now let’s prove $\deg{W_2}(x) \leq 36 n$. $\deg{W_2}(x)$ is the same as the degree of its columns. For $M: {{{\mathbb R}}}/{2{{\mathbb Z}}}\rightarrow {{{\mathbb R}}^2}$, we say $\deg{M}=k$ if $M$ is homotopic to $\left(\begin{matrix}\cos{k\pi x}\\ \sin{k\pi x}\end{matrix}\right)$.
For some constant $c>0$, we obviously have $$\begin{aligned}
\int_{{{\mathbb T}}}\|S(x)\|\ \mathrm{d}x+ \int_{{{\mathbb T}}}\|T(x)\|\ \mathrm{d}x \geq \int_{{{\mathbb T}}}\|S(x)+i T(x)\|\ \mathrm{d}x=\int_{{{\mathbb T}}} \|\tilde{U}_0(x)\|\ \mathrm{d}x \geq c.\end{aligned}$$ Without loss of generality we could assume $\int_{{{\mathbb T}}}\|S(x)\|\ \mathrm{d}x>\frac{c}{2}$. Also $$\begin{aligned}
\tilde{A}(x) S(x)=S(x+\alpha)\cos{2\pi\tilde{\theta}}-T(x+\alpha)\sin{2\pi\tilde{\theta}}+O(e^{-\frac{\epsilon_1}{45}N})\ \ \mathrm{on}\ {{\mathbb T}}.\end{aligned}$$ Then since $\|2\tilde{\theta}\|=L^{-1}$, $$\begin{aligned}
\tilde{A}(x) S(x)=S(x+\alpha)+O(L^{-\frac{1}{2}})\ \ \mathrm{on}\ {{\mathbb T}}.\end{aligned}$$ First we prove $\inf_{x\in {{{\mathbb T}}}} \|S(x)\|\geq e^{-2\epsilon_1 n}$. Suppose otherwise. Then there exists $x_0\in {{\mathbb T}}$, so that $\|S(x_0)\|<e^{-2\epsilon_1 n}$. Then $\|\mathrm{Re}\tilde{U}_0(x_0+l\alpha)\|<e^{-\frac{\epsilon_0}{8} n}$ for $|l|<e^{\frac{\epsilon_0}{4C}n}$, where $C$ is the constant that appeared in Theorem $\ref{polycontrol}$. We have already shown that $$\begin{aligned}
\|\tilde{U}_0(x)-[e^{-\pi i n_jx}\tilde{U}_0^{[9n]}]_{[-18n, 18n]}e^{\pi i n_jx}\|_{{{\mathbb T}}}<e^{-4\epsilon_1 n}.\end{aligned}$$ Thus $$\|\mathrm{Re}[e^{-\pi i n_jx}\tilde{U}_0^{[9n]}]_{[-18n, 18n]}(x_0+l\alpha)\|<e^{- \frac{\epsilon_0}{16}n}$$ for $|l|<e^{\frac{\epsilon_0}{4C}n}$. However $\mathrm{Re}[e^{-\pi i n_jx}\tilde{U}_0^{[9n]}]_{[-18n, 18n]}$ is a polynomial with essential degree at most $36n$. Using Lemma $\ref{polynomialestimate}$ we are able to get $\|\mathrm{Re}[e^{-\pi i nx}\tilde{U}_0^{[9n]}]_{[-18n, 18n]}e^{\pi i n_jx}\|_{{{\mathbb T}}}<e^{-\frac{\epsilon_0}{32}n}$, and thus $\|\mathrm{Re}\tilde{U}_0(x)\|_{{{\mathbb T}}}<e^{-\frac{\epsilon_0}{64}n}$ which is a contradiction to $\int_{{{\mathbb T}}}\|\mathrm{Re}\tilde{U}_0(x)\|\ \mathrm{d}x>\frac{c}{2}$. At the meantime, we also get $\|S(x)-\mathrm{Re}[e^{-\pi i n_jx}\tilde{U}_0^{[9n]}]_{[-18n, 18n]}(x)e^{\pi i n_jx}\|_{{{\mathbb T}}} \triangleq \|S(x)-h(x)\|_{{{\mathbb T}}} \leq e^{-4\epsilon_1 n}$. The first column of $W_2(x)$ is ${\det{W}_1(x)}^{-\frac{1}{2}}S(x)$. We have $$\begin{aligned}
&\|\frac{S(x)}{{\det{W}_1(x)}^{\frac{1}{2}}}-\frac{h(x)}{{w_0}^{\frac{1}{2}}}\|\\
\leq &\frac{1}{|{\det{W}_1(x)}^{\frac{1}{2}}|}\|S(x)-h(x)+(1-\frac{{\det{W}_1(x)}^{\frac{1}{2}}}{{w_0}^{\frac{1}{2}}})h(x)\|\\
\leq & L^{2C}(e^{-4\epsilon_1 n}+L^{8C} e^{-\frac{\epsilon_1}{100}N})\\
\leq & e^{-3\epsilon_1 n}< \|\frac{S(x)}{{\det{W}_1(x)}^{\frac{1}{2}}}\|\ \ \mathrm{on}\ {{\mathbb T}}.\end{aligned}$$ Thus by Rouch$\acute{e}$’s theorem $|\deg{W}_2(x)|=|\deg{h}(x)|\leq 19n$. Notice that $$\begin{aligned}
|\rho(\alpha, W^{-1}_2\tilde{A}W_2)+\tilde{\theta}|<C e^{-\frac{\epsilon_1}{200}N}.\end{aligned}$$ Then, by $\ref{rhoconju}$ for some $|m|\leq 19n$: $$\begin{aligned}
|\rho(\alpha, \tilde{A})-\frac{m}{2}\alpha+\tilde{\theta}|<C e^{-\frac{\epsilon_1}{200}N}.\end{aligned}$$
\
=
When $\lambda$ belongs to region II, let $\epsilon_2=\ln{\frac{\lambda_2+\sqrt{\lambda_2^2-4\lambda_1\lambda_3}}{\lambda_1+\lambda_3+\sqrt{(\lambda_1+\lambda_3)^2-4\lambda_1\lambda_3}}}>\epsilon_1$. Then $c(x)$ is analytic and nonzero on $|\mathrm{Im}(x)|<\frac{\epsilon_2}{2\pi}$. Furthermore, the winding number of $c(\cdot +i\epsilon)$ is equal to zero when $|\epsilon|< \frac{\epsilon_2}{2\pi}$.
\[c/|c|\] When $\lambda$ belongs to region II, we can find an analytic function $f(x)$ on $|\mathrm{Im}(x)|\leq \frac{\epsilon_1}{2\pi}$ such that $c(x)=|c|(x)e^{f(x+\alpha)-f(x)}$ and $\tilde{c}(x)=|c|(x)e^{-f(x+\alpha)+f(x)}$.
Since the winding numbers of $c(x)$ and $\tilde{c}(x)$ are $0$ on $|\mathrm{Im}(x)|\leq \frac{\epsilon_1}{2\pi}$, there exist analytic functions $g_1(x)$ and $g_2(x)$ on $|\mathrm{Im}(x)|\leq\frac{\epsilon_1}{2\pi}$, such that $c(x)=e^{g_1(x)}$ and $\tilde{c}(x)=e^{g_2(x)}$. Notice that $$\begin{aligned}
\int_{{{\mathbb T}}} \ln{|c(x)|}\ \mathrm{d}x=\int_{{{\mathbb T}}} \ln{|\tilde{c}(x)}|\ \mathrm{d}x\\
\int_{{{\mathbb T}}} \arg{c(x)}\ \mathrm{d}x=\int_{{{\mathbb T}}} \arg{\tilde{c}(x)}\ \mathrm{d}x,\end{aligned}$$ so there exists an analytic function $f(x)$ such that $2f(x+\alpha)-2f(x)=g_1(x)-g_2(x)$. Then $c(x)=|c|(x)e^{f(x+\alpha)-f(x)}$.
$\hfill{} \Box$
\[conjugate\] When $\lambda$ belongs to region II, there exists an analytic matrix $Q_{\lambda}(x)$ defined on $|\mathrm{Im}(x)|\leq \frac{\epsilon_1}{2\pi}$ such that $$\begin{aligned}
Q_{\lambda}^{-1}(x+\alpha)\tilde{A}_{\lambda, E}(x)Q_{\lambda}(x)=A_{\lambda, E}(x).\end{aligned}$$
$$\begin{aligned}
\tilde{A}_{\lambda, E}(x)=&
\frac{1}{\sqrt{|c|(x)|c|(x-\alpha)}}
\left(
\begin{matrix}
1 &0\\
0 &\sqrt{\frac{\tilde{c}(x)}{c(x)}}
\end{matrix}
\right)
\left(
\begin{matrix}
E-v(x) &-\tilde{c}(x-\alpha)\\
c(x) &0
\end{matrix}
\right)
\left(
\begin{matrix}
1 &0\\
0 &\sqrt{\frac{c(x-\alpha)}{\tilde{c}(x-\alpha)}}
\end{matrix}
\right)\\
=&
\frac{c(x)}{\sqrt{|c|(x)|c|(x-\alpha)}}
\left(
\begin{matrix}
1 &0\\
0 &\sqrt{\frac{\tilde{c}(x)}{c(x)}}
\end{matrix}
\right)
A(x)
\left(
\begin{matrix}
1 &0\\
0 &\sqrt{\frac{c(x-\alpha)}{\tilde{c}(x-\alpha)}}
\end{matrix}
\right)\\
=&
e^{f(x+\alpha)}\sqrt{|c|(x)}
\left(
\begin{matrix}
1 &0\\
0 &\sqrt{\frac{\tilde{c}(x)}{c(x)}}
\end{matrix}
\right)
A(x)
\left\lbrace e^{f(x)}\sqrt{|c|(x-\alpha)}
\left(
\begin{matrix}
1 &0\\
0 &\sqrt{\frac{\tilde{c}(x-\alpha)}{c(x-\alpha)}}
\end{matrix}
\right)
\right\rbrace^{-1}\\
=&Q_{\lambda}(x+\alpha)A_{\lambda, E}(x)Q_{\lambda}^{-1}(x).\end{aligned}$$
$\hfill{} \Box$
\[LEregion2\] If $\alpha$ is irrational, $\lambda$ belongs to region II, $E\in\Sigma (\lambda)$, then $L(\alpha, A_{\lambda, E}(\cdot+i\epsilon))=L(\alpha, \tilde{A}_{\lambda, E}(\cdot+i\epsilon))=0$ for $|\epsilon|\leq \frac{\epsilon_1}{2\pi}$.
$L(A(\cdot+i \epsilon))=L(D(\cdot +i\epsilon))-\int \ln{|c(x+i\epsilon)|}\mathrm{d}x$ $$\begin{aligned}
D(x+i\epsilon)
&=\left(
\begin{matrix}
E-e^{2\pi i (x+i\epsilon)}-e^{-2\pi i (x+i\epsilon)}\ \ &-\lambda_1 e^{2\pi i (x-\frac{\alpha}{2}+i\epsilon)}-\lambda_2-\lambda_3 e^{-2\pi i (x-\frac{\alpha}{2}+i\epsilon)}\\
\lambda_1 e^{-2\pi i(x+\frac{\alpha}{2}+i\epsilon)}+\lambda_2+\lambda_3 e^{2\pi i (x+\frac{\alpha}{2}+i\epsilon)}\ \ &0
\end{matrix}
\right)\\
&=e^{2\pi \epsilon}
\left(
\begin{matrix}
-e^{2\pi i x}+o(1)\ \ & -\lambda_3 e^{-2\pi i (x-\frac{\alpha}{2})}+o(1)\\
\lambda_1 e^{-2\pi i (x+\frac{\alpha}{2})}+o(1)\ \ &0
\end{matrix}
\right).\end{aligned}$$ Thus the asymptotic behaviour of $L(D(\cdot+i\epsilon))$ is: $$\begin{aligned}
&L(D(\cdot+i\epsilon))=\ln{|\frac{1+\sqrt{1-4\lambda_1\lambda_3}}{2}|}+2\pi \epsilon\ \ \mathrm{when}\ \epsilon\rightarrow \infty,\\
&L(D(\cdot+i\epsilon))=\ln{|\frac{1+\sqrt{1-4\lambda_1\lambda_3}}{2}|}-2\pi \epsilon\ \ \mathrm{when}\ \epsilon\rightarrow -\infty.\end{aligned}$$ Then it suffices to calculate $\int \ln{|c(x+i\epsilon)|}\mathrm{d}x$ in region II. We have $$\begin{aligned}
&\int \ln{|c(x+i\epsilon)|}\mathrm{d} x\\
=&\ln{\lambda_3}-2\pi \epsilon+\int \ln{|e^{2\pi i x}-y_{1,\epsilon}|}+\int \ln{|e^{2\pi i x}-y_{2,\epsilon}|}.\end{aligned}$$ where $y_{1,\epsilon}=\frac{-\lambda_2+\sqrt{\lambda_2^2-4\lambda_1\lambda_3}}{2\lambda_3}e^{2\pi \epsilon}$ and $y_{2,\epsilon}=\frac{-\lambda_2-\sqrt{\lambda_2^2-4\lambda_1\lambda_3}}{2\lambda_3}e^{2\pi \epsilon}$. $$\begin{aligned}
\int \ln{|c(x+i\epsilon)|}\mathrm{d}x=
\left\lbrace
\begin{matrix}
2\pi \epsilon+\ln{\lambda_1}\ \ \ \ \ \ \ \ \ \ \ \ \ \ &\epsilon>\frac{1}{2\pi}\ln{\frac{\lambda_2+\sqrt{\lambda_2^2-4\lambda_1\lambda_3}}{2\lambda_1}},\\
\\
\ln{\frac{\lambda_2+\sqrt{\lambda_2^2-4\lambda_1\lambda_3}}{2}}\ \ \ \ &\frac{1}{2\pi}\ln{\frac{\lambda_2-\sqrt{\lambda_2^2-4\lambda_1\lambda_3}}{2\lambda_1}}\leq \epsilon\leq \frac{1}{2\pi}\ln{\frac{\lambda_2+\sqrt{\lambda_2^2-4\lambda_1\lambda_3}}{2\lambda_1}},\\
\\
-2\pi \epsilon+\ln{\lambda_3}\ \ \ \ \ \ \ \ \ \ \ \ \ &\epsilon<\frac{1}{2\pi}\ln{\frac{\lambda_2-\sqrt{\lambda_2^2-4\lambda_1\lambda_3}}{2\lambda_1}}.
\end{matrix}
\right.\end{aligned}$$ Thus $L(A(\cdot +i\epsilon))=0$ when $|\epsilon|\leq \frac{1}{2\pi}\ln{\frac{\lambda_2+\sqrt{\lambda_2^2-4\lambda_1\lambda_3}}{\max{(1,\lambda_1+\lambda_3)}+\sqrt{\max{(1,\lambda_1+\lambda_3)}^2-4\lambda_1\lambda_3}}}=\frac{\epsilon_1}{2\pi}$.
Since $\tilde{A}_{\lambda, E}(x+i\epsilon)=Q_{\lambda}(x+\alpha+i\epsilon)A_{\lambda, E}(x+i\epsilon)Q_{\lambda}^{-1}(x+i\epsilon)$, the statement about $\tilde{A}_{\lambda, E}$ is also true.
$\hfill{} \Box$
Acknowledgement {#acknowledgement .unnumbered}
===============
I am deeply grateful to Svetlana Jitomirskaya for suggesting this problem and for many valuable discussions. This research was partially supported by the NSF DMS–1401204.
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[^1]: In general one cannot take $M\in C^{\omega}({{\mathbb T}}, SL(2,{{\mathbb R}}))$.
|
---
abstract: 'We have calculated long series expansions for self-avoiding walks and polygons on the honeycomb lattice, including series for metric properties such as mean-squared radius of gyration as well as series for moments of the area-distribution for polygons. Analysis of the series yields accurate estimates for the connective constant, critical exponents and amplitudes of honeycomb self-avoiding walks and polygons. The results from the numerical analysis agree to a high degree of accuracy with theoretical predictions for these quantities.'
address: 'ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia'
author:
- Iwan Jensen
title: Honeycomb lattice polygons and walks as a test of series analysis techniques
---
Introduction
============
Self-avoiding walks (SAWs) and polygons (SAPs) on regular lattices are among the most important and classic combinatorial problems in statistical mechanics. SAWs are often considered in the context of lattice models of polymers while SAPs are used to model vesicles. The fundamental problem is the calculation (up to translation) of the number of SAWs, $c_n$, with $n$ steps (SAPs, $p_n$, of perimeter $n$). As for many interesting combinatorial problems, SAWs have exponential growth, $c_n \sim A\mu^n n^{\gamma-1}$, where $\mu$ is the connective constant, $\gamma$ is a critical exponent, and $A$ is a critical amplitude. A major challenge (short of an exact solution) is the calculation, or at least accurate estimation of, $\mu$, critical exponents and amplitudes. Here our focus is on the numerical estimation of such quantities from exact enumeration data.
The success of series expansions as a numerical technique has relied crucially on several of Tony Guttmann’s contributions to the field a asymptotic series analysis. In pioneering the method of differential approximants (see [@AJG89a] for a review and ‘historical notes’) Tony Guttmann has given us an invaluable tool which over the years has been proved to be by far the best (in terms both of accuracy and versatility) method for analysing series. In this paper we use long series expansions for self-avoiding polygons and walks on the honeycomb lattice to test the accuracy of various methods for series analysis. For the honeycomb lattice the connective constant, critical exponents and many amplitude ratios are known exactly, making it the perfect test-bed for series analysis techniques.
The rest of the paper is organised as follows: In section \[sec:th\] we give precise definitions of the models and the properties we investigate and summarise a number of exact results. Section \[sec:enum\] contains a very brief introduction to the literature describing the algorithms used for the exact enumerations. In section \[sec:DA\] we give a brief introduction to the numerical technique of differential approximants and then proceed to analyse the SAP and SAW series clearly demonstrating how we can obtain very accurate estimates for the connective constant and critical exponents. Section \[sec:Ampl\] is concerned with the estimation of amplitudes. Not only do we obtain very accurate estimates for the amplitudes, but we also show how an analysis of the asymptotic behaviour of the series coefficients can be used to gain insight into corrections to scaling. Finally, in section \[sec:sum\] we discuss and summarise our main results.
Definitions and theoretical background \[sec:th\]
=================================================
An [*$n$-step self-avoiding walk*]{} $\bm{\omega}$ is a sequence of [*distinct*]{} vertices $\omega_0, \omega_1,\ldots , \omega_n$ such that each vertex is a nearest neighbour of it predecessor. SAWs are considered distinct up to translations of the starting point $\omega_0$. We shall use the symbol $\bm{\Omega}_n$ to mean the set of all SAWs of length $n$. A self-avoiding polygon of length $n$ is an $n-1$-step SAW such that $\omega_0$ and $\omega_{n-1}$ are nearest neighbours and a closed loop can be formed by inserting a single additional step between the two end-points of the walk. The two models are illustrated in figure \[fig:example\]. One is interested in the number of SAWs and SAPs of length $n$, various metric properties such as the radius of gyration, and for SAPs one can also ask about the area enclosed by the polygon. In this paper we study the following properties:
- the number of $n$-step self-avoiding walks $c_n$;
- the number of $n$-step self-avoiding polygons $p_n$;
- the mean-square end-to-end distance of $n$-step SAWs ${\langle R^2_e \rangle}_n$;
- the mean-square radius of gyration of $n$-step SAWs ${\langle R^2_g \rangle}_n$;
- the mean-square distance of a monomer from the end points of $n$-step SAWs ${\langle R^2_m \rangle}_n$;
- the mean-square radius of gyration of $n$-step SAPs ${\langle R^2 \rangle}_n$; and
- the $m^{\rm th}$ moment of the area of $n$-step SAPs ${\langle a^m \rangle}_n$.
![\[fig:example\] Examples of a self-avoiding walk (left panel) and polygon (right panel) on the honeycomb lattice. ](hcsapsaw_example.eps)
It is generally believed that the quantities listed above have the asymptotic forms as $n \to \infty$:
$$\begin{aligned}
c_n &=& A \mu^n n^{\gamma-1}[1 + o(1)], \label{eq:asympsaw} \\
p_n &=& B \mu^n n^{\alpha-3}[1 + o(1)], \label{eq:asympsap} \\
{\langle R^2_e \rangle}_n &=& Cn^{2\nu}[1 + o(1)], \label{eq:asympee} \\
{\langle R^2_g \rangle}_n &=& Dn^{2\nu}[1 + o(1)], \label{eq:asymprg}\\
{\langle R^2_m \rangle}_n &=& En^{2\nu}[1 + o(1)], \label{eq:asympmd} \\
{\langle R^2 \rangle}_n &=& Fn^{2\nu}[1 + o(1)], \label{eq:asympsaprg} \\
{\langle a^m \rangle}_n &=& G^{(m)}n^{2\nu m}[1 + o(1)]. \label{eq:asympmom} \end{aligned}$$ The critical exponents are believed to be universal in that they only depend on the dimension of the underlying lattice. The connective constant $\mu$ and the critical amplitudes $A$–$G^{(m)}$ vary from lattice to lattice. In two dimensions the critical exponents $\gamma = 43/32$, $\alpha =1/2$ and $\nu = 3/4$ have been predicted exactly, though non-rigorously [@Nienhuis82a; @Nienhuis84a]. In this work Nienhuis also predicted the exact value of the connective constant on the honeycomb lattice $\mu=\sqrt{2+\sqrt{2}}$. When analyzing the series data it is often convenient to use the associated generating functions such as $$\begin{aligned}
{\ensuremath{\mathcal{C}}}(x)& = &\sum_{n=0} c_n x^n \sim A(x)(1-\mu x)^{-\gamma}, \label{eq:sawgf} \\
{\ensuremath{\mathcal{P}}}(x)& = &\sum_{n=0} p_{2n+6 }x^n \sim B(x)(1-\mu^2 x)^{2-\alpha}. \label{eq:sapgf}\end{aligned}$$ In the polygon generating function we take into account that SAPs have even length and the smallest one has perimeter 6. The SAW (SAP) generating function has a singularity at the critical point $x=x_c = 1/\mu$ ($x=x_c^2=1/\mu^2$) with critical exponent $-\gamma$ ($2-\alpha$).
The metric properties for SAWs are defined by,
$$\begin{aligned}
{\langle R^2_e \rangle}_n &= & \frac{1}{c_n} \sum_{\bm{\Omega}_n} (\omega_0 - \omega_n)^2, \\
{\langle R^2_g \rangle}_n &= &\frac{1}{c_n} \sum_{\bm{\Omega}_n}\left [ \frac{1}{2(n+1)^2}
\sum_{i,j=0}^n (\omega_i - \omega_j)^2 \right ], \\
{\langle R^2_m \rangle}_n &= &\frac{1}{c_n} \sum_{\bm{\Omega}_n} \left [ \frac{1}{2(n+1)}
\sum_{i=0}^n \left [(\omega_0-\omega_j)^2+(\omega_n-\omega_j)^2 \right ] \right ],\end{aligned}$$
with a similar definition for the radius of gyration of SAPs.
While the amplitudes are non-universal, there are many universal amplitude combinations. Any ratio of the metric SAW amplitudes, e.g. $D/C$ and $E/C$, is expected to be universal [@CS89]. Of particular interest is the linear combination [@CS89; @CPS90] (which we shall call the CSCPS relation) $$\label{eq:CSCPS}
H \;\equiv\;
\left( 2 + \frac{y_t}{y_h} \right) \frac{D}{C}
\,-\, 2 \frac{E}{C} \,+\, \frac12,$$ where $y_t = 1/\nu$ and $y_h = 1 + \gamma/(2\nu)$. In two dimensions Cardy and Saleur [@CS89] (as corrected by Caracciolo, Pelissetto and Sokal [@CPS90]) have predicted, using conformal field theory, that $H = 0$. Cardy and Guttmann [@CG93] proved that $BF=\frac{5}{32\pi^2}\sigma a_0$, where $\sigma$ is an integer constant such that $p_n$ is non-zero when $n$ is divisible by $\sigma$, so $\sigma=2$ for the honeycomb lattice. $a_0=3\sqrt{3}/4$ is the area per lattice site on the honeycomb lattice. Richard, Guttmann and Jensen [@RGJ01] conjectured the exact form of the critical scaling function for self-avoiding polygons and consequently showed that the amplitude combinations $G^{(k)}B^{k-1}$ are universal and predicted their exact values. The exact value for $G^{(1)}=\frac{1}{4\pi}$ had previously been predicted by Cardy [@JLC94a].
The asymptotic form (\[eq:asympsaw\]) only explicitly gives the leading contribution. In general one would expect corrections to scaling so $$c_n= A\mu^n n^{\gamma-1}\left [1 + \frac{a_1}{n}+\frac{a_2}{n^2}+\ldots
+ \frac{b_0}{n^{\Delta_1}}+\frac{b_1}{n^{\Delta_1+1}}+\frac{b_2}{n^{\Delta_1+2}}+\ldots
\right]$$ In addition to “analytic” corrections to scaling of the form $a_k/n^k$, where $k$ is an integer, there are “non-analytic” corrections to scaling of the form $b_k/n^{\Delta_1+k}$, where the correction-to-scaling exponent $\Delta_1$ isn’t an integer. In fact one would expect a whole sequence of correction-to-scaling exponents $\Delta_1 < \Delta_2 \ldots$, which are both universal and also independent of the observable, that is, the same for $c_n$, $p_n$, and so on. Furthermore, there should also be corrections with exponents such as $n\Delta_i+m\Delta_j$, etc., with $n$ and $m$ positive integers. At least two different theoretical predictions have been made for the exact value of the leading non-analytic correction-to-scaling exponent: $\Delta_1 = 3/2$ based on Coulomb-gas arguments [@Nienhuis82a; @Nienhuis84a] and $\Delta_1 = 11/16$ based on conformal-invariance methods [@Saleur87a]. In a recent paper [@CGJPRS] the amplitudes and the correction-to-scaling exponents for SAWs on the square and triangular lattices were studied in great detail. The analysis provided firm numerical evidence that $\Delta_1=3/2$ as predicted by Nienhuis.
Enumerations \[sec:enum\]
=========================
The algorithm we used to enumerate SAPs on the honeycomb lattice is based on the finite-lattice method devised by Enting [@IGE80e] in his pioneering work, which contains a detailed description of the original approach for enumerating SAPs on the square lattice. A major enhancement, resulting in an exponentially more efficient algorithm, is described in some detail in [@JG99] while details of the changes required to enumerate area-moments and the radius of gyration can be found in [@IJ00a]. A very efficient parallel implementation is described in [@IJ03a]. The generalisation to enumerations of SAWs is straight forward as shown in [@IJ04a]. An implementation of the basic SAP enumeration algorithm on the honeycomb lattice can be found in [@EG89a]. Most of the enhancements we made to the square lattice case can also be readily implemented on the honeycomb lattice. The only slightly tricky part is the calculation of metric properties (though the changes are very similar to those required for the triangular lattice [@IJ04d]).
Using the a parallel version of our honeycomb lattice algorithms we have counted the number of self-avoiding walks and polygons to length 105 and 158, respectively. For self-avoiding walks to length 96 we also calculate series for the metric properties of mean-square end-to-end distance, mean-square radius of gyration and the mean-square distance of a monomer from the end points. In fact the algorithm calculates the metric generating functions with coefficients $c_n{\langle R^2_e \rangle}_n$, $n^2 c_n{\langle R^2_g \rangle}_n$, and $nc_n{\langle R^2_m \rangle}_n$, respectively, the advantage being that these quantities are integer valued. For self-avoiding polygons to length 140 we calculate series for the mean-square radius of gyration and the first 10 moments of the area. Again we actually calculate the series with integer coefficients $8n^2p_n{\langle R^2 \rangle}_n$ and $p_n{\langle a^k \rangle}_n$.
Differential approximants \[sec:DA\]
====================================
The majority of interesting models in statistical mechanics and combinatorics have generating functions with regular singular points such as those indicated in (\[eq:sawgf\]) and (\[eq:sapgf\]). The fundamental problem of series analysis is: Given a [*finite*]{} number of terms in the series expansion for a function $F(x)$ what can one say about the singular behaviour which after all is a property of the [*infinite*]{} series. Without a doubt the best series analysis technique when it comes to locating singularities and estimating the associated critical exponents is differential approximants (see [@AJG89a] for a comprehensive review of differential approximants and other techniques for series analysis). The basic idea is to approximate the function $F(x)$ by solutions to differential equations with polynomial coefficients. The singular behaviour of such ODEs is much studied (see [@Ince]) and the singular points and exponents are easily calculated.
A $K^{th}$-order differential approximant (DA) to a function $F(x)$ is formed by matching the coefficients in the polynomials $Q_i(x)$ and $P(x)$ of degree $N_i$ and $L$, respective, so that (one) of the formal solutions to the inhomogeneous differential equation $$\sum_{i=0}^K Q_{i}(x)(x\frac{{\rm d}}{{\rm d}x})^i \tilde{F}(x) = P(x)$$ agrees with the first $M=L+\sum_i (N_i+1)$ series coefficients of $F$. Singularities of $F(x)$ are approximated by the zeros $x_i$ of $Q_K(x)$ and the associated critical exponent $\lambda_i$ is estimated from the indicial equation. If there is only a single root at $x_i$ this is just $$\lambda_i=K-1-\frac{Q_{K-1}(x_i)}{x_iQ_K ' (x_i)}.$$ The physical critical point is the first singularity on the positive real axis.
In order to locate the singularities of the series in a systematic fashion we used the following procedure: We calculate all $[L;N_0,N_1,N_2]$ and $[L;N_0,N_1,N_2,N_3]$ second- and third-order inhomogeneous differential approximants with $|N_i - N_j| \leq 2$, that is the degrees of the polynomials $Q_i$ differ by at most 2. In addition we demand that the total number of terms used by the DA is at least $N_{\rm max}-10$, where $N_{\rm max}$ is the total number of terms available in the series. Each approximant yields $N_K$ possible singularities and associated exponents from the $N_K$ zeroes of $Q_K(x)$ (many of these are of course not actual singularities of the series but merely spurious zeros.) Next these zeroes are sorted into equivalence classes by the criterion that they lie at most a distance $2^k$ apart. An equivalence class is accepted as a singularity if it appears in more than 75% of the total number of approximants, and an estimate for the singularity and exponent is obtained by averaging over the included approximants (the spread among the approximants is also calculated). The calculation was then repeated for $k-1$, $k-2$, $\ldots$ until a minimal value of 8 was reached. To avoid outputting well-converged singularities at every level, once an equivalence class has been accepted, the data used in the estimate is discarded, and the subsequent analysis is carried out on the remaining data only. One advantage of this method is that spurious outliers, some of which will almost always be present when so many approximants are generated, are discarded systematically and automatically.
The polygon series \[sec:DAsap\]
--------------------------------
First we apply our differential approximant analysis to the self-avoiding polygon generating function. In table \[tab:sapDA\] we have listed the estimates for the critical point $x_c^2$ and exponent $2-\alpha$ obtained from second- and third-order DAs. We note that all the estimates are in perfect agreement (surely a best case scenario) in that within ‘error-bars’ they take the same value. From this we arrive at the estimate $x_c^2=0.2928932186(5)$ and $2-\alpha=1.5000004(10)$, where the error-bars reflect the spread among the estimates and the individual error-bars (note that DA estimates [*are not*]{} statistically independent so the final error-bars exceed the individual ones). The final estimates are in perfect agreement with the conjectured exact values $x_c^2=1/\mu^2=1/(2+\sqrt{2})= 0.292893218813\ldots$ and $2-\alpha=3/2$.
[lllll]{} $L$ & &\
& & & &\
0 & 0.29289321854(19)& 1.50000065(41) & 0.29289321865(12)& 1.50000040(28)\
5 & 0.29289321875(21)& 1.50000010(59) & 0.29289321852(48)& 1.50000041(99)\
10 & 0.29289321855(23)& 1.50000060(48) & 0.29289321878(32)& 1.49999999(97)\
15 & 0.29289321859(19)& 1.50000054(43) & 0.29289321861(37)& 1.50000035(67)\
20 & 0.29289321866(15)& 1.50000038(33) & 0.29289321860(21)& 1.50000049(43)\
Before proceeding we will consider possible sources of systematic errors. First and foremost the possibility that the estimates might display a systematic drift as the number of terms used is increased and secondly the possibility of numerical errors. The latter possibility is quickly dismissed. The calculations were performed using 128-bit real numbers. The estimates from a few approximants were compared to values obtained using MAPLE with 100 digits accuracy and this clearly showed that the program was numerically stable and rounding errors were negligible. In order to address the possibility of systematic drift and lack of convergence to the true critical values we refer to figure \[fig:sapDA\] (this is probably not really necessary in this case but we include the analysis here in order to present the general method). In the left panel of figure \[fig:sapDA\] we have plotted the estimates from third-order DAs for $x_c^2$ vs. the highest order term $N$ used by the DA. Each dot in the figure is an estimate obtained from a specific approximant. As can be seen the estimates clearly settle down to the conjectured exact value (solid line) as $N$ is increased and there is little to no evidence of any systematic drift at large $N$. One curious aspect though is the widening of the spread in the estimates around $N=140$. We have no explanation for this behaviour but it could quite possibly be caused by just a few ‘spurious’ approximants. In the right panel we show the variation in the exponent estimates with the critical point estimates. We notice that the ‘curve’ traced out by the estimates pass through the intersection of the lines given by the exact values. We have not been able to determine the reason for the apparent branching into two parts. However, we note that the lower ‘branch’ contain many more approximants than the upper one.
![\[fig:sapDA\] Plot of estimates from third order differential approximants for $x_c^2$ vs. the highest order term used and $2-\alpha$ vs. $x_c^2$. The straight lines are the exact predictions. ](hcsapDA.eps)
The differential approximant analysis can also be used to find possible non-physical singularities of the generating function. Averaging over the estimates from the DAs shows that there is an additional non-physical singularity on the negative $x$-axis at $x=x_-=-1/\mu_-^2 =-0.41230(2)$, where the associated critical exponent $\alpha_-$ has a value consistent with the exact value $\alpha_-=3/2$. In the left panel of figure \[fig:sapneg\] we have plotted $\alpha_-$ vs. the highest order term used by the DAs and we clearly see the convergence to a value consistent with $\alpha_-=3/2$. If we take this value as being exact we can get a refined estimate of $x_-$ from the plot in the right panel of figure \[fig:sapneg\], where we notice that the estimates for $\alpha_-$ cross the value $3/2$ for $x_-=-0.412305(5)$ which we take as our final estimate. From this we then get $\mu_- = 1.557366(10)$.
![\[fig:sapneg\] Plot of estimates from third order differential approximants for the location $x_-$ of the non-physical singularity and the associated exponent $\alpha_-$. The left shows $\alpha_-$ vs. the highest order term used and right panel $\alpha_-$ vs. $x_-$.](hcsapneg.eps)
The walk series \[sec:DAsaw\]
-----------------------------
Next we apply the differential approximant analysis to the self-avoiding walk generating function. In table \[tab:sawDA\] we have listed the estimates for the critical point $x_c$ and exponent $\gamma$ obtained from second- and third-order DAs. Firstly, we note that estimates are about an order of magnitude less accurate than in the polygon case. Secondly, there are now small but nevertheless seemingly systematic differences between the second- and third order DAs (in particular the second-order homogeneous ($L=0$) approximants are much less accurate than the other cases). On general theoretical grounds one would expect higher-order inhomogeneous approximants to be better in that they can accommodate more complicated functional behaviour. So based mainly on the third-order DAs we finally estimate that $0.541196102(4)$ and $\gamma=1.343758(8)$. This is consistent with the exact values $x_c=1/\mu=1/\sqrt{2+\sqrt{2}}=0.541196100146\ldots$ and $\gamma=43/32=1.34375$, though the central estimates for $\gamma$ are systematically a bit too high (and the second-order DAs are worse).
[lllll]{} $L$ & &\
& & & &\
0 & 0.541196097(19) & 1.34360(36) & 0.5411961075(19) & 1.3437685(54)\
5 & 0.5411961066(10) & 1.343770(18) & 0.5411961025(10) & 1.3437583(19)\
10 & 0.5411961065(12) & 1.3437669(53) & 0.54119610266(91) & 1.3437584(20)\
15 & 0.5411961069(16) & 1.343776(68) & 0.5411961011(17) & 1.3437551(38)\
20 & 0.5411961059(21) & 1.3437646(29) & 0.5411961022(26) & 1.3437580(59)\
In addition there is a singularity on the negative $x$-axis at $x=-x_c$ with a critical exponent consistent with the value $1/2$, and a pair of singularities at $x=\pm 0.64215(15){\rm i} $ with an exponent which is likely to equal $1/2$ (note that the value $0.64215(15)$ is consistent with $1/\mu_-$). These results help to at least partly explain why the walk series is more difficult to analyse than the polygon series. The walk series has more non-physical singularities and one of these (at $x=-x_c$) is closer to the origin than the non-physical singularity of the polygon series. Furthermore, as argued and confirmed numerically in the next section, the walk series has non-analytical corrections to scaling whereas the polygon series has only analytical corrections. All of these effects conspire to make the walk series much harder to analyse and it is indeed a great testament to the method of differential approximants that the analysis given above yields such accurate estimates despite all these complicating factors.
![\[fig:sawDA\] Plot of estimates from third order differential approximants for $\gamma$ vs. the number of terms used by the DA and $\gamma$ vs. $x_c$. The straight lines are the exact predictions. ](hcsawDA.eps)
In figure \[fig:sawDA\] we have plotted estimates for $\gamma$, obtained from third-order DAs, against the number of terms used by the DA (left panel) and against estimates for the critical point $x_c$ (right panel). The estimates for $\gamma$ display some rather curious and unexpected variations with the number of terms. Early on (around 80 terms) the estimates seems to settle at a value above the exact result. A little later the estimates start trending downwards so that around 95 terms they are in excellent agreement with the exact value. However, the estimates then unexpectedly start trending upwards again so that with more than a 100 terms the agreement with the exact result is only marginal. We also notice that in the right panel estimates for $\gamma$ vs. $x_c$ happen to just miss the intersection between the lines marking the exact predictions. This behaviour is curious and we have no ready explanation for it other than once again drawing attention to the quite complicated functional form of the generating function. The discrepancies between the estimates and exact values is marginal and certainly not significant enough to raise questions about the correctness of the theoretical predictions.
Metric properties \[sec:DAmetric\]
----------------------------------
Finally, we briefly turn our attention to the series for the metric properties of SAPs and SAWs. We actually study the metric generating functions with integer coefficients $8n^2p_n{\langle R^2 \rangle}_n$, $c_n {\langle R^2_e \rangle}_n$, $n^2 c_n{\langle R^2_g \rangle}_n$, and $nc_n{\langle R^2_m \rangle}_n$, which have critical exponents $-(\alpha-2\nu)=-2$, $-(\gamma+2\nu)=-91/32=-2.84375$, $-(\gamma+2\nu+2)=-155/32=-4.84375$, and $-(\gamma+\nu+1)=-123/32=-3.84375$, respectively (and as usual the polygon series use only the even terms). In table \[tab:metric\] we list the estimates obtained for the critical point and exponents using averages over third-order DAs. The exponent estimates from the SAP series are consistent with the expected value confirming $\nu=3/4$ as are the estimates from the SAW series. The only possible exception is the end-to-end distance series where the estimates for both $x_c$ and the exponent are systematically a little to high. However, the discrepancy is not very large and probably not significant.
[lllll]{} $L$ & &\
& & & &\
0 & 0.292893246(10) & 2.000176(35) & 0.5411961141(14) & 2.8438094(28)\
5 & 0.2928932440(70) & 2.000169(24) & 0.5411961136(31) & 2.8438080(64)\
10 & 0.292893245(24) & 2.000166(92) & 0.5411961124(37) & 2.8438054(82)\
15 & 0.292893235(61) & 2.00008(27) & 0.5411961133(33) & 2.8438072(66)\
20 & 0.292893262(42) & 2.00019(11) & 0.5411961113(25) & 2.8438031(59)\
& &\
& & & &\
0 & 0.541196111(22) & 4.843788(47) & 0.5411961013(28) & 3.8437852(95)\
5 & 0.541196115(12) & 4.843806(19) & 0.5411961014(21) & 3.8437843(90)\
10 & 0.5411961041(91) & 4.843789(21) & 0.5411961033(52) & 3.843789(11)\
15 & 0.5411961021(77) & 4.843784(21) & 0.5411961064(75) & 3.843794(22)\
20 & 0.5411961040(49) & 4.843794(10) & 0.5411961049(40) & 3.8437954(75)\
Amplitude estimates \[sec:Ampl\]
================================
Now that the exact values of $\mu$ and the exponents have been confirmed we turn our attention to the “fine structure” of the asymptotic form of the coefficients. In particular we are interested in obtaining accurate estimates for the leading critical amplitudes such as $A$ and $B$. Our method of analysis consists in fitting the coefficients to an assumed asymptotic form. Generally we must include a number of terms in order to account for the behaviour of the generating function at the physical singularity, the non-physical singularities as well as sub-dominant corrections to the leading order behaviour. As we hope to demonstrate, this method of analysis can not only yield accurate amplitude estimates, but it is often possible to clearly demonstrate which corrections to scaling are present.
Before proceeding with the analysis we briefly consider the kind of terms which occur in the generating functions, and how they influence the asymptotic behaviour of the series coefficients. At the most basic level a function $G(x)$ with a power-law singularity
$$\label{eg:Gbasic}
G(x) =\sum_n g_n x^n \sim A(x)(1-\mu x)^{-\xi},$$
where $A(x)$ is an analytic function at $x=x_c$, gives rise to the asymptotic form of the coefficients
$$\label{eq:CoAbasic}
g_n \sim \mu^n n^{\xi-1} \left [ \tilde{A}+ \sum_{i\geq 1} a_i/n^i \right ],$$
that is we get a dominant exponential growth given by $\mu$, modified by a sub-dominant term given by the critical exponent followed by analytic corrections. The amplitude $\tilde{A}$ is related to the function $A(x)$ in (\[eg:Gbasic\]) via the relation $\tilde{A}=A(1/\mu)/\Gamma(\xi)$. If $G(x)$ has a non-analytic correction to scaling such as $$\label{eq:Gcorr}
G(x) =\sum_n g_n x^n \sim (1-\mu x)^{-\xi}\left [A(x)+B(x)(1-\mu x)^{\Delta}\right ],$$ we get the more complicated form
$$\label{eq:CoAcorr}
g_n \sim \mu^n n^{\xi-1} \left [ \tilde{A}+ \sum_{i\geq 1} a_i/n^i+ \sum_{i\geq 0} b_i/n^{\Delta+i} \right ].$$
A singularity on the negative $x$-axis $\propto (1+\mu_- x)^{-\eta}$ leads to additional corrections of the form
$$\label{eq:CoAneg}
\sim (-1)^n \mu_-^n n^{\eta-1} \sum_{i\geq 0} c_i/n^i.$$
Singularities in the complex plane are more complicated. However, a pair of singularities in the complex axis at $\pm {\rm i}/\tau$, that is a term of the form $D(x)(1+\tau^2 x^2)^{-\eta}$, generally results in coefficients that change sign according to a $++--$ pattern. This can be accommodated by terms of the form
$$\label{eq:CoAcomp}
\sim (-1)^{\lfloor n/2 \rfloor}\tau^n n^{\eta-1} \sum_{i\geq 0} d_i/n^i.$$
All of these possible contribution must then be put together in an assumed asymptotic expansion for the coefficients $g_n$ and we obtain estimates for the unknown amplitudes by directly fitting $g_n$ to the assumed form. That is we take a sub-sequence of terms $\{g_n,g_{n-1},\ldots,g_{n-k}\}$, plug into the assumed form and solve the $k+1$ linear equations to obtain estimates for the first few amplitudes. As we shall demonstrate below this allows us to probe the asymptotic form.
Estimating the polygon amplitude $B$ \[sec:Bampl\]
--------------------------------------------------
![\[fig:sapampl\] Plots of fits for the self-avoiding polygon amplitude $B$ using in the left panel the asymptotic form () which ignores the singularity at $x=x_-$, and in the middle panel the asymptotic form () which includes the singularity at $x=x_-$. The right panel gives a closer look at the data form the middle panel. ](hcsapampl.eps)
The asymptotic form of the coefficients $p_n$ of the generating function of square and triangular lattice SAPs has been studied in detail previously [@CG96; @JG99; @IJ03a; @IJ04d]. There is now clear numerical evidence that the leading correction-to-scaling exponent for SAPs and SAWs is $\Delta_1=3/2$, as predicted by Nienhuis [@Nienhuis82a; @Nienhuis84a]. As argued in [@CG96] this leading correction term combined with the $2-\alpha=3/2$ term of the SAP generating function produces an [*analytic*]{} background term as can be seen from eq. (\[eq:Gcorr\]). Indeed in the previous analysis of SAPs there was no sign of non-analytic corrections-to-scaling to the generating function (a strong indirect argument that the leading correction-to-scaling exponent must be half-integer valued). At first we ignore the singularity at $x_-$ (since $|x_-| > x_c^2$ it is exponentially suppressed) and obtain estimates for $B$ by fitting $p_n$ to the form $$\label{eq:Bnoneg}
p_n = \mu^n n^{-5/2} \left [ B+ \sum_{i=1}^k a_i/n^i \right ].$$ That is we take a sub-sequence of terms $\{p_n,p_{n-2},\ldots,p_{n-2k}\}$ ($n$ even), plug into the formula above and solve the $k+1$ linear equations to obtain estimates for the amplitudes. It is then advantageous to plot estimates for the leading amplitude $B$ against $1/n$ for several values of $k$. The results are plotted in the left panel of figure \[fig:sapampl\]. Obviously the amplitude estimates are not well behaved and display clear parity effects. So clearly we can’t just ignore the singularity at $x_-$ (which gives rise to such effects) and we thus try fitting to the more general form
$$\label{eq:Bneg}
p_n = \mu^n n^{-5/2} \left [ B+ \sum_{i= 1}^k a_i/n^i \right ]+
(-1)^{n/2}\mu_-^n n^{-5/2} \sum_{i=0}^k b_i/n^i.$$
The results from these fits are shown in the middle panel of figure \[fig:sapampl\]. Now we clearly have very well behaved estimates (note the significant change of scale along the $y$-axis from the left to the middle panel). In the right panel we take a more detailed look at the data and from the plot we estimate that $B=1.2719299(1)$. We notice that as more and more correction terms are added ($k$ is increased) the plots of the amplitude estimates exhibits less curvature and the slope become less steep. This is very strong evidence that (\[eq:Bneg\]) indeed is the correct asymptotic form of $p_n$.
Estimating the walk amplitude $A$ \[sec:Aampl\]
-----------------------------------------------
From the differential approximant analysis we found that the walk generating function has non-physical singularities at $x=-x_c$ and $x=\pm \rmi/\mu_-$. In addition we expect from Nienhuis’s results (confirmed by extensive numerical work [@CGJPRS]) a non-analytic correction-to-scaling term with exponent $\Delta=3/2$, and since $\gamma=43/32$ this correction term does not vanish in the walk case. Ignoring for the moment the pair of complex singularities we first try with to fit to the asymptotic form
$$\label{eq:Aneg}
c_n = \mu^n n^{11/32} \left [ A + \sum_{i=2}^k a_i/n^{i/2} \right ]+
(-1)^{n}\mu^n n^{-3/2} \sum_{i=0}^k b_i/n^i,$$
where the first sum starts at $i=2$ because the leading correction is analytic. The resulting estimates for the leading amplitude $A$ are plotted in the top left panel of figure \[fig:sawampl\]. Clearly these amplitude estimates are not well behaved so we better not ignore the complex pair of singularities! Therefore we try again with the asymptotic form
![\[fig:sawampl\] Plots of fits for the self-avoiding walk amplitude $A$ using different asymptotic forms. In the top left panel we show plots using the form () which ignores the complex singularity. In the top right panel we include the complex singularity via the form () while the bottom left panel includes the complex singularity via the alternative form (). The bottom right panel is a more detailed look at this latter case. ](hcsawampl.eps)
$$\label{eq:Acomp}
c_n = \mu^n n^{11/32} \left [ A + \sum_{i= 2}^k a_i/n^{i/2} \right ]+
(-1)^{n}\mu^n n^{-3/2} \sum_{i=0}^k b_i/n^i+
(-1)^{\lfloor n/2 \rfloor}\mu_-^n n^{-3/2} \sum_{i=0}^k c_i/n^i.$$
The estimates for the leading amplitude $A$ are plotted in the top right panel in figure \[fig:sawampl\]. These amplitude estimates are not well behaved either and something is not quite right. Next we try to change the way we included the complex pair of singularities. In (\[eq:Acomp\]) we have assumed that all terms arising from the complex singularity have exactly the same sign-pattern. However, if we assume that the analytic correction terms arise from a functional form such as $D(x)(1+\mu_-^2 x^2)$, where $D(x)$ is a analytic, then the analytic correction terms would actually have a [*shifted*]{} sign-pattern. We therefore try fitting to the slightly modified asymptotic form
$$\label{eq:Afinal}
c_n = \mu^n n^{11/32} \left [ A + \sum_{i= 2}^k a_i/n^{i/2} \right ]+
(-1)^{n}\mu^n n^{-3/2} \sum_{i=0}^k b_i/n^i+
\mu_-^n n^{-3/2} \sum_{i=0}^k (-1)^{\lfloor (n+i)/2\rfloor }c_i/n^i,$$
where it should be noted that all we have done is change the way we include the terms from the complex singularities so as to shift the sign-pattern by a unit as $i$ is increased. The new estimates for the leading amplitude are plotted in the bottom left panel of figure \[fig:sawampl\] and quite clearly the convergence is now very much improved. In the bottom right panel of figure \[fig:sawampl\] we show a much more detailed look at the data and from this plot we can estimate that $ A=1.1449355(5)$.
The correction-to-scaling exponent \[sec:corr\]
-----------------------------------------------
In this section we shall briefly show how the method of direct fitting can be used to differentiate between various possible values for the leading correction-to-scaling exponent $\Delta_1$ (recall the two theoretical prediction $\Delta_1=3/2$ by Nienhuis and $\Delta_1=11/16$ by Saleur). As already stated there is now firm evidence from previous work that the Nienhuis result is correct. Here we shall present further evidence. Different values for $\Delta_1$ leads to different assumed asymptotic forms for the coefficients. For the SAP series we argued that a value $\Delta_1=3/2$ (or indeed any half-integer value) would result only in [*analytic*]{} corrections to the generating function and thus that $p_n$ asymptotically would be given by (\[eq:Bneg\]). If on the other hand we have a generic value for $\Delta_1$ we would get $$\label{eq:Bcoor}
p_n = \mu^n n^{-5/2} \left [ B+ \sum_{i= 1}^k a_i/n^i+ \sum_{i= 0}^k b_i/n^{\Delta_1+i} \right ]+
(-1)^{n/2}\mu_-^n n^{-5/2} \sum_{i=0}^k c_i/n^i.$$ Fitting to this form we can then estimate the amplitude $b_0$ of the term $1/n^{\Delta_1}$. We would expect that if we used a manifestly incorrect value for $\Delta_1$ then $b_0$ should vanish asymptotically thus demonstrating that this term is really absent from (\[eq:Bcoor\]). So we tried fitting to this form using the value $\Delta_1=11/16$. More precisely we fit to the generic form $$\label{eq:Bcoorfit}
p_n = \mu^n n^{-5/2}\sum_{i=0}^k a_i/n^{\alpha_i}+ (-1)^{n/2}\mu_-^n n^{-5/2} \sum_{i=0}^k b_i/n^i.$$ In the first instance we include only the leading term arising from $\Delta_1$, that is we use the sequence of exponents $\alpha_i=\{0, 11/16, 1, 2, 3, \ldots \}$. We also fit to a form in which the additional analytical corrections arising from $\Delta_1$ are included leading to the sequence of exponents $\alpha_i=\{ 0, 11/16, 1, 27/16, 2, 33/16, 3, 49/16, \ldots\}$. As stated in Section \[sec:th\] more generally one would also expect terms of the form $1/n^{m\Delta_1+i}$ with $m$ a non-negative integer. This leads to fits to the form given above but with $\alpha_i = \{0,0.6875, 1, 1.375, 1.6875, 2, 2.0625, 2.375, 2.6875, 2.75, 3 \ldots\}$. The estimates of the amplitude of the term $1/n^{\Delta_1}$ as obtained from fits to these forms are shown in figure \[fig:sapcorr\]. As can be seen from the left panel, where we fit to the first case scenario, the amplitude clearly seems to converge to 0, which would indicate the absence of this term in the asymptotic expansion for $p_n$. In the middle and right panels we show the results from fits to the more general forms. Again the estimates are consistent with the amplitude being 0. Though in this case the evidence is not quite as convincing. This is however not really surprising given that the incorrect value $\Delta_1=11/16$ gives rise to a plethora of absent terms which will tend to greatly obscure the true asymptotic behaviour.
![\[fig:sapcorr\] Plots of estimates for the amplitude of the term $1/n^{\Delta_1}$. The left panel shows results from fits to the form () where only the leading order term $1/n^{\Delta_1}$ is included (as well as analytical corrections). In the middle panel additional terms of the form $1/n^{\Delta_1+i}$ are included and in the right panel terms like $1/n^{m\Delta_1+i}$ are included. ](hcsapcorr.eps)
Amplitude ratios $D/C$ and $E/C$
--------------------------------
From fits to the coefficients in the metric series we find $AC=1.0141(1)$, $AD=0.14225(5)$ and $AE=0.4458(1)$ and thus the ratios are
$$D/C=0.14027(6) \mbox{\hspace{10mm} and\hspace{10mm}} E/C=0.43960(15)$$ $D/C$ and $E/C$ can also be estimated directly from the relevant quotient sequence, e.g. $r_n = {\langle R^2_g \rangle}_n/{\langle R^2_e \rangle}_n$, using the following method due to Owczarek [*et al.*]{} [@OPBG]: Given a sequence of the form $g_n \sim g_{\infty}(1 + b/n + \ldots)$, we construct a new sequence $\{h_n\}$ defined by $h_n = \prod_{m=1}^n g_m$. The associated generating function then has the behaviour $\sum h_n x^n \sim (1 - g_{\infty} x)^{-(1+b)}$, and we can now estimate $g_{\infty}$ form a differential approximant analysis. In this way, we obtained the estimates $$D/C = 0.1403001(2) \mbox{\hspace{10mm} and\hspace{10mm}} E/C = 0.439635(1)$$ These amplitude estimates leads to a high precision confirmation of the CSCPS relation $H=0.000003(13)$.
Amplitude combination $BF$
--------------------------
Next we study the asymptotic form of the coefficients $r_n=8n^2 p_n {\langle R^2 \rangle}_n$ for the radius of gyration. The generating function has critical exponent $-(\alpha+2\nu)=-2$, so the leading correction-to-scaling term no longer becomes part of the analytic background term. We thus use the following asymptotic form:
$$\label{eq:rgsasymp}
r_n \sim \mu^n n \left [ 8BF + \sum_{i\geq 0} a_i/n^{1+i/2} \right ] +
(-1)^{n/2}\mu_-^n n \sum_{i=0}^k b_i/n^i.$$
In figure \[fig:saprg\] we plot the resulting estimates for the amplitude $8BF$. The predicted exact value [@CG93] is $BF=\frac{5}{32\pi^2}\sigma a_0=\frac{15\sqrt{3}}{64\pi^2}=0.0411312745\ldots$, where for the honeycomb lattice $\sigma=2$ and $a_0=3\sqrt{3}/4$. Clearly extrapolation of these numerical results yield estimates consistent with the theoretical prediction.
![\[fig:saprg\] Plots of the estimates for the amplitude combination $8BF$. ](hcsaprg.eps)
Amplitude ratios of area-weighted moments
-----------------------------------------
The amplitudes of the area-weighted moments were studied in [@RJG03]. We fitted the coefficients to the assumed form $$\label{eq:momampl}
n p_n {\langle a^m \rangle}_n \sim \mu^n n^{2m\nu+\alpha-2} m!
\left[ G_m+\sum_{i\ge 0}^k a_i/n^{1+i/2} \right] +
(-1)^{n/2}\mu_-^n n^{2m\nu+\alpha-2} \sum_{i=0}^k b_i/n^i,$$ where the amplitude $G_m=G^{(m)}B/m!$ is related to the amplitude defined in equation (\[eq:asympmom\]). The scaling function prediction for the amplitudes $G_m$ is [@RGJ01] $$G_{2m}B^{2m-1} = -\frac{c_{2m}}{4\pi^{3m}} \frac{(3m-2)!}{(6m-3)!}, \qquad
G_{2m+1}B^{2m} = \frac{c_{2m+1}}{(3m)!\pi^{3m+1}2^{6m+2}},$$ where the numbers $c_m$ are given by the quadratic recursion $$c_m + (3m-4) c_{m-1} + \frac{1}{2}\sum_{r=1}^{m-1} c_{m-r}c_r=0, \qquad c_0=1.$$ In figure \[fig:momampl\] we have plotted the resulting estimates for some of the amplitude ratios. Clearly the numerical results are fully consistent with the theoretical predictions.
![\[fig:momampl\] Plots of the estimates for some of the amplitude combinations $G_m B^{m-1}$. ](hcsapmom.eps)
Summary and conclusions \[sec:sum\]
===================================
In this paper we have studied series for self-avoiding walks and polygons on the honeycomb lattice, including series for metric properties and moments of the area-distribution for polygons. We used various methods from Tony Guttmann’s tool-kit to analyse the series. The connective constant, critical exponents and many amplitude combinations are known exactly, making it the perfect test-bed for series analysis techniques.
In section \[sec:DA\] we used differential approximants to obtain estimates for the singularities and exponents of the SAP and SAW generating functions. Analysis of the SAP series (section \[sec:DAsap\]) yielded very accurate estimates for the critical point $x_c^2=0.2928932186(5)$ and exponent $2-\alpha=1.5000004(10)$. The estimates agree with the conjectured exact values $x_c^2=1/(2+\sqrt{2})= 0.292893218813\ldots$ and $2-\alpha=3/2$. In addition we found clear evidence of a non-physical singularity on the negative axis at $x=x_-=-0.412305(5)$ with an associated critical exponent $\alpha_-=3/2$. The analysis of the SAW series (section \[sec:DAsaw\]) also yielded estimates consistent with the predictions of the exact values. In this case there was a non-physical singularity at $x=-x_c$ as well as a pair of complex singularities at $x=\pm 0.64215(15) {\rm i}$. So the excellent agreement is particularly impressive in light of the quite complicated functional form of the generating function which has at least three non-physical singularities as well as non-analytical corrections to scaling. So the walk series is obviously much harder to analyse and it is a great testament to the method of differential approximants that the analysis nevertheless yields such accurate estimates.
In section \[sec:Ampl\] we looked closer at the asymptotic form of the coefficients. In particular we obtained accurate estimates for the leading critical amplitudes $A$ and $B$. Our method of analysis consisted in fitting the coefficients to an assumed asymptotic form. In section \[sec:Bampl\] we analysed the SAP series and demonstrated clearly that in fitting to the coefficients we cannot ignore the singularity at $x=x_-$ even though it is exponentially suppressed asymptotically. After inclusion of this term estimates for the leading amplitude $B$ were well behaved when including only analytic corrections and we found $B=1.2719299(1)$. We argued that this behaviour was consistent with a corrections-to-scaling exponent $\Delta_1$ being half-integer valued and in particular consistent with the prediction by Nienhuis that $\Delta_1=3/2$ (in section \[sec:corr\] we showed the absence of a term with $\Delta_1=11/16$). In the analysis of the SAW series we discovered some subtleties regarding the inclusion of the terms arising from the complex pair of singularities. Despite a quite complicated asymptotic form (\[eq:Afinal\]) taking into account all the singularities and the corrections-to-scaling exponent $\Delta_1=3/2$ we could still obtain a quite accurate amplitude estimate $A=1.1449355(5)$. This analysis clearly shows that it is possible to probe quite deeply into the asymptotic behaviour of the series coefficients and in particular to distinguish between different corrections to scaling.
E-mail or WWW retrieval of series {#e-mail-or-www-retrieval-of-series .unnumbered}
=================================
The series for the generating functions studied in this paper can be obtained via e-mail by sending a request to I.Jensen@ms.unimelb.edu.au or via the world wide web on the URL http://www.ms.unimelb.edu.au/ iwan/ by following the instructions.
Acknowledgments {#acknowledgments .unnumbered}
===============
The calculations in this paper would not have been possible without a generous grant of computer time from the Australian Partnership for Advanced Computing (APAC). We also used the computational resources of the Victorian Partnership for Advanced Computing (VPAC). We gratefully acknowledge financial support from the Australian Research Council.
References {#references .unnumbered}
==========
[10]{}
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|
---
abstract:
- 'Soit $f$ une application méromorphe dominante d’une variété Kählérienne compacte. Nous donnons une inégalité pour les exposants de Lyapounov d’une classe de mesures ergodiques de $f$ en utilisant l’entropie métrique et les degrés dynamiques de $f$. Nous en déduisons l’hyperbolicité de certaines mesures.'
- 'Let $f$ be a dominating meromorphic self-map of a compact Kähler manifold. We give an inequality for the Lyapounov exponents of some ergodic measures of $f$ using the metric entropy and the dynamical degrees of $f$. We deduce the hyperbolicity of some measures.'
author:
- Henry de Thélin
title: '[**Sur les exposants de Lyapounov des applications méromorphes**]{}'
---
[**[ ]{}**]{}
Mots-clefs: applications méromorphes, exposants de Lyapounov, entropie.
Classification: 37Fxx, 32H50, 58F15.
[**Introduction**]{} {#introduction .unnumbered}
====================
Soit $(X, \omega)$ une variété Kählérienne compacte de dimension $k$ et $f : X \mapsto X$ une application méromorphe dominante.
Nous désignerons par $C_f$ l’ensemble critique de $f$ et par $I_f$ son ensemble d’indétermination.
L’objet de cet article est de donner des formules générales pour les exposants de Lyapounov des mesures invariantes $\mu$ qui intègrent la fonction $\log d(x ,{\mathcal{A}})$ où $d$ est la distance dans $X$ et ${\mathcal{A}}=C_f \cup
I_f$. Remarquons que lorsqu’une mesure $\mu$ intègre la fonction précédente, elle ne charge pas l’ensemble ${\mathcal{A}}$: on peut donc définir $f_{*}
\mu$ et parler de mesure invariante. Par ailleurs, l’hypothèse d’intégrabilité de $\log d(x ,{\mathcal{A}})$ est vérifiée dès que $\mu$ intègre les fonctions quasi-psh. Les formules dépendront d’une part de l’entropie métrique $h(\mu)$ de $\mu$ et d’autre part des degrés dynamiques $d_q$ de $f$ (voir le paragraphe \[rappel\] pour leur définition).
Dans ce contexte nous avons le
[\[formule\]]{}
Soient $\mu$ une mesure invariante, ergodique telle que $\log d(x
,{\mathcal{A}}) \in L^1(\mu)$ et $\chi_1 \geq \dots \geq \chi_k$ les exposants de Lyapounov de $\mu$ (ils sont bien définis).
Fixons $1 \leq s \leq k$. On définit $l=l(s)$ et $l'=l'(s)$ par les formules suivantes:
$$\chi_1 \geq \dots \geq \chi_{s-l-1} > \chi_{s-l}= \dots = \chi_s =
\dots = \chi_{s+l'} > \chi_{s+l'+1} \geq \dots \geq \chi_k,$$ où $s-l$ est égal à $1$ si $\chi_1= \dots = \chi_s$ et $s+l'$ est égal à $k$ lorsque $\chi_s = \dots = \chi_k$.
Alors, on a les inégalités suivantes:
$$h(\mu) \leq \max_{0 \leq q \leq s-l-1} \log d_q +2 \chi_{s-l}^{+} +
\dots + 2 \chi_{k}^{+}$$
$$h(\mu) \leq \max_{s+l' \leq q \leq k} \log d_q -2 \chi_1^{-} - \dots -
2 \chi_{s+l'}^{-}$$
avec $\chi_i^{+}= \max ( \chi_i, 0)$ et $\chi_i^{-}= \min (\chi_i,0)$.
Signalons la ressemblance entre ces formules et celles de J. Buzzi pour les applications $C^{1 + \alpha}$ (voir [@Bu]).
Maintenant, à l’aide de notre théorème, on a:
[\[cor1\]]{}
Supposons que les degrés dynamiques vérifient $d_1 \leq
\dots \leq d_{s-1} < d_s > d_{s+1} \geq \dots \geq d_k$. Soit $\mu$ une mesure invariante, ergodique telle que $\log d(x
,{\mathcal{A}}) \in L^1(\mu)$ et $h(\mu) > \max ( \log d_{s-1}, \log
d_{s+1})$ (ou $h(\mu) > \log d_{k-1}$ si $s=k$). Alors $$\chi_1 \geq \dots \geq \chi_s \geq \frac{1}{2}(h(\mu) - \log d_{s-1}) > 0$$ et $$0 > \frac{1}{2} ( \log d_{s+1} - h(\mu)) \geq \chi_{s+1} \geq \dots \geq \chi_k.$$
En particulier la mesure $\mu$ est hyperbolique.
Lorsque dans le corollaire précédent la mesure $\mu$ est d’entropie $\log d_{s}$ (i.e. est d’entropie maximale par [@DS1] et [@DS2]), on obtient le:
[\[cor2\]]{}
Soit $\mu$ une mesure invariante, ergodique telle que $\log d(x
,{\mathcal{A}}) \in L^1(\mu)$. Alors, si $h(\mu) = \log d_s$ avec $d_1 \leq
\dots \leq d_{s-1} < d_s > d_{s+1} \geq \dots \geq d_k$ on a: $$\chi_1 \geq \dots \geq \chi_s \geq \frac{1}{2} \log
\frac{d_s}{d_{s-1}} > 0$$ et $$0 > \frac{1}{2} \log
\frac{d_{s+1}}{d_{s}} \geq \chi_{s+1} \geq \dots \geq \chi_k.$$ En particulier la mesure $\mu$ est hyperbolique.
Notons que les inégalités obtenues dans ce corollaire sont celles qui étaient conjecturées (voir [@Gu2] Conjecture 3.2).
Les hypothèses de ce corollaire sont vérifiées dans de nombreuses situations. En voici certaines.
Tout d’abord pour les endomorphismes holomorphes de ${\mathbb{P}}^k$ avec la mesure de Green $\mu$ (voir [@FS] et [@FS1] pour sa définition). En effet, la mesure $\mu$ est mélangeante et intègre $\log d(x,{\mathcal{A}})$ grâce à l’inégalité de Chern-Levine-Nirenberg. Pour ces endomorphismes, l’entropie métrique de $\mu$ vaut $k \log(d)$ et les $d_q$ valent $d^q$. En appliquant notre inégalité avec $s=k$, on a alors la minoration du plus petit exposant de $\mu$ par $\frac{\log(d)}{2}$ et on retrouve ainsi un résultat de J.-Y. Briend et J. Duval (voir [@BD1]).
De la même façon, lorsque $f$ est une application méromorphe sur une variété projective, avec son degré topologique strictement plus grand que les autres (i.e. $s=k$ dans le corollaire précédent), V. Guedj a construit une mesure $\mu$ mélangeante, d’entropie $\log d_k$ et qui intègre les fonctions quasi-psh (voir [@Gu] et [@DS1]). En utilisant notre formule, on retrouve alors la minoration du plus petit exposant de Lyapounov de $\mu$ par $\frac{1}{2} \log(d_k / d_{k-1})$ qu’il avait démontrée.
Lorsque $f$ est un automorphisme holomorphe d’une variété Kählérienne qui possède un degré dynamique strictement plus grand que les autres, T.-C. Dinh et N. Sibony ont construit une mesure de Green $\mu$ mélangeante qui intègre les fonctions quasi-psh et d’entropie $\max \log d_q$ (voir [@DS3]). Notre corollaire s’applique donc et on obtient ainsi un nouveau résultat pour ces automorphismes.
De la même façon, lorsque $f$ est une application birationnelle régulière de ${\mathbb{P}}^k$ (voir [@DS4]), T.-C. Dinh et N. Sibony ont construit une mesure de Green $\mu$ mélangeante et qui intègre la fonction $\log d(x,{\mathcal{A}})$ (ici $I_f=I^{+}$ et $C_f= f^{-1}(I^{-})$). On peut donc lui appliquer le corollaire.
Donnons une autre conséquence de notre théorème. Lorsque l’on applique la première formule du théorème avec $s=1$ on en déduit une formule de Ruelle ([@Ru]) pour les applications méromorphes:
Soit $\mu$ une mesure invariante, ergodique telle que $\log d(x
,{\mathcal{A}}) \in L^1(\mu)$. Alors:
$$h(\mu) \leq 2 \chi_1^{+} + \dots + 2 \chi_k^{+}.$$
Voici le plan de ce texte. Dans un premier paragraphe nous ferons des rappels sur les applications méromorphes et dans le second nous démontrerons les corollaires \[cor1\] et \[cor2\]. Dans le troisième nous parlerons de théorie de Pesin pour les applications méromorphes et dans le quatrième, nous ferons des rappels sur la transformée de graphe. Enfin, le cinquième paragraphe sera consacré à la démonstration de la première inégalité du théorème et le sixième à celle de la deuxième inégalité. Dans le dernier paragraphe nous donnerons un analogue de notre théorème pour les difféomorphismes de classe $C^{1 +
\alpha}$ dans les variétés Riemanniennes compactes.
[**Rappels sur les applications méromorphes**]{}
================================================
[\[rappel\]]{}
Commençons par rappeler la définition des degrés dynamiques de $f$ (voir [@RS] et [@DS2]).
On définit la forme $f^{*} (\omega^q)$ comme l’extension triviale de $(f_{|X
\setminus I_f})^{*} \omega^q$.
On pose $\delta_q(f):=
\int_X f^{*}( \omega^q) \wedge \omega^{k-q}$ pour $q=0, \dots, k$. Le degré dynamique d’ordre $q$ est alors: $$d_q:= \lim_{n \rightarrow \infty} ( \delta_q(f^n))^{1/n}.$$
Notons que $d_0=1$ et que la limite ci-dessus existe grâce à [@DS2].
Dans [@DS1] et [@DS2], T.-C. Dinh et N. Sibony ont défini, via un procédé de régularisation de courant, le pull-back par $f$ des courants positifs fermés de bidimension quelconque. En voici le procédé. Tout d’abord $f$ est une submersion sur un ouvert de Zariski $\Omega_{1,f}$ de $X$. Si $S$ est un courant positif fermé de bidegré $(l,l)$ sur $X$, alors, on peut définir $f^{*}S$ sur $\Omega_{1,f}$. C’est un courant positif fermé dont la masse $\int_{\Omega_{1,f}} f^{*}S \wedge
\omega^{k-l}$ est majorée par $c_X
\delta_l(f) \| S \|$ (voir le lemme 4 de [@DS1] et le corollaire 1.3 de [@DS2], ici $c_X$ est une constante qui ne dépend que de $X$). Ce courant admet donc un prolongement trivial $\widetilde{f^{*}S}$ à $X$ tout entier d’après un théorème de H. Skoda ([@S]). De plus $\widetilde{f^{*}S}$ est un courant positif fermé de masse majorée par $c_X \delta_l(f) \| S \|$. Ce courant sera appelé le pull-back de $S$ par $f$.
Nous allons maintenant donner deux lemmes qui serons utilisés dans la démonstration du théorème. Le premier est quasiment le même que le Lemme 5 de [@DS1]. On notera $\Omega_f= X \setminus
\cup_{n \in {\mathbb{Z}}} f^n(I_f)$.
[\[lemme1\]]{}
Soit $\epsilon > 0$. Il existe une constante $c_{\epsilon} > 0$ telle que pour tout $q=0, \dots , k$ on ait:
$$\int_{\Omega_f } \omega^{k-q} \wedge (f^{n_1})^{*} \omega \wedge
\dots \wedge (f^{n_q})^{*} \omega \leq c_{\epsilon} ( \max_{0 \leq j
\leq q }d_j + \epsilon)^{n_1}$$ pour tous les entiers naturels $n_1 \geq \dots \geq n_q \geq 0$.
La démonstration est la même que dans [@DS1]. Nous la donnons par confort pour le lecteur.
Soit $c > 0$ une constante telle que $\delta_j(f^n) \leq c (d_{j} +
\epsilon)^n$ pour tout $n \geq 0$ et tout $j=0, \dots , k$.
Soit $\Omega_{n,f} = X \setminus \cup_{0 \leq i \leq n-1} f^{-i}(C)$ où $C= X \setminus \Omega_{1,f}$. On va montrer par récurrence sur $q$, avec $0 \leq q \leq k$ que pour tous $n_1 \geq
\dots \geq n_q \geq 0$, on a $\| T_q \|= \int_{\Omega_{n_1 ,f}} T_q
\wedge \omega^{k-q} \leq c^q c_{X}^q (\max_{0 \leq j \leq q} d_j +
\epsilon)^{n_1}$ où $$T_q=(f^{n_1})^{*} \omega \wedge \dots \wedge (f^{n_q})^{*} \omega
,$$ et $T_0=1$.
C’est vrai pour $q=0$. Supposons la propriété vraie au rang $q-1$. Cela implique que $\| T_{q-1}' \| \leq c^{q-1} c_{X}^{q-1} (\max_{0 \leq j
\leq q-1} d_{j} +
\epsilon)^{n_1-n_q}$ avec $$T_{q-1}'=(f^{n_1-n_q})^{*} \omega \wedge \dots \wedge (f^{n_{q-1} -
n_q})^{*} \omega .$$ Le courant $T_{q-1}'$ est donc de masse finie sur $\Omega_{n_1-n_q,f}$. Il admet donc une extension triviale $\widetilde{T_{q-1}'}$ dans $X$ qui est un courant positif fermé de masse majorée par $c^{q-1} c_{X}^{q-1} (\max_{0 \leq j \leq q-1} d_{j} +
\epsilon)^{n_1-n_q}$. En utilisant la propriété du pull-back de T.-C. Dinh et N. Sibony énoncée avant le lemme avec $S=\widetilde{T_{q-1}'} \wedge \omega$, on obtient: $$\| T_q \|= \| (f^{n_q})^{*}(T_{q-1}' \wedge \omega) \| \leq c_X
\delta_{q}(f^{n_q}) \| T_{q-1}' \| \leq c^q c_{X}^q (\max_{1
\leq j \leq q} d_j + \epsilon)^{n_1}.$$ Cela démontre bien le lemme.
On utilisera aussi le lemme suivant:
[\[lemme2\]]{}
Soit $\epsilon > 0$. Il existe une constante $c_{\epsilon} > 0$ telle que pour tout $q=0 , \dots , k$ on ait:
$$\int_{\Omega_f } (f^n)^{*} \omega^{q} \wedge (f^{n_1})^{*} \omega \wedge
\dots \wedge (f^{n_{k-q}})^{*} \omega \leq c_{\epsilon} ( \max_{q \leq j
\leq k }d_j + \epsilon)^{n}$$ pour tous les entiers naturels $n \geq n_1 \geq \dots \geq n_{k-q} \geq 0$.
Soit $c > 0$ une constante telle que $\delta_j(f^n) \leq c (d_{j} +
\epsilon)^n$ pour tout $n \geq 0$ et tout $j=0, \dots , k$. Fixons $q$ entre $0$ et $k$. On peut supposer $q >0$ sinon on a le résultat par le lemme précédent.
On va montrer par récurrence sur $j$, avec $0 \leq j \leq k-q$ que pour tous $n \geq n_1 \geq
\dots \geq n_j \geq 0$, on a $\| T_j \|= \int_{\Omega_{n ,f}} T_j
\wedge \omega^{k-j-q} \leq c^{j+1} c_{X}^j (\max_{q \leq l \leq q+j} d_l +
\epsilon)^{n}$ où $$T_j=(f^n)^{*} \omega^q \wedge (f^{n_1})^{*} \omega \wedge \dots \wedge (f^{n_j})^{*} \omega
,$$ et $T_0= (f^n)^{*} \omega^q$.
C’est vrai pour $j=0$ par définition du $q$-ème degré dynamique. Supposons la propriété vraie au rang $j-1$. Cela implique que $\| T_{j-1}' \| \leq c^{j} c_{X}^{j-1} (\max_{q \leq l \leq q+j-1} d_l +
\epsilon)^{n-n_j}$ où $$T_{j-1}'= (f^{n-n_j})^{*} \omega^q \wedge (f^{n_1-n_j})^{*} \omega
\wedge \dots \wedge (f^{n_{j-1}-n_j})^{*} \omega.$$ Le courant $T_{j-1}'$ est donc de masse finie sur $\Omega_{n-n_j,f}$. Il admet donc une extension triviale $\widetilde{T_{j-1}'}$ dans $X$ qui est un courant positif fermé de masse majorée par $c^{j} c_{X}^{j-1} (\max_{q \leq l \leq q+j-1} d_l +
\epsilon)^{n-n_j}$. En utilisant encore la propriété du pull-back de T.-C. Dinh et N. Sibony avec $S=\widetilde{T_{j-1}'} \wedge \omega$, on obtient:
$$\| T_j \| = \| (f^{n_j})^{*}(T_{j-1}' \wedge \omega) \| \leq c_{X}
\delta_{q+j}(f^{n_j}) \|T_{j-1}'\| \leq c^{j+1} c_X^j (\max_{q \leq l \leq q+j} d_l +
\epsilon)^{n}.$$
Cela démontre la récurrence et quand on prend $j= k-q$, on obtient le lemme.
[**Démonstration des corollaires \[cor1\] et \[cor2\]**]{}
==========================================================
On suppose ici que les degrés dynamiques de $f$ vérifient $d_1 \leq
\dots \leq d_{s-1} < d_s > d_{s+1} \geq \dots \geq d_k$. Soit $\mu$ une mesure invariante, ergodique telle que $\log d(x
,{\mathcal{A}}) \in L^1(\mu)$ et $h(\mu) > \max ( \log d_{s-1}, \log
d_{s+1})$ (ou $h(\mu) > \log d_{k-1}$ si $s=k$). Montrons par l’absurde que $\chi_s >0$ et $\chi_{s+1} <0$.
Si $\chi_s \leq 0$ alors $\chi_{s-l}^{+}= \dots = \chi_s^{+}= \dots =
\chi_k^{+}=0$ et la première formule du théorème donnerait $\log
d_{s-1} < h(\mu) \leq \log d_{s-l-1}$ qui est absurde. De même si $\chi_{s+1} \geq 0$, on applique la deuxième formule avec $s=s+1$ et on obtient $\log
d_{s+1} < h(\mu) \leq
\log d_{s+1+l'}$ qui est une contradiction (ici $\chi_1^{-}=
\dots = \chi_{s+1}^{-}= \dots = \chi_{s+1+l'}^{-}=0$).
Passons à la minoration de $\chi_s$. Par la première formule du théorème et on a: $$h(\mu) \leq \log d_{s-l-1} + 2(l+1) \chi_s$$ car par ce que l’on a fait précédemment on a $\chi_{s+1}^{+}= \dots = \chi_k^{+} =0$ et $\chi_s^{+}=
\chi_s$. On obtient $\chi_s \geq \frac{1}{2} \left(
\frac{1}{l+1} h(\mu)- \frac{1}{l+1} \log d_{s-l-1} \right)$. La concavité de la fonction $q \rightarrow \log d_q$ (voir [@Gu] et [@Gr1]) implique $ \log d_{s-1} \geq \frac{1}{l+1} \log d_{s-l-1} + (1-
\frac{1}{l+1}) \log d_s$. On a donc $$\chi_s \geq \frac{1}{2}
\left(\frac{1}{l+1} h(\mu) - \frac{1}{l+1} \log d_s - \log d_{s-1} +
\log d_s \right).$$ Mais comme $h(\mu) \leq \log d_{s}$ (voir [@DS1] et [@DS2]), cette dernière quantité est supérieure à $\frac{1}{2} ( h(\mu) - \log d_s - \log d_{s-1} +
\log d_s)$, ce qui nous donne la minoration de $\chi_s$ que l’on cherche.
Pour la majoration de $\chi_{s+1}$ la méthode est exactement la même à condition d’utiliser la deuxième formule avec $s=s+1$.
[**Théorie de Pesin et applications**]{}
========================================
[\[Pesin\]]{}
Dans ce paragraphe, on considère une mesure de probabilité invariante, ergodique $\mu$ telle que $\log d(x, {\mathcal{A}}) \in L^1(\mu)$ (avec ${\mathcal{A}}=C_f \cup I_f$). On va voir que cette hypothèse permet de définir les exposants de Lyapounov pour $\mu$ et de faire de la théorie de Pesin.
Tout d’abord, on définit l’extension naturelle $\widehat{X}$ de $X$ par: $$\widehat{X}:= \{ \widehat{x}=( \dots, x_{-n} , \dots, x_0, \dots ,
x_n , \dots) \in
X^{{\mathbb{Z}}} \mbox{ , } f(x_{-n})=x_{-n+1} \}.$$ C’est l’ensemble des histoires des points de $X$. Dans cet espace $f$ induit une application $\widehat{f}$ qui est le décalage à droite et si on note $\pi$ la projection canonique $\pi(\widehat{x})=x_0$, alors $\mu$ se relève en une unique probabilité $\widehat{\mu}$ invariante par $\widehat{f}$ qui vérifie $\pi_{*}
\widehat{\mu}= \mu$.
Dans l’espace $\widehat{X}$, on ne gardera que les orbites qui ne visitent pas l’ensemble ${\mathcal{A}}$. On considère donc:
$$\widehat{X}^{*}= \{ \widehat{x} \in \widehat{X} \mbox{ , } x_n
\notin {\mathcal{A}}\mbox{ , } \forall n \in {\mathbb{Z}}\}.$$
Cet ensemble est invariant par $\widehat{f}$ et $\widehat{\mu}(\widehat{X}^{*})=1$ car $\mu({\mathcal{A}})=0$.
Maintenant, on peut munir $X$ d’une famille de cartes $(\tau_x)_{x \in
X}$ telles que $\tau_x(0)=x$, $\tau_x$ est définie sur une boule $B(0, \epsilon_0)$ avec $\epsilon_0$ indépendant de $x$ et la norme de la dérivée première et seconde de $\tau_x$ sur $B(0, \epsilon_0)$ est majorée par une constante indépendante de $x$. Pour construire ces cartes il suffit de partir d’une famille finie $(U_i, \psi_i)$ de cartes de $X$ et de les composer par des translations.
Dans toute la suite, on notera $f_x= \tau_{f(x)}^{-1} \circ f \circ
\tau_x$ qui est définie au voisinage de $0$ quand $x$ n’est pas dans $I_f$ et on posera aussi: $$f_x^n= f_{f^{n-1}(x)} \circ \dots \circ f_x$$ $$f_{\widehat{x}}^{-n}= f_{x_{-n}}^{-1} \circ \dots \circ f_{x_{-1}}^{-1}$$ où dans $f_{x_{-i}}^{-1}$ on a pris la bonne branche inverse de $f$ par rapport à $\widehat{x}$: celle qui envoie $x_{-i+1}$ sur $x_{-i}$.
Pour $\widehat{x} \in \widehat{X}^{*}$ on définit $D \widehat{f} (
\widehat{x})=Df_x(0)$ (où $\pi( \widehat{x})=x$). L’application $D
\widehat{f}$ va de $\widehat{X}^{*}$ dans $GL(k, {\mathbb{C}})$ et c’est à ce cocycle que nous allons appliquer la théorie de Pesin. Tout d’abord, nous avons le:
Les fonctions $\log^{+} \| D \widehat{f} ( \widehat{x}) \|$ et $\log^{+} \|( D \widehat{f} ( \widehat{x}))^{-1} \|$ sont dans $L^1(\widehat{\mu})$.
Commençons par montrer que $\log^{+} \| D \widehat{f} ( \widehat{x})
\|$ est dans $L^1(\widehat{\mu})$.
Si on applique le lemme 2.1 de l’article [@DiDu] de T.-C. Dinh et C. Dupont à $f$, on obtient:
Il existe $\tau > 0$ et $p \in {\mathbb{N}}^{*}$ tels que pour tout $x$ hors de $I_f$: $$\| Df(x) \| + \| D^2 f (x) \| \leq \tau d(x, I_f)^{-p}.$$
On en déduit: $$\log \| Df_x(0) \| \leq \log \tau + \log d(x, {\mathcal{A}})^{-p}$$ d’où $$\log^{+} \|Df_x(0) \| \leq \log \tau +\log d(x, {\mathcal{A}})^{-p}$$ car on peut supposer $\tau > 1$ et que le diamètre de $X$ est inférieur à $1$. Mais $$\int \log^{+} \| D \widehat{f} ( \widehat{x}) \| d
\widehat{\mu}(\widehat{x})= \int \log^{+} \| D f_{\pi(\widehat{x})}
(0) \| d \widehat{\mu}(\widehat{x})$$ qui est égal à $$\int \log^{+} \| D f_{x}(0) \| d (\pi_{*} \widehat{\mu})(\widehat{x})=\int \log^{+} \| D
f_{x}(0) \| d \mu(x).$$ Comme par hypothèse sur $\mu$, la fonction $\log d(x, {\mathcal{A}})$ est dans $L^1( \mu)$, on en déduit que la dernière intégrale ci-dessus est finie. Autrement dit, la fonction $\log^{+} \| D \widehat{f} ( \widehat{x}) \|$ est bien dans $L^1( \widehat{\mu})$.
Passons maintenant à $\log^{+} \|( D \widehat{f} ( \widehat{x}))^{-1}
\|$. Comme dans le lemme 2.1 de [@DiDu], nous allons utiliser l’inégalité de Lojasiewicz (voir [@Lo], IV.7.2 ). En effet cette inégalité nous donne:
Il existe $\tau' > 0$ et $p' \in {\mathbb{N}}^{*}$ tels que pour tout $x$ hors de $C_f$: $$| \mbox{Jac} f(x) |^2 \geq \tau' d(x, C_f)^{p'}.$$ Ensuite, grâce à l’estimée du lemme 2.1 de [@DiDu] donnée plus haut, on sait que les modules des valeurs propres de $Df(x)^H Df(x)$ sont majorés par $\tau^2 d(x,I_f)^{-2p}$ (ici la matrice $Df(x)^H$ est la transposée-conjuguée de $Df(x)$). Le minimum des modules des valeurs propres de la matrice $Df(x)^H Df(x)$ est donc minoré par $\frac{\tau'}{\tau^{2(k-1)}} d(x,C_f)^{p'} d(x,I_f)^{2p(k-1)}$. Cela implique que $\| (D f(x))^{-1} \|$ est majorée par $\tau'' d(x, C_f \cup
I_f)^{-p''}$ pour certains $\tau'' > 0$ et $p'' \in {\mathbb{N}}^{*}$.
On a donc: $$\log \| (Df_x(0))^{-1} \| \leq \log \tau'' + \log d(x, {\mathcal{A}})^{-p''}$$
qui est une fonction intégrable pour $\mu$.
Comme précédemment on en déduit que $\log^{+} \|( D \widehat{f} (
\widehat{x}))^{-1} \| $ est dans $L^1( \widehat{\mu})$.
Grâce à ce lemme on peut appliquer le théorème d’Oseledec et le théorie de Pesin au cocycle $D \widehat{f}$. On a (voir [@KH]):
(Oseledec)
Il existe un ensemble invariant $\widehat{Y} \subset \widehat{X}^{*}$ avec $\widehat{\mu}(\widehat{Y})=1$ tel que pour tout $\widehat{x} \in
\widehat{Y}$ on ait:
1\) L’existence d’une décomposition de ${\mathbb{C}}^k$: $${\mathbb{C}}^k= \oplus_{i=1}^{q} E_i(\widehat{x}),$$ où les $E_i(\widehat{x})$ vérifient $D \widehat{f}(\widehat{x})
E_i(\widehat{x})=E_i( \widehat{f}(\widehat{x}))$.
2\) L’existence de fonctions (exposants de Lyapounov de $f$): $$\lambda_1 > \dots > \lambda_q,$$ avec $$\lim_{m \rightarrow \pm \infty} \frac{1}{|m|} \log \|
A(\widehat{x},m)v \|= \pm \lambda_i$$ pour tout $v \in E_i(\widehat{x}) \setminus \{ 0 \}$. Ici les $A(\widehat{x},m)$ sont definis par: $$A(\widehat{x},m)= D \widehat{f}(\widehat{f}^{m-1}(\widehat{x})) \circ
\dots \circ D \widehat{f}(\widehat{x}) \mbox{ pour } m>0$$ $$A(\widehat{x},0)=Id$$ $$A(\widehat{x},m)=D \widehat{f}(\widehat{f}^{m}(\widehat{x}))^{-1}
\circ \dots \circ D \widehat{f}(\widehat{f}^{-1}(\widehat{x}))^{-1} \mbox{ pour } m<0.$$
De plus, nous avons le:
($\gamma$-réduction de Pesin)
Pour tout $\gamma >0$ il existe une fonction $C_{\gamma}: \widehat{X}^{*}
\rightarrow GL(k, {\mathbb{C}})$ telle que:
1\) $\lim_{m \rightarrow \infty} \frac{1}{m} \log \| C_{\gamma}^{\pm 1
}(\widehat{f}^m(\widehat{x})) \|=0$ (on parle de fonction tempérée).
2\) Pour presque tout $\widehat{x}$, la matrice $A_{\gamma}(\widehat{x})=C_{\gamma}^{-1}(\widehat{f}(\widehat{x})) D
\widehat{f}( \widehat{x}) C_{\gamma}(\widehat{x})$ a la forme suivante:
$$A_{\gamma}(\widehat{x})= \left(
\begin{array}{cccc}
A_{\gamma}^1(\widehat{x}) \\
& \ddots & \\
& & A_{\gamma}^{q}(\widehat{x})\\
\end{array}
\right)$$ où chaque $A_{\gamma}^i(\widehat{x})$ est une matrice carrée de taille $dim E_i(\widehat{x}) \times dim E_i(\widehat{x})$ et on a: $$e^{\lambda_i- \gamma} \leq \| A_{\gamma}^i(
\widehat{x})^{-1} \| ^{-1} \mbox{ et } \| A_{\gamma}^i(
\widehat{x}) \| \leq e^{\lambda_i+ \gamma}.$$
3\) Enfin, pour presque tout $\widehat{x}$ l’application $C_{\gamma}(\widehat{x})$ envoie la décomposition standard $\oplus_{i=1}^{q}
{\mathbb{C}}^{dim E_i(\widehat{x})}$ sur $\oplus_{i=1}^{q} E_i(\widehat{x})$.
Dans toute la suite nous noterons $\widehat{Y}$ l’ensemble des points de $\widehat{X}^{*}$ qui vérifient les conclusions de ces deux théorèmes. De plus, pour $\widehat{x} \in \widehat{Y}$, nous appellerons $\chi_1 \geq \dots \geq \chi_k$ les exposants de Lyapounov de $\widehat{\mu}$ notés avec répétition (contrairement aux $\lambda_i$ du théorème précédent).
Notons maintenant $g_{\widehat{x}}$ la lecture de $f_x$ dans ces cartes (i.e. $g_{\widehat{x}}=C_{\gamma}^{-1}(
\widehat{f}(\widehat{x})) \circ f_x \circ C_{\gamma}(\widehat{x})$ avec $\pi(\widehat{x})=x$).
Nous allons donner quelques propriétés de $g_{\widehat{x}}$ (avec $\widehat{x} \in \widehat{Y}$) qui nous seront utiles pour la suite.
[\[direct\]]{}
Il existe des constantes $\tau > 0$, $\epsilon_0 >0$ et $p \in
{\mathbb{N}}^{*}$ qui ne dépendent que de $f$ et $X$ telles que:
1\) $g_{\widehat{x}}(0)=0$
2\) $Dg_{\widehat{x}}(0)= \left(
\begin{array}{cccc}
A_{\gamma}^1(\widehat{x}) \\
& \ddots & \\
& & A_{\gamma}^{q}(\widehat{x})\\
\end{array}
\right)$
3\) Si on note $g_{\widehat{x}}(w)=Dg_{\widehat{x}}(0)w + h(w)$, on a: $$\| Dh(w) \| \leq \tau \| C_{\gamma}( \widehat{f}(\widehat{x}))^{-1} \| \|
C_{\gamma}(\widehat{x}) \|^{2} d(x, {\mathcal{A}})^{-p} \| w \|$$ pour $\|w\| \leq \epsilon_0 d(x, {\mathcal{A}})/ \|C_{\gamma}(\widehat{x})\|$.
Le premier point est évident. Le second provient immédiatement du théorème précédent. Il reste à prouver la troisième propriété.
On a $Dg_{\widehat{x}}(w)=Dg_{\widehat{x}}(0) + Dh(w)$, d’où: $$\| Dh(w) \| = \| Dg_{\widehat{x}}(w) - Dg_{\widehat{x}}(0) \| \leq
\| C_{\gamma}( \widehat{f}(\widehat{x}))^{-1} \| \| Df_x
(C_{\gamma}(\widehat{x})w) -Df_x(0) \| \|C_{\gamma}(\widehat{x})
\|.$$ Par l’estimée sur $\| D^2 f \|$ de T.-C. Dinh et C. Dupont qui se trouve au début de la démonstration du lemme précédent, on a:
$$\| Df_x (C_{\gamma}(\widehat{x})w) -Df_x(0) \| \leq \tau d( \tau_x
([0, C_{\gamma}(\widehat{x})w]), {\mathcal{A}})^{-p} \|
C_{\gamma}(\widehat{x})w \|$$ pour $\|C_{\gamma}(\widehat{x})w \| \leq \epsilon_0$. L’image par $\tau_x$ du segment $[0, C_{\gamma}(\widehat{x})w]$ vit dans la boule $B(x, K \|C_{\gamma}(\widehat{x})w \|)$ où $K$ est une constante qui ne dépend que de $X$. On en déduit que pour $\|C_{\gamma}(\widehat{x})w \| \leq \epsilon_0 d(x, {\mathcal{A}})$, on a: $$\| Dh(w) \| \leq \| C_{\gamma}( \widehat{f}(\widehat{x}))^{-1} \|
\|C_{\gamma}(\widehat{x}) \|^2 \tau (1-K \epsilon_0)^{-p} d(x, {\mathcal{A}})^{-p}
\|w \|.$$
Quitte à prendre $\epsilon_0$ petit, on peut supposer que $K
\epsilon_0$ est inférieur à $\frac{1}{2}$ et donc modulo un changement de la constante $\tau$ on a la majoration voulue de $\| Dh(w) \|$.
Dans la démonstration des formules du théorème nous utiliserons aussi $g_{\widehat{x}}^{-1}=C_{\gamma}^{-1}(\widehat{f}^{-1}(\widehat{x}))
\circ f_{\widehat{x}}^{-1} \circ C_{\gamma}(\widehat{x})=C_{\gamma}^{-1}(\widehat{f}^{-1}(\widehat{x}))
\circ f_{x_{-1}}^{-1} \circ C_{\gamma}(\widehat{x})$ et des estimées sur cette application qui sont données par la:
[\[inverse\]]{}
Il existe des constantes $\tau > 0$, $\epsilon_0 >0$ et $p \in
{\mathbb{N}}^{*}$ qui ne dépendent que de $f$ et $X$ telles que:
1\) $g_{\widehat{x}}^{-1}(w)$ est bien définie pour $\|w\| \leq \epsilon_0 d(x_{-1}, {\mathcal{A}})^p / \|C_{\gamma}(\widehat{x})\|$.
2\) $g_{\widehat{x}}^{-1}(0)=0$
3\) $Dg_{\widehat{x}}^{-1}(0)= \left(
\begin{array}{cccc}
(A_{\gamma}^1(\widehat{f}^{-1}(\widehat{x})))^{-1} \\
& \ddots & \\
& & (A_{\gamma}^{q}(\widehat{f}^{-1}(\widehat{x})))^{-1}\\
\end{array}
\right)$
4\) Si on note $g_{\widehat{x}}^{-1}(w)=Dg_{\widehat{x}}^{-1}(0)w + h(w)$, on a: $$\| Dh(w) \| \leq \tau \| C_{\gamma}( \widehat{f}^{-1}(\widehat{x}))^{-1} \| \|
C_{\gamma}(\widehat{x}) \|^{2} d(x_{-1}, {\mathcal{A}})^{-p} \| w \|$$ pour $\|w\| \leq \epsilon_0 d(x_{-1}, {\mathcal{A}})^p / \|C_{\gamma}(\widehat{x})\|$.
Commençons par démontrer que $g_{\widehat{x}}^{-1}(w)$ est bien défini pour $$\|w\| \leq \epsilon_0 d(x_{-1}, {\mathcal{A}})^p / \|C_{\gamma}(\widehat{x})\|.$$
Cela va reposer sur le lemme 2 de [@BD1] (construction de branches inverses).
Rappelons que grâce à l’estimée de T.-C. Dinh et C. Dupont qui se trouve au début de la démonstration du lemme précédent on a l’existence de $\tau' > 0$ et $p' \in {\mathbb{N}}^{*}$ tels que pour tout $x$ hors de $I_f$: $$\| Df(x) \| + \| D^2 f (x) \| \leq \tau' d(x, I_f)^{-p'}$$
D’autre part, dans la preuve de ce lemme, on a obtenu aussi l’existence de $\tau'' >0$ et $p'' \in {\mathbb{N}}^{*}$ avec $\| (D f(x))^{-1} \|
\leq \tau'' d(x, C_f \cup I_f)^{-p''}$ pour $x$ hors de ${\mathcal{A}}=C_f \cup I_f$.
Quitte à remplacer $p'$ et $p''$ par le maximum des deux, on pourra supposer dans la suite que $p'=p''$ et de même $\tau'=\tau''$. Ce sont des constantes qui ne dépendent que de $f$ et $X$.
Soit $w$ tel que $\| w \| \leq \epsilon_0' d(x_{-1}, {\mathcal{A}})$. On a $\tau_{x_{-1}}(w) \in B(x_{-1}, K \epsilon_0' d(x_{-1}, {\mathcal{A}}))$ (où $K$ ne dépend que de $X$) et alors: $$\| D f_{x_{-1}}(w) \| + \| D^2 f_{x_{-1}}(w) \| + \| (D
f_{x_{-1}}(w))^{-1} \| \leq 2 \tau' (1-K \epsilon_0')^{-p'} d(x_{-1}, {\mathcal{A}})^{-p'}\leq \tau' d(x_{-1}, {\mathcal{A}})^{-p'},$$ si on prend $\epsilon_0'$ petit et quitte à renommer $\tau'$.
L’inégalité ci-dessus combinée avec le lemme 2 de [@BD1] implique que $f_{\widehat{x}}^{-1}$ est définie sur une boule $B(0, \epsilon_0'
d(x_{-1},{\mathcal{A}})^{3p'}/ {\tau'}^3)=B(0,2 \epsilon_0 d(x_{-1}, {\mathcal{A}})^p)$. En particulier, $g_{\widehat{x}}^{-1}(w)=C_{\gamma}^{-1}(\widehat{f}^{-1}(\widehat{x}))
\circ f_{\widehat{x}}^{-1} \circ C_{\gamma}(\widehat{x})(w)$ est bien défini pour $\|w\| \leq 2 \epsilon_0 d(x_{-1},
{\mathcal{A}})^{p} / \|C_{\gamma}(\widehat{x})\|$.
Passons maintenant à la majoration de $\| D h(w) \|$. Pour cela, comme dans la proposition précédente, il faut contrôler $\| D^2
f_{\widehat{x}}^{-1} \|$.
L’image de $B(0, 2 \epsilon_0 d(x_{-1},{\mathcal{A}})^{p})$ par $f_{\widehat{x}}^{-1}$ est incluse dans $B(0,1)$ (toujours par le lemme 2 de [@BD1]). La formule de Cauchy nous donne donc:
$$\| D f_{\widehat{x}}^{-1}(w) \| + \| D^2 f_{\widehat{x}}^{-1}(w) \|
\leq \tau'' d(x_{-1}, {\mathcal{A}})^{-p''},$$ pour $\|w\| \leq \epsilon_0 d(x_{-1}, {\mathcal{A}})^{p}$.
Grâce à cette majoration, on peut maintenant contrôler $\| D h(w)
\|$. On a: $$\| Dh(w) \| = \| Dg_{\widehat{x}}^{-1}(w)- Dg_{\widehat{x}}^{-1}(0)
\| \leq \| C_{\gamma}( \widehat{f}^{-1}(\widehat{x}))^{-1} \| \|
Df_{\widehat{x}}^{-1}(C_{\gamma}(\widehat{x})w)-
Df_{\widehat{x}}^{-1}(0)\| \|C_{\gamma}(\widehat{x})\|,$$
d’où:
$$\| Dh(w) \| \leq \tau'' \| C_{\gamma}( \widehat{f}^{-1}(\widehat{x}))^{-1} \| \|
C_{\gamma}(\widehat{x}) \|^{2} d(x_{-1}, {\mathcal{A}})^{-p''} \| w \|,$$ pour $\|w\| \leq \epsilon_0 d(x_{-1}, {\mathcal{A}})^p / \|C_{\gamma}(\widehat{x})\|.$
Cela démontre bien la proposition quitte à prendre le maximum entre $p$ et $p''$.
Dans les deux propositions, on voit que la distance de $x$ à l’ensemble ${\mathcal{A}}$ joue un rôle crucial. Comme on les utilisera le long d’orbites de point, on aura besoin de savoir la distance entre $f^i(x)$ et ${\mathcal{A}}$. Celle-ci est donnée par le:
[\[distance\]]{}
Il existe un ensemble $\widehat{Y}$ dans $\widehat{X}$ de mesure pleine pour $\widehat{\mu}$ tel que pour tout $\widehat{x} \in
\widehat{Y}$ on ait: $$d(x_n, {\mathcal{A}}) \geq V(\widehat{x}) e^{- \gamma |n|}$$ pour tout $n \in {\mathbb{Z}}$. Ici $V$ est une fonction mesurable à valeur dans ${\mathbb{R}}^{*}_{+}$.
Il suffit d’appliquer le théorème de Birkhoff à la fonction $u(\widehat{x}) =\log d(\pi(\widehat{x}), {\mathcal{A}})$ qui est dans $L^1(\widehat{\mu})$ (voir par exemple le Lemme 2.3 de [@DiDu]).
Dans toute la suite, $\widehat{Y}$ désignera le sous-ensemble de points de $\widehat{X}^{*}$ qui vérifient les théorèmes d’Oseledec, de $\gamma$-réduction de Pesin et les conclusions du lemme précédent.
Nous allons maintenant faire des rappels sur la transformée de graphe.
[**Transformée de graphe**]{}
=============================
[\[graphe\]]{}
La cadre de ce paragraphe est ${\mathbb{C}}^k$. Dans toute la suite $\|.\|$ désignera la norme Euclidienne.
On considère $$g(X,Y)=(AX + R(X,Y), BY + U(X,Y))$$ avec $(X,0) \in E_1$ (abscisses), $(0,Y) \in E_2$ (ordonnées) et $A$, $B$ des matrices. On suppose aussi que $g(0,0)=(0,0)$ et $\max (\| D
R(Z) \|, \| D U (Z) \|) \leq \delta$ dans la boule $B(0,r)$. Enfin par hypothèse, on aura $\gamma \leq \| A \| \leq \| B^{-1}
\|^{-1}(1 - \gamma)$. Soit maintenant $\{ ( \Phi(Y),Y), Y \in D \}$ un graphe dans $B(0,r)$ au-dessus d’une partie $D$ de $E_2$ qui vérifie $\| \Phi(Y_1) - \Phi(Y_2) \| \leq \gamma_0 \| Y_1 - Y_2
\|$. Dans le théorème qui suit, on donne des conditions sur $\delta$, $\gamma$ et $\gamma_0$ pour que l’image de ce graphe par $g$ soit un graphe qui vérifie le même contrôle.
Si $\delta \| B^{-1} \| (1 + \gamma_0) <1$ alors l’image par $g$ du graphe précédent est un graphe au-dessus de $\pi_0(g(\mbox{graphe de }
\Phi))$ où $\pi_0$ est la projection sur les ordonnées. Par ailleurs, si $(\Psi(Y),Y)$ désigne ce nouveau graphe, on a: $$\| \Psi(Y_1) - \Psi(Y_2) \| \leq \frac{\| A \| \gamma_0 + \delta(1+
\gamma_0)}{\| B^{-1} \|^{-1} - \delta(1+ \gamma_0)} \| Y_1 - Y_2
\|$$ qui est inférieur à $\gamma_0 \| Y_1 - Y_2 \|$ si $\delta \leq
\epsilon(\gamma_0, \gamma)$.
Enfin, si de plus $B(0 , \alpha) \subset D$ et $\| \Phi(0) \|
\leq \beta$, alors $\pi_0(g(\mbox{graphe de } \Phi))$ contient $B(0,
(\| B^{-1} \|^{-1} - \delta(1 + \gamma_0)) \alpha - \| A \| \beta -
\delta \beta - \|
D^2 g \|_{B(0,r)} \beta^2)$ et $\| \Psi(0) \| \leq (1+ \gamma_0)( \| A \| \beta + \delta \beta +
\|D^2g \|_{B(0,r)} \beta^2) $ (si $\delta \leq \epsilon(\gamma_0, \gamma)$).
La démonstration est tirée essentiellement de [@KH] mais nous préférons la donner par confort pour le lecteur.
Soit $\lambda(Y)= BY + U(\Phi(Y),Y)$. C’est l’ordonnée de $g(\Phi(Y),Y)$. Pour démontrer que $g(\mbox{graphe de } \Phi)$ est un graphe au-dessus de $\pi_0(g(\mbox{graphe de } \Phi))$, il suffit de voir que $\lambda$ est une bijection de $D$ sur $\lambda(D)$ c’est-à-dire que $\lambda(Y)=Y_0$ a une unique solution pour $Y_0 \in
\lambda(D)$.
Posons $\gamma(Y)=B^{-1} Y_0 - B^{-1} U(\Phi(Y),Y)$. On a que $\lambda(Y)=Y_0$ est équivalent à $\gamma(Y)=Y$. Mais, si $Y_1 \mbox{
, } Y_2$ sont dans $D$, on a: $$\| \gamma(Y_1) - \gamma(Y_2) \| \leq \| B^{-1} \| \|
U(\Phi(Y_1),Y_1)- U(\Phi(Y_2),Y_2) \|$$ qui est inférieur à $ \| B^{-1} \| \delta (1 + \gamma_0) \|Y_1 -
Y_2 \|$. Autrement dit, quand $\delta \| B^{-1} \| (1 + \gamma_0) <1$, alors $\gamma(Y)=Y$ a bien une unique solution dans $D$.
Passons maintenant au contrôle de la pente du graphe $(\Psi(Y),Y)$ que l’on a obtenu.
On considère $(\Psi(Y_1'),Y_1')$ et $(\Psi(Y_2'),Y_2')$ deux points du graphe. Ils sont l’image de $(\Phi(Y_1),Y_1)$ et $(\Phi(Y_2),Y_2)$ par $g$. On notera $X_i'= \Psi(Y_i')$ ($i=1,2$). Alors, d’une part: $$\| X_1' - X_2' \| = \| A \Phi(Y_1) + R(\Phi(Y_1),Y_1) - A
\Phi(Y_2) - R(\Phi(Y_2),Y_2)\|$$ qui est inférieur à $$( \| A \| \gamma_0 + \delta(1 + \gamma_0)) \|Y_1 - Y_2 \|.$$ D’autre part:
$$\| Y_1' - Y_2' \| = \| B Y_1 + U(\Phi(Y_1),Y_1) - B Y_2 -
U(\Phi(Y_2),Y_2) \|$$ qui est supérieur à $$( \| B^{-1} \|^{-1} - \delta (1 + \gamma_0)) \| Y_1 - Y_2 \|$$ car $\| B(Y_1-Y_2) \| \geq \| B^{-1} \|^{-1} \| Y_1 - Y_2 \|$.
En combinant ces deux inégalités, on obtient:
$$\| \Psi(Y_1') - \Psi(Y_2') \| \leq \frac{\| A \| \gamma_0 + \delta(1
+ \gamma_0)}{ \| B^{-1} \|^{-1} - \delta (1 + \gamma_0)}\| Y_1' -
Y_2' \|$$ qui est l’inégalité cherchée.
Passons maintenant aux dernières estimations.
Tout d’abord on veut majorer la distance entre $(0,0)$ et l’image de $(\Phi(0),0)$ par $g$. Soit $v$ le vecteur $(\Phi(0),0)$ normalisé de sorte que $\| v \| = 1$. Si $P$ est un point du segment $[(0,0),
(\Phi(0),0)]$, on a: $$\| Dg(P)v - Dg(0)v \| \leq \|D^2g \|_{B(0,r)} \|P\|$$ Mais $\| Dg(0)v \| \leq \| A \| + \delta$ donc: $$\| Dg(P)v \| \leq \| A \| + \delta + \|D^2g \|_{B(0,r)} \beta.$$ On déduit de cette inégalité que la distance entre $(0,0)$ et l’image de $(\Phi(0),0)$ par $g$ est majorée par $ \| A \| \beta + \delta \beta +
\|D^2g \|_{B(0,r)} \beta^2$.
Maintenant, on a: $$\| \lambda(Y_1) - \lambda(Y_2) \| \geq \| B(Y_1 - Y_2) \| - \delta
(1+\gamma_0) \|Y_1 -Y_2 \|$$ qui est plus grand que $(\| B ^{-1} \|^{-1} - \delta(1 + \gamma_0)) \|
Y_1 - Y_2 \|$. En particulier, $\| \lambda(Y) - \lambda(0) \| \geq (\|
B ^{-1} \|^{-1} - \delta(1 + \gamma_0)) \alpha$ pour $Y \in
\partial B(0, \alpha)$. Comme $\| \lambda(0) \|$ est inférieur à $ \| A \| \beta + \delta \beta +
\|D^2g \|_{B(0,r)} \beta^2$, on en déduit que $\pi_0(g(\mbox{graphe de } \Phi))$ contient $B(0,
(\| B^{-1} \|^{-1} - \delta(1 + \gamma_0)) \alpha - \| A \| \beta -
\delta \beta - \| D^2 g \|_{B(0,r)} \beta^2)=B(0,r')$.
Il reste à majorer $\| \Psi(0) \|$. On a: $$\| \Psi(0) - \Psi(\lambda(0)) \| \leq \gamma_0 \| \lambda(0) \|$$ (si on suppose que $\delta \leq \epsilon(\gamma_0, \gamma)$). On obtient alors (toujours avec la majoration de la distance entre $(0,0)$ et $(\Psi(\lambda(0)), \lambda(0))=g(\Phi(0),0)$) $$\| \Psi(0) \| \leq (1 +\gamma_0)( \| A \| \beta + \delta \beta +
\|D^2g \|_{B(0,r)} \beta^2).$$ C’est l’estimée que l’on cherchait.
On va passer maintenant à la démonstration des deux formules.
[**Démonstration de la première inégalité du théorème**]{}
==========================================================
Commençons par rappeler la définition de l’entropie métrique.
Notons $d_n(x,y)= \max_{0 \leq i \leq n-1} \{ d(f^i(x), f^i(y)) \}$ et $B_n(x, \delta)$ la boule de centre $x$ et de rayon $\delta$ pour cette métrique. Par le théorème de Brin et Katok (voir [@BK]), l’entropie métrique de $\mu$ est donnée par la formule: $$h(\mu)= \lim_{\delta \rightarrow 0} \liminf_{n} - \frac{1}{n} \log \mu(B_n(x,
\delta))$$ pour $\mu$-presque tout $x$.
On va maintenant faire quelques uniformisations.
Soit $\Lambda_{\delta, n}= \{ x \mbox{ , } \mu (B_n(x,\delta)) \leq
e^{- h(\mu)n + \gamma n} \}$.
Si $\delta$ est pris petit on a $$\frac{4}{5} \leq \mu( \{ x \mbox{ , } \liminf_{n} - \frac{1}{n} \log \mu(B_n(x,\delta)) \geq
h(\mu) - \frac{\gamma}{2} \}) \leq \mu( \cup_{n_0} \cap_{n \geq n_0}
\Lambda_{\delta, n }).$$ En particulier, si on prend $n_0$ grand, on a: $\mu(\cap_{n \geq n_0} \Lambda_{\delta, n}) \geq 3/4$.
Rappelons que l’on note $\widehat{Y}$ l’ensemble des bons points de $\widehat{X}^{*}$ pour la théorie de Pesin.
Posons $$\widehat{Y}_{\alpha_0}=\{ \widehat{x} \in \widehat{Y} \mbox{
, } \alpha_0 \leq \| C_{\gamma} ( \widehat{x})^{\pm 1} \| \leq
\frac{1}{\alpha_0} \mbox{ , } V(\widehat{x}) \geq \alpha_0 \}$$ (voir le paragraphe \[Pesin\] pour les notations). Si $\alpha_0$ est suffisamment petit, on a $\widehat{\mu}(\widehat{Y}_{\alpha_0}) \geq 3/4$ d’où $\mu(A_{n_0}) \geq
1/2$ avec $A_{n_0}= \pi(\widehat{Y}_{\alpha_0}) \cap (
\cap_{n \geq n_0} \Lambda_{\delta,n})$.
Maintenant, comme pour les points $x$ de $A_{n_0}$ on a $\mu( B_n(x,
\delta)) \leq e^{-h(\mu)n + \gamma n}$, on peut trouver un ensemble $\{x_i \}_{1 \leq i \leq N}$ d’éléments de $A_{n_0}$ avec $N \geq
\frac{1}{2}e^{h(\mu)n - \gamma n }$ tels que $x_i =
\pi(\widehat{x_i})$ où $\widehat{x_i} \in \widehat{Y}_{\alpha_0}$ et avec les $B_n(x_i, \delta/2)$ disjointes (i.e. les points $x_i$ sont $(n,\delta)$-séparés).
$ $
Voici le plan de la démonstration de la formule. Dans celle-ci, on adapte des idées de J. Buzzi (voir [@Bu]) et S. E. Newhouse (voir [@Ne]) à notre contexte: celui des applications méromorphes. En chaque point $x_i$ nous allons construire une variété stable approchée $W_i$ de dimension $k-(s-l)+1$ (dans tout ce texte les dimensions seront des dimensions complexes). Cela signifiera en particulier que le diamètre de $f^{q}(W_i)$ restera inférieur à $\delta/4$ ($q=0, \dots , n-1$) et que les $W_i$ seront assez plates. Ensuite, dans un deuxième paragraphe nous minorerons le volume $k-s+l+1$-dimensionnel de ces variétés par à peu près $e^{-2
\chi_{s-l}^{+} n - \dots - 2 \chi_k^{+} n }$. Le volume total de toutes ces variétés est donc essentiellement supérieur à $e^{h(\mu)n-2 \chi_{s-l}^{+} n - \dots - 2 \chi_k^{+} n }$. Les $W_i$ étant assez plates, on pourra trouver un plan $P$ de dimension $k-s+l+1$ tel que d’une part la projection de tous les $W_i$ sur $P$ soit de volume minoré par la même quantité et d’autre part les $W_i$ seront des graphes au-dessus de $P$. Maintenant, si $\pi_1$ désigne la projection orthogonale sur $P$, la minoration de volume implique que les fibres de $\pi_1$ (qui sont des plans de dimension $s-l-1$) coupent $\cup W_i$ en moyenne en au moins $e^{h(\mu)n-2 \chi_{s-l}^{+} n -
\dots - 2 \chi_k^{+} n }$ points. Mais vu que les $x_i$ sont $(n,
\delta)$-séparés et que les diamètres des poussés en avant des $W_i$ restent petits, l’intersection d’une fibre de $\pi_1$ avec $\cup W_i$ donne des points $(n, \delta/2)$-séparés. Cela signifie qu’en moyenne le nombre de points $(n , \delta/2)$-séparés dans une fibre de $\pi_1$ est minoré par $e^{h(\mu)n-2 \chi_{s-l}^{+} n - \dots - 2 \chi_k^{+} n
}$. Enfin, dans le troisième paragraphe nous donnerons une majoration de cette moyenne par essentiellement $(\max_{0 \leq q \leq s-l-1} d_q)^n$ et cela prouvera l’inégalité.
[**Construction des variétés stables approchées**]{}
----------------------------------------------------
Rappelons que l’on a $\chi_1 \geq \dots > \chi_{s-l}= \dots = \chi_s
\geq \dots \geq \chi_k$. On notera $E_1(\widehat{x}) , \dots , E_{m}(\widehat{x})$ les $E_i(\widehat{x})$ du théorème d’Oseledec correspondant aux exposants $\chi_1 , \dots , \chi_{s-l-1}$ et $E_{m+1}(\widehat{x}) , \dots , E_{q}(\widehat{x})$ les $E_i(\widehat{x})$ de $\chi_{s-l} , \dots , \chi_k$ (voir le paragraphe \[Pesin\] pour les notations). Soit: $$E_u (\widehat{x}) = \oplus_{i=1}^{m} E_i(\widehat{x}) \mbox{ et }
E_s (\widehat{x}) = \oplus_{i=m+1}^{q} E_i(\widehat{x}).$$ Par ailleurs, $E_s (\widehat{x})$ sera dans la suite coupé en deux parties. Soit $n_1$ le nombre d’exposants parmi $\chi_{s-l}, \dots , \chi_k$ qui sont strictement négatifs (bien sûr $n_1$ peut être égal à $0$). Alors, nous noterons $E_s^1(
\widehat{x})$ la somme directe des $E_i(\widehat{x})$ ($i=m+1 , \dots ,
q$) correspondant aux $\chi_i$ strictement négatifs et $E_s^2(\widehat{x})$ la somme directe des autres $E_i(\widehat{x})$ de $E_s (\widehat{x})$. La dimension de $E_s^1(\widehat{x})$ est donc $n_1$ et celle de $E_s^2(\widehat{x})$ est $k-s+l+1-n_1$.
Soit $x$ un des $N$ points $x_i$ ($x=\pi(\widehat{x})$ avec $\widehat{x} \in \widehat{Y}_{\alpha_0}$). On va construire une variété stable approchée qui passe par $x$ en utilisant la transformée de graphe. Fixons $\gamma_0 >
0$ très petit devant $\alpha_0$. Dans toute la suite $n$ sera pris grand par rapport à des constantes qui dépendent de $\gamma_0$ et $\gamma$.
On se place maintenant dans $C_{\gamma}^{-1}(\widehat{f}^n(\widehat{x})) E_u (
\widehat{f}^n(\widehat{x})) \oplus
C_{\gamma}^{-1}(\widehat{f}^n(\widehat{x})) E_s ( \widehat{f}^n(\widehat{x}))$ et on part de $$\{ 0 \}^{s-l-1} \times B_2(0, e^{-4p \gamma n}) \times B_1(0, e^{-3p \gamma n}),$$ où $B_2(0, e^{-4p \gamma n})$ est la boule de ${\mathbb{C}}^{k-s+l+1-n_1}$ de centre $0$ et de rayon $e^{-4p \gamma n}$ et $B_1(0, e^{-3p \gamma n})$ est celle de ${\mathbb{C}}^{n_1}$ de centre $0$ et de rayon $e^{-3p \gamma n}$ . Cet ensemble est un graphe $(\Phi_n(Y),Y)$ au-dessus d’une partie de $C_{\gamma}^{-1}(\widehat{f}^n(\widehat{x}))
E_s ( \widehat{f}^n(\widehat{x}))$ (avec $\Phi_n(Y)=0$).
L’image du graphe $(\Phi_n(Y),Y)$ par $g_{\widehat{f}^n(
\widehat{x})}^{-1}$ est un graphe $(\Phi_{n-1}(Y),Y)$ au-dessus d’une partie de $C_{\gamma}^{-1}(\widehat{f}^{n-1}(\widehat{x}))
E_s ( \widehat{f}^{n-1}(\widehat{x}))$. Il vérifie de plus $\|
\Phi_{n-1}(Y_1) - \Phi_{n-1}(Y_2) \| \leq \gamma_0 \| Y_1 - Y_2 \|$.
Il s’agit d’utiliser le théorème du paragraphe \[graphe\].
Tout d’abord, si on prend pour abscisse $C_{\gamma}^{-1}(\widehat{f}^n(\widehat{x})) E_u (
\widehat{f}^n(\widehat{x}))$ et pour ordonnée $C_{\gamma}^{-1}(\widehat{f}^n(\widehat{x})) E_s (
\widehat{f}^n(\widehat{x}))$, on a $$g_{\widehat{f}^n( \widehat{x})}^{-1}(X,Y)=(AX + R(X,Y), BY +
U(X,Y))$$ avec $$\gamma \leq \|A\| \leq \| (A_{\gamma}^{m} ( \widehat{f}^{n-1}(\widehat{x})))^{-1}
\| \leq (1 - \gamma) \| (A_{\gamma}^{m+1} (
\widehat{f}^{n-1}(\widehat{x}))) \|^{-1}=(1 - \gamma) \| B^{-1} \|^{-1}$$ par la proposition \[inverse\], le théorème de $\gamma$-réduction de Pesin et le fait que $\gamma$ peut être supposé petit par rapport à des constantes qui ne dépendent que des exposants de Lyapounov de $\mu$.
De plus, toujours par cette proposition, $$\max(\| D R(X,Y) \|, \| D U(X,Y) \|) \leq \tau \|
C_{\gamma}(\widehat{f}^{n-1}(\widehat{x}))^{-1} \| \| C_{\gamma}(
\widehat{f}^{n}(\widehat{x})) \|^{2} d(f^{n-1}(x), {\mathcal{A}})^{-p} \|(X,Y)\|$$ avec $\|(X,Y) \| \leq \frac{\epsilon_0 d(f^{n-1}(x), {\mathcal{A}})^p}{ \|
C_{\gamma}(\widehat{f}^n(\widehat{x}))\|}$. Mais comme les fonctions $\|C_{\gamma}^{\pm 1} \|$ sont tempérées et que $\widehat{x}$ est dans $\widehat{Y}_{\alpha_0}$, on peut supposer que $ \|
C_{\gamma}(\widehat{f}^{n-1}(\widehat{x}))^{-1} \| \| C_{\gamma}(
\widehat{f}^{n}(\widehat{x})) \|^{2} \leq \frac{1}{\alpha_0^3}
e^{3 \gamma n }$ (voir [@KH] p. 668) et on a $d(f^{n-1}(x) , {\mathcal{A}})
\geq \alpha_0 e^{-n\gamma}$ (voir le lemme \[distance\]).
Pour $\|(X,Y) \| \leq e^{-2p \gamma n}$, on obtient:
$$\max(\| D R(X,Y) \|, \| D U(X,Y) \|) \leq e^{4 \gamma n}
e^{\gamma n p } e^{- 2 \gamma n p}$$ qui est très petit car $p$ peut être supposé supérieur à $5$. Cette quantité joue le rôle de $\delta$ dans le théorème du paragraphe \[graphe\]. Comme il est aussi petit que l’on veut pourvu que $n$ soit grand, on a bien démontré le lemme.
Maintenant, de ce graphe $(\Phi_{n-1}(Y),Y)$, on ne garde que la partie qui se trouve au-dessus de $\{ 0 \}^{s-l-1} \times B_2(0, e^{- 3
\gamma p n}) \times B_1(0, e^{- 3 \gamma p n}) $ (on fait un cut-off). Puis on prend son image par $g_{\widehat{f}^{n-1}(\widehat{x})}^{-1}$ qui de nouveau est un graphe $(\Phi_{n-2}(Y),Y)$ au-dessus d’une partie de $C_{\gamma}^{-1}(\widehat{f}^{n-2}(\widehat{x}))
E_s ( \widehat{f}^{n-2}(\widehat{x}))$ avec $\|
\Phi_{n-2}(Y_1) - \Phi_{n-2}(Y_2) \| \leq \gamma_0 \| Y_1 - Y_2 \|$ (la démonstration est exactement la même que dans le lemme précédent).
De ce graphe, on ne garde que la partie au-dessus $\{ 0 \}^{s-l-1} \times B_2(0, e^{- 3
\gamma p n}) \times B_1(0, e^{- 3 \gamma p n})$ et on continue ainsi le procédé jusqu’à obtenir un graphe $(\Phi_0(Y),Y)$ au-dessus d’une partie de $C_{\gamma}^{-1}(\widehat{x})
E_s (\widehat{x})$ qui vérifie $\|
\Phi_{0}(Y_1) - \Phi_{0}(Y_2) \| \leq \gamma_0 \| Y_1 - Y_2 \|$. Son image par $\tau_x \circ C_{\gamma}(\widehat{x})$ est la variété stable approchée que l’on voulait construire au point $x$.
On va maintenant minorer le volume de cette variété.
[**Minoration du volume des variétés stables**]{}
-------------------------------------------------
Dans un premier temps, on suppose que $n_1 >0$.
On repart du graphe $(\Phi_n(Y),Y)$ et cette fois-ci on va tirer en arrière des tranches de celui-ci. Plus précisément considérons: $$\{ 0 \}^{s-l-1} \times \{ a_{s-l} \} \times \dots \times \{
a_{k-n_1} \} \times B_1(0,e^{-3p \gamma n })$$ avec $(a_{s-l} , \dots , a_{k-n_1}) \in B_2(0 , e^{-4p \gamma n})$.
Cet élément est un graphe $(\Psi_n(Z),Z)$ au-dessus d’une partie de $C_{\gamma}^{-1}(\widehat{f}^n(\widehat{x})) E_s^1 (
\widehat{f}^n(\widehat{x}))$.
Pour les mêmes raisons que dans le paragraphe précédent, l’image de ce graphe par $g_{\widehat{f}^n( \widehat{x})}^{-1}$ est un graphe $(\Psi_{n-1}(Z),Z)$ au-dessus d’une partie de $C_{\gamma}^{-1}(\widehat{f}^{n-1}(\widehat{x}))
E_s^1 ( \widehat{f}^{n-1}(\widehat{x}))$. Il vérifie de plus $\|
\Psi_{n-1}(Z_1) - \Psi_{n-1}(Z_2) \| \leq \gamma_0 \| Z_1 - Z_2 \|$.
Par ailleurs, ce graphe vérifie aussi le (pour $\gamma_0 << \gamma$):
La projection du graphe $(\Psi_{n-1}(Z),Z)$ sur $C_{\gamma}^{-1}(\widehat{f}^{n-1}(\widehat{x}))
E_s^1 ( \widehat{f}^{n-1}(\widehat{x}))$ contient la boule $B_1(0, e^{-
3 p \gamma n})$ et $\| \Psi_{n-1}(0) \| \leq e^{- 4 p\gamma n +2 \gamma}$.
On applique la deuxième partie du théorème de la transformée de graphe du paragraphe \[graphe\].
Ici $\alpha=e^{-3 p \gamma n}$, $\beta= e^{-4 p \gamma
n}$ et $\|B^{-1} \|^{-1} \geq e^{\gamma}$ car on a sélectionné les exposants strictement négatifs.
Maintenant, on peut prendre $r=e^{-2 \gamma n p}$ (car $(\Phi_n(Y),Y)$ vit dans $B(0, e^{-2 \gamma n p})$) et en considérant les estimées obtenues à la fin de la démonstration de la proposition \[inverse\], on en déduit (pour $n$ grand): $$\| D^2 g_{\widehat{f}^n( \widehat{x})}^{-1} \|_{B(0,r)} \leq
e^{4 \gamma n} e^{\gamma n p}.$$
Comme dans le paragraphe précédent, le $\delta$ du théorème de la transformée de graphe est très petit. On en déduit donc bien que la projection du graphe $(\Psi_{n-1}(Z),Z)$ sur $C_{\gamma}^{-1}(\widehat{f}^{n-1}(\widehat{x}))
E_s^1 ( \widehat{f}^{n-1}(\widehat{x}))$ contient la boule $B_1(0, e^{-
3 p \gamma n})$.
Pour la majoration de $\| \Psi_{n-1}(0) \|$ il y a deux cas. Soit $n_1=k-s+l+1$ (i.e. $\chi_{s-l}, \dots , \chi_k$ sont tous strictement négatifs) et alors $\| \Psi_{n-1}(0) \| = \| \Phi_{n-1}(0) \|=0$. Soit $n_1 < k-s+l+1$ et la norme $\| A \|$ du théorème du paragraphe \[graphe\] est majorée par $e^{\gamma}$. L’estimée donnée dans ce théorème donne donc $\| \Psi_{n-1}(0) \| \leq e^{- 4 p\gamma n +2 \gamma}$ si $\gamma_0$ est très petit devant $\gamma$.
Maintenant, on ne garde que la partie de ce graphe qui se trouve au-dessus de $\{ 0 \}^{k-n_1} \times B_1(0, e^{-3 p \gamma
n})$. Remarquons que ce cut-off est finalement le même que celui du paragraphe précédent (où on gardait la partie au-dessus de $\{ 0 \}^{s-l-1} \times B_2(0, e^{- 3
\gamma p n}) \times B_1(0,e^{- 3 p \gamma n}) $). En effet, si on prend un point $Z$ dans $B_1(0, e^{-3 p \gamma n})$, on a $\| \Psi_{n-1}(Z) - \Psi_{n-1}(0) \| \leq
\gamma_0 \|Z \| \leq \gamma_0 e^{-3p \gamma n}$. On en déduit que $\|
\Psi_{n-1}(Z) \| \leq 2 \gamma_0 e^{-3p \gamma n}$ et donc que la projection de $(\Psi_{n-1}(Z),Z)$ sur $C_{\gamma}^{-1}(\widehat{f}^{n-1}(\widehat{x}))
E_s ( \widehat{f}^{n-1}(\widehat{x}))$ est incluse dans $B_2(0, e^{- 3 \gamma p n}) \times B_1(0,e^{- 3 p \gamma n}) $.
On recommence maintenant tout ce que l’on vient de faire en poussant en avant par $g_{\widehat{f}^{n-1}(\widehat{x})}^{-1}$ et ainsi de suite. A la fin, on obtient un graphe $(\Psi_0(Z),Z)$ au-dessus d’une partie de $C_{\gamma}^{-1}(\widehat{x}) E_s^1(\widehat{x})$ qui contient $ B_1(0, e^{- 3
\gamma p n})$ et avec $\| \Psi_0(0) \| \leq e^{-4 \gamma p n +
2 \gamma n}$. De plus ce graphe est assez plat car il vérifie $\| \Psi_{0}(Z_1) - \Psi_{0}(Z_2) \| \leq \gamma_0 \|Z_1 - Z_2 \|$.
En faisant varier $(a_{s-l} , \dots , a_{k-n_1}) \in B_2(0 , e^{-4p
\gamma n})$, on a donc feuilleté la variété stable approchée. Grâce à cette propriété, nous allons pouvoir minorer le volume $k-s+l+1$-dimensionnel de cette variété.
On se place dans $C_{\gamma}^{-1}(\widehat{x}) E_u(\widehat{x}) \oplus
C_{\gamma}^{-1}(\widehat{x}) E_s^2(\widehat{x}) \oplus
C_{\gamma}^{-1}(\widehat{x}) E_s^1(\widehat{x})$ et on considère un plan complexe de dimension $k-n_1$ de la forme $x_{k-n_1+1}=b_{k-n_1+1}, \dots , x_k=b_k$ avec $(b_{k-n_1+1}, \dots ,
b_k) \in B_1(0, e^{-3 \gamma p n})$. L’intersection $I_0$ de ce plan avec le graphe $(\Phi_0(Y), Y)$ de la variété stable est de dimension $k-n_1-s+l+1$. Nous allons minorer le volume $k-n_1-s+l+1$-dimensionnel de cette intersection par environ $e^{-2
\chi_{s-l}^{+} n - \dots - 2 \chi_k^{+} n }$. Cela impliquera que le volume $k-s+l+1$-dimensionnel du graphe $(\Phi_0(Y), Y)$ (on note $W_0$ cette variété) sera supérieur à $$e^{-2 \chi_{s-l}^{+} n - \dots - 2 \chi_k^{+} n - 6 \gamma p n
n_1}.$$ En effet, par la formule de la coaire (voir [@Fe] p. 258), ce volume est supérieur à: $$\int_{B_1(0 , e^{- 3 \gamma p n})} \int_{\pi_2^{-1}(Z) \cap W_0} d {\mathcal{H}}^{2(k-s+l+1-n_1)} d
{\mathcal{H}}^{2n_1}(Z),$$ (où $\pi_2$ est la projection orthogonale sur $C_{\gamma}^{-1}(\widehat{x}) E_s^1(\widehat{x})$). Et cette dernière quantité est plus grande que $e^{-2 \chi_{s-l}^{+} n - \dots - 2 \chi_k^{+} n - 6 \gamma p
n n_1}$.
Il reste donc à montrer que le volume $i_0=k-n_1-s+l+1$-dimensionnel de $I_0$ est supérieur à environ $e^{-2 \chi_{s-l}^{+} n - \dots - 2
\chi_k^{+} n }$.
Avant cela faisons une remarque. On a supposé jusqu’ici que $n_1
>0$. Lorsque $n_1=0$, on ne fait pas le tranchage du début de ce paragraphe et on passe directement à la minoration de $I_0$ qui est égal à $W_0$ que l’on va faire maintenant.
Dans un premier temps, on va évaluer le volume de $g_{\widehat{x}}(I_0)$ en fonction de celui de $I_0$. On a: $$\int_{I_0} \| \Lambda^{ i_0} D g_{\widehat{x}}(Z) \|^2 d
{\mathcal{H}}^{2i_0}=\mbox{volume}(g_{\widehat{x}}(I_0))$$ toujours par la formule de la coaire (ici on considère $g_{\widehat{x}}: I_0 \rightarrow g_{\widehat{x}}(I_0)$ comme fonction sur $I_0$ et $D g_{\widehat{x}}(Z)$ désigne toujours la différentielle complexe). Il s’agit de majorer $\| \Lambda^{ i_0} D g_{\widehat{x}}(Z) \|$ avec $Z \in I_0$. Tout d’abord cette quantité est inférieure à $\|
\Lambda^{ i_0} D g_{\widehat{x}}(Z) \|$ où cette fois-ci $g_{\widehat{x}}$ est considérée comme fonction sur $W_0$. Maintenant, $$\| \Lambda^{ i_0} D g_{\widehat{x}}(Z) \| = \| D g_{\widehat{x}}(Z)
v_1 \wedge \dots \wedge D g_{\widehat{x}}(Z) v_{i_0} \|$$ pour certains $v_1 , \dots , v_{i_0}$ tangents à $W_0$ (voir [@Ar] p. 119-120 pour les propriétés des produits extérieurs). Soient $u_1, \dots ,
u_{i_0}$ la projection sur $C_{\gamma}^{-1}(\widehat{x})
E_s(\widehat{x})$ de $v_1, \dots , v_{i_0}$. Cela signifie que pour $i=1, \dots , i_0$ on a $v_i= (D \Phi_0(P) \alpha_i,
\alpha_i)$ et $u_i=(0 , \alpha_i)$ où $P$ est la projection orthogonale de $Z$ sur $C_{\gamma}^{-1}(\widehat{x})
E_s(\widehat{x})$. D’après le contrôle sur le graphe $(\Phi_0(Y),Y)$, on a $\| D \Phi_0(P) \alpha_i \|\leq \gamma_0 \| \alpha_i \|$. Cela implique que $\| v_1 \wedge \dots \wedge v_{i_0} - u_1 \wedge \dots \wedge
u_{i_0}\| \leq \epsilon(\gamma_0)$ avec $\epsilon(\gamma_0)$ aussi petit que l’on veut pourvu que $\gamma_0$ le soit. En effet, si on note $G$ l’application linéaire $$G(X,Y)=(D \Phi_0(P) Y,Y)=
\left(
\begin{array}{cc}
0 & D \Phi_0(P)\\
0 & I
\end{array}
\right) \left(
\begin{array}{c}
X\\
Y
\end{array}
\right),$$ on a $$\| v_1 \wedge \dots \wedge v_{i_0} - u_1 \wedge \dots \wedge
u_{i_0}\| \leq \| (\Lambda^{i_0} G- \Lambda^{i_0} {\mathcal{I}})( u_1 \wedge \dots \wedge
u_{i_0})\|$$ où $${\mathcal{I}}(X,Y)=\left(
\begin{array}{cc}
0 & 0\\
0 & I
\end{array}
\right) \left(
\begin{array}{c}
X\\
Y
\end{array}
\right).$$ Enfin $\| \Lambda^{i_0} G- \Lambda^{i_0} {\mathcal{I}}\|$ est aussi petit que l’on veut pourvu que $\gamma_0$ le soit (par le théorème des accroissements finis).
Maintenant, $$\| \Lambda^{ i_0} D g_{\widehat{x}}(Z) \| = \| D g_{\widehat{x}}(Z)
v_1 \wedge \dots \wedge D g_{\widehat{x}}(Z) v_{i_0} \|$$ vérifie: $$\| \Lambda^{ i_0} D g_{\widehat{x}}(Z) \| \leq \|\Lambda^{ i_0} D
g_{\widehat{x}}(0)(u_1 \wedge \dots \wedge u_{i_0}) \| +A+B$$ avec $$A= \|(\Lambda^{ i_0} Dg_{\widehat{x}}(Z) - \Lambda^{ i_0} D
g_{\widehat{x}}(0))(v_1 \wedge \dots \wedge v_{i_0}) \|$$ $$B=\|\Lambda^{ i_0} D
g_{\widehat{x}}(0)(v_1 \wedge \dots \wedge v_{i_0}-u_1 \wedge \dots
\wedge u_{i_0}) \|.$$
$A$ est inférieur à $\|\Lambda^{ i_0} Dg_{\widehat{x}}(Z) - \Lambda^{ i_0} D
g_{\widehat{x}}(0)\|$ qui est aussi petit que l’on veut pourvu que l’on prenne $n$ grand. En effet d’une part comme à chaque étape on fait un cut-off, $Z$ vit ici dans la boule centrée en $(0,0)$ et de rayon $e^{- 2 \gamma p n}$, d’autre part on a un contrôle de la différentielle seconde qui est donnée par la proposition \[direct\].
Comme $B$ est inférieur à $\epsilon(\gamma_0)$, on obtient pour $n$ grand (en particulier devant des constantes qui dépendent de $\gamma_0$): $$\| \Lambda^{ i_0} D g_{\widehat{x}}(Z) \| \leq \|\Lambda^{ i_0} D
g_{\widehat{x}}(0)_{| C_{\gamma}^{-1} E_s(\widehat{x})} \| + \epsilon(\gamma_0).$$ Mais $$\|\Lambda^{ i_0} D
g_{\widehat{x}}(0)_{| C_{\gamma}^{-1} E_s(\widehat{x})} \| \leq e^{ \chi_{s-l} + \dots + \chi_{k-n_1}
+ \gamma k}$$ ce qui implique que $$\| \Lambda^{ i_0} D g_{\widehat{x}}(Z) \| \leq e^{ \chi_{s-l}^{+} + \dots + \chi_{k}^{+}
+ \gamma k}(1 + \epsilon(\gamma_0)).$$
Le volume de $g_{\widehat{x}}(I_0)$ est donc majoré par $\mbox{volume}(I_0) \times e^{2 \chi_{s-l}^{+} + \dots + 2 \chi_{k}^{+}
+2 \gamma k}(1 + \epsilon(\gamma_0))^2$.
Maintenant, on prend l’image de $g_{\widehat{x}}(I_0)$ par $g_{\widehat{f}(\widehat{x})}$ et ainsi de suite. En faisant les mêmes calculs que précédemment, on obtient alors une majoration du volume $i_0$-dimensionnel de $g_{\widehat{x}}^n(I_0)$ par $\mbox{volume}(I_0)
\times e^{2 \chi_{s-l}^{+}n + \dots + 2 \chi_{k}^{+} n
+ 2 \gamma kn}(1 + \epsilon(\gamma_0))^{2n}$.
Mais $g_{\widehat{x}}^n(I_0)$ rencontrent tous les $$\{ 0 \}^{s-l-1} \times \{ a_{s-l} \} \times \dots \times \{
a_{k-n_1} \} \times B_1(0,e^{-3p \gamma n })$$ avec $(a_{s-l} , \dots , a_{k-n_1}) \in B_2(0 , e^{-4p \gamma
n})$. Le volume $i_0$-dimensionnel de $g_{\widehat{x}}^n(I_0)$ est donc supérieur à $e^{-8p i_0 \gamma
n}$. Autrement dit, on a minoré le volume de $I_0$ par: $$e^{-2 \chi_{s-l}^{+}n - \dots - 2 \chi_{k}^{+} n
- 2 \gamma kn - 8p \gamma i_0 n}(1 + \epsilon(\gamma_0))^{-2n}$$ qui est supérieur à $$e^{-2 \chi_{s-l}^{+}n - \dots - 2 \chi_{k}^{+} n
- 10p \gamma k n}$$ (car $\gamma_0$ est très petit devant $\gamma$).
C’est la minoration que l’on cherchait car on obtient une minoration du volume de $W_0$ par $$e^{-2 \chi_{s-l}^{+}n - \dots - 2 \chi_{k}^{+} n
- 16p \gamma k n}.$$
On va passer à la majoration du volume de toutes ces variétés stables à l’aide des degrés dynamiques de $f$.
[**Majoration du volume**]{}
----------------------------
Nous avons construit des variétés stables pour chaque $x_i$ ($i=1,
\dots , N$) au-dessus de $C_{\gamma}^{-1}(\widehat{x_i}) E_s( \widehat{x_i})$. Considérons maintenant l’image de ces variétés par les $C_{\gamma}(\widehat{x_i})$. Chaque image est un graphe au-dessus de $E_s(\widehat{x_i})$ (pour le repère $E_u(\widehat{x_i}) \oplus
E_s(\widehat{x_i})$). De plus, si $(\Phi(Y),Y)$ est l’un d’eux, on a $\| \Phi(Y_1) - \Phi(Y_2) \| \leq \frac{\gamma_0}{\alpha_0^2} \| Y_1 -
Y_2 \|$ qui est aussi petit que l’on veut pourvu que $\gamma_0$ soit petit par rapport à $\alpha_0$ (ce que l’on a supposé). Quitte à remplacer $N$ par $N/K$ où $K$ est une constante qui ne dépend que de $X$, on peut supposer que tous ces graphes vivent dans une carte fixée $\psi: U \rightarrow
X$ et que les $x_i$ sont à distance au moins $\epsilon_0$ du bord de $U$ (cela signifie que $\tau_{x_i}$ est égal à $\psi$ modulo une translation). Toujours quitte à remplacer $N$ par $N/K$, on peut supposer que les graphes précédents sont des graphes au-dessus d’un plan complexe $P$ de dimension $k-s+l+1$ et que la projection de chaque graphe sur $P$ est de volume supérieur à $$e^{-2 \chi_{s-l}^{+}n - \dots - 2 \chi_{k}^{+} n
- 16p \gamma k n }$$ (éventuellement redivisé par une constante). Dans ce qui précède le plan $P$ peut être bougé un petit peu.
Le volume $k-s+l+1$-dimensionnel de la projection de tous les graphes $W(x_i)$ sur $P$ est donc supérieur à:
$$e^{h(\mu)n -2 \chi_{s-l}^{+}n - \dots - 2 \chi_{k}^{+} n
- 20p \gamma k n }.$$
Nous allons maintenant majorer ce volume à l’aide des degrés dynamiques.
Notons, $\pi_3$ la projection orthogonale sur $P$. $\pi_3(U)$ vit dans un compact $K$ de $P$ et pour $a \in K$, $F_a$ désignera la fibre $\pi_3^{-1}(a)$. Elle est de dimension $s-l-1$. Dans la suite $da$ sera la mesure de Lebesgue sur un voisinage de $K$ dans $P$.
Si ${\mathcal{W}}_s = \cup_{i=1}^{N} W(x_i)$ et $n(a)$ désigne le nombre d’intersection de $F_a$ avec ${\mathcal{W}}_s$, on a:
$$\int n(a) da = \mbox{volume de la projection de } {\mathcal{W}}_s \mbox{ sur
}P$$ est supérieur à $$e^{h(\mu)n -2 \chi_{s-l}^{+}n - \dots - 2 \chi_{k}^{+} n
- 20p \gamma k n }.$$
Cependant, pour $a$ fixé, les images par $\psi$ des points d’intersection entre $F_a$ et ${\mathcal{W}}_s$ sont $(n,
\delta/2)$-séparés. En effet considérons $y_1 \in \psi(F_a \cap
W(x_i))$ et $y_2 \in \psi(F_a \cap
W(x_j))$. Par définition des $x_i$, on a $d_n(x_i,x_j) \geq
\delta$. Cela signifie qu’il existe $l$ compris entre $0$ et $n-1$ avec $d(f^l(x_i),f^l(x_j)) \geq \delta$. Mais le diamètre de $f^l(\psi(W(x_i))$ et de $f^l(\psi(W(x_j))$ est inférieur à $\delta/4$ (car on a fait des cut-off) ce qui implique que $d(f^l(y_1), f^l(y_2))
\geq \delta/2$. Les points $y_1$ et $y_2$ sont $(n, \delta/2)$-séparés.
Si $\Omega_f= X \setminus \cup_{n \in {\mathbb{Z}}} f^n(I_f)$, on notera $\Gamma_n(a)$ l’adhérence de $\{ (z,f(z), \dots ,
f^{n-1}(z) ) \mbox{ , } z \in \psi(F_a \cap U) \cap \Omega_f \}$ dans $X^n$. C’est le multigraphe de $\psi(F_a \cap U)$. On munit $X^n$ de la forme de Kähler $\omega_n= \sum_{i=1}^{n} \Pi_i^{*} \omega$ où les $\Pi_i$ sont les projections de $X^n$ sur ses facteurs. Maintenant, on a:
$$\int \mbox{volume}(\Gamma_n(a))da \geq c(\delta) \int n(a) da.$$
La démonstration est la même que dans [@BD] (paragraphe 5) et [@Gr2]. Elle repose sur le théorème de Lelong (voir [@Le]). Nous la donnons par confort pour le lecteur.
On fixe $a$. Les $n$-orbites des points d’intersection entre $\psi(F_a \cap U)$ et $\psi({\mathcal{W}}_s)$ induisent un ensemble $F$ de $\Gamma_n(a)$ qui est $\delta/2$-séparé pour la métrique produit de $X^n$. Cela signifie que les $n(a)$ boules $B(y, \delta/4)$ avec $y
\in F$ sont disjointes. Par le théorème de Lelong, le volume de $\Gamma_n(a)
\cap B(y, \delta/4)$ est minoré par une constante $c(\delta)$. On en déduit donc que le volume de $\Gamma_n(a)$ est plus grand que $c(\delta) n(a)$. Cela démontre le lemme.
Maintenant,
$$\int \mbox{volume}(\Gamma_n(a))da = \int \int_{\Gamma_n(a)}
w_n^{s-l-1} da$$ est égal à $$\int \left( \sum_{0 \leq n_1,
\dots , n_{s-l-1 } \leq n-1} \int_{\psi(F_a \cap U) \cap \Omega_f}
(f^{n_1})^{*} \omega \wedge \dots \wedge (f^{n_{s-l-1}})^{*} \omega
\right) da$$ par définition de $\omega_n$ et le fait que l’on peut prendre $P$ générique. Soit $\Omega= \int [\psi(F_a \cap U)] da$ avec $[\psi(F_a \cap U)]$ le courant d’intégration sur $\psi(F_a \cap U)$. $\Omega$ est une forme de bidimension $(s-l-1,s-l-1)$.
De l’égalité précédente, on déduit que: $$\int \mbox{volume}(\Gamma_n(a))da = \sum_{0 \leq n_1,
\dots , n_{s-l-1 } \leq n-1} \int_{\Omega_f} \Omega \wedge
(f^{n_1})^{*} \omega \wedge \dots \wedge (f^{n_{s-l-1}})^{*}
\omega,$$ qui est inférieur à $$C_0 \sum_{0 \leq n_1,
\dots , n_{s-l-1 } \leq n-1} \int_{\Omega_f} \omega^{k-s+l+1} \wedge
(f^{n_1})^{*} \omega \wedge \dots \wedge (f^{n_{s-l-1}})^{*}
\omega,$$ où $C_0$ est une constante telle que $\Omega \leq C_0 \omega^{k-s+l+1}$. En utilisant le lemme \[lemme1\] avec $q=s-l-1$, on obtient: $$\int \mbox{volume}(\Gamma_n(a))da \leq c_{\epsilon} n ^{s-l-1} (\max_{0 \leq j \leq s-l-1}
d_j + \epsilon)^n.$$ Finalement, en combinant les inégalités obtenues, on a: $$c_{\epsilon} n ^{s-l-1} (\max_{0 \leq j \leq s-l-1}
d_j + \epsilon)^n \geq e^{h(\mu)n -2 \chi_{s-l}^{+}n - \dots - 2 \chi_{k}^{+} n
- 20p \gamma k n },$$ qui implique la première inégalité du théorème.
[**Démonstration de la deuxième inégalité du théorème**]{}
==========================================================
Comme la démonstration est à peu près la même que pour la première formule, on ne fera que l’esquisser.
Rappelons que l’on a $\chi_1 \geq \dots \geq \chi_s = \dots =
\chi_{s+l'} > \chi_{s+l'+1} \geq \dots \geq \chi_k$. Pour $\widehat{x} \in \widehat{Y}_{\alpha_0}$, on notera $E_1(\widehat{x}) , \dots , E_{m}(\widehat{x})$ les $E_i(\widehat{x})$ correspondant aux exposants $\chi_1 , \dots , \chi_{s+l'}$ et $E_{m+1}(\widehat{x}) , \dots , E_{q}(\widehat{x})$ les $E_i(\widehat{x})$ de $\chi_{s+l'+1} , \dots , \chi_k$. Soit: $$E_u (\widehat{x}) = \oplus_{i=1}^{m} E_i(\widehat{x}) \mbox{ et }
E_s (\widehat{x}) = \oplus_{i=m+1}^{q} E_i(\widehat{x}).$$ Par ailleurs, $E_u (\widehat{x})$ sera dans la suite coupé en deux parties. Soit $n_1$ le nombre d’exposants parmi $\chi_{1}, \dots , \chi_{s+l'}$ qui sont strictement positifs (bien sûr $n_1$ peut être égal à $0$). Alors, nous noterons $E_u^1(
\widehat{x})$ la somme directe des $E_i(\widehat{x})$ ($i=1, \dots,
m$) correspondant aux $\chi_i$ strictement positifs et $E_u^2(\widehat{x})$ la somme directe des autres $E_i(\widehat{x})$ de $E_u (\widehat{x})$. La dimension de $E_u^1(\widehat{x})$ est donc $n_1$ et celle de $E_u^2(\widehat{x})$ est $s+l'-n_1$.
On reprend les $N$ points $x_i$ du paragraphe précédent. En chaque $f^n(x_i)$, on peut construire des variétés instables approchées de dimension $s+l'$ par le procédé suivant. Soit $x$ un des $x_i$ (on a $x=\pi(\widehat{x})$ avec $\widehat{x} \in \widehat{Y}_{\alpha_0}$). On se place dans $C_{\gamma}^{-1}(\widehat{x}) E_u (
\widehat{x}) \oplus C_{\gamma}^{-1}(\widehat{x}) E_s ( \widehat{x})$ et on part de $$B_1(0, e^{-3p \gamma n}) \times
B_2(0, e^{-4p \gamma n}) \times \{ 0 \}^{k-s-l'},$$ où $B_1(0, e^{-3p \gamma n})$ est la boule de ${\mathbb{C}}^{n_1}$ de centre $0$ et de rayon $e^{-3p \gamma n}$ et $B_2(0, e^{-4p \gamma n})$ celle de ${\mathbb{C}}^{s+l'-n_1}$ de centre $0$ et de rayon $e^{-4p \gamma n}$. Cet ensemble est un graphe $(X, \Phi_0(X))$ au-dessus d’une partie de $C_{\gamma}^{-1}(\widehat{x}) E_u (
\widehat{x})$ (avec $\Phi_0(X)=0$). Toujours grâce à la transformée de graphe et le procédé de cut-off appliqués aux $g_{\widehat{f}^i(\widehat{x})}$, on obtient un graphe $(X, \Phi_n(X))$ au-dessus d’une partie de $C_{\gamma}^{-1}(\widehat{f}^n(\widehat{x})) E_u
(\widehat{f}^n(\widehat{x}))$. Par les mêmes arguments qu’au paragraphe précédent, le volume $s+l'$-dimensionnel de ce graphe est supérieur à $$e^{2 \chi_{1}^{-}n + \dots + 2 \chi_{s+l'}^{-} n
- 16p \gamma k n }.$$ Pour chaque $x_i$, on considère l’image du graphe construit au-dessus de $C_{\gamma}^{-1}(\widehat{f}^n(\widehat{x_i})) E_u
(\widehat{f}^n(\widehat{x_i}))$ par $C_{\gamma}(\widehat{f}^n(\widehat{x_i}))$. On notera $W(x_i)$ cette image et ${\mathcal{W}}_u = \cup_{i=1}^{N} W(x_i)$. Comme dans le paragraphe précédent, quitte à changer $N$ en $N/K$, on peut supposer que les $N$ variétés $W(x_i)$ vivent dans une carte $\psi:U \rightarrow X$ fixée, que les $W(x_i)$ sont des graphes au-dessus d’un plan $P$ de dimension $s+l'$ et que le volume $s+l'$-dimensionnel de la projection par $\pi_4$ des $W(x_i)$ sur $P$ est supérieur à $$e^{2 \chi_{1}^{-}n + \dots + 2 \chi_{s+l'}^{-} n
- 16p \gamma k n }.$$
Maintenant, $\pi_4(U)$ vit dans un compact $K$ de $P$ et pour $a \in
K$ on notera $F_a$ la fibre $\pi_4^{-1}(a)$. Si $n(a)$ désigne le nombre d’intersection entre $F_a$ et ${\mathcal{W}}_u$ et $da$ la mesure de Lebesgue sur un voisinage de $K$ dans $P$, on a: $$\int n(a) da \geq e^{h(\mu)n + 2 \chi_{1}^{-}n + \dots + 2 \chi_{s+l'}^{-} n
- 20 \gamma k n}.$$ Les points d’intersection entre $F_a$ et ${\mathcal{W}}_u$ induisent un ensemble $(n, \delta/2)$-séparé dans $f^{-n}(\psi(F_a \cap U))$. En effet les $g_{\widehat{f}^n(\widehat{x_i})}^{-j}(C_{\gamma}^{-1}(\widehat{f}^n(\widehat{x_i}))W(x_i))$ sont de diamètre très petit (pour $j=0, \dots , n$) et les $x_i$ sont $(n, \delta)$-séparés.
Si on note $\Gamma_n(a)$ le multigraphe de $f^{-n}(\psi(F_a \cap U))$, on a alors (toujours par le théorème de Lelong): $$\int \mbox{volume}(\Gamma_n(a))da \geq c(\delta) \int n(a) da.$$ Pour finir il reste à majorer $\int \mbox{volume}(\Gamma_n(a))da $.
Par un raisonnement équivalent à celui du paragraphe précédent, cette intégrale est inférieure à:
$$C_1 \sum_{0 \leq n_1,
\dots , n_{k-s-l' } \leq n-1} \int_{\Omega_f} (f^n)^{*} \omega^{s+l'} \wedge
(f^{n_1})^{*} \omega \wedge \dots \wedge (f^{n_{k-s-l'}})^{*}
\omega,$$ où $C_1$ est une constante qui ne dépend que de $X$. En utilisant le lemme \[lemme2\], on obtient: $$c_{\epsilon} n^{k-s-l'} ( \max_{s+l' \leq j \leq k}d_j + \epsilon)^n
\geq e^{h(\mu)n + 2 \chi_{1}^{-}n + \dots + 2 \chi_{s+l'}^{-} n
- 20 \gamma k n }.$$ Cela démontre la deuxième inégalité.
**[Le cas des difféomorphismes de classe $C^{1+ \alpha}$]{}**
=============================================================
Dans ce paragraphe, nous suivons la demande du referee en donnant une version de notre théorème pour les difféomorphismes de classe $C^{1 + \alpha}$ dans les variétés Riemanniennes compactes. Nous aboutirons ainsi à une inégalité plus faible que celle de J. Buzzi (voir [@Bu]). Commençons par préciser le cadre de ce paragraphe.
Soit $X$ une variété Riemannienne lisse compacte de dimension $k$ et $f$ un difféomorphisme de classe $C^{1 + \alpha}$.
J. Buzzi a introduit dans [@Bu] des notions d’entropie directionnelle. Dans ce paragraphe, nous considèrerons la suivante: pour $p$ compris entre $1$ et $k$, on note $${\mathcal{S}}^p:= \{ \sigma: ]-1,1[^p \mapsto X \mbox{, } \sigma \mbox{ de
classe } C^{\infty} \}.$$
On définit le $p$-volume de $\sigma \in {\mathcal{S}}^p$ par la formule:
$$v_p(\sigma)= \int_{]-1,1[^p} | \Lambda^p T_x \sigma | d
\lambda(x),$$
où $d \lambda$ est la mesure de Lebesgue sur $]-1,1[^p$ et $|
\Lambda^p T_x \sigma |$ est la norme de l’application linéaire $\Lambda^p T_x \sigma: \Lambda^p T_x (]-1,1[^p) \mapsto \Lambda^p
T_{\sigma(x)} X$ induite par la métrique Riemannienne sur $X$ (voir [@Ne]).
Nous désignerons par ${\mathcal{S}}^p(t)$ les éléments $\sigma$ de ${\mathcal{S}}^p$ pour lesquels le $p$-volume est inférieur ou égal à $t$.
L’entropie $p$-directionnelle de $f$ est alors définie par (voir [@Bu])
$$h_p(f):=\lim_{t \rightarrow 0} \lim_{\delta \rightarrow 0} \limsup_{n
\rightarrow + \infty} \frac{1}{n} \sup_{\sigma \in {\mathcal{S}}^p(t)} \log r(\delta,n, \sigma(]-1,1[^p)).$$
Ici $r(\delta,n, \sigma(]-1,1[^p))$ est le cardinal maximal d’un ensemble $(n, \delta)$-séparé inclus dans $\sigma(]-1,1[^p)$.
Alors, nous avons le:
Soient $\mu$ une mesure invariante, ergodique et $\chi_1 \geq \dots
\geq \chi_k$ les exposants de Lyapounov de $\mu$.
Fixons $1 \leq s \leq k$. On définit $l=l(s)$ par:
$$\chi_1 \geq \dots \geq \chi_{s-l-1} > \chi_{s-l} = \dots = \chi_s
\geq \chi_{s+1} \geq \dots \geq \chi_k,$$
où $s-l$ est égal à $1$ si $\chi_1 = \dots = \chi_s$.
Alors, on a l’inégalité suivante:
$$h(\mu) \leq h_{s-l-1}(f) + \chi_{s-l}^{+} + \dots + \chi_k^{+}$$
où $h(\mu)$ est l’entropie métrique de $\mu$ et $\chi_i^{+}= \max(\chi_i,0)$.
La preuve de ce théorème s’obtient en faisant des modifications mineures sur notre démonstration. Il s’agit d’utiliser l’introduction du paragraphe 5, le paragraphe 5.1 (qui suivent des idées de S. E. Newhouse et J. Buzzi (voir [@Ne] et [@Bu])) et enfin le paragraphe 5.2 (qui diffère de [@Ne] et [@Bu] et où on réalise la minoration du volume des variétés stables approchées en les feuilletant par des sous-variétés stables). Tous les autres paragraphes concernent les applications méromorphes et sont donc inutiles pour la preuve de cette inégalité.
Expliquons un certain nombre des petites modifications qu’il faut effectuer sur notre démonstration pour prouver l’inégalité ci-dessus.
Tout d’abord pour les rappels. La théorie de Pesin pour les difféomorphismes de classe $C^{1 + \alpha}$ est bien connue (voir par exemple [@KH]). Pour la preuve, on a besoin d’un analogue des propositions \[direct\] et \[inverse\]. On trouvera essentiellement la démonstration de cet analogue dans la preuve du théorème S.3.1 de [@KH]. Pour la transformée de graphe (voir le paragraphe \[graphe\]), il faut supposer $g$ de classe $C^{1 + a}$ (i.e. $\| D g(P) - D g(Q) \| \leq L \| P -
Q \|^{a}$) et se placer dans ${\mathbb{R}}^k$. La seule chose qui change dans le théorème c’est qu’il faut remplacer $\| D^2 g \|_{B(0,r)} \beta^2$ par $L \beta^{1 +
a}$ dans les formules.
Passons maintenant à la preuve de l’inégalité. Elle commence au début du paragraphe 5. Il s’agit ici et dans toute la suite d’enlever les extensions naturelles et la fonction $V$ (car on considère un difféomorphisme). Ensuite dans le plan de la preuve, on remplace la dimension complexe par la dimension réelle et les $2 \chi_i^{+}$ par $\chi_i^{+}$. A la fin on change la phrase “Enfin, dans le troisième paragraphe...” par la suivante: “Pour finir la démonstration, il suffit de majorer le nombre de points $(n, \delta/2)$-séparés dans une fibre de $\pi_1$ en utilisant $h_{s-l-1}(f)$ et on aboutit ainsi à l’inégalité annoncée.”
La démonstration continue avec le paragraphe 5.1. Ici il s’agit de tout recopier en supposant $p$ grand par rapport à $1/ \alpha$, en utilisant l’analogue de la proposition \[inverse\] et en changeant ${\mathbb{C}}$ par ${\mathbb{R}}$.
Ensuite, dans le paragraphe 5.2, il faut remplacer les $2 \chi_i$ par $\chi_i$, ${\mathcal{H}}^{2(k-s+l+1-n_1)}$ par ${\mathcal{H}}^{k-s+l+1-n_1}$ et ${\mathcal{H}}^{2
n_1}$ par ${\mathcal{H}}^{n_1}$. Il s’agit après de considérer la différentielle réelle et de changer $$\int_{I_0} \| \Lambda^{ i_0} D g_{\widehat{x}}(Z) \|^2 d
{\mathcal{H}}^{2i_0}=\mbox{volume}(g_{\widehat{x}}(I_0))$$ par $$\int_{I_0} \| \Lambda^{ i_0} D g_{x}(Z) \| d
{\mathcal{H}}^{i_0}=\mbox{volume}(g_{x }(I_0))$$ (formule de la coaire en réel). Enfin quand on parle de la différentielle seconde de $g_{\widehat{x}}$, il faut remplacer cet argument en utilisant le caractère Hölder de la différentielle de $f$.
[00]{} L. Arnold, *Random Dynamical Systems*, Springer Monographs in Mathematics, Springer-Verlag, (1998). J.-Y. Briend et J. Duval, *Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de ${\mathbb{C}}{\mathbb{P}}^k$*, Acta Math., [**182**]{} (1999), 143-157. J.-Y. Briend et J. Duval, *Deux caractérisations de la mesure d’équilibre d’un endomorphisme de ${\mathbb{P}}^k({\mathbb{C}})$*, IHES Publ. Math., [**93** ]{} (2001), 145-159. M. Brin et A. Katok, *On local entropy*, Geometric dynamics, Lect. Notes in Math., [**1007**]{} (1983), Springer Verlag, 30-38. J. Buzzi, *Entropy, volume growth and Lyapunov exponents*, preprint (1996). T.-C. Dinh et C. Dupont, *Dimension de la mesure d’équilibre d’applications méromorphes*, J. Geom. Anal., [**14**]{} (2004), 613-627. T.-C. Dinh et N. Sibony, *Regularization of currents and entropy*, Ann. Sci. Ecole Norm. Sup., [**37**]{} (2004), 959-971. T.-C. Dinh et N. Sibony, *Une borne supérieure pour l’entropie topologique d’une application rationnelle*, Ann. of Math., [**161**]{} (2005), 1637-1644. T.-C. Dinh et N. Sibony, *Dynamics of regular birational maps in ${\mathbb{P}}^k$*, J. Funct. Anal., [**222**]{} (2005), 202-216. T.-C. Dinh et N. Sibony, *Green currents for holomorphic automorphisms of compact Kähler manifolds*, J. Amer. Math. Soc., [**18**]{} (2005), 291-312. H. Federer, *Geometric measure theory*, Springer Verlag (1969). J.E. Forn[æ]{}ss et N. Sibony, *Complex dynamics in higher dimensions*, Complex Potential Theory (Montreal, PQ, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., [**439**]{}, Kluwer, Dordrecht (1994), 131-186. J.E. Forn[æ]{}ss et N. Sibony, *Complex dynamics in higher dimension I*, Astérisque, [**222**]{} (1994), 201-231. M. Gromov, *Convex sets and Kähler manifolds*, Advances in differential geometry and topology, World Sci. Publ., Teaneck, NJ, (1990), 1-38. M. Gromov, *On the entropy of holomorphic maps*, Enseign. Math., [**49**]{} (2003), 217-235. V. Guedj, *Ergodic properties of rational mappings with large topological degree*, Ann. of Math., [**161**]{} (2005), 1589-1607. V. Guedj, *Entropie topologique des applications méromorphes*, Ergodic Theory Dynam. Systems, [**25**]{} (2005), 1847-1855. A. Katok et B. Hasselblatt, *Introduction to the modern theory of dynamical systems*, Encycl. of Math. and its Appl., vol. 54, Cambridge University Press, (1995). P. Lelong , *Propriétés métriques des variétés analytiques complexes définies par une équation*, Ann. Sci. Ecole Norm. Sup., [**67**]{} (1950), 393-419. S. Lojasiewicz, *Introduction to Complex Analytic Geometry*, Birkhäuser, (1991). S. E. Newhouse, *Entropy and volume*, Ergodic Theory Dynam. Systems, [**8**]{} (1988), 283-299. D. Ruelle, *An inequality for the entropy of differentiable maps*, Bol. Soc. Brasil Mat., [**9**]{} (1978), 83-87. A. Russakovskii et B. Shiffman, *Value distribution for sequences of rational mappings and complex dynamics*, Ind. Univ. Math. J., [**46**]{} (1997), 897-932. H. Skoda, *Prolongement des courants positifs, fermés de masse finie*, Invent. Math., [**66**]{} (1982), 361-376 .
Henry de Thélin
Université Paris-Sud (Paris 11)
Mathématique, Bât. 425
91405 Orsay
France
|
---
abstract: 'Recently, neutron stars with very strong surface magnetic fields have been suggested as the site for the origin of observed soft gamma repeaters (SGRs). We investigate the influence of a strong magnetic field on the properties and internal structure of such strongly magnetized neutron stars (magnetars). The presence of a sufficiently strong magnetic field changes the ratio of protons to neutrons as well as the neutron appearance density. We also study the pion production and pion condensation in a strong magnetic field. We discuss the pion condensation in the interior of magnetars as a possible source of SGRs.'
address: 'Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA'
author:
- 'In-Saeng Suh and G. J. Mathews'
title: |
Nuclear Equation of State and\
Internal Structure of Magnetars
---
Introduction {#introduction .unnumbered}
============
Recently, observations of the soft gamma repeaters, SGR 0526-66, SGR 1806-20, SGR 1900+14, SGR 1627-41 and SGR 1801-23 (see [@hurley]) with BATSE, RXTE, ASCA, and BeppoSAX have confirmed the fact that these SGRs are a new class of $\gamma$-ray transients corresponding to strongly magnetized neutron stars (magnetars). Magnetars [@duncan; @thompson95] are newly born neutron stars with a surface magnetic field of $B \sim 10^{14} - 10^{15}$ G, probably created by a supernova explosion.
As relics of stellar interiors, the study of the magnetic fields in and around degenerate stars should give important information on the role such fields play in star formation and evolution. However, the origin and evolution of stellar magnetic fields remains obscure. The strength of the internal magnetic field in a neutron star in principle could be constrained by any observable consequences of a strong magnetic field. For example, rapid motion of neutron stars may be due to anisotropic neutrino emission induced by a strong magnetic field [@segre]. One could also consider the effect of magnetic fields on the thermal evolution [@heyl] and mass [@vshiv] of neutron stars. Recently, Chakrabarty et al. [@chakrabarty] have investigated the gross properties of cold nuclear matter in a strong magnetic field in the context of a relativistic Hartree model and have applied their equation of state to obtain masses and radii of magnetic neutron stars.
Since strong interior magnetic fields modify the nuclear equation of state for degenerate stars, their mass-radius relation also will be changed relative to that of nonmagnetic stars. Recently, we have obtained a revised mass-radius relation for magnetic white dwarfs [@SM20]. For strong internal magnetic fields of $B \sim 4.4 \times (10^{11} - 10^{13}$) G, we have found that both the mass and radius increase distinguishably and the mass-radius relation of some observed magnetic white dwarfs may be better fit if strong internal fields are assumed.
If ultrastrong magnetic fields exist in the interior of neutron stars as well, such fields will primarily affect the behavior of the residual charged particles. Standard internal properties such as the nuclear equation of state, neutron appearance, and the threshold density of muons and pions, would be modified by the magnetic field. Under charge neutrality and chemical equilibrium conditions, we calculate the ratio of protons to neutrons as well as the pion condensate equation of state in the presence of a sufficiently strong magnetic field. Here we shortly describe and summarize the results. The details of this work will be published elsewhere [@SM99].
Inverse $\beta$-decay and neutron appearance in a strong magnetic field {#inverse-beta-decay-and-neutron-appearance-in-a-strong-magnetic-field .unnumbered}
=======================================================================
Let us consider a homogeneous gas of free neutrons, protons, and electrons ($npe$) in $\beta$-equilibrium [@ST] in a uniform magnetic field. At high densities above $8 \times 10^6$ ${\rm {g \, cm^{-3}}}$, protons in nuclei are converted into neutrons via inverse $\beta$-decay: $e^{-} + p \longrightarrow n + \nu$. Since the neutrinos escape a star, energy is transport away from the system. Thus, the composition and structure of the star will be modified mainly by inverse $\beta$-decay. This reaction can proceed whenever the electron acquires enough energy to balance the mass difference between protons and neutrons, $Q = m_n - m_p = 1.293$ MeV. $\beta$-decay is blocked if the density is high enough that all energetically available electron energy levels in the Fermi sea are occupied.
In order to determine the equilibrium composition and equation of state, the coupled equations for chemical equilibrium and charge neutrality should now be solved simultaneously. Figure 1 shows the proton fraction $Y_p = n_p / n_B$, where $n_B$ is the baryon density, as a function of the neutron density $\rho_n$ for given field strength, $\gamma_e = B/B_{c}^{e}$, where $B_{c}^{e} \simeq 4.4 \times 10^{13}$ G. If the charged particles are in the lowest Landau level, inverse $\beta$ decay is not suppressed in magnetic fields. This means that rapid neutron-star cooling can occur in a strong magnetic field through the direct URCA process [@leinson]. However, electrons and protons, actually, are not in the lowest Landau level for higher densities above a critical density from which higher Landau levels begin to contribute to the chemical potential of electrons and protons, and hence, particle number densities. Therefore, discrete Landau levels become continuous and thus the proton concentration $Y_p$ goes back to the nonmagnetic case as the neutron density increases. As a result, inverse $\beta$ decay is still suppressed at high densities in strong magnetic fields.
=7.0cm 0.2cm
Finally, the equation of state, the mass-energy density $\rho = ({\cal E}_e + {\cal E}_p + {\cal E}_n)/c^2$, and the pressure $P = P_e + P_p + P_n$ (see [@lai] for a field strength less than log$\gamma_e$ = 2), are straightforwardly determined. Figure 2 shows the equation of state for a $npe$ gas in various magnetic fields. In this figure we can see that the neutron appearance density for an ideal $npe$ gas increases linearly with magnetic field strength.
=7.0cm 0.2cm
Pion production and condensation in a strong magnetic field {#pion-production-and-condensation-in-a-strong-magnetic-field .unnumbered}
============================================================
At very high density ($\rho {\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}}
\kern-.3em \raise.4ex \hbox{$>$}}}}\rho^{\ast}$), neutron-rich nuclear matter is no longer the true ground state of neutron-star matter. It will quickly decay by weak interactions into chemically equilibrated neutron star matter. Fundamental constituents, besides neutrons, may include a fraction of protons, hyperons, and possibly more massive baryons. In particular, if pion condensation exists in a magnetic field [@rojas], charged pion production and condensation through $n \rightarrow p + \pi^-$ is possible.
Figure 3 shows the equation of state for an ideal magnetic $npe\pi$ gas with pion condensation. We can see that magnetic fields reduce the pion condensation. However, we still have a distinguishable pion condensate equation of state in strongly magnetized neutron stars.
Discussion {#discussion .unnumbered}
==========
In this work, we have studied the nuclear equation of state for an ideal $npe$ gas in a strong magnetic field. Here, we show that the higher Landau levels are significant at high density in spite of the existence of very strong magnetic fields. In particular, at high density, the proton concentration approaches the same nonmagnetic limit. As a result, the inverse $\beta$ decay is still suppressed in intense magnetic fields. Therefore, neutron-star rapid cooling is not affected by the direct URCA process which is enhanced in strong magnetic fields. Finally, we see that the magnetic field reduces the amount of pion condensation. However, we have distinguishable effects of a pion condensate equation of state in strongly magnetized neutron stars.
It is generally accepted that neutrons and protons in a $npe$ gas are superfluid [@migdal; @yakovlev]. The charged pion condensate is also superfluid and superconductive [@migdal]. This pion formation and condensation in dense nuclear matter would have the significant consequence [@SM99] that the equation of state would be softened. First of all, softening the equation of state reduces the maximum mass of the stars. This softening effect with pion condensation also leads to detectable predictions [@migdal]. These are: (i) the rate of neutron star cooling via neutrinos would be enhanced, (ii) a possible phase transition of the neutron star to a superdense state. (iii) sudden glitches in the pulse period. In particular, if pion condensation occurs in a strong magnetic field, it may significantly affect starquakes.
According to the magnetar model by Duncan and Thompson [@duncan; @thompson95], SGRs are caused by starquakes in the outer solid crust of magnetars. In addition, Cheng and Dai [@cheng] recently suggested that SGRs may be rapidly rotating magnetized strange stars with superconducting cores. Although such models can explain some crucial features, there are still several unsettled issues [@liang]. Therefore, superconducting cores with a charged boson (pion, kaon) condensate in magnetars might be an alternative model to explain the energy source of soft gamma-rays from magnetars.
=7.0cm
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This work supported in part by NSF Grant-97-22086 and DOE Nuclear Theory Grant DE-FG02-95ER40934.
Hurley, K. 2000, review in this volume (astro-ph/9912061)
Duncan, R. C. & Thompson, C. 1992, [Astrophys. J.]{} 392, L9
Thompson, C. & Duncan, R. C. 1995, Mon. Not. R. Astron. Soc. 275, 255
Kusenko, A. & Segre, G. 1996, [Phys. Rev. Lett.]{} 77, 4872
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Suh, I.-S., & Mathews, G. J. 2000, [Astrophys. J.]{} in press (astro-ph/9906239)
Suh, I.-S. & Mathews, G. J. 1999, Submitted to ApJ. (astro-ph/9912301)
Shapiro, S. L. & Teukolsky, A. A. 1983, Black Holes, White Dwarfs, and Neutron Stars (New York: Wiley-Interscience)
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Liang, E. P. 1995, Astrophys. Space Sci. 231, 69
|
---
abstract: '0.5cm A novel semidiscrete Peierls-Nabarro model is introduced which can be used to study dislocation spreading at more than one slip planes, such as dislocation cross-slip and junctions. The strength of the model, when combined with [*ab initio*]{} calculations for the energetics, is that it produces essentially an atomistic simulation for dislocation core properties without suffering from the uncertainties associated with empirical potentials. Therefore, this method is particularly useful in providing insight into alloy design when empirical potentials are not available or not reliable for such multi-element systems. As an example, we study dislocation cross-slip and constriction process in two contrasting fcc metals, Al and Ag. We find that the screw dislocation in Al can cross-slip spontaneously in contrast with that in Ag, where the screw dislocation splits into two partials, which cannot cross-slip without first being constricted. The response of the dislocation to an external stress is examined in detail. The dislocation constriction energy and the critical stress for cross-slip are determined, and from the latter, we estimate the cross-slip energy barrier for straight screw dislocations.'
author:
- Gang Lu
- 'Vasily V. Bulatov'
- Nicholas Kioussis
title: 'A nonplanar Peierls-Nabarro model and its applications to dislocation cross-slip'
---
The past decades have witnessed a growing interest towards a quantitative understanding of deformation and strength of materials and, in particular, the effect of impurities and alloying elements on dislocation properties. While continuum elasticity theory describes well the long-range elastic strain of a dislocation for length scales beyond a few lattice spacings, it breaks down near the singularity in the region surrounding the dislocation center, known as the dislocation core. There has been a great deal of interest in describing accurately the dislocation core structure on an atomic scale because of its important role in many phenomena of crystal plasticity [@Duesbery1; @Vitek]. The core properties control, for instance, the mobility of dislocations, which accounts for the intrinsic ductility or brittleness of solids. The core is also responsible for the interaction of dislocations at close distances, which is relevant to plastic deformation. For example, by integrating the local rules derived from atomistic simulations of core interactions into dislocation-dynamics simulations, a connection between micro-to-meso scales can be established to study dislocation reactions and crystal plasticity [@Nature].
Cross-slip, the process by which a screw dislocation moves from one glide plane to another, is ubiquitous in plastic deformation of materials. For example, cross-slip is considered to be responsible for the onset of stage III in the stress-strain work-hardening curves. Furthermore, cross-slip can result in the formation of glide plane obstacles (sessile segments) in hcp metals and in L1$_2$-, B2- and L1$_0$- based intermetallic alloys, responsible for the anomalous high-temperature yield stress peak. However, theoretical studies of alloying and impurity effects on dislocation cross-slip have proved to be particularly difficult because one has to deal with both long-ranged elastic interactions (between dislocation segments) and short-ranged atomic interactions (due to the constriction process) that are inherent in a cross-slip process.
There are currently two theoretical approaches to study cross-slip. One is based on the line tension approximation which completely ignores atomic interactions, therefore it is not reliable [@Friedel; @Escaig]. The other approach is direct atomistic simulations employing empirical potentials [@Rasmussen; @Rao]. Although the second approach is quite powerful in determining cross-slip transition paths and estimating the corresponding activation energy barriers, it is time-consuming and, more importantly, it critically depends on the accuracy and availability of the empirical potentials employed in the simulations. For example, reliable interatomic potentials usually are not available for multi-elements materials. As a consequence, the possible hardening mechanisms due to alloying for most materials are still uncertain, and the design of new materials based on favorable cross-slip properties lacks guidance. Thus, the understanding at an atomic level of the chemistry effect on the dislocation core properties and cross-slip mechanisms is of great importance in predicting and controlling plastic deformation in structural materials, since the deformation behavior is often associated with the presence of substitutional or interstitial alloying elements. The ultimate goal of theoretical studies is then to use this information to direct alloying design for new materials with desired mechanical properties by tailoring the dislocation properties - in close collaboration with experimental efforts.
In this paper, we introduce a novel model based on the Peierls-Nabarro (P-N) framework which allows the study of dislocation cross-slip employing [*ab initio*]{} calculations. In fact, there has been a resurgence of interest recently in applying the simple and tractable P-N model to study dislocation core structure and mobility in conjunction with [*ab initio*]{} $\gamma$-surface calculations [@Joos; @Juan; @Bulatov; @Hartford; @Lu1; @Lu3]. This approach represents a combination of an atomistic ([*ab initio*]{}) treatment of the interactions across the slip plane and an elastic treatment of the continua on either side of the slip plane. Therefore, this approach is particularly useful for studying the interaction of impurities with dislocations when empirical potentials are either not available or not reliable to deal with such multi-element systems. Furthermore, it allows to study general trends in dislocation core properties and to correlate them with specific features of the underlying electronic structure. However, to date, all models based on the P-N framework are applicable only to a single slip plane while the important cross-slip process requires at least two active intersecting slip planes, i.e., the primary and cross-slip planes. In this work the semidiscrete variational P-N model[@Bulatov; @Lu1; @Lu3] is extended so as to take into account two intersecting slip planes. We shall apply this new model to study the dislocation constriction and cross-slip process in two fcc metals, Al and Ag, exhibiting different deformation properties. We are particularly interested in the evolution of the dislocation core structure under external stress and the interplay between the applied stress and the cross-slip process.
We begin by developing an appropriate energy functional for the Peierls dislocation at two intersecting slip planes. To facilitate presentation, we adopt the following conventions: In Fig. 1, a screw dislocation placed at the intersection of the primary (plane I) and cross-slip plane (plane II) is allowed to spread into the two planes simultaneously. The $X$ ($X'$) axis represents the glide direction of the dislocation at the plane I (II). For an fcc lattice, the two slip planes are (111) and ($\bar{1}11$), forming an angle $\theta \approx$ 71$^\circ$. The dislocation line is along the \[10$\bar{1}$\] ($Z$ axis) direction and $L$ represents the outer radius of the dislocation beyond which the configuration independent elastic energy is ignored.[@Lu1] In the spirit of the P-N model, the dislocation is represented as a continuous distribution of infinitesimal dislocations with densities of $\rho^{\rm I}(x)$ and $\rho^{\rm II}(x')$ on the primary and cross-slip planes, respectively, where, $x$ and $x'$ are the coordinates of the atomic rows at the two planes. Following the semidiscrete Peierls framework developed earlier [@Bulatov; @Lu1], the total energy of the dislocation is $$U_{tot}=U_{\rm I}+U_{\rm II}+\tilde{U}.$$ Here, $U_{\rm I}$ and $U_{\rm II}$ are the energies associated with the dislocation spread on planes I and II, respectively, and $\tilde{U}$ represents the elastic interaction energy between the dislocation densities on planes I and II. The expressions for $U_{\rm I}$ and $U_{\rm II}$ are identical to that given earlier for the single glide plane case [@Bulatov; @Lu1], while the new term $\tilde{U}$ can be derived from Nabarro’s equation for general parallel dislocations [@Nabarro], $$\begin{aligned}
U_{\rm I(II)}&=&\sum\limits_{i,j}\frac{1}{2}\chi_{ij}\{K_e[\rho^{\rm I(II)}_1(i)
\rho^{\rm I(II)}_1(j)+\rho^{\rm I(II)}_2(i)
\rho^{\rm I(II)}_2(j)]+
K_s\rho^{\rm I(II)}_3(i)\rho^{\rm I(II)}_3(j)\}\\
& & +\sum\limits_{i} \Delta x\gamma_3\left(f^{\rm I(II)}_1(i),f^{\rm I(II)}_2(i),
f^{\rm I(II)}_3(i)\right)
-\sum\limits_{i,l}\frac{x(i)^2-x(i-1)^2}{2}
\rho^{\rm I(II)}_l(i)\tau^{\rm I(II)}_l+Kb^2{\rm ln}L,\\
\tilde{U}&=&-\sum\limits_{i,j}K_s\rho^{\rm I}_3(i)\rho^{\rm p}_3(j)
A_{ij}-\sum\limits_{i,j}
K_e[\rho^{\rm I}_1(i)\rho^{\rm p}_1(j)+\rho^{\rm I}_2(i)\rho^{\rm p}_2(j)]A_{ij}\\
& &-\sum\limits_{i,j}
K_e[\rho^{\rm I}_2(i)\rho^{\rm p}_2(j)B_{ij}+\rho^{\rm I}_1(i)
\rho^{\rm p}_1(j)C_{ij}-
\rho^{\rm I}_2(i)\rho^{\rm p}_1(j)D_{ij}-\rho^{\rm I}_1(i)
\rho^{\rm p}_2(j)D_{ij}]~.\end{aligned}$$ Here, $f^{\rm I(II)}_1(i)$, $f^{\rm I(II)}_2(i)$ and $f^{\rm I(II)}_3(i)$ represent the edge, vertical and screw component of the general dislocation displacement at the $i$-th nodal point in plane I(II), respectively, while the corresponding component of dislocation density in plane I(II) is defined as $\rho^{\rm I(II)}(i) = \left(f^{\rm I(II)}(i)-
f^{\rm I(II)}(i-1)\right)/\left(x(i)-x(i-1)\right)$. The projected dislocation density $\rho^{\rm p}$(i) is the projection of the density $\rho^{\rm II}$(i) from plane II onto plane I in order to deal with the non-parallel components of the displacement. The $\gamma$-surface, $\gamma_3$, which in general includes shear-tension coupling can be determined from [*ab initio*]{} calculations. $\tau^{\rm I(II)}_l$ is the external stress component interacting with the corresponding dislocation densities, $\rho^{\rm I(II)}_l(i)$ ($l$ = 1,2,3). This term represents the contribution to the total energy from the elastic work done by the applied stress [@Bulatov; @Lu1]. The response of a dislocation to an applied stress is achieved by the minimization of the energy functional with respect to $\rho^{\rm I(II)}_l(i)$ at the given value of $\tau^{\rm I(II)}_l$. The dislocation core energy is defined as the configuration-dependent part of the total energy, which includes the density-dependent part of the elastic energy and the entire misfit energy, in the absence of stress [@Lu1]. $K_e$ and $K_s$ are the edge and screw components of the general prelogarithmic elastic energy factor $K$ [@Bulatov; @Lu1], while $\chi_{ij}$, $A_{ij}$, $B_{ij}$, $C_{ij}$ and $D_{ij}$ are double-integral kernels defined by $$\begin{aligned}
\chi_{ij}&=&\int\limits_{x_{j-1}}^{x_j}\int\limits_{x_{i-1}}^{x_i}
{\rm ln}|x-x'|dxdx',\\
A_{ij}&=&\int\limits_{x'_{j-1}}^{x'_j}\int\limits_{x_{i-1}}^{x_i}
\frac{1}{2}{\rm ln}(x_0^2+y_0^2)dxdx',\\
B_{ij}&=&\int\limits_{x'_{j-1}}^{x'_j}\int\limits_{x_{i-1}}^{x_i}
{\rm ln}\frac{x_0^2}{x_0^2+y_0^2}dxdx',\\
C_{ij}&=&\int\limits_{x'_{j-1}}^{x'_j}\int\limits_{x_{i-1}}^{x_i}
{\rm ln}\frac{y_0^2}{x_0^2+y_0^2}dxdx',\\
D_{ij}&=&\int\limits_{x'_{j-1}}^{x'_j}\int\limits_{x_{i-1}}^{x_i}
{\rm ln}\frac{x_0y_0}{x_0^2+y_0^2}dxdx',\end{aligned}$$ where $x_0 = L-x+x'\cos\theta$, and $y_0 = -x'\sin\theta$. The equilibrium structure of the dislocation is determined by minimizing the total dislocation energy functional energy with respect to the dislocation density.
In order to compare and understand the different cross-slip behavior in Al and Ag, we have carried out [*ab initio*]{} calculations for their $\gamma$-surfaces. In both calculations, we used a supercell containing six layers of atoms in the \[111\] direction. The [*ab initio*]{} calculations are based on the pseudopotential plane-wave method [@Payne] within the local density approximation. Owing to the planar nature of the dislocation core structure in fcc metals, we disregard in the $\gamma$-surface calculations the displacement perpendicular to the slip planes and consider partially the shear-tension coupling by performing volume relaxation along the \[111\] direction. The complete $\gamma$-surface for Al and Ag is shown in Fig. 2(a) and 2(b) respectively. The most striking difference between the two $\gamma$-surfaces is the large difference in intrinsic stacking fault energy, which is 165 mJ/m$^2$ for Al and 14 mJ/m$^2$ for Ag. This dramatic difference in $\gamma$-surface gives rise to very different dislocation core structures and cross-slip behavior that we are going to explore. The [*ab initio*]{} calculated Burgers vector, $b$, of a perfect 1/2\[101\] dislocation in Al and Ag is 2.85 and 2.83 Å, respectively.
The model calculation is set up by introducing a screw dislocation at the intersection of the two slip planes without applying external stress to the system at first. The initial configuration of the dislocation is specified by a step function for the screw displacement $f^{\rm I}_3(x)=0$ for $x<L$ and $f^{\rm I}_3(x)=b$ for $x \geq L$. All other displacement components including those on the cross-slip plane are set initially to zero. This corresponds to a pure screw dislocation with a zero width “spread” on the primary plane. We then relax the dislocation structure according to the energy functional. The results of the dislocation density $\rho(x)$ in the primary and cross-slip planes for Al and Ag are presented in Figs. 3(a) and 3(b), respectively. The screw dislocation in Al which starts out at the primary plane spontaneously spreads into the cross-slip plane, as the density peak at the cross-slip plane indicates. As expected, the edge component of the density is zero at the cross-slip plane because only the screw displacement can cross-slip. On the other hand, the screw dislocation in Ag dissociates into two partials, separated by 7.8 $b$ ($\approx$ 22 Å), in excellent agreement with the experimental value of 20 Å in TEM measurements [@Cockayne]. The left (right) partial has a positive (negative) edge component of the Burgers vector represented by the positive (negative) density. The integral of the edge density over all atomic sites is zero, corresponding to a pure screw dislocation. These partial dislocations cannot cross-slip, as the arrows in Fig. 3(a) indicate, without first annihilating their edge components, and the dislocation density on the cross-slip plane is essentially zero. Apparently, the lack of obvious dissociation in Al stems from the fact that Al has a much higher intrinsic stacking fault energy than Ag. The absence of obvious dissociation into partials in Al is also consistent with experiment [@Duesb0].
Next we apply an external Escaig stress to the dislocations and examine the evolution of the dislocation core structure under stress with the emphasis on the effect of stress on the dislocation cross-slip. The Escaig stress, defined as the edge component of the diagonal stress tensor, interacts only with the edge displacement of a dislocation, extending or shrinking its stacking fault width depending on the sign of the stress. We apply the Escaig stress only on the dislocation at the primary plane, and the stress components projected on the cross-slip plane are removed. The evolution of the dislocation core structure in Ag, represented by its displacement density distribution, is presented in Fig. 4 as the Escaig stress is varied. Upon application of positive (stretching) stress of $\tau_1^{\rm I}$ = 0.32 GPa, the separation of the partials rises rapidly from 7.8 $b$ (equilibrium separation in the absence of applied stress) to 12 $b$ (Fig. 4(a)). In Fig. 4(a) we show only the density at the primary plane, as the density at the cross-slip plane is essentially zero. For stress higher than 0.32 GPa, the partials move to the two ends of the simulation box and the lattice breaks down. To activate cross-slip, however, one needs to apply a negative (compressive) Escaig stress to the dislocation in order to annihilate the edge components of the partials’ displacement, known as a constriction process. Upon application of negative stress, the partials move towards each other and reduce the width of the stacking fault. During this process, the edge components of the displacement from the two partials annihilate each other at the primary plane while the screw component is being built up at both planes. One example of such structure is shown in Fig. 4(b). The left and right density peaks represent the original two partial dislocations, while the third peak at the center corresponds to the build-up of the screw density from the overlapping partials. Interestingly, the screw component of the dislocation density at the cross-slip plane is also accumulating, indicating the inception of the cross-slip process. However, further increasing the negative Escaig stress does not yield smaller separation between the partials. We find that the lower limit for the partials separation in Ag is 1.7 $b$, which is in agreement with the atomistic simulation result for Cu, reporting a corresponding value of 1.6 $b$ [@Duesbery]. In other words, no complete constriction can be achieved for a straight dislocation. The critical configuration in which the constriction is most developed is shown in Fig. 4(c). In this case, three partial dislocations with the same amount of screw component of the Burgers vector are formed. The cross-slipped screw dislocation density is about one third of the remaining screw density at the primary plane. In order to complete the cross-slip, either thermal fluctuations or other type of external stress have to be present. This is because the remaining edge component of the partials interacts with the Escaig stress, and as a result the partials exchange signs and move away from each other. For example, as shown in Fig. 4(d), the left partial now acquires a negative edge density and the right partial a positive edge density. Associated with the inversion of the edge component of the density for the partials, a run-on stacking fault is formed between them, with an energy of about 1.0 J/m$^2$. The run-on stacking fault is the most unstable stacking fault in an fcc lattice, with atoms from the neighboring (111) planes sitting right on top of each other. The distance between the two partials is more than 10 $b$. It is interesting to note that in the wake of the (partial) constriction process, a pure screw dislocation segment is formed at the intersection of the two planes, with an appreciable amount of cross-slip.
Next we examine the situation of Al. We first apply positive Escaig stress to the complete screw dislocation. We find that the screw dislocation remains unsplit until the stress reaches the threshold value of 0.96 GPa, required to separate the overlapping partials. The dislocation core structure corresponding to such a stress-driven dissociation is shown in Fig. 5(a). We find that the screw density component at the cross-slip plane is only reduced by a few percent due to the small splitting, leaving the density at the cross-slip plane approximately equal to that in the primary plane. Increasing the positive stress will further separate the partials and reduce cross-slip. The maximum positive Escaig stress, however, that the dislocation can sustain is 1.92 GPa (Fig. 5(b)). In this configuration, the partials are separated by 5 $b$. The central peak in the screw density plot corresponding to the original complete dislocation is reduced significantly, and the cross-slipped screw density amounts to only 1/3 of the screw density at the primary plane. Therefore, the application of positive Escaig stress in Al corresponds to an “inverse cross-slip” process that transfers displacement from the “cross-slip” plane to the “primary plane”. Interestingly, applying negative Escaig stress to the dislocation has no effect on the dislocation splitting or core width. The dislocation remains unsplit all the way until the stress is great enough to break down the lattice. Finally, it is important to point out that for all cases studied here in Al and Ag, upon removal of the applied external stress, the dislocation returns spontaneously to its equilibrium configuration.
We have also estimated the critical energetics that are relevant to cross-slip. For example, we calculated the constriction energy, defined as the difference in dislocation core energy between the normal and constricted states. By approximating the state with 1.7 $b$ separation between the partials as the constricted state, we were able to estimate the constriction energy for Ag to be 0.14 eV/$b$. A similar approach has been used to evaluate the constriction energy for a screw dislocation in Cu based on atomistic simulations, reporting a value of 0.17 eV/$b$, which is in a good agreement with our model calculations [@Duesbery]. Obviously, the constriction energy for Al is zero because its normal state is fully constricted. We have also calculated the critical stress for cross-slip, which is defined as the glide stress in the cross-slip plane to move a partially constricted dislocation from the primary to the cross-slip plane [@Duesbery]. We find that the critical stress for cross-slip in Ag is 1.68 GPa, compared to 0.32 GPa in Al. Finally, we estimated the cross-slip energy barrier, which in the context of our calculations, is defined as the difference in dislocation core energy before and after cross-slip takes place under the application of the above mentioned critical stress for cross-slip. In other words, we calculate the core energy difference for the dislocation between its normal state and the state that the dislocation just starts to cross-slip under the critical cross-slip stress. We find that the cross-slip energy barrier in Ag is 0.14 eV/$b$, much larger than that of 0.05 eV/$b$ in Al. One needs to be cautious when comparing our results for the cross-slip energy barrier directly with experiment, since the dislocations are assumed to be straight in our current implementation of the Peierls-Nabarro model. However, it is possible to extend the present formalism to deal with an arbitrarily curved dislocation where a more realistic cross-slip energy barrier can be obtained. Nevertheless, the present model is still capable to provide reliable energetics for straight dislocations.
In summary, we have presented a novel model based on the semidiscrete Peierls-Nabarro framework that allows the study of dislocation cross-slip and constriction. The $\gamma$-surface entering the model is determined from [*ab initio*]{} calculations which provide reliable atomic interactions across the slip plane. We find that the screw dislocation in Al can spontaneously spread into the cross-slip plane, while in Ag it dissociates into partials and can not cross-slip. We have also examined in detail the response of the dislocation core structure to an external Escaig stress and the effect of negative Escaig stress on the constriction of the Shockley partials. We find that one can not achieve 100% constriction for the case of straight partial dislocations considered in this work. By computing the dislocation core energy under stress we estimate the dislocation constriction energy for Al and Ag. The calculated values of the critical stress and the energy barrier for dislocation cross-slip demonstrate that dislocation cross-slip is much easier in Al than in Ag. Since our [*ab initio*]{} based model is much more expedient than direct [*ab initio*]{} atomistic simulations, it can serve as a powerful and efficient tool for alloy design, where the goal is to select the “right” elements with the “right” alloy composition to tailor desired mechanical, and in particular, dislocation properties, such as cross-slip properties.
Two of us (G.L. and N.K.) acknowledge the support from Grant No. DAAD19-00-1-0049 through the U.S. Army Research Office. G.L. was also supported by Grant No. F49620-99-1-0272 through the U.S. Air Force Office for Scientific Research.
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![Cartesian set of coordinates showing the directions relevant to the screw dislocation located at the intersection of the two slip planes. Plane I (II) denotes the primary (cross-slip) plane.[]{data-label="fig1"}](fig1.eps){width="300pt"}
![The $\gamma$-surfaces (J/$m^2$) for displacements along a (111) plane for (a) Al and (b) Ag . The corners of the plane and its center correspond to identical equilibrium configurations, i.e., the ideal lattice. The two energy surfaces are displayed in exactly the same perspective and on the same energy scale to facilitate comparison of important features.[]{data-label="fig2"}](fig2a.eps "fig:"){width="350pt"} ![The $\gamma$-surfaces (J/$m^2$) for displacements along a (111) plane for (a) Al and (b) Ag . The corners of the plane and its center correspond to identical equilibrium configurations, i.e., the ideal lattice. The two energy surfaces are displayed in exactly the same perspective and on the same energy scale to facilitate comparison of important features.[]{data-label="fig2"}](fig2b.eps "fig:"){width="350pt"}
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abstract: |
We consider the properties of the second order nonlinear differential equations $b''= g(a,b,b')$ with the function $g(a,b,b'=c)$ satisfying the following nonlinear partial differential equation $$g_{aacc}+2cg_{abcc}+2gg_{accc}+c^2g_{bbcc}+2cgg_{bccc}$$ $$+g^2g_{cccc}+(g_a+cg_b)g_{ccc}-4g_{abc}-4cg_{bbc} -cg_{c}g_{bcc}$$ $$-3gg_{bcc}-g_cg_{acc}+ 4g_cg_{bc}-3g_bg_{cc}+6g_{bb} =0\>.$$
Any equation $b''=g(a,b,b')$ with this condition on function $g(a,b,b')$ has the General Integral $F(a,b,x,y)=0$ shared with General Integral of the second order ODE’s $y''=f(x,y,y')$ with condition $\frac{\partial^4 f}{\partial y'^4}=0$ on function $f(x,y,y')$ or $$y''+a_{1}(x,y)y'^3+3a_{2}(x,y)y'^2+3a_{3}(x,y)y'+a_{4}(x,y)=0$$ with some coefficients $a_{i}(x,y)$.
author:
- |
Valerii Dryuma,\
Maxim Pavlov\
[*Institute of Mathematics and Informatics, AS RM,*]{}\
[*5 Academiei Street, 2028 Kishinev, Moldavia*]{}, [*MATI, Moskow, Russia*]{}
title: ON EQUATION FOR INITIAL VALUES IN THEORY OF THE SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS
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=-5mm
Introduction
============
The relation between the equations in form $$y''+a_{1}(x,y)y'^3+3a_{2}(x,y)y'^2+3a_{3}(x,y)y'+a_{4}(x,y)=0 \label{Cartan1}$$ and $$b''=g(a,b,b') \label{Dua1}$$ with function $g(a,b,b')$ satisfying the p.d.e $$g_{aacc}+2cg_{abcc}+2gg_{accc}+c^2g_{bbcc}+2cgg_{bccc}+$$ $$g^2g_{cccc}+(g_a+cg_b)g_{ccc}-4g_{abc}-4cg_{bbc} -cg_{c}g_{bcc}-$$ $$3gg_{bcc}-g_cg_{acc}+ 4g_cg_{bc}-3g_bg_{cc}+6g_{bb} =0 \label{Dua2}.$$ from geometrical point of view was studied by E.Cartan [@Cartan1].
In fact, according to the expressions on curvature of the space of linear elements (x,y,y’) connected with equation (1) $$\Omega^1_2=a[\omega^2 \wedge \omega^2_1]\,,\quad
\Omega^0_1=b[\omega^1 \wedge \omega^2]\,,\quad
\Omega^0_2=h[\omega^1 \wedge \omega^2]+k[\omega^2 \wedge \omega^2_1]\>.$$ where: $$a=-\frac{1}{6}\frac{\partial^4 f}{\partial y'^4}\,,\quad h=\frac{\partial b}{\partial y'}\,, \quad
k=-\frac{\partial \mu}{\partial y'}-\frac{1}{6}\frac{\partial^2 f}{\partial^{2} y'}\frac{\partial^3 f}{\partial^{3} y'}\,,$$ and $$\begin{aligned}
6b & = & f_{xxy'y'}+2y'f_{xyy'y'}+2ff_{xy'y'y'}+y'^2f_{yyy'y'}+2y'ff_{yy'y'y'} \\
& + & f^2f_{y'y'y'y'}+(f_x+y'f_y)f_{y'y'y'}-4f_{xyy'}-4y'f_{yyy'} - y'f_{y'}f_{yy'y'}\\
& - & 3ff_{yy'y'}-f_{y'}f_{xy'y'}+ 4f_{y'}f_{yy'}-3f_{y}f_{y'y'}+6f_{yy} \>.\end{aligned}$$ two types of equations by a natural way are evolved: the first type from the condition $a =0$ and second type from the condition $b =0$.
The first condition $a=0$ the equation in form (1) is detemined and the second condition lead to the equations (2) where the function $g(a,b,b')$ satisfies the above p.d.e. (3).
From the elementary point of view the relation between both equations (1) and (2) is a result of the special properties of their General Integral $$F(x,y,a,b)=0.$$ So we have the following fundamental diagramm: $$\begin{array}{ccccc}
& & F(x,y,a,b)=0 & & \\
& \swarrow \nearrow & & \searrow \nwarrow & \\
y''=f(x,y,y') & & & & b''=g(a,b,b') \\
& & & & \\
\Updownarrow & & & & \Updownarrow \\
& & & & \\
M^3(x,y,y') & & \Longleftrightarrow & & N^3(a,b,b')
\end{array}$$ which is presented the General Integral $F(x,y,a,b)=0$ (as some 3-dim orbifold) in form of the twice nontrivial fibre bundles on circles over corresponding surfaces: $$M^{3}(x,y,y')= U^{2}(x,y) \times S^1 \quad
\mbox{\rm and} \quad
N^{3}(a,b,b')= V^{2}(a,b) \times S^1\>.$$
An examples of solutions of dual equation
=========================================
Let us consider the solutions of equation (3).
It has many types of reductions and the symplest of them are $$g=c^{\alpha}\omega[ac^{\alpha-1}],\quad g=c^{\alpha}\omega[bc^{\alpha-2}],
\quad g=c^{\alpha}\omega[ac^{\alpha-1},bc^{\alpha-2}],
\quad g=a^{-\alpha}\omega[ca^{\alpha-1}],$$ $$\quad g=b^{1-2\alpha}\omega[cb^{\alpha-1}],
\quad g=a^{-1}\omega(c-b/a),
\quad g=a^{-3}\omega[b/a,b-ac],\quad
g=a^{\beta/\alpha-2}\omega[b^{\alpha}/a^{\beta},
c^{\alpha}/a^{\beta-\alpha}].$$
For any type of reduction we can write corresponding equation (2) and then integrate it.
As example for the function $$g=a^{-\gamma}A(ca^{\gamma-1})$$ we get the equation $$[A+(\gamma-1)\xi]^2 A^{1V}+3(\gamma-2)[A+(\gamma-1)\xi]A^{111}+(2-\gamma)A^1A^{11}+
(\gamma^2-5\gamma+6)A^{11}=0.$$
One solution of this equation is $$A-(2-\gamma)[\xi(1+\xi^2)+(1+\xi^2)^{3/2}]+(1-\gamma)\xi$$
This solution is coresponded to the equation $$b''=\frac{1}{a}[b'(1+b'^2)+(1+b'^2)^{3/2}]$$ with Gneral Integral $$F(x,y,a,b)=(y+b)^2+a^2-2ax=0$$ The dual equation has the form $$y''=-\frac{1}{2x}(y'^3+y')$$
Remark that the first examples of solutions of equation (3) was obtained in \[6-9\].
The
Equation (3) can be represent in form $$\begin{aligned}
g_{ac} + gg_{cc} - g_{c}^{2}/2 + cg_{bc} -2 g_{b} = h(a,b,c),\\\nonumber
h_{ac} + gh_{cc} - g_{c}h_{c} + ch_{bc} -3h_{b} = 0.\end{aligned}$$
From this is followed that exists the class of equations (\[Dua1\]) with function $g(a,b,c)$ satisfying the condition $$g_{ac} + gg_{cc} - g_{c}^{2}/2 + cg_{bc} -2 g_{b} = 0. \label{H}$$ which is more readily solved then equation (3).
Here we present some solutions of the equation (8) as function depending on two variables $g=g(a,c)$
In case when $g=g(a,c)$ and $h=0$ we have the equation $$g_{ac}+gg_{cc}-\frac{1}{2}{g_c}^2=0\>.$$
To integrate this equation we can transform its in more convenient form using variable $g_c=f(a,c)$. Then one obtains: $$2f_cf_{ac}+(f^2-2f_a)f_{cc}=0\>.$$ After the Legendre-transformation we obtain the equation: $$[(\xi\omega_{\xi}+\eta\omega_{\eta}-\omega)^2-
2\xi]{\omega}_{\xi\xi}-2\eta\omega_{\xi\eta}=0\>.$$ Using the new variable $\xi\omega_{\xi}+\eta\omega_{\eta}-\omega=R$ we have the new equation for $R$: $$R_{\xi}-\frac{1}{2}R^2\omega_{{\xi}{\xi}}=0\>$$ and the following relations: $$\omega_{\eta}=\frac{\omega}{\eta}+\frac{R}{\eta}+\frac{2{\xi}}{{\eta}R}-\frac{{\xi}A(\eta)}{\eta}\>,$$ $$\omega_{\xi}=-\frac{2}{R}+A(\eta)\>$$ with arbitrary function $A(\eta)$. From the conditions of compatibility is followed: $$2{\eta}R_{\eta}+R_{\xi}(2\xi-R^2)+\eta A_{\eta}R^2=0\>.$$ Integrating this equation we can obtain general integral.
In the particular case: $A=\frac{1}{\eta}$ we have: $$\frac{R^2}{R-2\eta}=-\frac{\xi}{\eta}+\Phi(\frac{1}{\eta}-\frac{2}{R})\>.$$ At the condition $A=0$ we obtain the equation: $$2{\eta}R_{\eta}+(2\xi-R^2)R_{\xi}=0\>,$$ which has the solution: $$R^2=2{\xi}+2{\eta}{\Phi(R)}\>,$$ were $\Phi(R)$ is arbitrary function.
After choosing the function $\Phi(R)$ we can find the function $\omega$ and then using the inverse Legendre transformation the function $g$ which is determined dual equation $b''=g(a,c)$.
The solutions of the equations of type $$u_{xy}=uu_{xx}+\epsilon u_{x}^2$$ was constructed in \[19\]. In work of \[20\] was showed that they can be present in form $$u=B'(y)+\int[A(z)-\epsilon y]^{(1-\epsilon)/ \epsilon} dz,$$ $$x=-B(y)+\int[A(z)-\epsilon y]^{1/ \epsilon} dz.$$
To integrate above equations we apply the parametric representation $$g=A(a)+U(a,\tau), \quad c=B(a)+V(a,\tau).\eqno(11)$$ Using the formulas $$g_c=\frac{g_{\tau}}{c_{\tau}}, \quad g_{a}=g_{a}+g_{\tau}\tau_{a}$$ we get after the substitution in (10) the conditions $$A(a)=\frac{d B}{d a}$$ and $$U_{a \tau}-\left(\frac{V_{a} U_{\tau}}{V_{\tau}}\right)_{\tau}+
U \left(\frac{ U_{\tau}}{V_{\tau}} \right)_{\tau} -
\frac{1}{2} \frac{U_{\tau}^2}{V_{\tau}}=0.$$
So we get one equation for two functions $U(a,\tau)$ and $V(a,\tau)$. Any solution of this equation are determined the solution of equation (10) in form (11).
Let us consider the examples. $$A=B=0, \quad U=2\tau-\frac{a\tau^2}{2}, \quad V=a\tau-2\ln(\tau)$$
Using the representation $$U=\tau \omega_{\tau}-\omega,\quad V=\omega_{\tau}$$ it is possible to obtain others solutions of this equation.
Equation $$g_{ac}=gg_{cc}-\frac{1}{2}{g_c}^2\>.$$ can be integrate in explicite form and solutions are $$g=-H'(a)+\int\frac{dz}{[A(z)+\frac{1}{2}a]^3}\>,$$ $$c=H(a)+\int\frac{dz}{[A(z)+\frac{1}{2}a]^2}\>,$$ with arbitrary functions $H(a)$ and $A(z)$.
In fact, for $A(z)=z$ we have $$g=-H'(a)+\int\frac{dz}{[z)+\frac{1}{2}a]^3}=-H'(a)-\frac{1}{2}\frac{1}
{[z)+\frac{1}{2}a]^2}\>,$$ and $$c=H(a)+\int\frac{dz}{[z+\frac{1}{2}a]^2}=H(a)-
\frac{1}{[z)+\frac{1}{2}a]^3}\>,$$
As result we get the solution
In general case the equation $$g_{acc}+gg_{ccc}=0,$$ is equivalent the equation $$g_{ac}+gg_{cc}-\frac{1}{2}{g_c}^2=B(a)\>.$$
It can be intgrate with help of Legender- transformation as in previous case.
Realy, we get $$[(\xi\omega_{\xi}+\eta\omega_{\eta}-\omega)^2-
2\xi+2B(\omega_{\xi})]{\omega}_{\xi\xi}-2\eta\omega_{\xi\eta}=0\>$$ and the relation $$2R_{\xi}=[R^2+2B(\omega_{\xi})\omega_{\xi \xi}.$$ It can be written in form $$2\frac{dR}{d\Omega}=R^2+2B(\Omega)$$ using the notation $$\omega_{\xi}=\Omega$$
In case $h\neq 0$ and $g=g(a,c)$ the system (3) is equivalent the equation $$\Theta_a(\frac{\Theta_a}{\Theta_c})_{ccc}-
\Theta_c(\frac{\Theta_a}{\Theta_c})_{acc}=1$$ where $$g=-\frac{\Theta_a}{\Theta_c}\quad h_c=\frac{1}{\Theta_c}$$
To integrate this equation we use the presentation $$c=\Omega(\Theta,a)$$
From the relations $$1=\Omega_{\Theta}\Theta_c, \quad 0=\Omega_{\Theta}\Theta_a+\Omega_c$$ we get $$\Theta_c=\frac{1}{\Omega_{\Theta}},\quad
\Theta_a=-\frac{\Omega_a}{\Omega_{\Theta}}$$ and $$\frac{\Omega_a}{\Omega_{\Theta}}(\Omega_a)_{ccc}+
\frac{1}{\Omega_{\Theta}}(\Omega_a)_{cca}=1$$
Now we get $$\Omega_{ac}=\frac{\Omega_{a \Theta}}{\Omega_{\Theta}}=
(\ln \Omega_{\Theta})_a=K,\quad
\Omega_{acc}=\frac{K_{\Theta}}{\Omega_{\Theta}},\quad$$ $$\Omega_{accc}=(\frac{K_{\Theta}}{\Omega_{\Theta}})_{\Theta}\frac{1}{\Omega_{\Theta}},\quad
(\Omega_{acc})_a=
(\frac{K_{\Theta}}{\Omega_{\Theta}})_a-\frac{\Omega_a}{\Omega_{\Theta}}
(\frac{K_{\Theta}}{\Omega_{\Theta}})_{\Theta}$$
As result the equation (6) take the form $$\left[\frac{(\ln\Omega_{\Theta})_{a\Theta}}{\Omega_{\Theta}}\right]_a=\Omega_{\Theta}$$ and can be integrate under the substitution $$\Omega(\Theta,a)=\Lambda_a$$
So we get the equation $$\Lambda_{\Theta\Theta}=\frac{1}{6}\Lambda_{\Theta}^3+
\alpha(\Theta)\Lambda_{\Theta}^2+
\beta(\Theta)\Lambda(\Theta)+\gamma(\Theta)$$ with arbitrary coefficients $\alpha,\beta,\gamma$.
This is Abel’s type of equation $$y'=A(x)y^3+B(x)y^2+C(x)y+D(x)$$
It can be rewriten in form $$y'=A(y-\phi)^3+\theta(y-\phi)^2+\lambda(y-\phi)+\phi'$$ or $$z'=A z^3+\theta z^2+\lambda z$$ Let us considere the examples.
1\. $\alpha=\beta=\gamma=0$
The solution of equation (8) is $$\Lambda=A(a)-6\sqrt{B(a)-\frac{1}{3}\Theta}$$ and we get $$c=A'-\frac{3B'}{\sqrt{B-\frac{1}{3}\Theta}}$$ or $$\Theta=3B-27\frac{B'^2}{(c-A')^2}$$ This solution is corresponded to the equation $$b''=-\frac{\Theta_a}{\Theta_c}=-\frac{1}{18B'}b'^3+\frac{A'}{6B'}b'^2+
(\frac{B''}{B'}-\frac{A'^2}{6B'})b'+A''+\frac{A'^3}{18B'}-\frac{A'B''}{B'}$$ cubical on the first derivatives $b'$ with arbitrary coefficients $A(a),B(a)$. This equation is equivalent to the equation $$b''=0$$ under the point transformation.
The following example is the solution of equation (8) in form $$g=b^{1-2\alpha}\omega[cb^{\alpha-1}]$$ Under this reduction one obtains the equation on the function $\omega(\xi=
cb^{\alpha-1})$ $$\omega\omega''-\frac{\omega'^2}{2}+(\alpha-1)\xi^2\omega''+
(2-3\alpha)\xi\omega'+2(2\alpha-1)\omega=0.$$ To make the new variable $\theta=\omega+(\alpha-1)\xi^2$ we obtain $$\theta\theta''-\frac{\theta'^2}{2}-\alpha\xi\theta'+2\alpha\theta=0.$$ This equation has solution in parametrical form $$\theta=\gamma\tau E(\tau), \quad \xi=\frac{\gamma E(\tau)}{\beta}$$ where $$E(\tau)=\exp[-\int \frac{\tau d\tau}{(\tau -1/2)^2 + \alpha/\beta^2-1/4}]$$ where $\beta, \gamma$ are parametrs and the explicit form of this integral depends on the value $$\epsilon=\alpha/\beta^2-1/4.$$
This solution is corresponed to the family of equations $$b''=b^{1-2\alpha}[\theta+(1-\alpha)\xi^2]$$ and (1) forming dual paar.
The values of coefficients $a_i(x,y,\alpha,\beta,\gamma)$ in corresponding equation (1) can be change radicaly at the variation of parameters as it is showed the calculation of integral (10).
Acknowledgement
===============
This work was supported by INTAS-93-0166. The V.D is grateful to the Physical Departement of Lecce University for support and kind hospitality.
[99]{} E. Cartan, [*Sur les variétés a connexion projective*]{}, Bulletin de la Société Mathémat. de France [**52**]{}, 205-241 (1924). E. Cartan, [*Sur un classe d’espace de Weyl*]{}, Ann. Sc. Ec.Normal Super., [**3e**]{}, ser. 60, 1-16, (1943). V. Dryuma, [*On Geometry of the second order differential equations*]{}, Proceedings of Conference Nonlinear Phenomena, ed. K.V. Frolov, Moskow, Nauka, 1991, 41-48. V. Dryuma, [*Projective duality in theory of the second order differential equations*]{}, Mathematical Researches, Kishinev, Stiintsa, 1990, v.112, 93–103. V. Dryuma, [*Geometrical properties of multidimensional differential equations and the Finsler metrics of dynamical systems*]{}, Theoretical and Mathematical Physics, Moskow, Nauka, 1994, v.99, no.2, 241-249. V. Dryuma, [*Geometrical properties of nonlinear dynamical systems*]{}, Proceedings of the First Workshop on Nonlinear Physics, Le Sirenuse, Gallipoli(Lecce), Italy, June 29-July 7, 1995, ed. E. Alfinito, M.Boiti, L. Martina and F. Pempinelli, World Scientific, Singapore, 1996, 83-93. Calogero F.[*A solvable nonlinear wave equation*]{}, Studies in Applied mathematics, [**LXX**]{}, N3, 189–199, (1984). Pavlov M.[*The Calogero equation and Liouville type equations*]{}, arXiv:nlin. SI/0101034, 19 Jan. 2001.
|
---
abstract: 'Previous observations with the *Rossi X-ray Timing Explorer* () have suggested that the power spectral density (PSD) of NGC 3783 flattens to a slope near zero at low frequencies, in a similar manner to that of Galactic black hole X-ray binary systems (GBHs) in the ‘hard’ state. The low radio flux emitted by this object, however, is inconsistent with a hard state interpretation. The accretion rate of NGC 3783 ($\sim 7\%$ of the Eddington rate) is similar to that of other AGN with ‘soft’ state PSDs and higher than that at which the GBH Cyg X-1, with which AGN are often compared, changes between ‘hard’ and ‘soft’ states ($\sim 2\%$ of the Eddington rate). If NGC 3783 really does have a ‘hard’ state PSD, it would be quite unusual and would indicate that AGN and GBHs are not quite as similar as we currently believe. Here we present an improved X-ray PSD of NGC 3783, spanning from $\sim 10^{-8}$ to $\sim 10^{-3}$ Hz, based on considerably extended (5.5 years) observations combined with two orbits of continuous observation by . We show that this PSD is, in fact, well fitted by a ‘soft’ state model which has only one break, at high frequencies. Although a ‘hard’ state model can also fit the data, the improvement in fit by adding a second break at low frequency is not significant. Thus NGC 3783 is not unusual. These results leave Arakelian 564 as the only AGN which shows a second break at low frequencies, although in that case the very high accretion rate implies a ‘very high’, rather than ‘hard’ state PSD. The break frequency found in NGC 3783 is consistent with the expectation based on comparisons with other AGN and GBHs, given its black hole mass and accretion rate.'
author:
- |
D. P. Summons$^{1,3}$[^1], P. Arévalo$^{1}$ , I. M. M$^{\mathrm{c}}$Hardy$^{1}$, P. Uttley$^{2}$ and A. Bhaskar$^{3}$\
$^1$School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK\
$^2$Astronomical Institute ‘Anton Pannekoek’, University of Amsterdam, Kruislaan 403, 1098 SJ, Amsterdam, the Netherlands\
$^3$School of Engineering Science, University of Southampton, Southampton SO17 1BJ, UK
bibliography:
- 'mn-jour.bib'
- 'references.bib'
date: 'Received /Accepted'
title: Timing evidence in determining the accretion state of the Seyfert galaxy NGC 3783
---
\[firstpage\]
galaxies: active – galaxies: Seyfert – galaxies: NGC 3783 – X rays: galaxies
INTRODUCTION
============
Super-massive black holes in active galactic nuclei (AGN) and Galactic stellar-mass black hole X-ray binary systems (GBHs) both display aperiodic X-ray variability which may be quantified by calculating the power spectral densities (PSDs) of the X-ray light curves. The PSDs can typically be represented by red-noise type power laws (i.e. $P(\nu)$, the power at frequency $\nu$, $\propto \nu^{\alpha}$ where $\alpha \sim$ -1) with a bend or break (to $\alpha \leq$ -2) at a characteristic PSD frequency. The time-scale, corresponding to the bend-frequency, scales approximately linearly with black hole mass from AGN to GBHs [@McHardy:1988kx; @Edelson:1999uq; @Uttley:2002zw; @Uttley:2005tq; @Markowitz:2003gm; @McHardy:2004hv; @McHardy:2005pe], albeit with some scatter. However, the scatter is entirely accounted for by variations in accretion rate, allowing scaling between AGN and GBHs on time-scales from $\sim$ years to $\sim$ ms [@McHardy:2006fk].
GBHs are observed in a number of distinct X-ray spectral states which also have distinct X-ray timing properties. Two common states are the low/hard (hereafter ‘hard’) and high/soft (hearafter ‘soft’) states. In the hard state, the energy-spectrum is dominated by a highly variable power law component and the PSDs are well fitted by multiple broad Lorentzians. For use in AGN, where signal/noise is lower than in GBHs, this PSD shape can be approximated by a double-bend power law with slopes $\alpha=0$, -1 and -2, from low to high frequency, where the high- and low-frequency bends correspond to the strongest peaks in the Lorentzian parameterisation. The breaks are typically separated by only one to two decades in frequency. In the soft state, the energy spectrum is dominated by an approximately constant thermal disc component which extends into the X-ray band in GBHs but which in AGN is shifted down to the optical/UV band. Therefore, a meaningful comparison between the PSDs of soft state GBH and AGN can only be made in cases where the GBH power-law emission is strong enough to show significant variability. Such GBHs are rare and the best example is Cyg X-1 which shows a ‘1/f’ PSD over many decades of frequencies [@Reig:2002ys]. The soft state is distinguished from the hard state by having only one, high frequency, break in this power law, from slope -1 to -2.
It has been suggested that this pure simple broken or cut-off power-law PSD shape is unique to the soft state of Cyg X-1, which is a persistent source. However in transient GBHs with similar X-ray spectra, the power law PSD component may be seen in combination with broad Lorentzian features [@Done:2005fk]. [@Axelsson:2006fk] also note that a mixed power law plus Lorentzian PSD is also present in Cyg X-1 in lower luminosity, harder spectral states, but as the luminosity rises the Lorentzian features weaken and the power law PSD component strengthens until, in the softest state, it completely dominates. Since the softest spectral states of transient GBHs are dominated by constant disc emission we cannot determine whether they show a similar PSD shape to Cyg X-1.
However, a direct comparison of transient GBHs and Cyg X-1 is complex, since the transients show much larger luminosity changes, and complex hysteresis effects in spectral hardness versus luminosity (e.g. @Homan:2001fk, @Belloni:2005uq) which are not seen in Cyg X-1. Therefore it is not clear that one can compare timing properties between Cyg X-1 and transient GBHs simply as a function of observed X-ray spectrum.
None the less, it is still interesting that the X-ray spectrum of Cyg X-1 never becomes totally disc-dominated, and always contains a relatively strong variable component whose PSD resembles that of X-ray bright AGN. If variability originates, at least partly, in the disc, so power spectral shape is related to the disc structure, that structure might be severely disrupted during outbursts, thereby suggesting a possible difference between the persistent Cyg X-1 and the transient GBH sources. The similarities between the PSDs of Cyg X-1 and AGN may also be related to the possible similarities in accretion flows between AGN and Cyg X-1 noted by [@Done:2005fk].
To date, NGC 3783 and the Narrow Line Seyfert 1 Galaxy (NLS1) Ark 564 are the only AGN with suggested second, low-frequency breaks in their PSDs (i.e. similar to low/hard GBHs) and are both commonly referred to as being unusual (e.g. @Done:2005fk). The power spectral evidence for a second break in the case of Ark 564 is very strong (@Pounds:2001nx [@Papadakis:2002fk; @Markowitz:2003gm], M$^{\rm
c}$Hardy et al. in prep.). Of all the AGN with good timing data, Ark 564 shows the highest accretion-rate (possibly super-Eddington) so it would not be surprising if it were in an unusual state, e.g. the ‘very high’ state where the PSD, in GBHs, also displays two distinct breaks. The properties of NGC 3783, on the other hand, are similar to those of AGN with proven soft-state PSDs (e.g. NGC 3227, NGC 4051 @McHardy:2004hv, MCG-6-30-15 @McHardy:2005pe), and in particular it is radio quiet (e.g. @Reynolds:1997vn). In the hard state, GBHs are strong radio sources whereas in the soft state the radio emission is quenched [@Corbel:2000zr; @Fender:2001ys; @Kording:2006ly]. We also note that NGC 3783 has a more moderate accretion rate than Ark 564 ($\sim 7
\%$), and more similar to the other AGN mentioned above, and Cyg X-1 changes from the hard to the soft state at around 2% of the Eddington accretion rate (i.e. =0.02) (@Pottschmidt:2003kx [@Wilms:2006uq; @Axelsson:2006fk]). These two facts do not lie easily with a hard state identification of NGC 3783. Thus it would be surprising, and might indicate that our current ideas regarding the scaling between AGN and GBHs are not entirely correct, if NGC 3783 were proven to have a hard state PSD. It is therefore important to determine whether NGC 3783 does have a second, low frequency, break in its PSD or not.
[@Markowitz:2003gm] recognised the presence of a break in the $2-10$ keV PSD of NGC 3783 at $4\times 10^{-6}$ Hz and found provisional evidence for a second lower-frequency break at $\sim$ 2$\times 10^{-7}$ Hz. Specifically, [@Markowitz:2003gm] rejected the possibility that the PSD is described by a single-break power law with low-frequency slope -1, similar to other AGN, at the 98% confidence level. In this paper we re-investigate the evidence for the second break in the PSD of NGC 3783, using new long-term monitoring data that covers the frequency range where the break appears to be. By including additional archival data spanning several years, along with short time-scale observations by , we will demonstrate that the improved PSD is perfectly compatible with a single-bend power law, consistent with the behaviour of the other moderately-accreting Seyferts. In Section 2 we describe the observations and the methods by which we extract the and light curves. In Sections 3 we discuss the PSD of NGC 3783 as produced from the and observations, and compare it with various PSD models. In Section 4 we briefly review the implications of our observations.
OBSERVATIONS AND DATA REDUCTION
===============================
Data Reduction
---------------
From 1999 to 2006, NGC 3783 has been the target of various monitoring campaigns with . These campaigns have consisted of short, $\sim$1 ks duration, observations with the proportional counter array (PCA, @Zhang:1993ly). We have analysed the archival PCA STANDARD-2 data and our own proprietary data with FTOOLS v6.0.2 using standard extraction methods. We use data from the top layers of PCUs 0 and 2 up to 2000 May 12 and only top layer PCU 2 data from observations after this date. The remaining PCUs were not used due to repeated breakdowns.
Data were selected according to the standard ‘goodtime’ criteria, i.e. target elevation $<10^{\circ}$, offset pointing $<0.02^{\circ}$, and electron contamination $<0.1$. The background was simulated with the L7 model for faint sources using PCABACKEST v3.0. The response matrices for each PCA observation were calculated using PCARSP v10.1. The final $2-10$ keV fluxes were calculated using XPSEC v12.2.1 by fitting a power law with galactic absorption to the PHA data.
The data used in our analysis, together with the sampling patterns, are listed in Table \[sample\] and displayed in Fig. \[longlc\]. The early data (to MJD 52375) with 4 d sampling, together with the 20 d period of 3 h sampling already presented by [@Markowitz:2003gm] are followed, after a 2 year gap, by our new long-term monitoring, with 2 d sampling. As the gap is large compared to the duration of each monitoring campaign we will include data from each monitoring campaign as separate lightcurves in our fits.
[**Light curve** ]{} [**Sampling interval**]{} [**Observation length**]{} [**Date Range \[MJD\]**]{}
-------------------------- --------------------------- ---------------------------- ----------------------------
Long-term 1 $\sim$4.36 days 1194.6 days 51180.5–52375.1
Long-term 2 $\sim$2.1 days 928.3 days 53063.4–53991.6
Intense monitoring $\sim$3.2 hours 19.9 days 51960.1–51980.1
observations (2 orbits) 200-s 3.2 days 52260.8–52264.0
Data Reduction
---------------
NGC 3783 was observed by during revolutions 371 and 372, between 2001 December 17 and 2001 December 21. Temporal analysis of these data were first presented by [@Markowitz:2005zb] who discusses the coherence, frequency-dependent phase lags, and variation of high frequency PSD slope with energy. Here we use these data to constrain the high frequency part of the overall long and short timescale PSD. We used data from the European Photon Imaging Cameras (EPIC) PN and MOS2 instruments, which were operated in imaging mode. MOS1 was operated in Fast Uncompressed Mode and we do not use those data here. The PN camera was operated in Small Window mode, using the medium filter. Source photons were extracted from a circular region of $40 \arcsec$ radius and the background was selected from a source-free region of equal area on the same chip. We selected single and double events, with quality flag=0. The MOS2 camera was operated in the Full Window mode, using the medium filter. We extracted source and background photons using the same procedure as for the PN data and selected single, double, triple and quadruple events. These data showed no serious pile-up when tested with the [*XMM-SAS*]{} task [*epatplot*]{}.
We constructed light curves, for each detector and orbit, in the 0.2–2, 2–10 and 4–10 keV energy bands. We filled in the $\sim 5$ ks gap in the middle of orbit 371 light curves, and some other much smaller gaps, by interpolation and added Poisson noise. The resulting PN and MOS2 continuous light curves were then combined to produce the final light curves for each orbit. The combined, background subtracted, average count rates in the 0.2–10 keV band were 11.8 c/s for orbit 371 and 15.8 c/s for orbit 372, and the 0.2-2 keV combined light curve is shown in Fig. \[xmmlcfig\]. Poisson noise dominates the PSD on timescales shorter than 1000s, so the light curves were binned into 200s bins.
Power spectral Modelling
========================
Combining and data
--------------------
To determine the PSD over the largest possible frequency range we combine the and data. In GBHs the break-frequency and slope of the PSD below the break appear to be independent of the chosen energy band [@Cui:1997zr; @Churazov:2001le; @Nowak:1999ys; @Revnivtsev:2000vn; @McHardy:2004hv]. On the other hand, the PSD normalisation and the slope above the break are often energy-dependent [@Markowitz:2005zb]. Therefore, when combining data from different instruments, it is preferable to use similar energy ranges. The data are in the 2–10 keV band and, for NGC 3783, that band has a median photon energy of 5.7 keV. The band with the same median photon energy is 4.1–10 keV. However the count rate in that band is low (2 c/s) so we only detect significant source power above the Poisson noise level at frequencies below $10^{-4}$Hz. To probe higher frequencies we can use the 0.2–2 keV data (8.8 c/s) but we must re-scale its PSD normalisation to that of the 4–10 keV PSD. We determined the scaling correction by producing PSDs in both energy bands and fitting the same bending power law model to the noise-subtracted data. On the assumption that the PSD shape below the high frequency break is energy-independent, the combined 2–10 keV and 0.2–2 keV PSD will then have the shape of the 0.2–2 keV PSD.
Monte Carlo simulations {#MCsims}
-----------------------
We use the Monte Carlo technique of [@Uttley:2002zw] (PSRESP), to estimate the underlying PSD parameters in the presence of sampling biases. In this method we first calculate the observed (or ‘dirty’) PSD, in parts, from the observed lightcurves, using the Discrete Fourier Transform. Here the PSD estimates are binned in bins of width 1.3$\nu$, where $\nu$ is the starting frequency by taking the average of the log of power [@Papadakis:1993xl]. We require a minimum of 4 PSD estimates per bin. We then compare simultaneously the dirty PSDs from each lightcurve with various model PSDs derived from lightcurves simulated with the same sampling pattern as the real observations. We alter the model parameters to obtain the best fit for any given model. We refer the reader to [@Uttley:2002zw] for a full discussion of the method.
For each set of chosen underlying-PSD model parameters, we simulate red-noise light curves, as described by [@Timmer:1995qj]. The light curves are simulated with time resolutions of 10.5 h, 5.0 h and 18 m for the first and second long time-scale and the medium time-scale light curves respectively. The simulated resolution, which is 10 times shorter than the typical sampling intervals of the real observations, given in column 2 of Table \[sample\], is to take into account the effect of aliasing. These simulated light curves were resampled and binned to match the real NGC 3783 observations. light curves were simulated with 200-s resolution, as at shorter time-scales the underlying varability power is negligible compared to the Poisson noise, so aliasing does not play a role. The Poisson noise level was not subtracted from the observed PSD but was added to the simulated PSDs. To reproduce the effect of red-noise leak, each light curve was simulated to be $\sim$300 times longer than the real observation, and was then split into sections, constituting 300 simulated light curves for each observed lightcurve. The simulated model average PSD is evaluated from this ensemble of PSD realisations, and the errors are assigned from the rms spread of the realisations within a frequency bin.
We present the results of several PSD model fits in an attempt to quantify the underlying model shape that best describes the PSD of NGC 3783, and associate an acceptance probability with each model. We initially test a simple unbroken power law model. We next fit a power law with a single-bend in the PSD, and then a model incorporating a double-bend. We also fit a single-bend power law with a Lorentzian component.
Unbroken power law model
------------------------
To begin, we fitted a simple power law model to the data of the form:
$$P(\nu)=A\left(\frac{\nu}{\nu_0}\right)^{\alpha}$$
where $A$ is the normalisation at a frequency $\nu_0$, and $\alpha$ is the power law slope. We made 900 simulations and in Fig. \[novb\] we show the best fit plotted in $\nu \times P_{\nu}$, which has a PSD slope of -$2.1$. However, the fit is poor and this model can be rejected with a probability $>$ 99 %, or $\gtrsim$ 3 $\sigma$.
Single-bend power law model
---------------------------
Here we fit a single-bend power law to the data. This model best describes the PSD of Cyg X-1 in the high/soft state, and provides a good fit to the PSDs of the AGN NGC 4051 and MCG–6-30-15 [@McHardy:2004hv; @McHardy:2005pe].
$$P(\nu)=\frac{A\:\nu^{\alpha_L}}{1+\left(\frac{\nu}{\nu_B}\right)^{\alpha_L-\alpha_H}}$$
Fig. \[sinpsd\] presents the observed PSD fitted with a single-bend power law model, for which a good likelihood of acceptance is obtained (P = 44 %). The best fit bend-frequency is $\nu_B=6.2^{+40.6}_{-5.6}\times 10^{-6}$ Hz, the high-frequency slope is $\alpha_H=-2.6^{+0.6}_{-*}$, and the low-frequency slope is $\alpha_L=-0.8^{+*}_{-0.5}$. The errors are 90% confidence limits, an asterisk indicates that the limit is unconstrained. For $\alpha_H$ the best fit value is well within the searched parameter space but the degeneracy produced by red-noise leak in the probability at high values of $\alpha_H$, means that the upper limit is not constrained at the 90% confidence level. The confidence contours for the main interesting parameters are plotted in Figs. \[singlecon13\] and \[singlecon14\]. Table \[results\] shows the single-bend power law best fit parameters to the data. The best fit single-bend frequency obtained here is consistent with the value found by [@Markowitz:2003gm]
--------------- ----------------------- ---------------------- ---------------------- ---------------------- ------------------------------------ ------------------------------- --------------------
[**Model**]{} [**Normalisation**]{} [**$\alpha_{H}$**]{} [**$\alpha_{I}$**]{} [**$\alpha_{L}$**]{} [**$\nu_H$**]{} [**$\nu_L$**]{} [**Acceptance**]{}
[($a$)]{} [(Hz)]{} [(Hz)]{} [(%)]{}
Single-bend $1.5\times 10^{-4}$ $-2.6^{+0.6}_{-1.0}$ NA $-0.8^{+0.8}_{-0.5}$ $6.2^{+40.6}_{-5.6}\times 10^{-6}$ NA 44.4
Double-bend $1.0\times 10^{2}$ $-3.2^{+1.2}_{-*}$ $-1.3^{+*}_{-*}$ 0.0 $2.6^{+*}_{-*}\times 10^{-5}$ $1.7^{+*}_{-*}\times 10^{-7}$ 63.9
--------------- ----------------------- ---------------------- ---------------------- ---------------------- ------------------------------------ ------------------------------- --------------------
Double-bend power law model
---------------------------
[@Markowitz:2003gm] provide tentative evidence that a second, lower, frequency break exists in the PSD of NGC 3783. Thus, we also fitted a more complex double-bend power law model to see if the goodness-of-fit is improved significantly. The double-bend power law model is given by:
$$P(\nu)=\frac{A\:\nu^{\alpha_L}}{\left[1+\left(\frac{\nu}{\nu_L}\right)^{\alpha_L-\alpha_I}\right]\left[1+\left(\frac{\nu}{\nu_H}\right)^{\alpha_I-\alpha_H}\right]},$$
where $\alpha_I$ is the intermediate-frequency slope and $\nu_L$ and $\nu_H$ are the low and high bend-frequencies respectively. We fixed the low-frequency slope to zero, to avoid making the simulations computationally prohibitive, and because a low-frequency slope of zero would allow the best qualitative comparison to the low state of Cyg X-1 [@Nowak:1999ys].
Fig. \[doupsd\] presents the same observed PSD as in Fig. \[sinpsd\], but fitted with the double-bend power law model. A good likelihood of acceptance is obtained (P=64 %). The best-fitting high bend-frequency is $\nu_H=2.6^{+*}_{-*}\times 10^{-5}$ Hz, the high-frequency slope is $\alpha_H=-3.2^{+1.2}_{-*}$, the intermediate-frequency slope is $\alpha_I=-1.3^{+*}_{-*}$, the low-frequency bend is $\nu_L=1.7^{+*}_{-*}\times 10^{-7}$ Hz. As before, we use 90% confidence limits. The added parameters allow extra freedom to find better fit probabilities for any given set of double-bend parameters. For this reason, the contour levels cover larger ranges in the parameter space and therefore, most of the 90% contours in our double-bend fit remain unbounded over the fitted parameter space. The high-frequency slope is subject to the same problems as in the single-bend model. Table \[results\] contains a summary of the best-fitting model parameters.
The best-fitting low-frequency bend is found close to the lowest frequencies probed by the data and, as seen in Fig. \[doublecon12\], it is essentially unbounded down to the lowest measurable frequency at the 68% confidence level. These facts suggest that the second, low-frequency, bend might not be required by the data and that the improvement in the fit might be only due to the increased complexity of the model fitted.
The likelihood of acceptance is better in the double-bend model than in the single-bend model, 64 versus 44 % respectively, but there are more free parameters. In order to determine the significance of this improvement, we performed the following test. Using the best-fitting single-bend PSD parameters, we generated 300 realisations of the sets of and lightcurves. Each realisation was then fitted with the best-fitting double-bend parameters, exactly as was done with the real data, and the distribution of their fit probabilities was constructed. We found that 121 out of the 300 single-bend simulations have a higher fit probability than the real data, when fitted with the double-bend model. Therefore, we conclude that the improvement in fit probability is no more than may be expected from fitting a model which is more complicated than required by the data: the double-bend model does not represent a significant improvement.
Single-bend power law with a Lorentzian component
-------------------------------------------------
We finally consider whether the observed PSD might be best-described by adding a Lorentzian component, such as are commonly used to describe broad-band noise components in GBHs [e.g. @Nowak:2000qf], to the single-bend power law. We are motivated to consider this possibility because the PSD of the intense-sampling light curve is not very well described by either the single- or double-bend power law model. Visual inspection of this light curve, shown in Fig. \[intlc\], suggests that the variability is strongly concentrated on time-scales of around a day, or equivalently, frequencies around $10^{-5}$ Hz, which is confirmed by the peak seen in the corresponding section of the PSD, and the drop in the same PSD at lower frequencies ($\sim 10^{-6}$Hz). The long-term monitoring PSDs, however, do not show a dip at $10^{-6}$Hz, creating a large discrepancy in the PSD measurements at this frequency. A strongly peaked component in the underlying PSD, at $\sim10^{-5}$ Hz, could produce the observed features. Such a component would appear as a peak in a PSD that covered frequencies above and below its peak-frequency, but would be insufficiently sampled by the long-term monitoring campaigns; thus, its power would be aliased into the highest frequencies of the longer time-scale data, making them rise above the underlying model level and causing the apparent disparity.
The Lorentzian profile is described by:
$$P_{\rm Lor}(\nu)=\frac{A Q \nu_c}{\nu_c^2+4Q^2(\nu_c-\nu)^2},$$
where the centroid frequency $\nu_c$ is related to the peak-frequency $\nu_p$ by $\nu_p=\nu_c\sqrt{1+1/4Q^2}$ and the quality factor Q is equal to $\nu_c$ divided by the full width at half maximum of the Lorentzian. The variable $A$ parameterizes the relative contribution of the power law and Lorentzian components to the total rms. Fitting a Lorentzian component in addition to a single-bend power law provides a good fit (P=52 %). The best-fitting Lorentzian contributes 20% of the variance in the frequency range probed and its best-fitting parameters are quoted in Table \[results\_lor\]. Fig. \[lorvfv\] shows the observed PSD compared with the best-fitting single-bend power law model plus a Lorentzian component. The Lorentzian feature in the model can reproduce qualitatively the spurious power at the high frequency end of the long-term monitoring data and the turn down effect observed in the intensive-sampling data.
--------------------------------- ----------------------------------- ------------------- ------------------- ---------------------- ---------------------- --------------------
[**$\nu_p$**]{} [**$\nu_B$**]{} [**$Q$**]{} [**$A$**]{} [**$\alpha_{L}$**]{} [**$\alpha_{H}$**]{} [**Acceptance**]{}
[(Hz)]{} [(Hz)]{} [(%)]{}
$4.8^{+*}_{-0.8}\times 10^{-6}$ $1.1^{+0.6}_{-0.4}\times 10^{-5}$ $5.1^{+*}_{-3.6}$ $0.9^{+*}_{-0.7}$ $-1.0^{+*}_{-*}$ $-2.6^{+*}_{-*}$ [52.3]{}
--------------------------------- ----------------------------------- ------------------- ------------------- ---------------------- ---------------------- --------------------
To determine the significance of the Lorentzian component fit we repeated the procedure used in determining the significance of the double-bend model. We found that 222 of the 300 single-bend simulated PSDs have a higher fit probability than the data, when fitted with the single-bend power law plus Lorentzian model. This result indicates that the increase in fit probability could be due to the added complexity of the model, and that the improvement in the fit over a simple bending power law is not significant.
Discussion and Conclusions
==========================
We have combined our own new monitoring data with archival and observations to construct a high-quality PSD of NGC 3783 spanning five decades in frequency.
We find that a ‘soft’ state model, with a single bend at $6.2\times10^{-6}$ Hz, similar to that found earlier by [@Markowitz:2003gm], a power law of slope approximately -$0.8$ extending over almost three decades in frequency below the bend, and slope above the bend of approximately -2.6 is a good fit to the data. We also find that a ‘hard’ state model, with a double bend, fits the data, as does a model with a single bend plus an additional Lorentzian component. However the improvement in fit is marginal and, given the additional free parameters, is not significant. Thus we conclude that a simple ‘soft’ state model provides the most likely explanation of the data.
Assuming a mass of $3\times 10^{7} M_\odot$ for NGC 3783 [@Peterson:2004ve], and an accretion rate of 7% of the Eddington limit (@Uttley:2005bh, based on @Woo:2002cr), then NGC 3783 is still in good agreement with the scaling of PSD break timescale as $\sim M/\dot{m}_{E}$ between AGN and GBHs found by [@McHardy:2006fk].
Our new fits, show that the PSD of NGC 3783 is perfectly consistent with a single-bend power law with low-frequency slope of -1, in contrast with the earlier result of [@Markowitz:2003gm], who found that a similar model was rejected tentatively at $\sim98$% confidence. The difference can be understood in terms of the improved long-term data. Our new monitoring observations occur every 2 days, compared to 4 days previously, thereby increasing the long term data set by a factor 2.6 and, in particular, providing overlap at high frequencies with the intensive monitoring data. The drop in long-timescale variability power, evident in the older long term monitoring data is not reproduced by the new long-term monitoring data, showing that this drop could be just a statistical fluctuation. In addition, the very high frequencies are better constrained by the 2 orbits of data than by the earlier data used by [@Markowitz:2003gm].
The classification of the PSD as being ‘soft’ state means that NGC 3783 is no longer considered unusual amongst AGN. The fact that this AGN is radio-quiet strongly supports the analogy with GBHs in the soft state. Also the accretion rate of NGC 3783 (=0.07) (@Uttley:2005bh, based on @Woo:2002cr) is similar to that of other AGN with soft-state PSDs (e.g. NGC 3227 @Uttley:2005bh, NGC 4051 @McHardy:2004hv, MCG-6-30-15 @McHardy:2005pe). This accretion rate is above the rate at which the persistent GBH Cyg X-1 transits between hard and soft states in either direction and at which other GBHs transit from the soft to hard state (=0.02) [@Maccarone:2003uq; @Maccarone:2003kx]. We note that other transient GBHs in outburst, where the variable power law emission in the soft state PSD is weak, can remain in the hard state to much higher accretion rates ($\sim$ 2–50% @Homan:2005fk) but it is not clear whether we should expect similar PSD shapes to AGN for such outbursting sources. Thus NGC 3783 remains compatible with other moderately accreting AGN in being analogous to Cyg X-1 in the soft state. It is, of course, possible that the transition rate might not be independent of mass. Observations do not yet greatly constrain the transition rate as a function of mass but the abscence of large deviations from the so-called ‘fundamental’ plane of radio luminosity, X-ray luminosity and black hole mass [@Merloni:2003kx; @Falcke:2004ko] argues against a large spread in the transistion accretion-rates (e.g. see @Kording:2006fk). In the case of Seyfert galaxy NGC 3227, the accretion rate is $\sim$ 1–2% and a ‘soft’ state PSD is measured [@Uttley:2005bh], which suggests that the transition accretion-rate in AGN should be at or below that value.
Our observations, which show that NGC 3783 does not have a highly unusual PSD, therefore confirm the growing similarities between AGN and Galactic black hole systems and leave only Arakelian 564, which is probably a very high state object, as the only AGN showing clear double breaks (or multiple Lorentzians) in its PSD (e.g. @Arevalo:2006dq, M$^{\rm c}$Hardy et al. in prep.).
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank the referee, Chris Done, for useful comments and suggestions. This research has made use of the data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA’s Goddard Space Flight Center. We would like to thank Information Systems Services (ISS) at the University of Southampton for the use of their Beowulf cluster, *Iridis2*. PU acknowledges support from a Marie Curie Inter-European Research Fellowship.
\[lastpage\]
[^1]: E-mail: dps@astro.soton.ac.uk
|
---
abstract: 'We have conducted a systematic investigation of the origin and underlying physics of the line–line and line–continuum correlations of AGNs, particularly the Baldwin effect. Based on the homogeneous sample of Seyfert 1s and QSOs in the SDSS DR4, we find the origin of all the emission-line regularities is Eddington ratio ([$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{}). The essential physics is that [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{} regulates the distributions of the properties (particularly column density) of the clouds bound in the line-emitting region.'
author:
- 'Xiaobo Dong, Jianguo Wang, Tinggui Wang, Huiyuan Wang, Xiaohui Fan, Hongyan Zhou, Weimin Yuan, and Qian Long'
title: The origin and physical mechanism of the ensemble Baldwin effect
---
Baldwin effect and its three 2nd-order effects
==============================================
Active galactic nuclei (AGNs; including QSOs) are essentially a kind of radiation and line-emitting systems powered by gravitational accretion onto suppermassive black holes. The global spectra of AGNs are remarkably similar, emission lines with similar intensity ratios sitting atop a blue continuum that has a similar slope among AGNs regardless of their luminosities and redshifts (Davidson & Netzer 1979, Korista 1999). However, this similarity is only a zeroth-order approximation; in 1977, Baldwin found that, among the QSO ensemble, the equivalent width (EW) of the [CIV]{} $\lambda$1549 emission line correlates negatively with the continuum luminosity (Baldwin 1977). From then on, such a negative EW–L correlation has been found for almost all emission lines in the ultraviolet and optical bands, and has been termed as “Baldwin effect” (hereafter BEff; see Osmer & Shields 1999, Sheilds 2007 for reviews).[^1] Furthermore, also found are the three kinds of 2nd-order effects of the BEff, namely, the dependence of the BEff slope (the slope of the logEW–logL relation) on luminosity, on ionization energy, and, particularly, on velocity. The [CIV]{} BEff is stronger in the peak and red wing than in the blue (Francis & Koratkar 1995, Richards et al. 2002)! The velocity dependence betrays the nature of the BEff: this kind of negative correlation relates to the line-emitting gas gravitationally bound in the broad-line region (BLR).
The origin and the physical mechanism
=====================================
In order to explore the origin and the underlying mechanism of the BEff, based on the homogeneous sample of 4178 $z\leq 0.8$ Seyfert 1s and QSOs with median spectral S/N $\gtrsim 10$ per pixel in the SDSS DR4, we have conducted a systematic investigation of the line–line and line–continuum correlations for broad and narrow emission lines in the near-UV and optical, from [MgII]{} $\lambda 2800$ to [\[OIII\]]{} $\lambda 5007$. Our findings are as follows:\
(i) The strongest correlations of almost all the emission-line intensity ratios and EWs are with [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{}, either positively (e.g. [FeII]{} EW) or negatively (e.g. [MgII]{} EW), rather than with $L$ or [$M_\mathrm{BH}$]{}; besides, generally intensity ratios have tighter correlations with [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{} than the EWs of the related lines.\
(ii) The intensity ratios of [FeII]{} emissions – both narrow and broad – to [MgII]{} have very strong, positive correlations with [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{}; interestingly enough, (narrow [FeII]{} $\lambda 4570$)/[MgII]{} has a stronger correlation with [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{} than the optical and UV (broad [FeII]{})/[MgII]{}, with Spearman [$r_{\rm \scriptscriptstyle S}$]{}= 0.74 versus 0.58 (optical) and 0.46 (UV); see Fig. 1 (Dong et al. 2009a).
These findings argue that Eddington ratio ($\ell \equiv {\ensuremath{L/{\ensuremath{L\mathrm{_{Edd}}}}}}$)[^2] is the origin of the BEff, as of other regularities of almost all emission lines (e.g., the [FeII]{}–[\[OIII\]]{} anticorrelation, Boroson & Green 1992). This once has been suggested by Baskin & Laor (2004) and Bachev et al. (2004) for the [CIV]{} BEff. We propose that the underlying physics is certain self-regulation mechanisms caused by (or corresponding to) [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{}; these mechanisms maintain the normal dynamically quasi-steady states of the gas surrounding the central engine of AGNs (Dong et al. 2009a,b). Briefly, the essential one is that *there is a lower limit on the column density ([$N_\mathrm{H}$]{}) of the clouds gravitationally bound in the AGN line-emitting region, set by [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{}* (hereafter the [$N_\mathrm{H}$]{}–[$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{} mechanism; see also Fig. 1 of Fabian et al. 2006, Marconi et al. 2008). As [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{} increases, the emission strength decreases for high-ionization lines (e.g. [CIV]{}) and optically thick lines that are emitted at the illuminated surface (e.g. [Ly$\alpha$]{}) or in the thin transition layer (e.g. [MgII]{}) of the BLR clouds; for low-ionization, optically thin lines such as [FeII]{} multiplets that originate from the volume behind the Hydrogen ionization front (i.e., from the ionization-bounded clouds only), as [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{} increases the emission strength increases.[^3] This is schematically sketched in Fig. 2.
The implications
================
$\mathcal{I.}$ An implication is that BG92’s PC1, if only the spectral correlations in the UV–optical are concerned, shares the same origin with PC2 that is exactly the HeII BEff. A lesson is that we should be more cautious about the premises of blind source separation methods such as Principal Component Analysis.
$\mathcal{II.}$ As suggested insightfully by G. Richards (e.g. Richards 2006), *the [CIV]{} line blueshifting (in other words, blue asymmetry) is the same phenomenon of BEff*. The underlying physical picture is clear now: There are two components in the [CIV]{} emission, one arising from outflows and the other from the clouds gravitationally bound in the BLR; the fraction of bound clouds that optimally emit [CIV]{} line decreases with increasing [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{} according to the [$N_\mathrm{H}$]{} – [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{} mechanism.
$\mathcal{III.}$ If the observed large scatter of [FeII]{}/[MgII]{} at the same redshift is caused predominately by the diversity of [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{}, then once this systematic variation is corrected according to the tight [FeII]{}/[MgII]{} – [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{} correlation, it is hopeful to still use [FeII]{}/[MgII]{} as a measure of the Fe/Mg abundance ratio and thus a cosmic clock (*at least in a statistical manner*).
Appendix: Not *Baldwin Effect*, but *ell Effect*? {#appendix-not-baldwin-effect-but-ell-effect .unnumbered}
=================================================
This Appendix is to present more results taken from Dong et al. (2009b) that did not appear in the proceedings paper due to page limit. The aim is to show that the traditional BEff – the dependence of emission-line EWs on luminosity – is likely not to be fundamental, but derived from the dependence on Eddington ratio ($\ell$). To avoid any confusion, below we call the latter ell effect (*Ell* stands for $\ell$, Eddingtion ratio).
The ell effect is the 1st-order small variation to the 0th-order global similarity of QSO spectra that is well explained by “locally optically-emitting clouds” (LOC) photoionization modeling (Baldwin et al. 1995). As discussed above, in the study of the QSO emission-line correlations, the focus seems to be shifting from the physics (*microphysics*) mainly of the accretion process to the ‘statistical physics’ (*macrophysics*) of the surrounding clouds (Korista 1999; Korista, private communication). With more realistic constraints to be accounted for \[e.g., the distribution function of the cloud number with $\ell$ and ${\ensuremath{N_\mathrm{H}}}$, $N_{\rm c}(\ell, {\ensuremath{N_\mathrm{H}}}(\ell)\,)$; cf. Fig. 2\], the 1st-order regularities and even the 2nd-order effects of QSO emission lines (see §1) may be reproduced exactly by future LOC modeling.
Fig. 3 confirms that $\ell$ is the primary driver of the BEff of [MgII]{} $\lambda 2800$. Fig. 4 shows that, at the 0th-order approximation, [MgII]{} luminosity is directly proportional to the continuum luminosity, exactly as predicted by photoionization theory; at the 1th-order approximation, for different $\ell$ the proportional coefficient is different, $ \log k \simeq \log k_0 + k' \cdot \log \ell $ — this is just the ell effect illustrated in Fig. 3c.
Some researchers once have found that the slopes of the emission line versus continuum luminosity relations in the log–log scale is not unity (see references in §2 of Shields 2007). We must note that this is likely not to be intrinsic (see Dong et al. 2009b for a detailed investigation). It is caused partly by selection effect inherent in any magnitude-limited sample, with high-luminosity objects having higher $\ell$ and thus smaller EWs. For optical emission lines particularly (e.g. [H$\beta$]{}), this is mainly caused by the contamination of the host-galaxy starlight (cf. Croom et al. 2002). The starlight contamination aggravates gradually towards longer wavelengths; moreover, within a fixed aperture, it aggravates with decreasing AGN luminosity.
In one word, a sole fundamental parameter, $\ell$, well regulates the ordinary state of the surrounding gas that is inevitably inhomogeneous and clumpy (‘clouds’).
![ Plots of the intensity ratios of the broad and narrow [FeII]{}emissions to [MgII]{} $\lambda 2800$ versus Eddington ratio for the homogeneous sample of 4178 Seyfert 1s and QSOs. Also plotted are the best-fitted linear relations in the log–log scale (Dong et al. 2009a). ](Xiaobo_Dong_fig1.eps){width="100.00000%"}
![Schematic sketches of the distribution functions of the total illuminated surface area $S_\mathrm{c}$ with [$N_\mathrm{H}$]{} (panel a) and of the total volume $V_\mathrm{c}$ with [$N_\mathrm{H}$]{} (panel b) of the clouds bound in the broad-line region and inner narrow-line region of AGNs. The sketches are plotted assuming the distribution of the cloud number to be $\mathrm{d}\,N_\mathrm{c}\,/\,\mathrm{d}\,{\ensuremath{N_\mathrm{H}}}= A(\ell)\, {{\ensuremath{N_\mathrm{H}}}}^{-\gamma}$ with $\gamma = 3.2$; the lower-limit [$N_\mathrm{H}$]{} cutoff is set by the [$N_\mathrm{H}$]{}–[$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{} mechanism, roughly scaling with [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{}. Note that the strengths of high-ionization lines (e.g. [CIV]{}) and optically-thick lines (e.g. [Ly$\alpha$]{} and [MgII]{}) are roughly proportional to $S_\mathrm{c}$ while that of low-ionization, optically-thin lines such as narrow-line and broad-line [FeII]{}roughly proportional to $V_\mathrm{c}$ (Dong et al. 2009b). ](Xiaobo_Dong_fig2.eps){width="100.00000%"}
![ The relation between the equivalent width of [MgII]{} $\lambda 2800$ of the 2092 type-1 AGNs in the $z>0.45$ subsample and their $\lambda L_{\lambda}$(2500Å), black hole mass ([$M_\mathrm{BH}$]{}), and Eddington ratio ($\ell \equiv {\ensuremath{L\mathrm{_{bol}}}}/{\ensuremath{L\mathrm{_{Edd}}}}$).](Xiaobo_Dong_fig3.eps){width="100.00000%"}
![ The log–log plot of [MgII]{} versus continuum luminosity relation for the 2092 objects. The green line is the 1-to-100 relation to aid the eye. Inset is the plot of the same data, with objects having ${\ensuremath{M_\mathrm{BH}}}< 5 \times 10^7$ [$M_{\odot}$]{} denoted as pink and objects having ${\ensuremath{M_\mathrm{BH}}}> 5 \times 10^8$ [$M_{\odot}$]{} navy-blue. Note that $L_{\rm MgII} = k(\ell) \, L_{2500}
\simeq k_0 L_{2500} \, \ell ^{k'} $. The 0th-order term, $k_0$, can be calculated by the classical LOC photoionization of Baldwin et al. (1995); the 1st-order term, $\ell ^{k'}$, by the ell effect (cf. Fig. 3c). See Dong et al. (2009b) for details. ](Xiaobo_Dong_fig4.eps){width="100.00000%"}
DXB thanks Kirk Korista, Martin Gaskell, Aaron Barth, Alessandro Marconi, Philippe Véron, Daniel Proga and Xueguang Zhang for the helpful discussions and comments, thanks Zhen-Ya Zheng for the help in improving IDL figures, and thanks Sheng-Miao Wu, Lei Chen and Fu-Guo Xie for their warm hospitality and brainstorming discussions during my visits in Shanghai Observatory. This work has made use of the data of the Sloan Digital Sky Survey (SDSS). The SDSS Web Site is http://www.sdss.org/. This work is supported by Chinese NSF grants NSF-10533050, NSF-10703006 and NSF-10728307, the CAS Knowledge Innovation Program (Grant No. KJCX2-YW-T05), and a National 973 Project of China (2007CB815403).
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Dong, X., Wang, J., Wang, T., Wang, H., Fan, X., Zhou, H., Yuan, W. 2009a, arXiv:0903.5020 Dong, X., Wang, J., Wang, T., Wang, H., Fan, X., Zhou, H., Yuan, W., Qian, L. 2009b, to be submitted
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Richards, G. T. 2006, arXiv:astro-ph/0603827 (talk presented at the “AGN Winds in the Caribbean” Workshop, St. John, USVI; 28 November - 2 December, 2005; http://www.nhn.ou.edu/ leighly/VImeeting/ )
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[^1]: To distinguish the BEff in the ensemble from a similar correlation present in individual variable AGNs, the former is usually called “the ensemble BEff” that is the topic of this paper.
[^2]: Eddington ratio is the ratio between the bolometric and Eddington luminosities. Eddington luminosity ([$L\mathrm{_{Edd}}$]{}), by definition, is the luminosity at which the gravity of the central source acting on an electron–proton pair (i.e. fully ionized gas) is balanced by the radiation pressure due to electron Thomson scattering; ${\ensuremath{L\mathrm{_{Edd}}}}= 4 \pi G c M {\ensuremath{m_{\rm p}} }/ {\ensuremath{\sigma_{\rm \scriptscriptstyle T}}}$, where $G$, $c$, $M$, [$m_{\rm p}$ ]{}, [$\sigma_{\rm \scriptscriptstyle T}$]{}are the gravitational constant, speed of light, mass of the central source, proton mass, Thomson scattering cross-section, respectively. In accretion-powered radiation systems, [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{} is often referred to as *dimensionless accretion rate $\dot{m}$* (the relative accretion rate normalized by Eddington accretion rate $\dot{M}_{\rm Edd}$, $\dot{m} \equiv \dot{M}/\dot{M}_{Edd} = \eta c^2 \dot{M}/ {\ensuremath{L\mathrm{_{Edd}}}}$, $\dot{M}$ being mass accretion rate and $\eta$ the accretion efficiency) as $\dot{m}$ is not an observable; yet the two notations are different both in meaning and in scope of application. Even in the accretion-powered radiation systems like AGNs, [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{} ($L$) is not equivalent to $\dot{m}$ ($\dot{M}$) except in the simple thin accretion disk model of Shakura & Sunyaev (1973). Therefore, we would rather call [$L/{\ensuremath{L\mathrm{_{Edd}}}}$]{}*dimensionless luminosity* ($\ell$).
[^3]: Certainly, for particular radiation systems that are powered by gravitational accretion (e.g. AGNs), $\ell$ is linked tightly to $\dot{m}$ anyway (cf. Footnote 2; Merloni & Heinz 2008). Thus there is an additional effect associated with $\ell$ yet directly via $\dot{m}$ as follows. The increase in $\ell$ means, meanwhile, the increase in $\dot{m}$, the gas supply. Reasonably, it is from the supplied gas spiraling into the central engine that (at least a significant part of) the line-emitting clouds originate; this is particularly true for the BLR and inner NLR clouds that are located between the torus (as the fuel reservoir) and the accretion disk (e.g. Gaskell & Goosman 2008). Hence, as $\ell$ increases, the total mass of line-emitting gas increases.
|
---
abstract: 'Single pion leptoproduction in the region of the $(3,3)$ resonance is currently of high interest for at least two reasons: (i) These reactions constitute an important part of the total cross section in low energy reactions and are utilized to detect neutrino oscillations in current and future long baseline experiments. (ii) Intranuclear rescattering of the pions in heavy nuclei results in interesting and sizable modifications of the free nucleon cross sections which are testable in electroproduction experiments. In this article we give a basic introduction to the pion multiple scattering model of Adler, Nussinov, and Paschos (ANP) with special emphasis on pion absorption. We also estimate the probability of multiple scattering.'
address:
- 'Theoret. Physik II, Univ. Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany'
- 'Theoret. Physik III, Univ. Dortmund, 44221 Dortmund, Germany'
author:
- 'I. Schienbein and J.-Y. Yu'
title: Pion absorption and rescattering in the ANP model revisited
---
Introduction {#sec:intro}
============
Neutrino nucleus scattering will be an important tool in future long baseline (LBL) experiments to precisely measure neutrino parameters like neutrino masses and mixing angles. These experiments will use neutrino beams with energies of the order ${\ensuremath{{\cal O}}}(1\ {\ensuremath{\mathrm{GeV}}})$ where quasi-elastic reactions (QE), single pion resonance production (RES) and deep inelastic scattering (DIS) are all important. Moreover, due to the small neutrino cross sections heavy targets like oxygen, argon or iron have to be used.
In the following we will deal with single pion resonance production, i.e., with the reactions $$\nu +T\rightarrow l +T^\prime +\pi^{\pm,0}
\label{eq:reaction}$$ where $T$ is a nuclear target ($_8O^{16},\, _{18} Ar^{40},\, _{26} Fe^{56}$) and $T^\prime$ a final nuclear state.
At this stage we make the basic assumption that the reaction can be described by two [*independent*]{} steps [@Adler:1974qu]:
- \
In this first step a pion is produced by the scattering of the incoming neutrino off a nucleon in the target, such that nuclear corrections due to the Pauli principle and the Fermi motion of the nucleons should be taken into account.
- \
Once the pion has been produced it will travel through the nucleus. During this journey the pion can have several rescatterings in which it can change its charge or be absorbed.
Step 2 can be described by a $3 \times 3$ charge exchange matrix $M$ which depends only on the properties of the target \[modeled by a charge density profile $\rho(r)$\] but which is independent of the identities of the leptons in step 1. It should be noted that the above assumption of two independent steps generates predictive power, since the formalism can be applied to charged current (CC) and neutral current (NC) neutrino- and electroproduction of pions in the resonance region.
Absorbing the Pauli suppression factor of step 1 into the normalization of $M$, the measurable final distributions (cross sections) of pions ${\ensuremath{(\pi^+,\pi^0,\pi^-)_f}}$ can be related to the initial distributions ${\ensuremath{(\pi^+,\pi^0,\pi^-)_i}}$ for a [*free*]{} target in the simple form $${\underbrace{\left(\begin{array}{c}\displaystyle
{{\ensuremath{{\operatorname{d}}}}\sigma(_ZT^A;{\pi^+})\over {\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(_ZT^A;{\pi^0})\over
{\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(_ZT^A;{\pi^-})\over
{\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}
\end{array}\right)}_{\rm nuclear \,target}}
={{M}
{\underbrace{\left(\begin{array}{c}\displaystyle
{{\ensuremath{{\operatorname{d}}}}\sigma(N_T;{\pi^+})\over {\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(N_T;{\pi^0})\over {\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}\\
\displaystyle{{\ensuremath{{\operatorname{d}}}}\sigma(N_T;{\pi^-})\over {\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}
\end{array}\right)}_{\rm free\, nucleon}}}
\label{eq:fac}$$ with $${{\ensuremath{{\operatorname{d}}}}\sigma({N_T};\pm 0)\over {\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}
= {{Z}{{\ensuremath{{\operatorname{d}}}}\sigma({p};\pm 0)
\over {\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}
+ {(A-Z)}{{\ensuremath{{\operatorname{d}}}}\sigma({n};\pm 0)
\over {\ensuremath{{\operatorname{d}}}}Q^2{\ensuremath{{\operatorname{d}}}}W}}
$$ where the free nucleon cross sections are averaged over the Fermi momentum of the nucleons. The matrix $M$ depends on the target material and the final state kinematic variables, i.e. $M = M[T; Q^2,W]$. In the following we will restrict the discussion to isoscalar targets and refer the reader to ref. [@Adler:1974wu] for targets with a neutron excess. The charge exchange matrix $M$ for isoscalar targets can be parameterized by three independent parameters $A_p$, $d$, and $c$: $${M} = { A_p} \left(\begin{array}{ccc}
{1-{ c} - { d}} & { d} & { c} \\
{ d} & {1-2 { d}} & { d} \\
{ c} & { d} & {1-{ c} - { d}}
\end{array}\right).
\label{eq:M1}$$ So far there is no direct experimental information on these three parameters available. On the other hand a model for pion multiple scattering by Adler, Nussinov and Paschos (ANP model) [@Adler:1974qu] allows to calculate these parameters and (averaged) charge exchange matrices for oxygen, argon and iron targets can be found in [@Paschos:2000be].
These matrices are useful to calculate CC and NC single pion production in the (3,3) resonance region induced by neutrinos [@Paschos:2000be; @Paschos:2001np; @Paschos:2002mb; @prsy] and electrons [@psy] and the rescattering effects are predicted to be quite large. For example, the pion energy spectra for $\pi^0$’s produced on oxygen targets in NC neutrino production will be reduced by up to $40 \%$ whereas the analogous spectra for the charged pions remain almost unchanged compared to the free nucleon cross sections [@Paschos:2000be]. Similar results are predicted for the $W$-distributions for 1-pion resonance production in electron-oxygen scattering [@psy]. For the $\pi^0$ the $W$-distribution will be reduced by about $50 \%$ while the production of $\pi^+$ remains almost unchanged compared to the free case since the decrease due to absorption is widely compensated by an increase due to charge exchange. This marked signal of rescattering effects should be easily testable in low energy electron scattering experiments, for example at JLAB.
The rest of this paper is organized as follows. In the next section we give an introduction to the ANP model. In Sec. \[sec:norm\] we discuss the normalization of pion absorption which we fix by experimental information. In Sec. \[sec:dynamics\] we discuss the dynamics of the ANP model in more detail. Finally in Sec. \[sec:summary\] we summarize the main results.
The ANP model
=============
In this section we introduce the multiple scattering model by Adler, Nussinov and Paschos [@Adler:1974qu]. Due to space limitations the presentation will be short and we refer the reader to Ref. [@Adler:1974qu] for a thorough discussion of the omitted details.
The main ingredients of the model are as follows: (i) The target nucleus is taken to be a collection of independent nucleons distributed spatially according to a charge density profile $\rho(r)$. (ii) The nucleons are regarded to be fixed within the nucleus neglecting Fermi motion and nucleon recoil effects. This implies that the pion energy $E_\pi$ does not change in the elastic scatterings. Furthermore the target is assumed to stay isotopically neutral ($Z=N$) during the multiple scattering processes. (iii) The pion interactions in the nucleus are assumed to be incoherent $\pi N$ reactions taking place in the $\Delta$ resonance region. Since pion production and more complex channels are closed in this region the rescattering processes can be described by two cross sections, the pion absorption cross section per nucleon ${\ensuremath{\sigma_{\rm abs}}}(W)$ and the usual elastic $\pi N$ cross sections. The relevant $\pi N$ reactions read $$\begin{gathered}
\pi^+ + N \to \pi^+ + N ,\quad \pi^+ + n \to \pi^0 + p ,
\\
\pi^0 + N \to \pi^0 + N ,\quad
\pi^0 + p \to \pi^+ + n ,
\\
\pi^0 + n \to \pi^- + p ,\quad
\pi^- + N \to \pi^- + N ,
\\
\pi^- + p \to \pi^0 + n,\\
\pi^{\pm,0} + N \to X ,\quad (\pi \not\in X)
{(\rightarrow {absorption})}\ .\end{gathered}$$ (iv) In the $\Delta$ resonance region the $\pi N$ cross section is dominated by the isospin $I=3/2$ amplitude such that all elastic cross sections are related to $\sigma_{\pi^+ p}(W)$ by Clebsch-Gordan coefficients. Let $${\ensuremath{\vec{q}_i}}=
\big(n_i({ \pi^+}),n_i({ \pi^0}),n_i({ \pi^-})\big)^T$$ denote the initial multiplicity distribution of pions in the medium (with fixed pion energy). In a single elastic scattering this distribution will be modified to a final distribution ${\ensuremath{\vec{q}_f}}$: $${\ensuremath{\vec{q}_f}}= Q\ {\ensuremath{\vec{q}_i}}$$ with a known $3 \times 3$ matrix $Q$ [@Adler:1974qu] which follows easily from a Clebsch-Gordan analysis of isospin.
Accordingly, taking into account all possible multiple scatterings the final pion charge distribution is given by $${\ensuremath{\vec{q}_f}}= M\ {\ensuremath{\vec{q}_i}}, \qquad M = \sum_{n=0}^\infty {P_n} Q^n$$ where $P_n$ is the probability that the pion exits the nucleus after exactly $n$ $\pi N$ scatterings. It should be noted that the matrix $Q$ does not include any absorption which is taken into account by $\sum_{n=0}^\infty P_n < 1$. The eigenvalues and eigenvectors of the matrix $Q$ are given by $$\begin{aligned}
\lambda_1 &=& 1\, , \, q_1 = (1,1,1)^T,
\\
\lambda_2 &=& \tfrac{5}{6}\, , \, q_2 = (1,0,-1)^T,
\\
\lambda_3 &=& \tfrac{1}{2}\, , \, q_3 = (1,-2,1)^T \end{aligned}$$ and one finds immediately the three eigenvalues of the charge exchange matrix $M$ in dependence of the three eigenvalues $\lambda_k$ of $Q$ $$f(\lambda_k)=\sum_{n=0}^\infty P_n \lambda_k^n \, , \, k=1,2,3\ .
\label{eq:flam}$$
The connection between the eigenbasis and the canonical basis is given by $$\begin{gathered}
A_p\,(1-c-d) = \tfrac{1}{3} f(1)+\tfrac{1}{2} f(\tfrac{5}{6})
+\tfrac{1}{6} f(\tfrac{1}{2}),
\nonumber\\
A_p\, d = \tfrac{1}{3} f(1)-\tfrac{1}{3} f(\tfrac{1}{2}),
\nonumber\\
A_p\, c = \tfrac{1}{3} f(1)-\tfrac{1}{2} f(\tfrac{5}{6})
+\tfrac{1}{6} f(\tfrac{1}{2})\end{gathered}$$ or inversely $$\begin{aligned}
A_p & = & f(1),
\nonumber\\
c & = & \tfrac{1}{3} - \tfrac{1}{2}{f(\tfrac{5}{6})/f(1)}
+ \tfrac{1}{6}{f(\tfrac{1}{2})/f(1)},
\nonumber\\
d &=& \tfrac{1}{3}{\left[1 - {f(\tfrac{1}{2})/f(1)}\right] }\ .\end{aligned}$$ As already mentioned in the introduction, the Pauli suppression factor of step 1 will be conveniently absorbed into the normalization of the charge exchange matrix $M$ $$A_p \rightarrow A_p = g(W,Q^2) f(1)\ .$$ The function $f(\lambda)$ contains the dynamical details of pion multiple scattering in the nucleus. An outline of how this function is calculated by solving a transport problem for pions in the nucleus will be given in the next section.
Calculation of $ f(\lambda)$
----------------------------
To a very good approximation [@Adler:1974qu] the three-dimensional problem can be reduced to a one-dimensional transport problem by projecting the forward-hemisphere of the scattered pion onto the forward direction and the backward-hemisphere onto the backward direction \[see Eq. \]. In this approximation the pion scatters back and forth along its initial direction of motion until it is absorbed or leaves the nucleus. As is explained in more detail in Refs. [@Adler:1974qu; @Adler:1974wu] the nucleon density profile along the scattering line can be scaled out of the problem by an appropriate change in length variable such that it is equivalent to consider a [*uniform*]{} one-dimensional nuclear medium extending from $x=0$ to $x=L$ with effective length (optical thickness) $$L = L(b) = \frac{1}{\rho(0)} \int_{-\infty}^{+\infty} {\ensuremath{{\operatorname{d}}}}z\
\rho(\sqrt{z^2+b^2})\ ,
\label{eq:optical}$$ where $b$ is the impact parameter. Furthermore, to simplify the discussion we only consider forward scattering of the pions (and also neglect Pauli suppression in this step 2). The solutions taking into account forward and backward scattering, $f_\pm(\lambda)$, can be found in Appendix A of [@Adler:1974qu]. Apart from being more realistic, they are important for estimates of the background to proton decay by neutrino production of pions in the $\Delta$ resonance region [@Gaisser:1986vn].
The transport process can be described in terms of the basic probabilities for a pion density to propagate from $y$ to $x$ and to a) interact (scattering or absorption) b) scatter c) be absorbed in $[x,x+dx]$ $$\begin{aligned}
a)\, {\ensuremath{\left\langle{x}| \right.}} {\ensuremath{P_{\rm tot}}}{\ensuremath{\left. |{y}\right\rangle}} &=& \kappa e^{-\kappa |y-x|} \Theta(y-x)
\label{eq:pa}\\
b)\, {\ensuremath{\left\langle{x}| \right.}} {\ensuremath{P_{\rm cex}}}{\ensuremath{\left. |{y}\right\rangle}} & =& \mu {\ensuremath{\left\langle{x}| \right.}} {\ensuremath{P_{\rm tot}}}{\ensuremath{\left. |{y}\right\rangle}}
\label{eq:pb}\\
c)\, {\ensuremath{\left\langle{x}| \right.}} {\ensuremath{P_{\rm abs}}}{\ensuremath{\left. |{y}\right\rangle}} & =& {\ensuremath{a}}{\ensuremath{\left\langle{x}| \right.}} {\ensuremath{P_{\rm tot}}}{\ensuremath{\left. |{y}\right\rangle}}
\label{eq:pc}\end{aligned}$$ with $ \kappa = \rho(0) {\ensuremath{\sigma_{\rm tot}}}$ (’inverse interaction length’), $\mu = {\ensuremath{\sigma_{\rm cex}}}/{\ensuremath{\sigma_{\rm tot}}}$ being the probability that the pion is scattered and ${\ensuremath{a}}= {\ensuremath{\sigma_{\rm abs}}}/{\ensuremath{\sigma_{\rm tot}}}$ the probability that the pion is absorbed (in a single scattering process). The choice of $\rho(0)$ as density of the uniform one-dimensional medium (i.e. $\kappa = \rho(0) {\ensuremath{\sigma_{\rm tot}}}$) is related to the normalization of the effective length in Eq. . Note also that $\mu + {\ensuremath{a}}= 1$ since ${\ensuremath{\sigma_{\rm tot}}}= {\ensuremath{\sigma_{\rm cex}}}+ {\ensuremath{\sigma_{\rm abs}}}$. The cross sections for charge exchange (cex) and absorption (abs) will be specified below.
The density of pions in the medium after $n$ scatterings, ${\ensuremath{\left. |{{\ensuremath{\rho_{\rm in}^{(n)}}}}\right\rangle}}$, is related to the initial density ${\ensuremath{\left. |{{\ensuremath{\rho_{\rm in}^{(0)}}}}\right\rangle}}$ by $${\ensuremath{\left. |{{\ensuremath{\rho_{\rm in}^{(n)}}}}\right\rangle}} = {\ensuremath{P_{\rm cex}}}^n {\ensuremath{\left. |{{\ensuremath{\rho_{\rm in}^{(0)}}}}\right\rangle}}$$ where the initial density is normalized to ${\ensuremath{\left\langle{x}|{{\ensuremath{\rho_{\rm in}^{(0)}}}}\right\rangle}} = 1/L$ such that the number of pions in the medium is one: $${\ensuremath{N_{\rm in}^{(0)}}} \equiv \int_0^L {\ensuremath{{\operatorname{d}}}}x\ {\ensuremath{\left\langle{x}|{{\ensuremath{\rho_{\rm in}^{(0)}}}}\right\rangle}} = 1 \ .$$ The density of pions leaving the medium after $n$ scatterings, $ {\ensuremath{\left. |{{\ensuremath{\rho_{\rm out}^{(n)}}}}\right\rangle}}$, is given by $$\begin{aligned}
{\ensuremath{\left. |{{\ensuremath{\rho_{\rm out}^{(n)}}}}\right\rangle}} & =& (1-{\ensuremath{P_{\rm tot}}}) {\ensuremath{\left. |{{\ensuremath{\rho_{\rm in}^{(n)}}}}\right\rangle}}
\\
& =& (1-{\ensuremath{P_{\rm tot}}}) {\ensuremath{P_{\rm cex}}}^n {\ensuremath{\left. |{{\ensuremath{\rho_{\rm in}^{(0)}}}}\right\rangle}}\ .\end{aligned}$$ The probability that the pion leaves the medium after exactly $n$ rescatterings is then $$P_n \equiv {\ensuremath{N_{\rm out}^{(n)}}} = \int_0^L\ {\ensuremath{{\operatorname{d}}}}x\ {\ensuremath{\left\langle{x}|{{\ensuremath{\rho_{\rm out}^{(n)}}}}\right\rangle}}\ .
\label{eq:pn}$$ The dynamical function $f(\lambda)$ we wish to calculate is then formally given by $$f(\lambda) =\sum_{n=0}^\infty P_n \lambda^n
= \int_0^L\ {\ensuremath{{\operatorname{d}}}}x\ {\ensuremath{\left\langle{x}|{{\ensuremath{\psi_{\rm out}}}}\right\rangle}}
\label{eq:f}$$ with $${\ensuremath{\left. |{{\ensuremath{\psi_{\rm out}}}}\right\rangle}} := \sum_{n=0}^\infty \lambda^n {\ensuremath{\left. |{{\ensuremath{\rho_{\rm out}^{(n)}}}}\right\rangle}}\ .$$ Using $1-{\ensuremath{P_{\rm tot}}}= 1-\tfrac{1}{\sigma} + \tfrac{1}{\sigma} (1 -\lambda {\ensuremath{P_{\rm cex}}})$ with $\sigma := \lambda \mu$ and $\sum_{n=0}^\infty \lambda^n {\ensuremath{P_{\rm cex}}}^n = (1- \lambda {\ensuremath{P_{\rm cex}}})^{-1} =: 1+F$ we can write ${\ensuremath{\left. |{{\ensuremath{\psi_{\rm out}}}}\right\rangle}}$ as $${\ensuremath{\left. |{{\ensuremath{\psi_{\rm out}}}}\right\rangle}} = \left[1 + (1-\tfrac{1}{\sigma}) F\right]{\ensuremath{\left. |{{\ensuremath{\rho_{\rm in}^{(0)}}}}\right\rangle}}\ .
\label{eq:out}$$ Inserting Eq. into Eq. we find $$f(\lambda) = 1 + (1-\tfrac{1}{\sigma})
\int_0^L\ {\ensuremath{{\operatorname{d}}}}x\ {\ensuremath{{\operatorname{d}}}}y\ {\ensuremath{\left\langle{x}| \right.}}F{\ensuremath{\left. |{y}\right\rangle}} \frac{1}{L}
\label{eq:f2}$$ where we have inserted the unit operator $$1 = \int_0^L {\ensuremath{{\operatorname{d}}}}y\ {\ensuremath{\left. |{y}\right\rangle}}{\ensuremath{\left\langle{y}| \right.}}$$ and used ${\ensuremath{\left\langle{x}|{{\ensuremath{\rho_{\rm in}^{(0)}}}}\right\rangle}} = 1/L$.
Using the definition of the operator $1+F=(1-\lambda {\ensuremath{P_{\rm cex}}})^{-1}$ and $(1-\lambda {\ensuremath{P_{\rm cex}}})^{-1} (1 -\lambda {\ensuremath{P_{\rm cex}}})=1$ we can furthermore write $$F = \lambda {\ensuremath{P_{\rm cex}}}+ F\ \lambda {\ensuremath{P_{\rm cex}}}$$ which leads us, using Eqs. and , to the transport integral equation $$f(y) = \sigma (1 - e^{-\kappa y})
+ \sigma \kappa \int_0^y {\ensuremath{{\operatorname{d}}}}z\ f(z) e^{-\kappa (y-z)}
\label{eq:transport}$$ with $$f(y) := \int_0^L {\ensuremath{{\operatorname{d}}}}x\ {\ensuremath{\left\langle{x}| \right.}} F {\ensuremath{\left. |{y}\right\rangle}}\ .
\label{eq:fy}$$ The integral equation can be transformed into a differential equation by evaluating $\tfrac{{\ensuremath{{\operatorname{d}}}}}{{\ensuremath{{\operatorname{d}}}}y} [e^{\kappa y} f(y)]$ for the left and the right side of Eq. resulting in $$\frac{{\ensuremath{{\operatorname{d}}}}}{{\ensuremath{{\operatorname{d}}}}y}f = \kappa (\sigma - 1) f(y) + \kappa \sigma
\, , \, f(0) = 0\ .
\label{eq:dgl}$$ This differential equation is solved by $$f(y) = \frac{\sigma}{1-\sigma} \left[1-h(y)/h(0) \right]\ ,
\label{eq:solution}$$ where $h(y) = e^{\kappa (\sigma - 1)y}$ is a solution of the homogeneous equation $h^\prime = \kappa (\sigma - 1) h$. The final solution for the dynamical function $f(\lambda)$ is now easily found to be $$\begin{aligned}
f(\lambda) &=& 1 + (1-\tfrac{1}{\sigma})
\frac{1}{L} \int_0^L\ {\ensuremath{{\operatorname{d}}}}y\ f(y)
\nonumber\\
&=& \frac{1 - e^{-\kappa L(1-\sigma)}}{\kappa L (1-\sigma)}\, , \,
\sigma = \lambda \mu\ .
\label{eq:final}\end{aligned}$$ Finally, we average the solution in Eq. over impact parameters: $$f(\lambda) =
\frac{\int_0^{\infty} b\ {\ensuremath{{\operatorname{d}}}}b\ L(b)
f(\lambda,L(b))}{\int_0^{\infty} b\ {\ensuremath{{\operatorname{d}}}}b\ L(b)}\ .
\label{eq:impactav}$$
ANP model: Input
----------------
In this subsection we summarize the input quantities entering the ANP model.
\(i) First of all the nucleon density $\rho(r)$ enters the calculation of the effective length $L(b)$ via $ L(b) = \tfrac{1}{\rho(0)} \int {\ensuremath{{\operatorname{d}}}}z \ \rho[r(z,b)]$. The charge density profiles have been determined in electron scattering experiments. For lighter targets like $_{8}O^{16}$ the charge density is parameterized by a ’harmonic oscillator form’ $$\rho(r) = \rho(0)\exp(-r^2/R^2) (1+C \frac{r^2}{R^2} +
C_1 \frac{r^4}{R^4})$$ whereas for heavier targets like $_{18}Ar^{40}$ and $_{26}Fe^{56}$ a ’two parameters Fermi model’ is utilized: $$\rho(r) = \rho(0)\big[1+\exp{((r-C)/C_1)}\big]^{-1} \ .$$ The parameters $\rho(0)$, $R$, $C$, and $C_1$ can be found in Table 1 of [@Paschos:2000be]. The corresponding charge density profiles $\rho(r)$ are shown in Fig. \[fig:rho\].
\(ii) The second input is the usual cross section for elastic $\pi N$ scattering in the $(3,3)$ resonance region. This region is dominated by the $I = \tfrac{3}{2}$ amplitude. The elastic cross sections per nucleon for a $\pi^k$ ($k=\pm,0$) in the initial state are given by $Z/A\ \sigma_{\pi^k p} + N/A\ \sigma_{\pi^k n}$. For an isoscalar target ($Z=N$) and neglecting the $I=\tfrac{1}{2}$ channel they are independent of the pion charge and will be denoted by ${\ensuremath{\sigma_{\rm cex}}}$. Moreover, the elastic cross section ${\ensuremath{\sigma_{\rm cex}}}$ is proportional to the cross section $\sigma_{\pi^+ p}(W)$ $$\begin{aligned}
\frac{{\ensuremath{{\operatorname{d}}}}\sigma}{{\ensuremath{{\operatorname{d}}}}\Omega} &\propto& \sigma_{\pi^+ p}(W)\
(1+3 \cos^2 \phi)\ ,
\label{eq:diff}\\
{\ensuremath{\sigma_{\rm cex}}}&=&
\tfrac{2}{3} \sigma_{\pi^+ p}(W)
\label{eq:sigcex}\end{aligned}$$ with $$\sigma_{\pi^+ p}(W) = \sigma_{(3,3)}(W) + 20\ {\rm mb}$$ where $\sigma_{(3,3)}(W)$ is a resonant cross section which can be found in [@Adler:1974qu] and the second term is a constant non-resonant background. In Eqs. and we have omitted for simplicity a possible Pauli suppression factor taking into account the Pauli exclusion principle in step 2. Note also, that it is debatable whether such a factor should be included here since we are working in a picture with fixed nucleons.
\(iii) The third ingredient is the cross section per nucleon for pion absorption, ${\ensuremath{\sigma_{\rm abs}}}(W)$. For this quantity Sternheim and Silbar have published two parametrizations [@Sternheim:1972ad; @Silbar:1973em] which they have extracted from data on single pion production in $p A$ scattering. These parametrizations can also be found in [@Adler:1974qu]. In Fig. \[fig:xs\] these two absorption cross sections, model (A) and model (B), are shown together with the elastic and the corresponding total cross sections. As can be seen the models (A) and (B) for $ {\ensuremath{\sigma_{\rm abs}}}$ are quite different in [*shape*]{} and [*normalization*]{}.
Fixing the normalization of ${\ensuremath{\sigma_{\rm abs}}}(W)$ {#sec:norm}
================================================================
The two models for the absorption cross section, model (A) [@Sternheim:1972ad] and model (B) [@Silbar:1973em] result in quite different total amounts of absorbed pions: for example, model (A) predicts that about $19\%$ of the pions will be absorbed in oxygen compared to $43\%$ in model (B).
These numbers have been obtained in an ’averaging approximation’ [@Adler:1974qu] in which the $W$-dependence of $f(\lambda, W)$ is averaged over the $(3,3)$ region. In this case the dynamical functions (the charge exchange matrix) are mainly sensitive to the region around $W \simeq m_\Delta$ and thus mainly sensitive to the [*normalization*]{} of ${\ensuremath{\sigma_{\rm abs}}}(W \simeq m_\Delta)$.
In the following we fix this normalization by using experimental data for 1-pion production in CC $\nu_\mu$-deuteron \[$\leftrightarrow$ free case\] and $\nu_\mu$-neon scattering. These data have been weighted to the [*same*]{} atmospheric $E_\nu$ spectrum in a paper by Merenyi et al. [@Merenyi:1992gf] allowing for direct comparison. In Table 1 of [@Merenyi:1992gf] relative charged current populations (w.r.t. the total cross section) are provided for deuteron (D) and neon (Ne). The fractional contributions of the 1-pion production channels $(\pi^+,\pi^0,\pi^-)$ are given by: $$\begin{aligned}
\text{D}:\, {\ensuremath{\vec{r}_i}}&=& (0.165,0.09,0)^T
\\
\text{Ne}:\, {\ensuremath{\vec{r}_f}}&=& (0.11,0.05,0.01)^T
\nonumber\\
& & \pm (0.014,0.02,0.01)^T\, .\end{aligned}$$
The fractional populations ${\ensuremath{\vec{r}_i}}$ and ${\ensuremath{\vec{r}_f}}$ are related in the ANP model by: $${\ensuremath{\vec{r}_f}}= M {\ensuremath{\vec{r}_i}}\times K\, ,\quad
K = \frac{10 \sigma_{\rm tot}(\text{D})}{\sigma_{\rm tot}(\text{Ne})}\ .$$ Unfortunately the total cross sections, weighted to an atmospheric neutrino flux, have not been published in [@Merenyi:1992gf]. However, the correction factor $K$ should be close to one since the total cross section for Ne is not affected by charge exchange and pion absorption. In the following exercise we assume $K=1$ and later assign a normalization uncertainty of $3\%$.
Furthermore, it is not viable to solve ${\ensuremath{\vec{r}_f}}\overset{!}{=} {M[A_p,d,c]}\ {\ensuremath{\vec{r}_i}}$ for the three parameters $ A_p,d,c$ since the solution [*strongly*]{} varies within the errors of ${\ensuremath{\vec{r}_f}}$. Instead, we have made the reasonable assumption $0 \le c < d$ which means that the probability for $\pi^- \to \pi^+$ is smaller than the probability for $\pi^0 \to \pi^+$. Under this assumption one can fit the two parameters $A_p$ and $d$ for a fixed value of $c$ as long as $c < d$:
$c$ $A_p$ $d$
-------------------- -------------------- ------- -------
$c=0\phantom{.00}$ $\rightarrow$ Fit: 0.695 0.147
$c=0.01$ $\rightarrow$ Fit: 0.696 0.128
$c=0.02$ $\rightarrow$ Fit: 0.696 0.109
$c=0.03$ $\rightarrow$ Fit: 0.696 0.091
$c=0.04$ $\rightarrow$ Fit: 0.697 0.072
$c=0.05$ $\rightarrow$ Fit: 0.698 0.053
\[$ \chi^2/d.o.f \simeq 0.4$\] As can be seen the parameter $ A_p$ is well constrained: $ A_p = 0.696 \pm 0.002$. More conservatively we use in the following $A_p = 0.70 \pm 0.02$ taking into account the normalization uncertainty due to the correction factor $K$. On the other hand the parameters $d$ and $c$ are correlated and adopt values in the range $c \in [0,0.05]$, $d \in [0.15,0.05]$.
Inspecting Eq. we find that $f(\lambda = 1) = \sum_{n=0}^\infty P_n = 1-A$ where $A$ is the fraction of absorbed pions. Therefore, the averaged fraction $\bar A$ of absorbed pions is given by $\bar A = 1-{\ensuremath{\bar f}}(\lambda=1)$ where ${\ensuremath{\bar f}}(\lambda)$ is the averaged dynamical function which is related to the above extracted parameter $A_p$ via $A_p = g(\bar W,\bar Q^2) {\ensuremath{\bar f}}(1)$. In order to estimate the effect of the Pauli suppression factor we take $ \bar W \simeq m_\Delta$ and $\bar Q^2 \simeq 0.1\ {\ensuremath{\mathrm{GeV^2}}}$ leading to $g(\bar W,\bar Q^2) = 0.93 \pm 0.05$ \[see Table 3 in [@Paschos:2000be]\]. Using these values we finally find ${\ensuremath{\bar f}}(1) = 0.75 \pm 0.05$ implying that about $25\%$ of the pions have been absorbed in the neon target: $\bar A = 0.25 \pm 0.05$.
A fraction of $ 25 \%$ pion absorption can be obtained by renormalizing absorption model (B) \[model (A)\] by a factor $\simeq 0.3$ \[$\simeq 1.4$\]. With this renormalization we can compare the ANP model with the above fitted results. The ANP model gives for both renormalized absorption models for a neon target ${\ensuremath{\bar f}}(1) =0.75$ (by construction), $d = 0.15$, and $c = 0.05$. This result compares favorably with the fitted values.
Dynamics of the ANP model {#sec:dynamics}
=========================
Linearisation {#sec:linear}
-------------
The simple solution in Eq. is well-suited for further analytical investigations of pion multiple scattering. For example, the forward solution at $\lambda = 1$, $$f(\lambda=1,L,W)=
\frac{1-e^{-\rho_0 L {\ensuremath{\sigma_{\rm abs}}}}}{\rho_0 L {\ensuremath{\sigma_{\rm abs}}}}\ ,$$ can be considered in the limit $ \rho_0 L {\ensuremath{\sigma_{\rm abs}}}\ll 1$ relating the fraction of absorbed pions $A$ to the absorption cross section per nucleon $ {\ensuremath{\sigma_{\rm abs}}}(W)$: $$A(L,W)= 1 - f(1,L,W) \simeq \tfrac{1}{2} \rho_0 L {\ensuremath{\sigma_{\rm abs}}}(W)\ .
\label{eq:abs1}$$
Averaging $L=L(b)$ over the impact parameters $b$ for oxygen as target material gives $\bar L \simeq 1.9 R$ with radius $ R\simeq 1.833\ {\rm fm}$. The nuclear density for oxygen, $ \rho_0 = 0.141\ {\rm fm}^{-3}$, has been taken from Table 1 in [@Paschos:2000be]. Replacing $\rho_0 L$ by $\rho_0 \bar L \simeq 0.05\ {\rm mb}^{-1}$ in Eq. and taking into account the averaging over the $(3,3)$ resonance by the replacement $W \to m_\Delta$ we arrive at the following rule of thumb for the fraction of absorbed pions in oxygen $$A \simeq 0.025\ {\ensuremath{\sigma_{\rm abs}}}(W \simeq m_\Delta) [{\rm mb}]\ .
\label{eq:abs2}$$
The absorption cross sections can be taken from Fig. \[fig:xs\] and we find the following fractions of absorbed pions:
model ${\ensuremath{\sigma_{\rm abs}}}(W=m_\Delta)$ $A$
------------------ ----------------------------------------------- --------
(A) $\simeq 6.0\ {\rm mb}$ $15\%$
(B) $\simeq 28.4\ {\rm mb}$ $71\%$
0.3 $\times$ (B) $\simeq 8.5\ {\rm mb}$ $21\%$
Here the renormalization factor $0.3$ for model (B) has been taken from the preceding section. The same result is found by renormalizing model (A) with a factor $1.4$. Note also that the $71\%$ for (the original) model (B) is not realistic because the condition $\rho_0 \bar L {\ensuremath{\sigma_{\rm abs}}}\ll 1$ is not satisfied.
’How many’ multiple scatterings? {#sec:prob}
--------------------------------
Figure \[fig:3\] shows the dynamical function $f(\lambda,W)$ for oxygen in dependence of $\lambda$ for several values of $W$. Writing $f(\lambda) = \sum_{n=0}^\infty P_n \lambda^n$ with the probabilities $P_k$ ($k=0,1,2,\ldots$) that the pion is observed after $k$ $\pi N$ scatterings it is obvious that $ f(\lambda=0) = P_0$ and $ f(\lambda=1)= 1-A$ where $A$ is the probability for pion absorption. \[Note that $\sum_{k=0}^\infty P_k + A = 1$.\] It is an interesting question, how large the probabilities $P_n, n>0$ for multiple scattering are. Qualitatively, a stronger curvature of the function $f(\lambda)$ indicates a higher probability for multiple scattering. As can be seen, in the vicinity of $W=m_\Delta$ the probability that the pion rescatters several time is largest whereas at $W=1.1\ {\ensuremath{\mathrm{GeV}}}$ the curve is almost linear such that only $P_0$ and $P_1$ contribute appreciably.
$W [{\ensuremath{\mathrm{GeV}}}]$ 1.1 1.2 1.25 1.5
----------------------------------- ---------- ------ ---------- ----------
$A$ [0.0]{} 11.8 [25.7]{} 48.4
$P_0$ [90.8]{} 33.6 28.8 43.4
$P_1$ 8.0 14.1 12.2 6.5
$P_2$ 1.0 10.5 8.9 1.3
$P_3$ 0.13 7.8 6.5 0.26
$P_4$ $\ldots$ 5.8 4.8 $\ldots$
$P_5$ 4.3 3.5
$P_6$ 3.2 2.6
$P_7$ 2.4 1.9
$P_8$ 1.8 1.4
$P_9$ 1.3 1.0
$P_{10}$ 1.0 0.7
: Probabilities in $\%$ for multiple scattering.
\[tab:prob\]
Of course the probabilities $P_n$ can be calculated exactly within the ANP model according to Eq. or by differentiating the solution for $f(\lambda)$ in Eq. : $P_n = \tfrac{1}{n!} \tfrac{{\ensuremath{{\operatorname{d}}}}^n}{{\ensuremath{{\operatorname{d}}}}\lambda^n} f_{|\lambda=0}$.
On the other hand, inspired by the fact that the pion energy remains constant, it is interesting to make the assumption (which is only asymptotically correct) that $P_{k+1} \simeq r P_k$ for $k \ge 1$ or equivalently $P_k \simeq P_1 r^{k-1}$ for $k \ge 1$. Under this assumption $f(\lambda)$ is a geometrical series and is given by the simple solution $$f(\lambda) = \sum_{n=0}^\infty P_n \lambda^n \simeq
P_0 + \frac{P_1 \lambda}{1 - r \lambda}\ .
\label{eq:series}$$ The constant $r$ can be fixed from $f(\lambda=1) = 1 - A$ resulting in $r =1-\frac{P_1}{1-A-P_0}$.
Thus, once $A$, $P_0$ and $P_1$ are known, all higher probabilities and the function $f(\lambda)$ can be easily estimated. The result of such a procedure is listed in the following table where $P_0$, $P_1$ and $A$ have been determined from Fig. \[fig:3\].
Conclusions {#sec:summary}
===========
In this article, we have given a basic introduction to the pion multiple scattering model of Adler, Nussinov, and Paschos [@Adler:1974qu] focusing on the input parameters of the model, particularly on the cross section for pion absorption ${\ensuremath{\sigma_{\rm abs}}}(W)$ which is the least well determined ingredient. Using data for $\nu_\mu$-deuteron and $\nu_\mu$-neon scattering [@Merenyi:1992gf] we could fix the normalization of ${\ensuremath{\sigma_{\rm abs}}}(W)$ corresponding to a fraction of $(25 \pm 5)\%$ of absorbed pions in neon. The parameters $A_p$, $d$, and $c$ have been determined to be $A_p = 0.70 \pm 0.02$, $c \in [0,0.05]$, and $d \in [0.15,0.05]$ which compares quite favorably with the predictions of the ANP model for neon.
In order to test and improve the ANP model it will be necessary to make detailed measurements of single pion electroproduction in the region of the $(3,3)$ resonance using different heavy targets and to compare it with the corresponding cross sections on free nucleons [@psy].
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---
abstract: 'We construct families of high performance quantum amplitude damping codes. All of our codes are nonadditive and most modestly outperform the best possible additive codes in terms of encoded dimension. One family is built from nonlinear error-correcting codes for classical asymmetric channels, with which we systematically construct quantum amplitude damping codes with parameters better than any prior construction known for any block length $n\geq 8$ except $n=2^r-1$. We generalize this construction to employ classical codes over $GF(3)$ with which we numerically obtain better performing codes up to length $14$. Because the resulting codes are of the codeword stabilized (CWS) type, easy encoding and decoding circuits are available.'
author:
- 'Peter W. Shor, Graeme Smith, John A. Smolin, Bei Zeng [^1] [^2] [^3]'
title: 'High performance single-error-correcting quantum codes for amplitude damping'
---
Introduction
============
Quantum computers offer the potential to solve certain classes of problems that appear to be intractable on a classical machine. For example, they allow for efficient prime factorization [@shor], breaking modern public-key cryptography systems based on the assumption that factorization is hard. Quantum computers may also be useful for simulating quantum systems [@Feynman; @lloyd].
However, quantum computers are particularly subject to the deleterious effects of noise and decoherence. It was thought, for a time, that quantum error-correction would be precluded by the no cloning theorem [@nocloning] which seems to rule out redundancy as usually employed in error correction. The discovery of quantum error-correcting codes [@shorECC; @Steane0] that allow for fault-tolerant quantum computing [@shorfault] significantly bolstered the hopes of building practical quantum computers.
For the most part, people have concentrated on dealing with the worst case—arbitrary (though hopefully small) noise. This turns out to be equivalent to correcting Pauli-type errors, ${\sigma_x}={ 0\ 1 \choose
1\ 0},{\sigma_y}={0\ -i \choose i\ \ 0},{\sigma_z}={1 \ \ 0\choose 0\ -\!1}$, acting on a bounded-weight subset of the qubits in the code. Since the Pauli operators form a basis of $2\times 2$ matrices, a code that can correct all Pauli errors can in also protect against any general qubit noise [@bigpaper; @chiara].
However, as first demonstrated by Leung et al. [@Debbie], designing a code for a particular type of noise can result in codes with better performance. In practice the types of noise seen are likely to be unbalanced between amplitude (${\sigma_x}$-type) errors and phase (${\sigma_z}$-type) errors, and recently a lot of attention has been put into designing codes for this situation and in studying their fault tolerance properties [@Lev] [@Panos] [@Evans] [@Peter1].
In this paper, we will focus on [*amplitude damping*]{} noise, another type of noise seen in realistic settings. Amplitude damping noise is asymmetric, with some chance of turning a spin up ${|{1}\rangle}$ qubit into a spin down ${|{0}\rangle}$ state but never transforming ${|{0}\rangle}$ to ${|{1}\rangle}$. This models, for example, photon loss in an optical fiber: A photon in the fiber may leak out or absorbed by atoms in the fiber, but to good approximation photons do not spontaneously appear in the fiber. Several people have considered this type of noise [@Debbie; @Peter1; @ChuangLeungYamamoto] but there is no systematic method for constructing such codes. In general it is a difficult problem to design codes for any particular noise model.
In this paper we present a method for finding families of codes correcting one amplitude-damping error. We begin with an ansatz relating a restricted type of amplitude-damping code to classical codes for the binary asymmetric (or $Z$-) channel. The $Z$-channel is the classical channel that takes 1 to 0 with some probability, but never vice versa[^4]. The amplitude damping channel is its natural quantum generalization. The problem of designing codes for the amplitude damping channel is thus reduced to a finding classical codes for the $Z$-channel, subject to a constraint. This lets us carry over many known results from classical coding theory.
We further simplify the problem by using a novel mapping between binary and ternary codes. This allows us to find quantum amplitude-damping codes by studying ternary codes on a greatly reduced search space.
The rest of the paper is organized as follows. In section \[prelims\] we describe quantum channels and the quantum error-correction conditions. In section \[AD\] we define what it means to correct amplitude damping errors and show how they relate to classical symmetric codes. In section \[symm\] we show how a particular class of amplitude-damping codes arises from classical codes for the asymmetric channel, and give some new codes based on powerful extant results on classical $Z$-channel codes [@CR]. In section \[GF3\] we define a mapping from binary to ternary codes (and back) and use this to construct new and better amplitude damping codes. Finally, in section \[summary\] we summarize our results and give a table of the best amplitude-damping codes and how they compare to previous work.
Preliminaries\[prelims\]
========================
Pure quantum states are represented by vectors in a complex vector space. We will be concerned with finite-dimensional systems. The simplest quantum system (called a qubit) can be described by an element of $\CC^2$, and $n$ qubits together are described by an elements of $\CC^2 \otimes \ldots \otimes \CC^2= (\CC^2)^{\otimes n}.$ Such pure states are always chosen to be normalized to unity. More generally a quantum system can be described by a density matrix, a trace one linear operator from $(\CC^2)^{\otimes n}$ to $(\CC^2)^{\otimes n}$, usually denoted $\rho$.
The most general physical transformations allowed by the quantum mechanics are completely positive, trace preserving linear maps which can be represented by the Kraus decomposition: $$\cN(\rho)=\sum_k A_k \rho A_k^\dag\ {\rm where\ }\sum_k A_k^\dag A_k = {1 \hspace{-1.0mm} {\bf l}}.$$ For example the the Kraus operators for the depolarizing channel, the natural quantum analogue of the binary symmetric channel, are the Pauli matrices. The Kraus operators for the amplitude damping channel with damping rate $\epsilon$ are $$A_0=\begin{pmatrix} 1 & 0 \\0 &
\sqrt{1-\epsilon} \end{pmatrix}{\rm\ and\ } A_1=\begin{pmatrix} 0 &
\sqrt{\epsilon} \\0 & 0 \end{pmatrix} .$$
A quantum error correcting code is subspace of $(\CC^2)^{\otimes n}$ which is resilient to some set of errors acting on the individual qubits such that all states in that subspace can be recovered. For a $d$-dimensional codespace spanned by the orthonormal set ${|{\psi_i}\rangle}$, $i=1 \ldots d$ and a set of errors $\cE$ there is a physical operation correcting all elements $E_\mu \in \cE$ if the error correction conditions [@KnillLF; @bigpaper] are satisfied: $$\forall_{ij,\mu\nu}\ \ {\langle{\psi_i}|} E_\mu^\dag E_\nu {|{\psi_j}\rangle} = C_{\mu\nu}\delta_{ij},$$ where $C_{\mu \nu}$ depends only on $\mu$ and $\nu$.
Correcting amplitude damping \[AD\]
===================================
For small $\epsilon$, we would like to correct the leading order errors that occur during amplitude damping. Letting $A={\sigma_x}+i{\sigma_y}$, $B=I-{\sigma_z}$, we have $$A_1=\frac{\sqrt{\epsilon}}{2}A,\ \ A_0=I-\frac{\epsilon}{4}(I-{\sigma_z})+O(\e^2).$$ It can be shown if we wish to improve fidelity through an amplitude damping channel from $1-\epsilon$ to $1-\epsilon^t$ it is sufficient to satisfy the error-detection conditions for $2t$ $A$ errors and $t$ ${\sigma_z}$ errors. We will say the such a code corrects $t$ amplitude damping errors since it improves the fidelity, to leading order, just as much as a true $t$-error-correcting code would for the same channel. We will use the notation $\lfloor\lfloor n,K,t \rfloor\rfloor$ to mean an $n$-qubit code protecting a $K$-dimensional space and correcting $t$ amplitude damping errors, sometimes referring to this as a $t$-AD code. Our notation descends from the traditional coding-theory notation of $[n,k,d]$ to mean an $n$-bit classical code of distance $d$ protecting $k$ bits and $[[n,k,d]]$ to mean an $n$-qubit quantum code of distance $d$ protecting $k$ qubits. Note that our AD notation uses $K$ as the full dimensions of the protected space, [*not*]{} $k$, the $\log$ of the dimension. This is in preparation for the codes we will design which do not protected an integral number of qubits.
Since the amplitude damping channel is not a Pauli channel the usual tools for designing quantum codes cannot be directly used. One possible approach would be to design CSS [@shorECC; @Steane0; @Steane2] codes with different ${\sigma_x},{\sigma_z}$ distances [@Steane]. For the particular case of single-error-correcting AD code, we then would like to have CSS code of ${\sigma_x}$ distance $3$ (correcting a single ${\sigma_x}$ error) and ${\sigma_z}$ distance $2$ (detecting a single ${\sigma_z}$ error). Gottesman gives a construction of this kind of CSS code in Chapter 8.7 of [@Daniel]. We summarize his result as follows:
\[th:ADCSS\] If there exists a binary $[n,k,3]$ classical code ${\mathcal C}$ and ${\mathbf 1}$ (the all $1$ string of length $n$) is in the dual code of ${\mathcal C}$, then there exists an $\lfloor\lfloor n,2^{k-1},1\rfloor\rfloor$ code.
These codes indeed have better performance than codes designed for depolarizing channels. For instance, a $\lfloor\lfloor 7,2^3,1 \rfloor\rfloor$ exists while only $[[7,1,3]]$ single-error-correcting stabilizer codes exist for the depolarizing channel. In general, the classical Hamming bound for $[n,k,3]$ codes gives $k\leq n-\log(n+1)$, which gives a bound for $[[n,k]]$ single-error-correcting AD codes constructed by Theorem \[th:ADCSS\], [*i.e.*]{} $$k\leq n-1-\log(n+1),$$ while the quantum Hamming bound ([*cf.*]{} [@Daniel]) gives $$k\leq n-\log(3n+1)$$ for $[[n,k,3]]$ stabilizer codes for the depolarizing channel.
However, one expects that these codes cannot be optimal; since we only need to correct ${\sigma_x}+i{\sigma_y}$, correcting both ${\sigma_x}$ and ${\sigma_y}$ is excessive and would seem to lead to inefficient codes. Fletcher et al. took the first step toward making AD codes based on the non-Pauli error model, [*i.e.*]{} codes correcting ${\sigma_x}+i{\sigma_y}$ error, not both ${\sigma_x}$ and ${\sigma_y}$ errors [@Peter1]. Their codes are stabilizer codes with parameters $[[2n,n-1]]$ and correct a single amplitude damping error. Later another work [@Peter2] took a further step toward making AD codes correcting ${\sigma_x}+i{\sigma_y}$ error. These works constructed some nonadditive codes correcting a single amplitude damping error, and via numerical search for short block length found AD codes with better performance than codes given by the CSS construction of Theorem \[th:ADCSS\].
The construction of [@Peter2] consists of codewords ${|{\psi_u}\rangle}$ of the self-complementary format [@family], which is $${|{\psi_u}\rangle}=\frac{1}{\sqrt{2}}\left(|u\rangle+|\bar{u}\rangle\right),
\label{SC}$$ where $u$ is a binary string of length $n$ and $\bar{u}={\mathbf 1}\oplus u$.
As observed in [@family], which focused on nonadditive single-error-detecting codes, codes consisting of codewords given by Eq. (\[SC\]) automatically detect a single ${\sigma_z}$ error, so we have, as shown in [@Peter2]:
\[th:ADSC\] A self-complementary code corrects a single amplitude damping error if and only if no confusion arises assuming the decay occurs at no more than one qubit.
We will take the above observation as a starting point for making amplitude damping codes, by choosing classical self-complimentary codes which correct single errors arising from the classical asymmetric channel (or $Z$-channel).
Systematic construction from classical asymmetric codes \[symm\]
================================================================
Now we would like to relate the self-complementary construction to classical error correcting codes for the asymmetric channel. Before doing that we first briefly review the classical theory of those codes.
The **binary asymmetric channel** (denoted by ${\mathcal Z}$ in Fig. \[fig:channel\]) is the channel with $\{0,1\}$ as input and output alphabets, where the crossover $1\rightarrow 0$ occurs with positive probability $p$, whereas the crossover $1\rightarrow 0$ never occurs.
![The binary asymmetric channel ${\mathcal Z}$ and the ternary channel ${\mathcal T}$.[]{data-label="fig:channel"}](channel.pdf)
We will call a classical code that protects against one error in the binary asymmetric channel ${\mathcal Z}$ a [*$1$-code*]{} and use the notation $\lfloor n,K,t\rfloor$ analogous to our notation for the quantum amplitude damping code.
We can then formalize our observation as:
\[th:ADtoAS\] If ${\mathcal C}$ is a classical $\lfloor n,K,1 \rfloor$ code and $\forall u\in
{\mathcal C}$, $\bar{u}\in {\mathcal C}$, then $Q=\{|u\rangle+|\bar{u}\rangle,\ u\in {\mathcal C}\}$ is a single-error correcting amplitude damping code, $\lfloor\lfloor
n,K/2,1\rfloor\rfloor$.
This theorem is almost a direct corollary of Theorem \[th:ADSC\] so we omit a detailed proof. The main idea is that a classical code ${\mathcal C}$ that contains both $u$ and $\bar{u}$ takes care of correcting amplitude damping errors while the self-complementary form of ${|{\psi_u}\rangle}$ takes care of detecting the phase errors. And the size of the quantum code $Q$ is of course $K=|{\mathcal C}|/2$. This theorem allows us to use any classical self-complimentary $1$-code to construct self-complementary amplitude damping codes. The question that remains is how to find classical self-complimentary $1$-codes.
Varshamov showed almost all linear codes that are able to correct $t$ asymmetric errors are also able to correct $t$ symmetric errors [@Varshamov1]. Therefore, to go beyond $t$-symmetric-error correcting codes, we will look to non-linear constructions. Note that the quantum codes we construct from these non-linear codes are codeword stabilized codes, so these nonlinear classical codes will typically result in nonadditive quantum codes [@CWS].
Constantin-Rao Codes
--------------------
Constantin-Rao ([*CR*]{}) Codes [@CR] are the best known nonlinear $1$-codes. These beat the best symmetric single-error-correcting codes for all $n\neq 2^r-1$. An $n$-bit CR codes is constructed based on an abelian group $G$ of size $n+1$. The group operation is written as ‘$+$’ for abelian groups.
The Constantin-Rao code ${\mathcal C}_g\ \forall g\in G$ is given by $${\mathcal C}_g=(\{(x_1,x_2,...,x_n)|\sum_{i=1}^n x_ig_i=g\ \mod\ n+1\}),$$ where $x_i\in\{0,1\}$ and $g_1,g_2,...,g_n$ are the non-identity elements of $G$.
The cardinality of ${\mathcal C}_g$ is lower bounded by $$|{\mathcal C}_g|\geq \frac{2^n}{n+1}$$ for some $g\in G$.
Let $o(g)$ be the order of $g$, then it is known $$|{\mathcal C}_0|\geq |{\mathcal C}_g|,$$ with equality if and only if $o(g)$ is a power of $2$.
For a given nonprime $n+1$, there may be many abelian groups of size $n+1$. If the group $G$ is a cyclic group of order $n+1$, then the corresponding codes are called Varshamov-Tenengol’ts codes [@Varshamov2]. It is known that the largest Constantin-Rao code of length $n$ is the code ${\mathcal C}_0$ based on the group $G=\bigoplus_{p|n+1}\bigoplus_{i=1}^{n_p}\mathbb{Z}_p$, where $n+1=\Pi_{p|n+1}p^{n_p}$ [@Klove].
An exact expression for the size of a CR code based on the group properties is known, and a basic result is that for any group $G$ and any group element $g$, $|{\mathcal C}_g|$ has size approximately $\frac{2^n}{n+1}$ (for a review, see [@Klove]). Note $\frac{2^n}{n+1}$ is the Hamming bound for $1$-error correcting codes over the binary symmetric channel. Thus, CR codes provide excellent performance compared to symmetric codes and, indeed, outperform the best known symmetric codes for all block-lengths but $n = 2^r-1$.
Amplitude damping codes from Constantin-Rao codes
-------------------------------------------------
To build quantum codes from ${\mathcal C}_g$, we need to find CR codes which are self-complimentary (and preferably large). We will show these exist for all $n > 1$.
\[fact:even\] For even $n$, the Constantin-Rao code ${\mathcal C}_0$ is self-complementary.
This is based on a simple observation that all the nonzero group elements add up to zero for any abelian group of even size.
The case of odd lengths $n$ is more complicated. We first consider the case where $n=4k+3$. Recall that the largest Constantin-Rao code of length $n$ is the code ${\mathcal C}_0$ based on the group $G=\bigoplus_{p|n+1}\bigoplus_{i=1}^{n_p}\mathbb{Z}_p$, where $N=\Pi_{p|n+1}p^{n_p}$. Then further note that for an abelian group $\mathbb{Z}_2\oplus\mathbb{Z}_2\oplus\mathbb{G}$, where the group $\mathbb{G}$ is of odd size, all the nonzero group elements add up to zero. This leads to the following
For $n=4k+3$, the Constantin-Rao code ${\mathcal C}_0$ of the maximal cardinality is self-complementary.
Since $|{\mathcal C}_0|\geq |{\mathcal C}_g|\geq \frac{2^n}{n+1}$, AD codes constructed from Fact \[fact:even\] and Fact \[fact:4k3\] outperform the CSS AD codes of even length and odd length $n=4k+3$ constructed by Theorem \[th:ADCSS\].
Note we also have
\[fact:4k3\] For $n=4k+3$, the Varshamov-Tenengol’ts code ${\mathcal V}_\frac{n+1}{4}$ of the maximal cardinality is self-complementary.
The case for $n=4k+1$ is more tricky. We cannot directly get a self-complementary code of length $n$ from some Constantin-Rao codes ${\mathcal C}_g$ of the same length $n$. But instead we can construct self-complementary AD codes of length $n$ from the Varshamov-Tenengol’ts codes ${\mathcal V}_g$ of length $n+1$.
\[fact:4k1\] For $n=4k+1$, the shortened Varshamov-Tenengol’ts code ${\mathcal V}'_{\frac{n+2-r}{2}}$ obtained by deleting an odd coordinate $r$ from Varshamov-Tenengol’ts code ${\mathcal V}_{\frac{n+2-r}{2}}$ of length $n+1$ is self-complementary.
The codewords of this shortened Constantin-Rao code are given by $$\sum_{i=1,i\neq r}^{n+1} ix_i=\frac{n+2-r}{2} \mod\ n+2.
\label{nequals4kplus1}$$ Since $\sum_{i=1,i\neq r}^{n+1} i \mod\ n+2 = n+2-r$, for any set of $x_i$s we have $$\sum_{i=1,i\neq r}^{n+1} ix_i + i\bar{x}_i \mod\ n+2 = n+2-r$$ where $x_i \in \{0,1\}$ and $\bar{x}_i=1 \oplus x_i$. If the $x_i$s satisfy (\[nequals4kplus1\]) then so do the $\bar{x}_i$s. Therefore ${\mathcal V}'_{\frac{n+2-r}{2}}$ is self-complementary.
It is known that the size of these shortened Varshamov-Tenengol’ts codes are approximately $\frac{2^n}{n+2}$ [@Klove]. But we know that the size of binary symmetric codes for length $n=4k+1$ is upper bounded by $\frac{2^n}{n+2}$ [@MacWilliams], so the construction of AD codes given by Fact \[fact:4k1\] also outperforms the CSS AD codes of length $n=4k+1$ constructed by Theorem \[th:ADCSS\].
\[eg:AD816\] For $n=8$, choose the abelian group of size $n+1=9$ be $\mathbb{Z}_3\oplus\mathbb{Z}_3$. The codewords of the Constantin-Rao code ${\mathcal C}_0$ are given by a linear code ${\mathcal C}_1$ generated by $$\{00000011, 00001100, 00110000\};$$ and four pairs $P_i$ (i=1…4): $$\begin{aligned}
{\mathcal P}_1&=&\{10100001, 10101101\},\nonumber\\
{\mathcal P}_2&=&\{10000110, 10110110\},\nonumber\\
{\mathcal P}_3&=&\{01100100, 01100111\},\nonumber\\
{\mathcal P}_4&=&\{00101010, 11101010\};\end{aligned}$$ and all the complements of $\bigcup_{i=1}^{4}{\mathcal P}_i\bigcup {\mathcal C}$.
The weight distribution of this code is given by (for definition of weight distribution, see [@Shor-laflamme-1997; @Rains-weight-1998]) $A_0=1; A_1=0; A_2=1/4; A_3=0; A_4=9/2; A_5=0; A_6=9/4; A_7=0; A_8=8.$ Some of them are non-integers, so this code is nonadditive.
The size of the quantum code is $16$, so this is a $\lfloor\lfloor 8,2^4,1\rfloor\rfloor$ code. Note the CSS AD code constructed by Theorem \[th:ADCSS\] for $n=8$ gives parameters $\lfloor\lfloor 8,2^3,1\rfloor\rfloor$. And the best single-error-correcting stabilizer code for the depolarizing channel is $[[8,3,3]]$. Therefore, this nonadditive AD code encodes one more logical qubit than the best known stabilizer code with the same length and is capable of correcting a single amplitude damping error.
For short block length ($\leq 16$), a comparison of the code dimensions given by this Constantin-Rao construction with other constructions will be listed in Table \[table:ADcodes\] in Sec. \[sec:summary\]. One can see that this Constantin-Rao construction outperforms all the other constructions apart from the $GF(3)$ construction given in Sec. \[GF3\]. However, since the $GF(3)$ construction is not systematic (those codes given by the $GF(3)$ construction in Table \[table:ADcodes\] are found by numerical search), this Constantin-Rao construction is the best known systematic construction for single-error-correcting AD codes.
The $GF(3)$ construction and the ternarization map \[GF3\]
==========================================================
We will begin by defining a channel ${\mathcal T}$ which acts on a three letter alphabet and find ternary codes on this channel. We will then show that such codes are related to binary codes for the asymmetric channel and since the binary codes will be self-complimentary by construction that they will yield quantum amplitude damping codes as well.
The ternarization map
---------------------
The **ternary channel** (denoted ${\mathcal T}$ in the figure) has $\{0,1,2\}$ as input and output alphabets, where the crossovers $0\rightarrow 0$, $0\rightarrow 1$, $0\rightarrow 2$, $1\rightarrow 0$, $1\rightarrow 1$, $2\rightarrow 0$, and $2\rightarrow 2$ all occur with nonzero probability, but $1\rightarrow 2$ and $2\rightarrow 1$ never occur.
We define a map that takes pairs of binary coordinates into a single ternary coordinate. There are four possible values of binary pairs, and only three ternary coordinates, so it cannot be one-to-one.
The ternarization map $\tilde{\mathfrak{S}}: \mathbb{F}^2_2\rightarrow \mathbb{F}_3$ is defined by: $$\tilde{\mathfrak{S}}:\ \{00,11\}\rightarrow 0,\ 01\rightarrow 1,\ 10\rightarrow 2.$$
This is not a one to one map. So the inverse map needs to be specified carefully, that is, a ternary symbol $0$ after the inverse map gives two binary codewords $00$ and $11$.
The map $\mathfrak{S}: \mathbb{F}_3\rightarrow \mathbb{F}^2_2$ is defined by: $$\mathfrak{S}:\ 0\rightarrow \{00,11\},\ 1\rightarrow 01,\ 2\rightarrow 10.$$
For a binary code of length $n=2m$, by choosing a pairing of coordinates, the map $\tilde{\mathfrak{S}}^m: \mathbb{F}^{2m}_{2}\rightarrow \mathbb{F}_3^m$ then takes a given binary code of length $2m$ to a ternary code of length $m$.
The optimal $1$-code ${\mathcal C}^{(4)}$ of length $n=4$ and dimension $4$ has four codewords $\{0000,1100,0011,1111\}$. By pairing coordinates $\{1,2\}$ and $\{3,4\}$, the ternary image under $\tilde{\mathfrak{S}}^2$ is then $\{00\}$.
On the other hand, $\mathfrak{S}^m: \mathbb{F}_3^m\rightarrow \mathbb{F}^{2m}_2$ takes a given ternary code of length $m$ to a binary code of length $2m$.
\[eg:ternary423\] By starting from the linear ternary code $[4,2,3]_3$, with generators $\{0111,1012\}$, we get the binary image code ${\mathcal C}^{(8)}$ under $\mathfrak{S}^4$: $$\begin{array}{llll}
00000000 & 00000011 & 00001100 & 00001111 \\
00110000 & 00110011 & 00111100 & 00111111 \\
11000000 & 11000011 & 11001100 & 11001111 \\
11110000 & 11110011 & 11111100 & 11111111 \\
00010101 & 00101010 & 11010101 & 11101010\\
01000110 & 10001001 & 01110110 & 10111001 \\
01011000 & 10100100 & 01011011 & 10100111 \\
10010001 & 01100010 & 10011101 & 01101110 \\
\end{array}$$ which is of dimension $32$ and corrects one asymmetric error. Note this gives exactly the same binary $1$-code as the one given in Example \[eg:AD816\], which is the Constantin-Rao code ${\mathcal C}_0$ of length $n=8$ constructed from the group $\mathbb{Z}_3\oplus\mathbb{Z}_3$. This example hints at some relationship between the $GF(3)$ construction and the Constantin-Rao codes. \[Eg832\]
The $GF(3)$ construction for $1$-codes \[sec:GF3classic\]
---------------------------------------------------------
### Even block length
Example \[Eg832\] suggests that good $1$-codes may be obtained from ternary codes under the map $\mathfrak{S}^m$. We would like to know the general conditions under which a ternary code gives a $1$-code via the map $\mathfrak{S}^m$. The main result of this section states that any single-error-correcting code for the ternary channel ${\mathcal T}$ gives a $1$-code under the map $\mathfrak{S}^m$ [@gf3classical].
It will be useful in what follows to define an asymmetric distance between two codewords:
Letting $N(\mathbf{x},\mathbf{y})=\#\{i|x_i=0\ \text{and}\ y_i=1\}$, we define the [*asymmetric distance*]{} between $\mathbf{x}$ and $\mathbf{y}$ as $$\Delta(\mathbf{x},\mathbf{y}):=\max\{N(\mathbf{x},\mathbf{y}),N(\mathbf{y},\mathbf{x})\}.$$
It is easy to see that a set of codewords with minimum asymmetric distance $2$ is a $1$-code.
If ${\mathcal C}'$ is a single-error-correcting ternary code for the channel ${\mathcal T}$ of length $m$, then ${\mathcal C}=\mathfrak{S}^{m}({\mathcal C}')$ is a $1$-code of length $2m$. \[ChannelT\]
**Proof** For any two ternary codewords $\mathbf{c}'_1,\mathbf{c}'_2\in \mathcal{C}'$, we need to show that the asymmetric distance between $\mathfrak{S}^m(\mathbf{c}'_1)$ and $\mathfrak{S}^m(\mathbf{c}'_2)$ is at least two.
First, we cover the case when $\mathbf{c}'_1=\mathbf{c}'_2$. Distinct binary codewords may arise from the same ternary codeword due to the two different actions of $\mathfrak{S}$ on $0$. Such codewords have $\Delta \ge 2$ since $\Delta(00,11)=2$.
Next, if the Hamming distance between $\mathbf{c}'_1$ and $\mathbf{c}'_2$ is three, then the distance between $\mathfrak{S}^m(\mathbf{c}'_1)$ and $\mathfrak{S}^m(\mathbf{c}'_2)$ is also three since $\Delta(00,01), \Delta(11,01),\Delta(00,10),\Delta(00,01),$ and $\Delta(01,10)$ are all one and three such $\Delta$s occur.
Finally, the following Hamming distance two pairs are allowed in a single-error-correcting ternary code for $\mathcal T$: $$\begin{array}{lllll}
01,22 & 10,22 & 01,12 & 10,21 & 02,11\\
20,11 & 02,21 & 20,12 & 11,22 & 12,21 \\
\end{array}
\label{pairs}$$ It is straightforward to verify that $\mathfrak{S}$ on these pairs also results in binary codes with $\Delta \ge 2$. $\square$
The following corollary is straightforward.
If ${\mathcal C}'$ is a linear $[n,k,3]_3$ code (the subscript indicates that the code is over a three-letter alphabet rather than a binary alphabet), then $\mathfrak{S}^{m}({\mathcal C}')$ is a $1$-code of length $2m$. \[linear\]
### Odd block length \[oddonecode\]
Theorem \[ChannelT\] only works for designing $1$-codes of even length. Now we generalize this construction to the odd length situation, starting from ‘adding a bit’ to the ternary code [@gf3classical].
We call a code acting on $\mathbb{F}_2 \times \mathbb{F}_3^m$ a [*generalized ternary code*]{} of length $m+1$. We further adopt the conventions that $\mathfrak{S}^m({\mathcal C}')$ gives a $(2m+1)$-bit binary code by acting on the $m$ trits of a generalized ternary code ${\mathcal C}'$ and $\tilde{\mathfrak{S}}^{2m}({\mathcal C})$ when ${\mathcal C}$ has length $2m+1$ gives a generalized ternary code by acting on the last $2m$ bits of ${\mathcal C}$.
If ${\mathcal C}'$ is a single-error-correcting generalized ternary code for the channel ${\mathcal Z}\times{\mathcal T}^{m}$ of length $m+1$, then ${\mathcal C}=\mathfrak{S}^{m}({\mathcal C}')$ is a $1$-code of length $2m+1$. \[ChannelZT\]
**Proof**
As in the proof of Theorem \[ChannelT\] we need to show that for any two codewords $\mathbf{c}'_1,\mathbf{c}'_2\in \mathcal{C}'$, we need to show that the asymmetric distance between $\mathfrak{S}^m(\mathbf{c}'_1)$ and $\mathfrak{S}^m(\mathbf{c}'_2)$ is at least two. If the Hamming distance between codewords on [*just the ternary*]{} part of the code is at least two, then the situation reduces to the previous proof.
We need only worry about the case where the Hamming distance between $\mathbf{c}'_1$ and $\mathbf{c}'_2$ is two, and one of the differences in on the binary coordinate. Assume the first coordinate is a bit and the second is a trit, then since ${\mathcal C}'$ is a single-error-correcting generalized ternary code the only allowed pairs are $01,12$; and $ 12,11$. The corresponding images of each pair under $\mathfrak{S}^m$ give binary codewords of asymmetric distance $\Delta=2$. $\square$
To illustrate this generalized ternary construction, let us look at the following example.
The code $\{0000,0111,0222,1012,1120,1201\}$ corrects a single error from the channel ${\mathcal Z}\times{\mathcal T}^{3}$. Under the map $\mathfrak{S}^3$ it gives the binary code $$\begin{array}{llll}
0000000 & 0000011 & 0001100 & 0001111 \\
0110000 & 0110011 & 0111100 & 0111111 \\
0010101 & 0101010 & 1000110 & 1110110 \\
1011000 & 1011011 & 1100001 & 1101101 \\
\end{array}$$ which is a binary code of length $7$, dimension $16$ which corrects one asymmetric error. \[gt7\]
The following corollary is straightforward, but gives the most general situation of the ternary construction.
If ${\mathcal C}'$ is a ternary single error correcting code of channel ${\mathcal Z}^{{m_1}}\times{\mathcal T}^{{m_2}}$ of length $m_1+m_2$, then ${\mathcal C}=\mathfrak{S}^{m_2}({\mathcal C}')$ is a $1$-code of length $m_1+2m_2$. \[general\]
The $GF(3)$ construction for AD codes
-------------------------------------
### Even block length
We first examine under which conditions the image of a ternary code under $\mathfrak{S}$ could be self-complementary.
A ternary code ${\mathcal C}'$ is self-complementary if for any $\mathbf{c}'\in {\mathcal C}'$, $\bar{\mathbf{c}}'\in {\mathcal C}'$, where $\bar{\mathbf{c}}=({\mathbf 3}\ominus \mathbf{c})\ \mod 3$ (${\mathbf 3}=33\ldots 3$, [*i.e.*]{} the all ‘3’ string).
The ternary code ${\mathcal C}'=\{000,111,222\}$ is self-complementary. For $111\in {\mathcal C}'$, $\overline{111}=333\ominus 111=222$.
We say that binary code ${\mathcal C}$ of even length $n=2m$ has [*ternary form*]{} if $\mathfrak{S}^{m}(\tilde{\mathfrak{S}}^m({\mathcal C}))={\mathcal C}$.
The properties of $\mathfrak{S}$ gives the following
\[fact:tecomp\] If a ternary code ${\mathcal C}'$ of length $m$ is self-complementary, then its binary image under $\mathfrak{S}$, ${\mathcal C}=\mathfrak{S}^m({\mathcal C}')$, is self-complementary. On the other hand, if a binary code ${\mathcal C}$ of length $2m$ is of ternary form and is self-complementary, then its ternary image $\tilde{\mathfrak{S}}^{2m}({\mathcal C})$ is self-complementary.
To use Fact \[fact:tecomp\] to construct good single-error-correcting AD codes for even block length, first recall Example \[eg:AD816\] (and Example \[eg:ternary423\]):
\[eg:GF3\] The code given in Example \[eg:AD816\] under the $\mathfrak{S}$ map (pairing up coordinates $\{1,2\},\{3,4\},\{5,6\},\{7,8\}$) gives a linear code over $GF(3)$ generated by $\{0111,1012\}$.
We know that all the linear ternary codes are self-complementary, so the $1$-codes constructed from linear ternary codes of distance $3$ can directly used to construct single-error-correcting AD codes [@gf3classical]. Since in general we search for self-complementary ternary codes ${\mathcal C}'$ with largest possible size of ${\mathcal C}=\mathfrak{S}({\mathcal C}')$, those AD codes obtained from linear ternary codes of distance $3$ are sub-optimal.
We now show that the AD codes given by the Constantin-Rao construction are actually a special case of the $GF(3)$ construction.
\[th:ternary\] For $n$ even, the Varshamov-Tenengol’ts code ${\mathcal V}_0$, and the Constantin-Rao code ${\mathcal C}_0$ of largest cardinality has ternary form.
**Proof** We only need to prove that there exists a choice of pairing, such that for any codeword $v\in \mathcal{V}_0\ (\mathcal{C}_0)$, if $v$ restricts on one chosen pair $\alpha$ is $00$, then there exists another codeword $v'\in \mathcal{V}_0\ (\mathcal{C}_0)$ such that $v'=v|_{\tilde{\alpha}}$ and $v'|_{\alpha}=11$. Here $\tilde{\alpha}$ denotes all the other coordinates apart from $\alpha$.
For the Varshamov-Tenengol’ts code $\mathcal{V}_0$ of even length $n$, choose the pairing $\{i,n-i+1\}_{i=1}^{n/2}$, then the above condition is satisfied. This is because $i+n-i+1=n+1\ mod\ n+1=0$.
For the Constantin-Rao code $\mathcal{C}_0$ of largest cardinality, which is given by the group $G=\bigoplus_{r}\bigoplus_{i=1}^{n_{r}}\mathbb{Z}_{p_r}$, note $n$ is even, so $n+1$ is odd. Therefore all $p_r$ are odd for $p_r|n+1$, where $n+1=\Pi_{p_r|n+1}p_r^{n_{r}}$. Write any group element as $(s_{11},...,s_{1{n_1}},s_{21},...,s_{2{n_2}}... )$. Then we can pair it with $(p_1-s_{11},...,p_1-s_{1{n_1}},p_2-s_{21},...,p_2-s_{2{n_2}}... )$, $mod\ (p_1,...,p_1,p_2,...,p_2,...) $, where $s_{rj_r}\in\{0,...,p_r-1\}$ and $j_r=1,...,n_r$. $\square$
From both Fact \[fact:even\] and Theorem \[th:ternary\] we learn that for even block length, the Constantin-Rao code ${\mathcal C}_0$ of maximal cardinality is both self-complementary and has ternary form. Therefore, the AD codes given by the Constantin-Rao construction is actually a special case of the $GF(3)$ construction.
### Odd block length
For $n$ odd, we need to generalize the $GF(3)$ construction. As already discussed in Sec. \[oddonecode\], for $n=2m+1$, we design codes correcting a single error of the channel ${\mathcal Z}\times{\mathcal T}^{m}$. And we call these codes ‘generalized ternary.’
We need to examine under which condition the image of a generalized ternary code under $\mathfrak{S}$ is self-complementary.
A generalized ternary code ${\mathcal C}'$ of length $2m+1$ is self-complementary if for any $\mathbf{c}'\in {\mathcal C}'$, $\bar{\mathbf{c}}'\in {\mathcal C}'$. Here $\bar{{c}}'_1=1\oplus {c}'_1$, $\bar{{c}}'_i=3\ominus {c}'_i\ \mod 3$, for $i=2,\ldots,m+1$.
The generalized ternary code ${\mathcal C}'=\{000,100,011,122\}$ is self-complementary, because $\overline{000}=100$ and $\overline{011}=122$.
The properties of $\mathfrak{S}$ give the following:
\[fact:getecomp\] If a generalized ternary code ${\mathcal C}'$ of length $m+1$ is self-complementary, then its binary image under the map ${\mathcal C}=\mathfrak{S}^m({\mathcal C}')$ is self-complementary. On the other hand, if a binary code ${\mathcal C}$ of length $2m+1$ has generalized ternary form and is self-complementary, then its image $\tilde{\mathfrak{S}}^{2m}({\mathcal C})$ is self-complementary.
We now show that the AD codes given by the Constantin-Rao construction are actually a special case of the generalized ternary construction.
A binary code ${\mathcal C}$ of odd length $n=2m+1$ has [*generalized ternary form*]{} if $\mathfrak{S}^{m}(\tilde{\mathfrak{S}}^m({\mathcal C}))={\mathcal C}$.
Based on this definition, if a binary code ${\mathcal C}$ of odd length $2m+1$ has generalized ternary form, then it can be constructed from some codes correcting a single error of the channel ${\mathcal Z}\times{\mathcal T}^m$ via the ternarization map. The following theorem then shows that certain Varshamov-Tenengol’ts-Constantin-Rao codes are a special case of asymmetric codes constructed from single-error-correcting codes for the channel ${\mathcal Z}\times{\mathcal T}^m$ [@gf3classical].
\[th:geternary\] For $n$ odd, the Varshamov-Tenengol’ts code ${\mathcal V}_g$ has generalized ternary form.
**Proof** We only need to prove that there exists a choice of pairing which leaves a single coordinate as a bit, such that for any codeword $v\in \mathcal{V}_g$, if $v$ contains the paired bits $00$, then there exist another codeword $v'\in \mathcal{V}_g$ which is identical except that the $00$ pair is replaced by $11$, and vice versa.
For the Varshamov-Tenengol’ts code $\mathcal{V}_g$ of odd length, choose the pairing $\{i,n-i+1\}_{i=1}^{(n-1)/2}$, leave the coordinate $(n+1)/2$ as a bit, then the above pairing condition is satisfied. This is because $i+(n-i)+1=(n+1)\!\mod (n+1)=0$. $\square$
Now recall Fact \[fact:4k3\], which states that for block length $n=4k+3$, ${\mathcal V}_{\frac{n+1}{4}}$ is self-complementary. We further show the following:
For $n=4k+3$, ${\mathcal V}_{\frac{n+1}{4}}$ is of generalized ternary form.
To see this, do the pairing $\{i,n-i+1\}_{i=1}^{(n-1)/2}$. Here we leave the coordinate $(n+1)/2$ unpaired so it is unchanged under the map $\tilde{\mathfrak{S}}^m$.
For length $4k+1$, recall Fact \[fact:4k1\] that the shortened Varshamov-Tenengol’ts code ${\mathcal V}'_{\frac{n+2-r}{2}}$ obtained by deleting any ‘odd’ coordinate $r$ from Varshamov-Tenengol’ts code ${\mathcal V}_{\frac{n+2-r}{2}}$ of length $n+1$ is self-complementary. We further show the following:
For $n=4k+1$, the shortened Varshamov-Tenengol’ts code ${\mathcal V}'_{\frac{n+2-r}{2}}$ obtained by deleting any ‘odd’ coordinate $r$ from Varshamov-Tenengol’ts code ${\mathcal V}_{\frac{n+2-r}{2}}$ of length $n+1$ has generalized ternary form.
To see this, for the shortened Varshamov-Tenengol’ts code given by $$\sum_{i=1,i\neq r}^{n+2} ix_i=\frac{n+2-r}{2} \mod\ n+2,$$ do the pairing $\{i,n-i+2\}_{i=1}^{n/2}$. Here we leave the coordinate $n-r+2$ unpaired so it is unchanged under the map $\tilde{\mathfrak{S}}^m$.
Summary of new constructions for amplitude damping codes \[summary\] {#sec:summary}
====================================================================
For short block length we summarize the results of single-error-correcting AD codes obtained from the $GF(3)$ construction in Table \[table:ADcodes\], and compare them with AD codes obtained from other constructions.
$$\nonumber
\begin{array}{c c c c c c}
\\
\hline
n & GF(4) & \cite{Daniel} & \cite{Peter2} & {\mathcal C}_g & GF(3)\\
\hline
4 & 1 & 1 &2 & 2 & 2\\
5 & 2 & 2 &2 & 2 & 2\\
6 & 2 & 4 &5 & 5 & 5\\
7 & 2 & 8 &8 & 8 & 8\\
8 & 8 & 8 &12 & 16 & 16\\
9 & 8 & 16 &18 & 23 & 24 \\
10 & 16 & 32 &41 & 47 & 49\\
11 & 32 & 64 & 78 & 86 & 89 \\
12 & 64 & 128 &146 & 158 & 168\\
13 & 128 & 256 & 273 & 274 & 291 \\
14 & 256 & 512 & 515 & 548 & 572\\
15 & 512 & 1024 & 931 & 1024 & * \\
16 & 1024 & 1024 & 1716 & 1928 & * \\
\hline
\end{array}$$
\[table:ADcodes\]
Note the $\lfloor\lfloor 12,168,1\rfloor\rfloor$ code in Table \[table:ADcodes\] is cyclic, which can be obtained by the classical $1$-code $\lfloor 12,336,1\rfloor$ given in [@gf3classical]. The $\lfloor\lfloor 10,49,1\rfloor\rfloor$ code is ‘almost cyclic’, from which (deleting $4$ classical codewords then add another $2$) we can obtain a cyclic code $\lfloor\lfloor 10,47,1\rfloor\rfloor$, with classical codewords $$00000\ 11111\ 22222\ 21100\ 20111$$ and their cyclic shift, plus all the complements. There is another cyclic code $((10,47))$, with classical codewords $$00000\ 11111\ 22222\ 21100\ 21011$$ and their cyclic shift, plus all the complements.
Table \[table:ADcodes\] shows that the Constantin-Rao construction ${\mathcal C}_g$ outperforms other constructions apart from the (generalized) $GF(3)$ construction. This is reasonable since we know that the Constantin-Rao construction is actually a special case of the (generalized) $GF(3)$ construction. For all lengths up to $14$, the (generalized) $GF(3)$ construction indeed gives AD codes of best parameters. Lengths $>14$ are out of reach of the current computational power we have. As we know that the Constantin-Rao construction outperform the CSS construction for all lengths except $n=2^r-1$, where the binary Hamming codes are ‘good’, it is very much desired to know whether the (generalized) $GF(3)$ construction can give us something outperforms the CSS construction for the length $n=2^r-1$. From [@gf3classical] we know this is possible for classical $1$-codes, but it remains a mystery for the quantum case, which we leave for future investigation.
Finally, numerical search also found a $\lfloor\lfloor 9,26,1\rfloor\rfloor$ single-error-correcting AD code (exhaustively found to be optimal among all the self-complementary codes), which cannot be obtained from any of the above constructions. Also we have found, via random search, a $\lfloor\lfloor 10,51,1\rfloor\rfloor$ code, which also cannot be obtained from any of the above constructions.
Acknowledgements {#acknowledgements .unnumbered}
================
GS and JAS received support from the DARPA QUEST program under contract no. HR0011-09-C-0047.
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[^1]: PW Shor is with the Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
[^2]: G. Smith and JA Smolin are with the IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA
[^3]: B. Zeng is with the Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA and was with the IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA
[^4]: Not to be confused with quantum ${\sigma_z}$ errors, the channel takes its name from its diagram resembling the letter ’Z.’ See Figure \[fig:channel\].
|
---
author:
- |
Artur Bekasov\
University of Edinburgh\
<artur.bekasov@ed.ac.uk> Iain Murray\
University of Edinburgh\
<i.murray@ed.ac.uk>
bibliography:
- '/Users/artur/git/papers/bibliography.bib'
title: 'Bayesian Adversarial Spheres: Bayesian Inference and Adversarial Examples in a Noiseless Setting'
---
Introduction
============
Modern deep neural network models suffer from *adversarial examples*, i.e. confidently misclassified points in the input space [@szegedy2014]. Recently @gilmer2018 introduced *adversarial spheres*, a toy set-up that simplifies both practical and theoretical analysis of the problem.
It has been shown that Bayesian neural networks are a promising approach for detecting[^1] adversarial points [@rawat2017; @li2017; @bradshaw2017; @gal2018]. Bayesian methods explicitly capture the *epistemic* (or *model*) uncertainty, which we hope will detect parts of the input space that are not covered by training data well enough to justify confident predictions.
In this work, we use the adversarial sphere set-up to understand the properties of approximate Bayesian inference methods in a noiseless setting, where the only relevant type of uncertainty is epistemic uncertainty. We compare predictions of Bayesian and non-Bayesian methods, showcasing the strength of Bayesian methods, although revealing open challenges for deep learning applications.
**Contribution** Following our experiments we highlight the following observations:
Even a linear model suffers from adversarial examples in the adversarial sphere setup, while careful regularization proves unhelpful.
An accurate Bayesian method (MCMC) makes the model uncertain for adversarial examples, while keeping it reasonably confident for validation points.
The setup presents an example where model uncertainty estimated with bootstrap ensembling is insufficient.
MCMC results could be improved by using a more flexible prior that enables the model to exploit the symmetry in the problem.
A cheaper variational approximation does not result in an accurate posterior approximation, but demonstrates surprisingly good results on the benchmark. Using a richer variational family does not necessarily result in improved performance down-stream.
Adversarial spheres
===================
The adversarial sphere dataset is defined as two concentric hyperspheres with different radii. Each sphere constitutes a manifold for one of the two classes, with points distributed uniformly on the surface. The goal is to learn a decision boundary that would separate the two classes, which we know is itself a hypersphere with a certain radius.
In the original paper, @gilmer2018 show that in high dimensions ($D>60$) we can optimize to find points *on one of the spheres* that the model confidently misclassifies, even if the model demonstrates 100% accuracy on a huge validation set. Visualisations presented in the paper hint at local overfitting as an explanation for such behavior, which motivates the use of Bayesian methods.
We note that the labels in the dataset are deterministic, i.e. there is no inherent *alleatoric* uncertainty in the data, given the correct model. Such setup is interesting because a typical motivation for regularization methods is to avoid fitting the noise in the data. When the problem is noiseless, overfitting could be caused by uncaptured epistemic uncertainty, which is not addressed by standard regularization methods. At the same time, lack of alleatoric uncertainty is inherent to many real-world problems, such as natural image classification.
Bayesian Logistic Regression
============================
The target decision boundary in the adversarial sphere problem is non-linear. However, it can be represented using logistic regression applied to squared features: $$\begin{gathered}
P\left(y\!=\!1\,|\,\bm{x},\bm{w}\right) = \sigma\left(\bm{w}^\top\boldsymbol{\phi}(\bm{x})\right),
\qquad \boldsymbol{\phi}(\bm{x}) = [x_1^2~~\dots~~x_D^2]^\top .\end{gathered}$$
This model is able to learn axis-aligned ellipsoidal decision boundaries in $D$ dimensions. Learned NN basis could also be used, as explored by @snoek2015, where Bayesian logistic regression could be framed as being Bayesian about the last layer of a neural network. Inference gets more difficult for all parameters of a neural network, hence we must convince ourselves that we fully understand properties of inference methods for this simpler model.
Choosing a prior for Bayesian logistic regression, especially in a linearly separable setting, is not trivial [@gelman2008]. In such setting we are seeking large weights values, in order to make confident predictions. This implies the use of broad, uninformative (or weakly informative) priors. (In our experiments we use an isotropic Gaussian with large width $\sigma_w = 100$.)
Experiments
===========
[.325]{} ![ **(left)** Posterior density for one of the weights in a 500-dimensional setting, conditioned on the MAP values for all the other weights. The overall shape is Gaussian, but the region around the mode is flat. This is due to a broad prior used, and linear separability of the data. **(middle)** Background: likelihood for a 2D sphere problem. Red circles: samples from a variational posterior. White crosses: MCMC samples. Variational posterior captures the uncertainty about the “direction” in the weight space, but rules out large weights that are typical under the true posterior. **(right)** Background: likelihood for a 2D sphere problem. Red circles: samples from a variational posterior for a *hierarchical parameterization*. While most of the mass is still near the origin, higher weight values no longer have zero density. []{data-label="fig:fig"}](fig/posterior "fig:"){width="\textwidth"}
[.325]{} ![ **(left)** Posterior density for one of the weights in a 500-dimensional setting, conditioned on the MAP values for all the other weights. The overall shape is Gaussian, but the region around the mode is flat. This is due to a broad prior used, and linear separability of the data. **(middle)** Background: likelihood for a 2D sphere problem. Red circles: samples from a variational posterior. White crosses: MCMC samples. Variational posterior captures the uncertainty about the “direction” in the weight space, but rules out large weights that are typical under the true posterior. **(right)** Background: likelihood for a 2D sphere problem. Red circles: samples from a variational posterior for a *hierarchical parameterization*. While most of the mass is still near the origin, higher weight values no longer have zero density. []{data-label="fig:fig"}](fig/2d_samples "fig:"){width="\textwidth"}
[.325]{} ![ **(left)** Posterior density for one of the weights in a 500-dimensional setting, conditioned on the MAP values for all the other weights. The overall shape is Gaussian, but the region around the mode is flat. This is due to a broad prior used, and linear separability of the data. **(middle)** Background: likelihood for a 2D sphere problem. Red circles: samples from a variational posterior. White crosses: MCMC samples. Variational posterior captures the uncertainty about the “direction” in the weight space, but rules out large weights that are typical under the true posterior. **(right)** Background: likelihood for a 2D sphere problem. Red circles: samples from a variational posterior for a *hierarchical parameterization*. While most of the mass is still near the origin, higher weight values no longer have zero density. []{data-label="fig:fig"}](fig/2d_samples_hier "fig:"){width="\textwidth"}
Results for a 500-dimensional adversarial sphere dataset are summarized in Table \[tab:results\].
**Maximum Likelihood / MAP** We first train a logistic regression model using maximum likelihood. For a large training set the model becomes immune to adversarial attacks, but on a smaller training dataset (with 1000 datapoints) the model demonstrates perfect accuracy on the validation set (error rate below $10^{-5}$), but at the same time we can find points which the model misclassifies with more than $99\%$ confidence. Switching to simple penalized ML / MAP with a Gaussian prior on the weights (i.e. adding L2 regularization) does not resolve the issue. We note the following contradiction: we know that in the “true” solution weight magnitudes are large, as the dataset is linearly separable, yet we are penalizing such values.
**MCMC sampling** We then attempt to understand what an accurate Bayesian method [slice sampling, @neal2003] would do. The results show that these approximate Bayesian predictions become uncertain for adversarial points. While we also get reasonably confident predictions on points from the validation set, the confidence is reduced when compared to ML/MAP results. Not all validation samples are close to the (limited) training data, hence the model cannot be completely confident in its predictions without exploiting symmetries/invariances in the data.
**Bootstrap** Can we get similar results from a simpler method? We train an ensemble of models using bootstrap sampling, and use an averaged prediction during test time. Bootstrap ensembles are often claimed to be an approximation to the Bayesian posterior [@hastie2001 §8.4]. In our setup, however, the uncertainty estimated in this way is insufficient — the worst adversarial error is hardly reduced. Moreover, the rightmost column of Table \[tab:results\] shows that the adversarial points found are transferable to other ensembles trained in the same way. In other words, the adversarial procedure learns to exploit the training method, and not a particular ensemble, which is not the case for MCMC.
**Laplace approximation** Sampling, while accurate, is often impractical. We look into a cheaper method of Laplace approximation, that fits a Gaussian to the posterior by matching the curvature at the mode [@mackay2003 Chapter 27].We observe that while the method also detects adversarial examples, it becomes just as uncertain on validation points. The issue is caused by the non-Gaussian shape of the posterior near the mode, as illustrated in Figure \[fig:fig\], which stems from multiple steep decision boundaries having the same likelihood given the data. This results in an unrealistically wide Gaussian approximation and uncertain predictions for all points. This phenomenon is also discussed by @kuss2005.
**Variational approximation** Variational inference is an alternative way of fitting a simple distribution family to the true posterior. We implement and evaluate Stochastic Variational Inference [SVI, @hoffman2013; @ranganath2013] with a full-covariance Gaussian family in our experiments. Variational inference makes probabilistic predictions that are surprisingly close to the ones of MCMC. At the same time, when we look at the samples from the true posterior, as shown in Figure \[fig:fig\], the fit is clearly not perfect. In particular, the variational posterior assigns near-zero density to higher weight values, ruling out the steepest decision boundaries, which are closest to the truth.
**Hierarchical model** An alternative model could be used to enable more accurate variational approximation. We reparameterize the model using a hierarchical approach: $w_i \sim N(0, e^v); v \sim N(0, \sigma_{v}^2)$. Equivalently, using a “non-centered” parameterization: $w_i = e^{v/2} z_i; z_i \sim N(0, 1); v \sim N(0, \sigma_{v}^2)$. We can then fit two variational distributions, $q_{{\bm{z}}}({\bm{z}})$ and $q_v(v)$. Intuitively, we are defining a distribution over the “direction” in the weight space using ${\bm{z}}$, and the positive “distance” in that direction using $v$. This is related to the work by @ranganath2016, but where we also update the prior to match the variational family. Figure \[fig:fig\] shows that such parameterization results in a more sensible posterior fit, where we no longer assign zero density to larger weight values. However, in this case the down-stream performance is not improved.
**Exploiting symmetry** Similarly, we can use a hierarchical parameterization to allow the model to exploit the symmetry in the problem. We know that the max margin decision boundary is a sphere, hence the “true” weight values must be close to one another. We thus assume the weights come from a prior distribution $w_i \sim N(\mu, \sigma_w^2)$, where the mean has its own hyper-prior $\mu \sim N(0, \sigma_{\mu}^2)$. This results in a model with well calibrated uncertainty, as seen in the last row of Table \[tab:results\]. Some of the earliest work on Bayesian neural networks recognized the importance of choosing hierarchical priors carefully [@neal1994]. The prior we use here is favored by the data, but trying it was guided by our knowledge of the problem. It is likely that in realistic problems with deeper networks, the choice of prior will only have a stronger effect on the uncertainties reported by Bayesian methods. Exploring the families of priors that can capture symmetries and invariances in real problems is an important direction in Bayesian deep learning.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the EPSRC Centre for Doctoral Training in Data Science, funded by the UK Engineering and Physical Sciences Research Council (grant EP/L016427/1) and the University of Edinburgh. Authors thank James Ritchie for his proof-of-concept implementation for this work.
Details of training
===================
We use 1000 training samples and 100k validation samples in a 500-dimensional setting for our experiments.
We normalize the input features of the logistic regression after applying the basis functions, as our preliminary experiments have revealed that normalization plays an important role in the performance of some methods.
We use the Pytorch implementation of the LBFGS optimizer for MAP, bootstrap and Laplace experiments.[^2] SGD with momentum is used for variational methods, with batch size of 100, learning rate of 0.01 and momentum coefficient of 0.98. Limited hyperparameter exploration was performed. We run optimization for 50 thousand iterations.
A spherical Gaussian prior with $\sigma_{w} = 100$ is used for experiments with a standard model. Given feature normalization, this represents a reasonably broad belief. For the hierarchical model, we use $\sigma_{v} = 100$.
Details of adversarial optimization {#sec:adv_opt}
===================================
The goal of adversarial optimization could be formalized as follows: $$\begin{gathered}
{\bm{x}}_{\mathit{adv}} = \operatorname*{argmax}_{{\bm{x}}} p(y=y_{\mathit{target}}|{\bm{x}}); \quad y_{\mathit{target}} = 1 - y_{\mathit{true}}\end{gathered}$$
In our work, we would also like to restrict ${\bm{x}}$ to lie on a surface of a sphere with a given radius. In line with @gilmer2018, we solve the constrained optimization problem using Projected Gradient Descent, which works by projecting the current point onto the required sphere after every gradient step.
If we attempt to use a gradient based optimizer with such objective, however, we are likely to run into numerical problems. Logistic regression model is defined as $$\begin{gathered}
p(y=1|{\bm{x}}) = \sigma(a),\end{gathered}$$ where $\sigma$ is a sigmoid function and $a = {\bm{w}}^{\top}{\bm{x}}$. For a model trained with MLE or MAP, a random point $x$ on one of the spheres will typically result in a large magnitude of $a$, as model’s predictions are extremely confident. This, in turn, will select a point at one of the tails of $\sigma(a)$. The sigmoid function saturates for relatively small activation magnitudes, especially when using single floating-point precision. This means that $\nabla_a \sigma(a) = 0$, and we will not be able to make any meaningful optimization steps.
To fix this, we use the fact that the sigmoid is a monotonically increasing function, hence: $$\begin{gathered}
\operatorname*{argmax}_{{\bm{x}}} p(y=y_{\mathit{target}}|{\bm{x}}) = \operatorname*{argmax}_{{\bm{x}}} \sigma(a) = \operatorname*{argmax}_{{\bm{x}}} a.\end{gathered}$$ In other words, we can optimize the logit of a prediction, rather the prediction itself, hence avoiding numerical issues outlined above.
The same trick is not applicable to ensembles, however. Ensemble prediction is defined as $$\begin{gathered}
p(y=1|{\bm{x}}) = \frac{1}{M} \sum_{i=1}^M \sigma(a_m), \end{gathered}$$ where $a_m$ is an activation of $m$-th model in the ensemble. Intuitively, we can not maximize this objective by maximizing the mean of activations. The mean of activations could be maximized by pushing one of the activations to infinity. This would clearly not maximize the original objective, however, as saturation of the sigmoid limits the contribution of each model’s prediction.
Instead, we use another trick. We know that log probabilities have better numerical properties than probabilities themselves, and that log is also a monotonically increasing function. Rewriting the objective in terms of log probabilities we get $$\begin{gathered}
\log p(y=1|{\bm{x}}) = \log \frac{1}{M} \sum_{i=1}^M \sigma(a_m) = \log \sum_{i=1}^M \sigma(a_m) + const. \end{gathered}$$ Then, we can apply Jensen’s inequality to set a lower bound on the log sum: $$\begin{gathered}
\log \sum_{i=1}^M \sigma(a_m) \ge \sum_{i=1}^M \log \sigma(a_m),\end{gathered}$$ where $\log \sigma(a_m)$ could be expressed as $-\text{softplus}(-a_m)$, which has a numerically stable implementation. We could then optimize this lower bound instead of the original objective, at least to guide the optimization to a more numerically stable region during early steps.
We note that it has also been proposed to use outlined numerical issues as defense mechanism, to make it difficult for an attacker to obtain meaningful gradients for adversarial optimization. Recent work by @athalye2018 discusses these issues further, and outlines various ways of defeating such defense in different models.
In our experiments, we optimize the more numerically stable lower-bound of the loss for 300 iterations before switching to the real optimization criterion. Step size of 0.01 is used for most experiments, but we lower it to 0.0001 for Monte-Carlo SVI and sampling experiments, due to observed instability during optimization. Optimization is terminated if the best achieved loss has not noticeably improved for 10 iterations. We consider an absolute difference in loss value of more than $1\mathrm{e}{-4}$ to be a noticeable improvement. This number was picked empirically, to strike the balance between good convergence and the amount of computation.
[^1]: We note that *detecting* is not equivalent to *fixing*. Ideally, we would like our models to classify all in-sample points confidently and correctly. It has been suggested that this might only be achievable by modeling all the invariances present in the data [@gal2018].
[^2]: Note that this version of LBFGS does not implement line search.
|
---
abstract: 'Debris disks are usually thought to be gas-poor, the gas being dissipated by accretion or evaporation during the protoplanetary phase. HD141569A is a 5 Myr old star harboring a famous debris disk, with multiple rings and spiral features. I present here the first PdBI maps of the $^{12}$CO(2-1), $^{13}$CO(2-1) gas and dust emission at 1.3 mm in this disk. The analysis reveals there is still a large amount of (primordial) gas extending out to 250 au, i.e. inside the rings observed in scattered light. HD141569A is thus a ‘hybrid’ disk with a huge debris component, where dust has evolved and is produced by collisions, with a large remnant reservoir of gas.'
---
A debris disk still containing gas {#a-debris-disk-still-containing-gas .unnumbered}
==================================
[**Inclination ($^{\circ}$)**]{} [**Postion Angle ($^{\circ}$)**]{} [**R$_{out}$ (au)**]{} [**R$_{in}$ (au)**]{} [**T$_0$ (K)**]{} [**q**]{}
----------- ---------------------------------- ------------------------------------ ------------------------ ----------------------- ------------------- -----------------
$^{12}CO$ 54.4 $\pm$ 0.4 86.1 $\pm$ 0.2 254 $\pm$ 3 22 $\pm$ 1 44 $\pm$ 2 0.35 $\pm$ 0.05
$^{13}CO$ 57 $\pm$ 2 88 $\pm$ 1 253 $\pm$ 15 21 $\pm$ 7 16 $\pm$ 4 0.2 $\pm$ 0.3
: Best fit parameters from DiskFit gas modeling
![Integrated intensity of the CO and dust emission at 1.3 mm, superimposed to the $HST$ scattered emission (Clampin et al. 2003). The cross indicates the position angle and aspect ratio as determined from gas modeling. Left: $^{12}$CO J=2-1 emission, contour spacing: 6$\sigma$, i.e. 5.5$\times$10$^{-1}$ Jy/beam.km.s$^{-1}$. Beam size: 2.48$\times$1.45”. Middle: $^{13}$CO J=2-1 emission, contour spacing: 3$\sigma$, i.e. 7.8$\times$10$^{-2}$ Jy/beam.km.s$^{-1}$. Beam size: 1.76$\times$1.32”. Right: continuum emission, contour spacing: 3$\sigma$, i.e. 2.0$\times$10$^{-1}$ mJy/beam. Beam size: 2.57$\times$1.31”.[]{data-label="fig:maps"}](maps-clampin.eps){width="4.6in"}
HD141569A is a 5$\pm$3 Myr old star (Merín et al. 2004), of spectral type B9.5V/A0Ve, located 116$\pm$8 pc away (van Leeuwen 2007). With a stellar mass of 2 M$_{\odot}$, the HD141569A system appears to be in an intermediate evolutionary stage between protoplanetary and debris disks.
A debris disk has been discovered first by $IRAS$, with an infrared excess of the same order of magnitude as $\beta$ Pictoris (L$_{disk}$/L$_{\star}$=8$\times$10$^{-3}$; Sylvester et al. 1996). Optical images reveal that the debris disk around HD141569A is very complex, with a double-ring architecture, a large inner depletion within 125 au, and arc and spiral features (e.g. Augereau et al. 1999, Biller et al. 2015). The dust appears to be of second generation origin, i.e. produced by collisions, as indicated by the timescale for collisions of $\sim$10$^{4}$ years which is $100$ times less than the age of the star (Boccaletti et al. 2003).
In addition to its impressive debris disk, CO gas has been detected around HD141569A (Zuckerman et al. 1995; Dent et al. 2005). NIR CO and other atomic lines have also been observed (Goto et al. 2006; Thi et al. 2014). The inferred total remnant mass of gas has thus been estimated in the range 80-135 M$_{\oplus}$ (Jonkheid et al. 2006).
We present here the first resolved maps of the $^{12}$CO J=2-1 and $^{13}$CO J=2-1 emission lines, which we obtained in 2014/2015 with the Plateau de Bure Interferometer array. The integrated intensity maps of the gas are displayed in Fig.1, as well as the continuum emission at 1.3 mm. We have modeled the data in the uv-plane using the code DiskFit (Piétu et al. 2007), based on a power-law description of the physical parameters, e.g. T(r)=T$_{0}$(r/R$_{0}$)$^{-q}$. Table 1 shows the parameters determined from this modeling for the $^{12}$CO and $^{13}$CO. The disk extends from $\sim$20 au to 250 au. From the $^{12}$CO/$^{13}$CO line ratio, the $^{12}$CO appears to be still optically thick while the $^{13}$CO is optically thin. The temperature is thus best determined from the $^{12}$CO modeling ($\sim$45 K at 100 au, a typical value for an A star). The $^{13}$CO better probes the surface density, which is here $\sim$30 times less than around typical HAeBe disks, like MWC480.
HD141569A is thus a ‘hybrid’ disk with a large gas component, likely primordial, and an impressive evolved debris disk. The flux at 1.3 mm is 3.5$\pm$0.1 mJy, a low value in agreement with fast evolution of the dust. The links between gas and dust properties in this and other star/disk systems have to be studied in more detail, in particular to better understand the disk dissipation/evolution mechanisms which influence the shaping of young planetary systems.
References {#references .unnumbered}
==========
Augereau, J.-C.; Lagrange, A. M.; Mouillet, D. et al., 1999, A&A, 350,51\
Biller, B. A.; Liu, M. C.; Rice, K. et al., 2015, MNRAS, 450, 4446\
Boccaletti, A.; Augereau, J.-C.; Marchis, F. et al. 2003, ApJ, 585, 494\
Dent, W. R. F.; Greaves, J. S.; Mannings, V. et al. 2005, MNRAS,277,25\
Goto, M.; Usuda, T.; Dullemond, C. P. et al., 2006, ApJ, 652, 758\
Jonkheid, B.; Kamp, I.; Augereau, J.-C. et al., 2006, A&A, 453,163\
Merín, B.; Montesinos, B.; Eiroa, C. et al, 2004, A&A, 419, 301\
Piétu, V.; Dutrey, A.; Guilloteau, S., 2007, A&A, 467, 163\
Sylvester, R. J.; Skinner, C. J.; Barlow, M. J. et al., 1996, MNRAS, 279, 915\
Thi, W.-F.; Pinte, C.; Pantin, E. et al., 2014, A&A, 561, 50\
van Leeuwen, F., 2007, A&A, 474, 653\
Zuckerman, B.; Forveille, T.; Kastner, J. H, 1995, Nature, 373, 494\
|
---
abstract: 'By implementing a dynamic wind-tunnel model in a smoothed-particle chemodynamic/hydrodynamic simulation suite, we have investigated the effects of ram pressure and tidal forces on dwarf galaxies similar to the Magellanic Clouds, within host galaxies with gas and dark matter halos that are varied, to compare the relative effects of tides and ram pressure. We concentrate on how the distributions of metals are affected by interactions. We find that while ram pressure and tidal forces have some effect on dwarf galaxy outflows, these effects do not produce large differences in the metal distributions of the dwarf disks other than truncation in the outer regions in some cases, and that confinement from the host galaxy gas halo appears to be more significant than ram pressure stripping. We find that stochastic variations in the star formation rate can explain the remaining variations in disk metal properties. This raises questions on the cause of low metallicities in dwarf galaxies.'
author:
- David Williamson
- Hugo Martel
bibliography:
- 'ram.bib'
title: 'Chemodynamics of dwarf galaxies under ram-pressure'
---
Introduction
============
According to the observed mass-metallicity relation [@2004ApJ...613..898T], the lowest-mass galaxies have the lowest metallicities. These dwarf galaxies should be strongly affected by interactions with more massive galaxies, which could affect the slope and scatter of the mass-metallicity relation at the low-mass end. Although the role of interactions on the morphology of dwarf galaxies has been thoroughly explored [e.g. @2003MNRAS.345.1329M; @2004MNRAS.352..363M; @2005MNRAS.364..607M; @2010MNRAS.405.1723S; @2011ApJ...740L..24K; @2014ApJ...780..119K], the [*chemodynamical*]{} effects of interactions on a dwarf galaxy have not been as directly or thoroughly examined [although see @2013MNRAS.436.1191T; @2018MNRAS.474.2194E] .
We have examined the role of tidal effects on the chemodynamical evolution of dwarf galaxies in a previous paper [@2016ApJ...831....1W hereafter Paper II]. We expected that tidal stripping would preferentially remove high-metallicity outflows and thus act as a drain of metallicity from dwarf galaxies. However, we found that tidal stripping can actually [*enhance*]{} the metallicity of our simulated galaxies. A metallicity gradient is produced by ongoing centrally-concentrated star formation, so that when the outer low-metallicity regions of the dwarfs are stripped, the high-metallicity core is preferentially retained. However, this effect is mild enough that differences in star formation rates are also a major (or even dominant) contributor, especially as tidal forces can either trigger or suppress the gravitational instabilities that lead to star formation.
We might assume that ram pressure would have a stronger effect than tides. Indeed it is often shown that ram pressure can produce dramatic stripping in dwarf galaxies [e.g. @2010AdAst2010E..25M for a review]. However, comparing observations to cosmological simulations, @2014MNRAS.442.1396W found that among high-mass dwarf galaxies (stellar mass $M_s>10^{8.5}$), only $30$% were quenched. Similarly, @2018MNRAS.474.2194E found that environmental effects did not significantly affect the width of the metal distribution function (MDF) in zoom-in simulations from the FIRE-2 and LATTE simulation suites. These simulations had a mass resolution of $\sim7000$ M$_\odot$. Ram pressure stripping of metals may be sensitive to resolution, as low-density metal-rich outflows may escape with less mixing at higher resolution. Pushing a greater fraction of metals into the wind would cause the dwarf galaxy metallicities to be more sensitive to large-scale environmental effects. These authors also only examined the total MDF of the dwarf galaxies, and did not consider the spatial distribution of the metals.
The simulations of @2013MNRAS.436.1191T also examined tidal stripping of dwarf galaxies, concluding that AGB stars are critical for reducing the N/O ratio in interacting dwarf galaxies. However, these simulations did not include explicit chemodynamics (that is, the production and mixing of metals is not explicitly tracked in the simulations), and thus their results rely on assumptions on the metal content of galaxy winds.
The simulations of @2015ApJ...815...77S explored the effects of ram-pressure on a Magellanic Cloud model. They found that a front is formed, but that ram pressure stripping was not strong. Again, these simulations did not explicitly track metallicities, and also did not include a wind produced by star formation. Such a wind could transport metals further from the center of the dwarf galaxy potential, where they can be more easily stripped.
In this work, we perform high-resolution simulations with explicit chemodynamics from idealized initial conditions within both a tidal field and a wind tunnel based on a gas halo model. We examine the effects of tidal forces and ram pressure on the metal contents of dwarf galaxy winds and disks, and on the spatial distribution of metals within a dwarf galaxy. We focus on a dwarf galaxy with properties intermediate between the Large and Small Magellanic Clouds (section \[section\_galaxymodel\]), within a gas halo and tidal field similar to that of the Milky Way. Simulations have shown [@2012MNRAS.421.2109B] that the Milky Way’s interactions with the Magellanic Clouds are weak, and that the observed interaction effects are the result of the Magellanic Clouds interacting with each other. Thus, in addition to being broadly applicable to massive dwarf galaxies in general, our results will shed light on how the Magellanic Clouds might have evolved chemodynamically if they interacted only with the Milky Way and not each other.
The remainder of this paper is organized as follows. In Section \[section\_method\] we describe our numerical method and simulation set-up. In Section \[section\_results\], we describe the results of these simulations. In Section \[section\_discussion\] we compare our results to other work and discuss potential numerical issues. We then summarize our conclusions in Section \[section\_conclusions\].
Method {#section_method}
======
Simulation code
---------------
We use a version of the GCD+ smoothed-particle hydrodynamics (SPH) code [@2003MNRAS.340..908K; @2012MNRAS.420.3195B; @2013MNRAS.428.1968K; @2014MNRAS.438.1208K]. This code includes a stochastic star formation model that relaxes the single stellar population assumption, allowing different star particles to represent stars of different masses. Star particles return energy and metals to the ISM through supernovae and stellar winds. The Plummer-equivalent force softening length is $2$ pc. Smoothing lengths are calculated dynamically through an iterative method, so that each particle has $\approx58$ neighboring particles. The minimum smoothing length is $2$ pc, which means that particles in very dense regions have $>58$ neighbors. The metal content of particles is tracked throughout the simulation, and a sub-grid diffusion model allows metals to spread between particles. Our version of GCD+ includes modified algorithms for metal deposition and diffusion, as described in @2016ApJ...822...91W [hereafter Paper I], and a variable background potential to represent the varying tidal forces on a satellite galaxy moving through a host galaxy potential, as described in Paper II. In this paper, we have further extended the code to include the gas component of this host galaxy to model ram pressure, as we will describe in section \[section\_rammodel\].
[lcccccccc]{} Run & $M_h$ & $n_0$ & $v_0$ & $R_p$ & $f_R$ & $f_T$ & $f_{RT}$ & $P_C$\
& (M$_\odot$) & cm$^{-3}$ & km s$^{-1}$& kpc & & & & \
\
A & $10^{12}$ & $0.460$ & $190.0$ & $100$ & $1.000$ & $1.00$ & $1.00$ & $1.00$\
A\* & $10^{12}$ & $0.460$ & $190.0$& $100$ & $1.000$ & $1.00$ & $1.00$ & $1.00$\
B & $10^{12}$ & $0.046$ & $190.0$& $100$ & $0.100$ & $1.00$ & $0.10$ & $0.10$\
C & $10^{11}$ & $0.460$ & $72.8$& $100$ & $0.147$ & $0.17$ & $0.86$ & $0.15$\
D & $10^{11}$ & $0.046$ & $72.8$& $100$ & $0.015$ & $0.17$ & $0.09$ & $0.02$\
\
A- & $10^{12}$ & $0.460$ & $150.0$ & $60$ & & & & \
A-- & $10^{12}$ & $0.460$ & $100.0$ & $29$ & & & & \
A--- & $10^{12}$ & $0.460$ & $50.0$ & $11$ & & & & \
B- & $10^{12}$ & $0.046$ & $150.0$ & $60$ & & & & \
B-- & $10^{12}$ & $0.046$ & $100.0$ & $29$ & & & & \
B--- & $10^{12}$ & $0.046$ & $50.0$ & $11$ & & & & \
Galaxy model {#section_galaxymodel}
------------
We use the same dwarf galaxy model as in Papers I & II. This model consists of a disk of gas and stars within a dark matter halo, with properties similar to those of disk-like irregular or Magellanic-type galaxies such as the Magellanic Clouds. We briefly summarize the dwarf galaxy models here, but further details and motivations are provided in Papers I & II. The total disk mass is $5\times10^8$ M$_\odot$, with a gas fraction of $f_g=0.5$. The stellar disk has a scale height of $100$ pc and a scale length of $540$ pc. The gas disk has a scale length of $860$ pc, and the vertical distribution of gas is initially set by the criterion of hydrodynamic equilibrium, although stellar feedback and radiative cooling cause the gas to rapidly move away from its initial equilibrium state. The initial metal abundances are $[\alpha/\mathrm{H}]=-2$ for all $\alpha$ species, and $[\mathrm{Fe}/\mathrm{H}]=-3$, giving $[\alpha/\mathrm{Fe}]=1$. The metallicity gradient is initially flat, and so any metallicity gradient produced in the simulations is a result of explicitly-modelled evolution.
The disk consists of $5\times10^5$ particles, giving a mass resolution of $1000$ M$_\odot$. This is placed inside an active dark matter halo with an NFW profile [@1997ApJ...490..493N] of mass $9.5\times10^9$ M$_\odot$ and concentration parameter $c=10$, which consists of $9.5\times10^5$ particles.
Tidal forces and ram pressure {#section_rammodel}
-----------------------------
We perform each simulation in the center-of-mass frame of the dwarf galaxy as it orbits through the halo of a host galaxy. The tidal field of the host galaxy is calculated analytically (see Paper II). However, an analytic method is not sufficient to model ram pressure, which relies on the complex hydrodynamic interactions between the dwarf’s gas and the host galaxy’s gas halo. On the other hand, directly modelling the entire host gas halo as a system of particles with sufficient mass resolution to simultaneously resolve the dwarf galaxy would be prohibitively expensive, requiring billions of particles for each run. Much of this computational expense would also be unnecessary, as here we are not interested in the evolution of the host galaxy itself. To solve this issue, we place the galaxy in a cubic box of width $160$ kpc which follows the dwarf galaxy as it moves through the host halo, and only follow the evolution of material within this region. This is essentially a wind tunnel model, with gas particles entering on one end and exiting on the other.
To build this wind tunnel box, we divide the host galaxy gas halo into $100^3$ cubic cells of width $5$ kpc, where the density and temperature of the center of each cell is set by an analytic function (given below). These cells are defined in the host galaxy frame. If the motion of the $160$ kpc cubic box through this grid of cells causes the center of a cell to enter this box, the cell is populated with particles. If a cell center leaves this region, all of the gas particles it contains are deleted. This effectively produces inflow-conditions on the “forward” edge of the box, and outflow-conditions on the “outward” edge of the box. We also freeze the temperature and halo-frame velocity of particles in a boundary zone two cells thick on each wall of this box.
We populate a cell by randomly generating particles evenly throughout the cell. The initial temperatures are linearly interpolated between the cell-centered values of the surrounding cells. We determine the temperature and density of the gas halo cells from profiles which model a Milky Way gas halo. Following @2015ApJ...815...77S, we use the $\beta$ profile parameters of @2013ApJ...770..118M for the distribution of hot gas, and use the large distance limit for the density distribution,
$$\label{eq_ndist}
n(r) = n_0 \frac{r_c}{r}^{3\beta},$$
where $n_0=0.46$ cm$^{-3}$, $r_c=0.35$ kpc, and $\beta=0.71$. We use this model as a base Milky Way model, but set $n_0=0.046$ cm$^{-3}$ in some runs, to isolate the evolution in a system with reduced ram pressure. We note that the scatter of halo gas masses for galaxies of similar properties was found to be quite large in the Illustris simulations [@2016MNRAS.462.3751K], and hence it is not unusual for two galaxies with the same dark matter halo mass to have gas halos whose masses differ by a factor of ten.
The gravitational potential of the host galaxy is set as an NFW halo. We use two different halo masses to investigate the change in tidal forces, using $M_h=10^{12}$ M$_\odot$ as a Milky Way model, and $M_h=10^{11}$ M$_\odot$ for the lower-mass model with weak tides. We assume a fixed concentration parameter of $c=12$. The concentration parameter only varies slowly with halo mass and with a large scatter [@2007MNRAS.381.1450N], and so again it is reasonable to change the halo mass by a factor of ten without changing the concentration parameter.
To set the temperature of the gas halo, we again follow @2015ApJ...815...77S and use the temperature profile of @1998ApJ...497..555M,
$$\label{eq_Tdist}
T(r) = \gamma \frac{G\mu m_p M(r)}{3rk_B}$$
where $M(r)$ is dominated by the mass of the NFW halo, and $\gamma$ is the adiabatic index. The abundances of the halo are the solar values scaled down by $10^{-2}$.
Simulations
-----------
We have produced eleven runs, each with the same dwarf galaxy model, but with various host galaxy and orbital properties. Runs A, A-, A--, A---, B, B-, B--, and B---, have a Milky-Way mass background potential, while Runs C and D have a lower-mass background potential. Runs A (and A- etc) and C have a Milky-Way mass host galaxy gas halo, while Runs B (and B- etc) and D have a lower-mass host galaxy gas halo. We have varied these values so that we can determine the effects of tides and ram pressure in a Milky Way environment by comparing with runs where tides and ram pressure should be much weaker. In particular, both tidal forces and ram pressure should be very weak in Run D, and so this run forms a basis we can compare with the simulations with stronger tides and/or ram pressure. The simulation properties are summarized in Table \[ictable\].
We produced five runs with circular orbits to investigate where the strengths of ram pressure and tidal stripping are constant over time, to facilitate direct comparison between the runs. These runs are labelled A-D without any ‘-’ suffix, and have a constant orbital radius of $r=100$ kpc.
We also produced six runs with elliptical orbits to investigate the more impactful effects of ram pressure and tides at low pericentres. These runs have the same host galaxy properties as the runs with circular orbits, but with a different initial velocity. These are indicated with a number of ‘-’ signs in the suffix of the run name, where the greater the number of ‘-’ signs, the closer the orbital pericenter is to the host galaxy centre. These runs have the same host galaxy conditions as Runs A and B. The initial velocities $v_0$ and radii at pericenter $R_p$ are given in Table \[ictable\].
Additionally, we performed an additional run with identical initial conditions to Run A, except with a different random seed. This run, which we call Run A\*, is used to quantify the level of variation caused by stochastic variations in the star formation rate.
The magnitude of ram pressure is proportional to $\rho v^2$, where $\rho$ is the density of the medium (proportional to $n_0$) and $v$ is the speed of the galaxy relative to the medium. For the runs with circular orbits, we use this to determine the strength of ram pressure of all runs relative to Run A, defined as $f_R = \rho_i/\rho_A (v_{0i}/v_{0A})^2$ for each run $i$.
The dwarf galaxies in the circular runs orbit at a constant distance through a potential of the same concentration parameter, and so we can characterize the magnitude of tidal forces with a single parameter $f_T$, which can be calculated analytically from the NFW potential. If we normalize the tide strength so that $f_T=1$ for runs A and C, we find that $f_T=0.17$ in runs B and D. We can then define the ratio between the tide and ram pressure strengths relative to Run A as $f_{RT} = f_R/f_T$.
Finally, although ram pressure stripping is insensitive to gas temperature [e.g. @2009ApJ...694..789T], the temperature of the gas halo is important for thermal pressure [*confinement*]{} of the dwarf galaxy’s gas. For a circular orbit, using equations and , and assuming an ideal gas law, the pressure of the halo gas that the dwarf galaxy encounters is proportional to the product of $M_h(r=100\mathrm{~kpc})$ and $n_0$. We can calculate the relative confinement pressure $P_C$ from this, normalising the pressure so that for Run A, $P_C=1$.
All of these values are given in Table \[ictable\] for the circular runs to aid in the interpretation of our results.
{width="\textwidth"}
{width=".49\textwidth"} {width=".49\textwidth"}
{width=".49\textwidth"} {width=".49\textwidth"}
{width="\textwidth"}
{width="\textwidth"}
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![\[ramMDF\] MDFs at t = 2 Gyr, for all runs. Top panel: gas component in runs with circular orbits; middle panel: gas component in runs with elliptical orbits; bottom panel: stellar component in all runs. ](ramMDF.pdf){width="\columnwidth"}
Results {#section_results}
=======
Figure \[sfr\] shows the star formation rates and cumulative star formation for all runs. The cumulative star formation is defined as the total mass of gas that has been transformed into stars by this point in time. As in Papers I & II, at first the star formation rate gradually increases as gravitational instabilities drive gas inwards. The star formation reaches a peak rate and then starts to decline as gas is consumed.
Runs C and D have peaks that are earlier and greater in magnitude than the peaks in Runs A and B. Run C also has the greatest cumulative mass of star formation for most of the simulation time. These two runs have the weakest tidal forces, $f_T$. However, there is no monotonic trend between $f_{RT}$ or $f_R$ and the total star formation, so there is no clear connection between ram pressure and star formation here. The final star formation mass of Run D is also very close to that of Run A, despite the large difference in $f_R$. Given the chaotic nature of N-body dynamics [@2009MNRAS.398.1279S], the reliance of star formation on gravitational instability, and our stochastic star formation algorithm, many of the differences in star formation between the runs with circular orbits can be attributed to psuedo-random perturbations amplified by gravitational instabilities, as in Paper II. We note that, despite having the same initial conditions, the different random seed of Run A\* causes its cumulative star formation to significantly differ from Run A, with differences in cumulative star formation at any particular time often larger than the differences between Run A\* and Run D. The peaks in star formation rate also occur at different times and with different magnitudes in Run A and Run A\*. Hence our simulations do not show evidence that mild and constant ram pressure and tides have a direct influence on star formation, beyond being an additional source of perturbations.
The more dramatic interaction effects in the elliptical runs tell a different story. Overall, the star formation rates are generally larger, due to tidal stirring generating strong star-forming instabilities. Tides do have a significant effect here. The peaks in star formation appear near pericenter, and the runs with more circular orbits that reach pericenter later generally show peaks that occur later. Runs A--- and B--- have the same orbit and experience the same tidal forces, and have an initial peak of star formation at a similar time and a similar magnitude. Runs B- and A- show a similar correspondence. As the B runs have a halo gas density ten times smaller than the A runs, and should thus have ram pressure effects that are ten times weaker, these agreements shows that these bursts of star formation rate are caused by tidal stirring, and not by ram pressure compression.
The exceptions are Runs A-- and B--, which share an orbit, but have dramatically different star formation rates – Run A-- has a particularly large burst of star formation. Examining the evolution of that run in detail, we found that, solely in this run, the ram pressure, outflow rate, tidal stretching, and dwarf galaxy orbit happen to be arranged such that the outflow produced during the first $\sim800$ Myr of star formation is rapidly driven back into the galaxy as it dives into the denser parts of the halo (but before it reaches pericenter), stimulating a large burst in star formation. Again, the main effect of the halo gas is through confining the outflows, and not through ram pressure stripping, and it just happens that in this particular setup the confinement happens to drive a strong burst of star formation.
Figure \[evolution\] shows snapshots of the column density and metallicity of the circular orbit runs at $t=1$ Gyr and $t=2$ Gyr. Outflows are visible in the column density plots for all runs, and the effect of tides is also clear, stretching and bending the outflows. This is most visible in Runs A, A\*, and B where the $M_h$ is greatest, and less visible in Runs C and D where tidal forces are smaller. The effect of the gas halo is also clear here. Rather than being a source of stripping through ram pressure, the background gas halo [*confines*]{} the outflows and keep the gas closer to the dwarf galaxy. Runs A and C both have more confined outflows than Runs B and D. The extent of the wind appears to depend more on the gas halo density $n_0$ than on the ram pressure strength $f_R$ or the ram-pressure-to-tides ratio $f_{RT}$. We also note that Runs B and C have similar values of $P_C$, but that the outflows are more confined in Run C where $n_0$ is higher. This suggests that the confinement is not only caused by thermal pressure, but also by the amount of mass the outflow must plough through.
However, these outflows do not appear to be efficient at transporting metals out of the dwarf galaxy. The metallicity plots in Figure \[evolution\] show that while these outflows have a greater metallicity than the environment of the dwarf galaxy, they are not metal-rich compared with the dwarf galaxy core, and most of the metals remain concentrated into the inner few kiloparsecs of the dwarfs. The outflows mix with and entrain low-metallicity gas, and do not consist only of metal-rich supernova ejecta. Additionally, continuous star formation in the center of the dwarf galaxy further increases the central metallicity but outflowing gas can no longer be enriched after being launched, and thus the metallicity of outflows lags behind the metallicity of the dwarf galaxy center. Hence, while interactions have a clear effect on the [*morphology*]{} of the outflows, this does not appear to have an impact on the metallicity of the dwarf galaxy itself, as the outflows are ineffective at removing metals from the dwarf galaxy.
More significant effects are found in the elliptical runs plotted in Figure \[evolution-ellipse\]. The effects of pressure confinement is clear in the density plots, as the B runs with lower gas density show more freely flowling winds than the A runs. The effects of tides are also clearly visible, particularly in Runs A--- and B---, where the outflows are very disturbed. In the elliptical runs, the tidal shocks do disturb the galaxies enough to redistribute metals beyond the disk (as well as processing the dwarf disks into a more elliptical configuration in Runs A--- and B---), and it appears that this effect increases as tidal forces increase. However, pressure effects only appear to be significant on the outflow at large distances, as the metal distributions in the A runs are similar to those of the B runs with corresponding orbits.
To gain more insight into the cause of these effects, we plot the metallicity profiles in Figure \[zslopes\]. Here we have plotted the gas-phase metallicity ratios \[Fe/H\] and \[O/Fe\] as a function of $|z|$, the distance from the disk plane (the vertical profile), and as a function of $R$, the radial distance within the disk plane (the radial profile). The vertical profile only includes gas within $R<5$ kpc, and the radial profile only includes gas within $|z|<2$ kpc. The central concentration of metals is clear by the sharp peak in \[Fe/H\] at low $z$ and $R$. Unlike \[Fe/H\], \[O/Fe\], increases with distance. This is because regions of recent star formation have a greater relative abundance of iron due to Type-II supernovae. Hence we see \[O/Fe\] essentially follows the inverse of \[Fe/H\], dropping at low $z$ and $R$.
Much of the variations between runs can be explained by examining the star formation rates. The star formation is bursty, and the metal distribution depends on whether there has been a recent burst of star formation, and how dramatic it was. Dramatic bursts of star formation can produce outflows that rapidly transport metals large distances from the dwarf galaxy center, although as stated above, the metallicity of these outflows does not reach that of the star-forming galaxy center.
Run D has the earliest bursts of star formation and hence the earliest outflows, producing a higher metallicity at $z>2$ kpc compared to other runs, as seen in Figure \[zslopes\]. The large burst of star formation at $t\approx1$ Gyr also pushes gas out radially along the disk plane, enriching the gas at $R>2$ kpc by $t=1.5$ Gyr and $t=2$ Gyr. Similarly, the bursts of star formation in Run A around $t=1.25$ Gyr enrich the gas at $z>2$ kpc for $t\geq1.5$ Gyr. Run B has the lowest total star formation, and has only a single burst of moderately rapid star formation, and so the metals remain more concentrated in the core. But the dependence on stochastic star formation is most clear when we compare Run A and Run A\*, where ram pressure and tidal forces should be identical, but the metallicity gradient varies greatly between the two runs, depending on the details of recent star formation.
Interestingly, Run C has a series of strong bursts of star formation and has the greatest total star formation, but the metallicity beyond $z=2$ kpc or $R=2$ kpc is among the lowest out of all the runs, especially at $t=1.5$ Gyr and $t=2$ Gyr. This can again be explained as primarily a result of star formation. In Run C, there are several bursts of rapid star formation from $t=1$ Gyr to $t=1.35$ Gyr. As is the general case in these runs, the star formation is centrally concentrated, and thus consumes the most highly enriched gas. In this case, the star formation is sufficiently rapid that this depletion is more significant than the enrichment of distant gas through outflows. We find that at $t=2$ Gyr, Run C has the lowest mass of metals in disk gas, but the greatest mass of metals in disk stars.
Confinement further contributes to this. The greater halo pressure and the greater density of the ambient medium in Run C stops winds from propagating as far as in Run D, keeping metals centrally concentrated in Run C, but allowing metals in Run D to reach higher distances. However, ram pressure [*stripping*]{} does not appear to have a significant effect, nor tidal stripping in the models with circular orbits.
By contrast, in the runs with elliptical orbits (Figure \[zslopes-ellipse\]), the effects of tides is significant. By $t=2$ Gyr, it is clear that beyond the dwarf center, \[Fe/H\] generally becomes higher (and \[O/Fe\] generally lower) with increasing tide strengths. Although there is some variation due to recent star formation, the metallicity slopes are largely paired by orbit, with no clear difference between runs with different halo gas densities. This confirms that ram pressure does not have a significant effect here, even in the inner part of the gas halo, but that tidal stripping and stirring do have a large role.
The metallicity evolution can be summarized by the evolution of the effective yield plotted in Figure \[metevolve\]. We define the effective yield as the increase in total disk ($R<5$ kpc, $|z|<2$ kpc) metal mass divided by the total quantity of mass consumed so far by star formation. We include the metal mass of both stars and gas in this quantity so that the effective yield only varies when metals escape the disk region. We note that this differs from other conventional definitions of the term ‘effective yield’. We see an early drop in the effective yield as the disk reaches equilibrium, and then all runs reach a similar yield from $0.5-1$ Gyr. After this point, the effective yield drops as bursts of star formation expel enriched gas (although as we note above, the most heavily enriched central gas is mostly retained). In the final stages ($t>1.5$ Gyr), the yield increases steadily because continuous star formation enriches the gas and produces enriched stars, but does not provide the dramatic bursts of feedback required to effectively expel gas. In this period, the rate of increase of the effective yield is similar in all simulations.
For the runs with circular orbits, the final yield at $t=2$ Gyr then largely depends on whether and to what extent the recent star formation history of the galaxy is dominated by yield-reducing dramatic bursts or by yield-increasing continuous star formation. This relates to both the timing and magnitude of the bursty phase. For example, Run B has a large final yield because its star formation is weak, producing only a small decrease in effective yield. Run A has a longer period of strong star formation, and has a much lower final effective yield. Run D also has a deeper drop in effective yield due to its bursty star formation, but this burst occurs earlier, and so continuous star formation begins earlier, with the result that it reaches a final effective yield intermediate between A and B. Run C has a similar final effective yield, despite having the most star formation, but as noted above, this results from its rapid star formation consuming high-metallicity gas and thus reducing the gas supply for outflows. This is most plainly seen by the fact that the final yield of Run A\* is closer to that of Run D than Run A, despite Runs A and A\* having identical orbital and halo properties. Again, for the runs with circular orbits there is no clear correlation between the final yield and the strength of tidal stripping or ram pressure stripping – the yields appear to be dominated by stochastic variations in the star formation rate.
The runs with elliptical orbits do show environmental effects on the effective yield, though not always dominating over star formation effects. The two runs with the greatest tidal forces – A--- and B--- – show significantly lower effective yields than all other runs. However, the rest of the elliptical runs show no more variation in final effective yield than the circular runs. This is because the total mass of metals is dominated by the large mass of metals in the central region, which is often not strongly effected by environmental effects.
We can disentangle the role of the high-metallicity region by examining the metallicity distribution functions (MDF). The MDFs for the disk gas and formed stars at $t=2$ Gyr is plotted in Figure \[ramMDF\]. The stellar MDFs agree between all runs, showing no significant sensitivity on environmental effects. This is because stars are generally formed in the center of the galaxy, where environmental effects are at their weakest.
The gas MDFs do show an environmental dependence. The metallicities of the highest metallicity peaks vary between the runs, but as stated above these peaks represent gas within a very small central region ($\lesssim200$ pc), and are not strongly affected by environmental effects. Instead, it is the low-metallicity gas that shows evidence of tidal stripping. The elliptical runs are again paired by orbit rather than gas halo density, with a lower quantity of low-metallicity gas as tidal strength increases, independent of the density of the gas halo. This shows that tidal stripping is again dominant over ram pressure.
There may also be an indication of a pressure confinement effect in the circular orbit runs. The runs with greater halo densities (A,A\*,C) retain higher quantities of low-metallicity gas than the runs with lower halo densities (B,D). However, this effect is not large compared to the difference between Run A\* and Run A, and may not be significant.
Discussion & Comparison with other work {#section_discussion}
=======================================
The effects of interactions on dwarf galaxies have been investigated in cosmological simulations, with work that is typically focused on the role of interactions in producing a population of quenched red dwarf galaxies. These simulations typically have mass resolutions of $10^4-10^6$ M$_\odot$, and can only resolve the most massive dwarfs, typically with $M_s>10^8$ M$_\odot$. Both the Illustris simulation and semi-analytic models based on the Millennium simulations produce quenching that is too rapid compared with SDSS observations [@2014MNRAS.442.1363W; @2015MNRAS.447L...6S] despite these simulations being performed with very different numerical methods – the Millennium simulations were performed with the SPH code Gadget [@2005MNRAS.364.1105S] and the Illustris simulation was performed with the moving-mesh code AREPO [@2010MNRAS.401..791S]. However, without a semi-analytic model to post-process the simulation, it has been reported that the Millennium simulation produces ‘surprisingly inefficient’ quenching [@2014MNRAS.442.1396W]. This particular study only considered dwarf galaxies with a gas greater than $10^{8.5}$ M$_\odot$, that are perhaps too massive to be quenched by ram pressure and tidal stripping. A study of metal stripping and star formation of more massive satellites ($M>10^9$ M$_\odot$ in Illustris [@2016ApJ...822..107G] found that star formation rates are concentrated and disks are truncated (but not rapidly quenched) by interactions, producing a higher observed metallicity. However, inefficient interaction effects (ram-pressure in particular) has also been found in a high-resolution simulation of the lower-mass dwarf galaxy Leo T [@2016ApJ...826..148E]. As for an intermediate-mass galaxy such as the LMC, truncation rather than dramatic tidal stripping is also found in simulation [@2005MNRAS.363..509M]. The zoom-in simulations of @2018MNRAS.478..548S investigated the effect of dwarf mass at higher resolution, and found a steep trend in quenching from a $90$% quenching fraction for $M_s=10^6$ M$_\odot$ dwarfs down to $30$% for $M_s=10^8$ M$_\odot$ dwarfs. In the EAGLE simulation, it was found that only $10$% of LMC mass galaxies in Milky Way mass haloes were quenched [@2018MNRAS.479..284S]. The general consensus appears to be that the effects of quenching and ram pressure on Magellanic Cloud mass galaxies is small, and this is consistent with our results.
However, ram pressure is a phenomenon that can be very sensitive to numerical methodologies. This was most famously demonstrated in the ‘blob test’ of @2007MNRAS.380..963A, where a sphere of dense cold fluid is impacted by a high-velocity hot low-density medium – analogous to molecular gas in a satellite galaxy being stripped by a hot host halo. It was found that traditional SPH methods preserved the stability of the blob far longer than grid-based methods. This was explained by the improper calculation of pressure gradients at boundaries causing a surface-tension effect, along with a small contribution from artificial viscosity. This led to the development of new SPH paradigms, such as the pressure-entropy paradigm [e.g. @2013MNRAS.428.2840H]. This form shows a greater agreement with grid codes, and is the form used in our version of GCD+ [for numerical tests of this code, see @2013MNRAS.428.1968K]. Moving-mesh or ‘mesh-free’ methods [@2010MNRAS.401..791S; @2015MNRAS.450...53H] have also gained attention in recent years. However, comparisons between hydrodynamic methods in galaxy models and cosmological simulations have found little dependence on hydrodynamic method [@2012MNRAS.423.1726S; @2015MNRAS.450...53H; @2015MNRAS.454.2277S], other than traditional SPH methods producing spurious over-densities and higher star-formation rates. In general, it is found that the variations between simulations is utterly dominated by the variations in sub-grid models, and not in hydrodynamics methods. Hence, as we are using a ‘modern’ SPH method, we can be reasonably confident that our results are not affected by numerical problems with our hydrodynamics implementation.
Summary & Conclusions {#section_conclusions}
=====================
To investigate the effects of ram pressure and tidal stripping on dwarf galaxies, we have performed chemodynamical simulations of dwarf galaxies with similar properties to the Magellanic Clouds orbiting within host galaxies with dark matter and gas halos similar to the Milky Way, investigating circular and elliptical orbits, and varying both the gas and dark matter content separately to have either the Milky Way mass or one tenth the mass of the Milky Way. We have found that the effects of ram pressure on the metallicity of the dwarf galaxy disks are not significant, and that the differences between the models can be explained by stochastic variations in the star formation rate, and by the effects of tides, which are only significant at low pericenters. Ram pressure and tidal forces do affect the morphology and metallicity of outflows at large distances from the dwarf galaxy, even on dwarf galaxies circular orbits at large distances, but this does not significantly affect the metallicities of the dwarf galaxy disk if the tides are not very strong. We do find that tidal effects can truncate the dwarf galaxy in plunging orbits, but instead of lowering the metallicity of the dwarf galaxies, the effect is to remove low-metallicity outer gas. We also find that the host galaxy’s halo pressure can [*confine*]{} outflows rather than strip them, slightly enhancing the dwarf galaxy’s metallicity.
This raises two important issues. Firstly, given that our model dwarf galaxy is essentially a lone Magellanic Cloud, the lack of strong interaction effects in our simulations supports the conclusions of @2012MNRAS.421.2109B that the interaction effects observed in the Magellanic Clouds are primarily the result of interactions between the two Clouds, and not of those between the Clouds and the Milky Way – although we have concentrated on metallicity distributions rather than on morphology.
Secondly, and more critically, how are the observed low metallicities and low gas fractions of dwarf galaxies generated if ram pressure stripping is ineffective? A clue here may be the surprisingly low metallicities of our outflows. While we use a modern hydrodynamics scheme, the wind could still be sensitive to resolution. Although our mass resolution of $1000$ M$_\odot$ is finer than that of e.g. @2018MNRAS.474.2194E, it may be that high-metallicity outflows are still suppressed at current resolutions. We plan to produce higher-resolution simulations in a future paper.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was supported by the Canada Research Chair program and NSERC. The authors acknowledge the use of the [*Guillimin*]{} and [*Colosse*]{} computing clusters supported by Calcul-Québec/Compute-Canada, as well as the IRIDIS High Performance Computing Facility supported by the University of Southampton, in the completion of this work. DW is supported by European Research Council Starting Grant ERC-StG-677117 DUST-IN-THE-WIND. We thank our anonymous reviewer for advice that improved the content and presentation of this paper.\
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---
abstract: 'We study uncertainty relations for pairs of conjugate variables like [number ]{}and angle, of which one takes integer values and the other takes values on the unit circle. The translation symmetry of the problem in either variable implies that measurement uncertainty and preparation uncertainty coincide quantitatively, and the bounds depend only on the choice of two metrics used to quantify the difference of [number ]{}and angle outputs, respectively. For each type of observable we discuss two natural choices of metric, and discuss the resulting optimal bounds with both numerical and analytic methods. We also develop some simple and explicit (albeit not sharp) lower bounds, using an apparently new method for obtaining certified lower bounds to ground state problems.'
author:
- Paul Busch
- Jukka Kiukas
- 'R.F. Werner'
title: 'Sharp uncertainty relations for [number ]{}and angle '
---
Introduction
============
The study of uncertainty relations has experienced a major boost in recent years. As more and more experiments reach quantum limited accuracy, sharp quantitative uncertainty and error bounds become more relevant. It has also become evident that the subject of quantum uncertainty cannot be reduced to the classic standard uncertainty relation that was first made rigorous by Kennard [@Kennard], Robertson [@Robertson] and others and is now found in every textbook. The purpose of this paper is to present a new case study that exemplifies the three main directions of generalization currently being pursued. The example at hand is given by the number-angle pair of observables, where by “number” we understand an observable whose spectrum is the set of all integers.
The first extension of the uncertainty principle concerns the set of scenarios to which quantitative uncertainty relations apply. The Kennard relation is a “preparation uncertainty relation”, i.e., a quantitative expression of the observation that there is no state preparation for which the distributions of two observables under consideration are both sharp. This relation can be tested, in principle, by separate runs of precise measurements of the two observables, performed on ensembles of systems in the same state. In contrast to this, one may consider attempted joint measurements of noncommuting observables, as already intuitively envisaged by Heisenberg [@Heisenberg27]; one finds that such joint measurements are constrained by “measurement uncertainty relations” [@Wer04; @BP07; @BLWprl] which describe the unavoidable error bounds. The number-angle pair also highlights the need to consider uncertainty measures other than standard deviations. The second direction of generalization to be considered is thus in the concrete mathematical expressions measuring the “sharpness” of distributions, or the error of an approximate measurement. We will consider two alternative types of uncertainty measures and associated error measures for each of the two observables concerned. Finally, the third direction of generalization concerns the forms of preparation and measurement uncertainty relations that are applicable to arbitrary pairs of observables. The Robertson relation involving the expectations of the commutator in the lower bound fails to provide preparation uncertainty relation in the above sense because the only state-independent lower bound one can get from it is zero. Nevertheless, non-trivial state-independent bounds usually do exist. Uncertainty is really a ubiquitous phenomenon, joint measurability or simultaneous sharp preparability are the exceptions rather than the rule. Accordingly, the problem of establishing tight uncertainty relations for pairs of observables amounts to the task of establishing their (preparation or measurement) *uncertainty regions*, defined as the set of pairs of uncertainty values in all states (for preparation uncertainty) and the set of pairs of error values in all possible joint measurements (for measurement uncertainty), respectively.
It is a pleasing property of the case at hand—[number ]{}and angle—that an essentially complete treatment can be given. This is due to the phase space symmetry which implies, exactly as for standard position and momentum [@BLWjmp], that “metric distance” and “calibration distance” for the error assessment of observables satisfy the same relations and also that they are quantitatively the same as corresponding preparation uncertainty relations. In this paper we deduce the ensuing measurement uncertainty relations for [number ]{}and angle.
Our paper is organized according to the methods employed. In Section \[sec:basics\] we review the conceptualization of the preparation uncertainty measures and associated error measures to be used throughout the paper. This is followed by an overview of our main results (Section \[sec:overview\]). In Section \[sec:sym\] we show how the phase space symmetry can be exploited to find optimal joint measurements among the covariant phase space observables. Here we show the identity of preparation uncertainty regions and measurement uncertainty (or error) regions in the case at hand, introducing calibration uncertainty regions as a mediating construct. The lower boundary of the uncertainty regions is characterized as a ground state problem. Numerical estimates of the optimal tradeoff curves are shown in Section \[sec:numerics\]. Next, Sections \[sec:exact\] and \[sec:lower\] give determinations of the exact ground states and lower boundaries for the various combinations of deviation measures, and we show how existing uncertainty relations for [number ]{}and phase can be reproduced or strengthened using our systematic approach. We conclude with an outlook in Section \[sec:outlook\].
Conceptual Uncertainty Basics {#sec:basics}
=============================
[Number ]{}and angle observables
--------------------------------
The complementarity between [number ]{}and angle appears in physics in various guises. Essentially, this is for any parameter with a natural periodicity. Geometric angles are one case, with the complementary variable given by a component of angular momentum. Another important case is quantum optical phase, which is complementary to an harmonic oscillator Hamiltonian. At least for preparation uncertainty this is no difficulty, since a relation valid for all states also holds for states supported on the subspace of positive integers. One does get additional or sharper relations from building in this constraint, however. A further important field of applications is quasi-momentum with values in the Brillouin-Zone for a lattice system (of which we only consider the one-dimensional case here). This is also related to form factors arising in the discussion of diffraction patterns and fringe contrast from periodic gratings [@BBK].
The literature on angle-angular momentum uncertainty is almost exclusively concerned with the preparation scenario, although the lack of an error disturbance relation has been noted [@tanimu]. In the preparation case a major obstacle was the Kennard and Robertson [@Robertson] relation and their role as a model how uncertainty relations should be set up. There is nothing wrong with an observable with outcomes on a circle. But much work was wasted on the question of how to represent “angle” measurements by a selfadjoint operator [@JudgeLewis; @Kraus]. Additional unnecessary confusion in the case of semibounded [number ]{}and quantum optical phase was generated by the ignorance or lack of acceptance of generalized (“POVM”) observables. On the positive side, an influential paper by Judge [@judge63] produced a relation (for the arc metric), and conjectured an improvement, which was proved shortly afterwards [@EvettMahmoud]. In this context the role of ground state problems for finding optimal bounds, which is also the basis of our methods, seems to have appeared for the first time [@vanLeuven]. The appearance of the chordal metric grew out of the approach of avoiding the “angle” problem, replacing $\theta$ by the two selfadjoint operators $\cos\theta$ and $\sin\theta$.
Measures of uncertainty and error
---------------------------------
Both in preparation and in measurement uncertainty we have to assess the difference of probability distributions: For preparation uncertainty it is the difference from a sharp distribution concentrated on a single point. This is also the basis of calibration error assessment. For metric error we also need to express the distance of two general distributions. We want to express the distance between distributions on the same scale as the distance of points. For example, in the case of position and momentum errors, for all error measures one considers $\Delta Q$ to be measured in length units and $\Delta P$ in momentum units, and this is satisfied by the choice of standard deviation for these measures.
So let us assume that the outcomes of some observable (represented by a POVM, a positive operator valued measure) lie in a space $X$ with metric $d$. For real valued quantities like a single component of position or momentum this usually means $X={\mathbb{R}}$ and $d(x,y)=|x-y|$.
We now extend the distance function on the points to a distance between a probability measure $\mu$ on $X$ and a point $x\in X$. It will just the be the mean distance from $x$: $$\label{meandist}
d_\alpha(\mu,x)=\left(\int\!\mu(dy) d(y,x)^\alpha\right)^{\frac1\alpha}.$$ Here the exponent $\alpha\in[1,\infty)$ gives some extra flexibility as to how large deviations are weighted relative to small ones. The $\alpha^{\rm th}$ root ensures that the result is still in the same units as $d$, and also that for a point measure $\mu$ concentrated on a point $y$ we have $d_\alpha(\mu,x)=d(y,x)$ for all $\alpha$. It is also true for all $\alpha$ that $d_\alpha(\mu,x)=0$ happens only for the point measure at $x$. We will later mostly choose $\alpha=2$ and drop the index $\alpha$; in this case $d_2(\mu,x)$ is the root mean square distance from $x$ for points distributed according to $\mu$.
With this measure of deviation of a distribution $\mu$ from the point we can introduce the generalized standard deviation, $$\label{stdev}
d_\alpha(\mu,*)=\min_{x\in X}d_\alpha(\mu,x).$$ Note that for $\alpha=2$, $X={\mathbb{R}}$ and $d(x,y)=|x-y|$ this recovers exactly the usual standard deviation, with the minimum being attained at the mean of $\mu$. The symbol $*$ is just a reminder of the minimization, and emphasizes that $d_\alpha(\mu,*)$ is just the distance of $\mu$ from the set of point measures.
For this interpretation to make sense we must also let the second argument of $d_\alpha$ be a general probability distribution $\nu$, resulting in a metric on the set of probability measures. The canonical definition here is the transport distance [@Villani] $$\label{meanmindist}
d_\alpha(\mu,\nu)=\inf_\gamma\left\lbrace\int\!\gamma(dx\,dy) d(x,y)^\alpha \Bigm\vert \couples\mu\gamma\nu\right\rbrace^{\frac1\alpha},$$ where “$\couples\mu\gamma\nu$” is a shorthand for $\gamma$, the variable in this infimum, being a “coupling” of $\mu$ and $\nu$, i.e., it is a measure on $X\times X$ with $\mu$ and $\nu$ as its marginal distributions. One should think of $\gamma$ as a plan for converting the distribution $\mu$ into $\nu$, maybe for some substance rather than for probability. The cost of transferring a mass unit from $x$ to $y$ is supposed to be $d(x,y)^\alpha$, and the plan $\gamma$ records just how much mass is to be moved from $x$ to $y$. The marginal property means that the initial distribution is $\mu$ and the final one $\nu$. Then $d_\alpha(\mu,\nu)^\alpha$ is the optimized cost. When the final distribution is a point measure, there is not much to plan, and we recover . Therefore there is little danger of confusion in using the same symbol for the metrics of points and of probability measures.
Now we can use these notions of spread and distance for expressing uncertainties related to observables $A,B$ with outcome spaces $X$ and $Y$, each with a suitably chosen metric and error exponent. Let us denote by $\rho\mby A$ the probability measure of outcomes in $X$ upon measuring $A$ on systems prepared according to $\rho$. To express [*preparation uncertainty*]{}, let us consider the set $\PU$ of generalized variance pairs $$\label{devpairs}
\PU=\Bigl\{ \bigl(d_\alpha(\rho\mby A,*)^\alpha,\,d_\beta(\rho\mby B,*)^\beta\bigr)\bigm\vert \rho \text{ a state}\Bigr\}.$$ A preparation uncertainty relation is some inequality saying that the uncertainty region does not extend to the origin: the two deviations cannot both be simultaneously small. If this set is known, we consider it as the most comprehensive expression of preparation uncertainty. Its description by inequalities for products or weighted sums or whatever other expression is a matter of mathematical convenience, and we will, of course, develop appropriate expressions. A lower bound for the product is useful [*only*]{} for position and momentum and its mathematical equivalents. In this case the dilatation invariance $(q,p)\mapsto(\lambda q,\lambda^{-1}q)$ forces the uncertainty region to be bounded by an exact hyperbola. But if one of the observables considered can take discrete values, the set will reach an axis, making every state-independent lower bound on the product trivial.
We should note that the set is in general not convex, and can have holes (for examples, see [@AMU]). However, in order to express lower bounds, the essence of uncertainty, it makes no difference if we fill in these holes, and include with every point also those for which both coordinates are larger or the same. The resulting set, the ‘monotone hull’ $\PU^+$ of $\PU$, is bounded below by the graph of a non-increasing function, the [*tradeoff curve*]{} (see Fig. \[fig:basic4\] for examples). It still need not be convex in general, but we will see that convexity holds in the examples we study.
For [*measurement uncertainty*]{} we again consider two observables (POVMs) $A,B$ with the same outcomes, metrics and error exponents. Now the question is: can $A,B$ be measured jointly? The claim is, usually, that no matter how we try there will be an error in our implementation. So let $A',B'$ be the margins of some joint measurement with outcomes $X\times Y$. Then $A'$ must exhibit some errors relative to $A$, i.e., some output distributions $\rho\mby{A'}$ must be different from $\rho\mby{A}$. We define as the error of $A'$ with respect to $A$ the quantity $$\label{dAA}
d_\alpha(A',A)=\sup_\rho d_\alpha(\rho\mby{A'},\rho\mby{A}).$$ Note that we are using here a worst case quantity with respect to the input state. This is what we should do for a figure of merit for a measuring instrument. If a manufacturer claims that his device $A'$ will produce distributions $\varepsilon$-close to those of $A$ for any input state, he is saying that $d_\alpha(A',A)\leq\varepsilon$. Making such a claim for just a single state is as useless as advertising a clock which measures “the time $12{:}00$” very precisely (but maybe no other). Now we can look at the uncertainty region $$\label{MUpairs}
\MU=\Bigl\{ \bigl(d_\alpha(A',A)^\alpha,\,d_\beta(B',B)^\beta\bigr)\bigm\vert \couples{A'}{\!}{B'}\Bigr\},$$ where the notation $\couples{A'}{\!}{B'}$ indicates that $A'$ and $B'$ are jointly measurable (that is, they are margins of some POVM that serves as a joint observable). All general remarks made about the preparation uncertainty region $\PU$ also hold for $\MU$.
The supremum in is rather demanding experimentally. Good practice for testing the quality of a measuring device is [*calibration*]{}, i.e., testing it on states with known properties, and seeing whether the device reproduces these properties. In our case this means testing the device $A'$ on states $\rho$ whose $A$-distribution is sharply concentrated around some $x$, and looking at the spread of the $A'$-distribution around the same $x$. We define as the calibration error of $A'$ with respect to $A$ the quantity $$\label{dAAmin}
\Delta_\alpha^c(A',A)=\lim_{\varepsilon\to0}\sup\left\lbrace d_\alpha(\rho\mby{A'},x)
\Bigm\vert d_\alpha(\rho\mby{A},x)\leq\varepsilon\right\rbrace.$$ Here the limit exists because the set (and hence the sup) is decreasing as $\varepsilon\to0$. This definition only makes sense if there actually are sufficiently many sharp states for $A$, so we will use this definition only when the reference observable $A$ is projection valued. Since the calibration states in this definition are also contained in the supremum , it is clear that $\Delta_\alpha^c(A',A)\leq d_\alpha(A',A)$, so the set $\CU$ of calibration error pairs will generally be larger than $\MU$.
Setting and Overview of Results {#sec:overview}
===============================
We now consider systems with Hilbert space $\HH=L^2({\mathbb{T}},d\theta)$, where ${\mathbb{T}}$ denotes the unit circle, with $d\theta$ the integration over angle. The notation derives from “torus” and is customary in group theory. We use it here to emphasize the group structure (of multiplying phases or adding angles mod$2\pi$) but also to avoid a fixed coordinatization such as ${\mathbb{T}}\cong[-\pi,\pi)$, which would misleadingly assign a special role to the cut point $\pm\pi$. We will refer to ${\mathbb{T}}$ as our “position space”. The corresponding “momentum” space is ${\mathbb{Z}}$, and changing to the momentum representation in $\ell^2({\mathbb{Z}})$ is done by the unitary operator of expanding in a Fourier series. With $\{e_n\}_n\in{\mathbb{Z}}$ denoting the Fourier basis, this means that $$\label{Fourier}
(\Fou\psi)(n)=\langle{e_n}\vert{\psi}\rangle=\frac1{\sqrt{2\pi}}\int\!\! d\theta\, e^{i n \theta} \psi(\theta).$$ We have two natural projection valued observables, the angle (=position, phase) observable $\Theta$ taking values on ${\mathbb{T}}$, and the [number ]{}(=angular momentum observable) $N$ with values in ${\mathbb{Z}}$. That is, if $f:{\mathbb{T}}\to{\mathbb{C}}$ is some function of the angle variable, $f(\Theta)$ denotes the multiplication operator $(f(\Theta)\psi)(\theta)=f(\theta)\psi(\theta)$, and similarly $g(N)$ denotes the multiplication by a function $g(n)$ in the momentum representation. The outcome proability densities of these observables on an input state $\rho$ are denoted by $\rho\mby\Theta$ and $\rho\mby N$, respectively. Thus, $$\begin{aligned}
\int\!\!d\theta\,\rho\mby\Theta(\theta)\,f(\theta)&=&\tr\rho f(\Theta), \\
\sum_n \rho\mby N(n)\, g(n)&=& \sum_n \langle e_n|\rho|e_n\rangle\ g(n).\end{aligned}$$ Then the basic claim of preparation uncertainty is that $\rho\mby\Theta$ and $\rho\mby N$ cannot be simultaneously sharp, and the basic claim of measurement uncertainty is that there is is no observable with pairs of outcomes $(\theta,n)$ for which the marginal distributions found on input state $\rho$ are close to $\rho\mby\Theta$ and $\rho\mby N$.
In order to apply the ideas of the previous section, we need to choose a metric in each of these spaces. For discrete values ($X={\mathbb{Z}}$) we can naturally take the standard distance or a discrete metric: $$\begin{aligned}
\label{donZ}
\dst(n,m)&=&|n-m| \qquad\text{or}\nonumber\\
\ddi(n,m)&=&1-\delta_{nm}.\end{aligned}$$ Similarly, there are two natural choices for angles, depending on whether the basis for the comparison is how far we have to rotate to go from $\theta$ to $\theta'$ (“arc distance”) or else the distance of phase factors $\exp(i\theta)$ and $\exp(i\theta')$ in the plane (“chordal distance”): $$\begin{aligned}
\label{donT}
\dar(\theta,\theta')&=&\min_{n\in{\mathbb{Z}}}|\theta-\theta'-2\pi n| \qquad\text{or}\nonumber\\
\dch(\theta,\theta')&=&\left|e^{i\theta}-e^{i\theta'}\right|=2\left|\sin\frac{\theta-\theta'}2\right|.\end{aligned}$$ The variances based on these bounded metrics will have an upper bound. Since the minimum in makes $d_\alpha(\mu,*)^\alpha$ a concave function of $\mu$, so that we find, by averaging over translates, that the equidistribution has the maximal variance for all translation invariant metrics and all exponents. Both metrics, or rather their quadratic ($\alpha=2$) variances have been discussed before. The $\dar$-variance was used by Lévy [@Levy] and Judge [@judge63], the $\dch$-variance seems to have appeared first in von Mises [@vMises]. In fact, the quadratic chordal variance can also be written as $$\begin{aligned}
\label{vMis}
d_{\mch,2}(\mu,*)^2&=&\inf_\alpha\int\!\!\mu(d\theta)\left|e^{i\theta}-e^{i\alpha}\right|^2 \nonumber\\
&=&2\Bigl(1-\sup_\alpha{{\rm Re\,}}\ e^{-i\alpha} \int\!\!\mu(d\theta)e^{i\theta} \Bigr)\nonumber\\
&=&2\Bigl(1-\bigl|\langle e^{i\theta}\rangle_\mu\bigr|\Bigr),\end{aligned}$$ which is von Mises’ “circular variance”. For a review of these choices see Ref. . The only property needed in our approach is that the metric should not break the rotation invariance, i.e., it should be a function of the difference of angles. We will therefore use every metric $d$ also as a single variable function, i.e., $d(x)=d(x,0)$ and $d(x,y)=d(x-y)$. Functions which do not come from a metric have been considered in Ref. .
{width="75.00000%"}
Then we have the following result.
\[mainthm\] For all error exponents and choices of translation invariant metric the three uncertainty regions $\PU^+=\MU^+=\CU^+$ coincide. They are depicted for $\alpha=\beta=2$ in Figure \[fig:basic4\]. Every point on one of the tradeoff curves belongs to a unique pure state (resp. a unique extremal phase space covariant joint measurement).
The proof of this Proposition is based entirely on the corresponding proof for standard position and momentum [@BLWjmp]. We will sketch the main steps in the next section, and also show how the computation of the tradeoff curve can be reduced to solving ground state problems for certain Hamiltonians. The detailed features of these diagrams are then developed in the subsequent sections, sorted by the methods employed. The tradeoff curves in Figure 1 for the generalized variances, denoted simply $\Delta^2(N),\Delta^2(\Theta)$, are determined numerically (see Section \[sec:numerics\]). Since the algorithms employed provide optimal bounds, the figures are correct within pixel accuracy (which can be easily pushed to high accuracy). In fact, the problem is very stable, in the sense that near minimal uncertainty implies that the state (or joint observable) is close to the minimizing one. The bounds for this are in terms of the spectral gap of the Hamiltonian and are also discussed in Section \[sec:numerics\].
However, the only case in which the exact tradeoff curves can be described in closed form is (see Section \[sec:discrete\]) $$\label{dischord}
\Delta_\mdi(N)\geq 1-\frac12\sqrt{\Delta^2_\mch(\Theta)\bigl(4-\Delta^2_\mch(\Theta)\bigr)}\ ,$$ even though, in all cases, the optimizing states can be expressed explicitly in terms of standard special functions (see Section \[sec:exact\]). Therefore simple and explicit lower bounds are of interest. A problem here is that computing the variances for some particular state always produces a point inside the shaded area, i.e., an upper bound to the lower bound represented by the tradeoff curve. This is useless for applications, so in Section \[sec:lower\] we develop a procedure proving lower bounds, and thus correct (if suboptimal) uncertainty relations.
Phase Space Symmetry and Reduction to a Ground State Problem {#sec:sym}
============================================================
In this section we briefly sketch the arguments leading to the equality of preparation und measurement uncertainty regions. The full proof is directly parallel to the one given in Ref. for position and momentum. The basic reference for phase space quantum mechanics is Ref. . The theory there is developed for phase spaces of the form ${\mathbb{R}}^n\times{\mathbb{R}}^n$, but all results we need here immediately carry over to the general case $X\times\widehat X$, where $X$ is a locally compact abelian group and $\widehat X$ is its dual, in our case $X={\mathbb{T}}$ and $\widehat X={\mathbb{Z}}$. A systematic extension of Ref. to the general case, including the finer points, is in preparation in collaboration with Jussi Schultz.
Covariant phase space observables
---------------------------------
The phase space in our setting is the group $\Omega={\mathbb{T}}\times{\mathbb{Z}}$. We join the position translations and the momentum translations to phase space translations, which are represented by the displacement or [*Weyl operators*]{} $$\label{Weyl}
(W(n,\theta)\psi)(x)= e^{-\frac 12 n\theta}e^{-inx}\psi(x-\theta).$$ These operators commute up to a phase, so that the operators $T_\omega(A)=W(\omega)^*AW(\Omega)$, i.e., the action of the Weyl operators on bounded operators $A\in\BB(\HH)$, is a representation of the abelian group $\Omega$. A crucial property is the square integrability of the matrix elements of the Weyl operators, which we will use in the form that for any two trace class operators $\rho,\sigma$ on $\HH$ the formula $$\label{sqint}
\int d\omega \tr\bigl( \rho\,T_\omega(\sigma)\bigr)=\tr \rho\,\tr \sigma,$$ where $\int\!\!d\omega=\sum_n\int\!\frac{d\theta}{2\pi}$. Hence, when both $\rho$ and $\sigma$ are density operators the integrand in this equation is a probability density on $\Omega$. In summary [@QHA]:
Every density operator $\sigma$ serves an observable $F_\sigma$ with outcomes $\omega$ in the phase space $\Omega$, via $$\label{covobs}
F_\sigma(A)=\int_{\omega\in A}\mskip-20mu d\omega\ T_\omega(\sigma).$$ Here $\sigma$ figures as the operator valued Radon-Nikodym density with respect to $d\omega$ of $F_\sigma$ at the origin, and by translation also at arbitrary points. The observables obtained in this way are precisely the [*covariant*]{} ones, i.e., those satisfying the equation $T_\omega(F(A))=F(A-\omega)$.
For the discussion of uncertainties we need the margins of such observables. One can guess their form from a fruitful classical analogy [@QHA], by which the integrand in can be read as a convolution of $\rho$ and $\sigma$. For classical probability densities on a cartesian product it is easily checked that the margin of the convolution is the convolution of the margins. The same is true for operators, only that the margins of a density operator are the classical distributions $\rho\mby\Theta$ and $\rho\mby N$ described above. If $p$ is a probability density on phase space we will also denote by $p\mby\Theta$ and $p\mby N$ the respective margins on ${\mathbb{T}}$ and ${\mathbb{Z}}$. In particular, for the output distribution of the covariant observable $F_\sigma$, i.e., $p(\theta,n)=\tr\rho W(\theta,n)^*\sigma W(\theta,n)$, One checks readily checks the marginal relations $$\begin{aligned}
p\mby N&= \rho\mby N \ast \sigma\mby N, \\
p\mby\Theta&= \rho\mby\Theta \ast \sigma\mby\Theta,\end{aligned}$$ where “$\ast$” means the convolution of probability densities on ${\mathbb{Z}}$ and ${\mathbb{T}}$. Since this is the operation associated with the sum of independent random variables we arrive at the following principle:
[Both margins of a covariant phase space measurement $F_\sigma$ can be simulated by first making the corresponding ideal measurement on the input state, and then adding some independent noise, which is also independent of the input state. The distribution of this noise is the corresponding margin of the density operator $\sigma$]{}.
This principle is responsible for the remarkable equality of the preparation uncertainty region $\PU^+$ and the measurement uncertainty regions $\MU^+$ and $\CU^+$. Indeed the added noise is what distinguishes the margins of an attempted joint measurement from an ideal measurement, and this has precisely the distribution relevant for preparation uncertainty for $\sigma$.
$\MU$ and $\CU$: Reduction to the covariant case
------------------------------------------------
While this principle explains quite well what happens in the case of covariant observables $F_\sigma$, Proposition \[mainthm\] makes no covariance assumption. The key for reducing the general case is the observation that our quality criteria in terms of $d(\Theta',\Theta)$ do not single out a point in phase space. Thus, let $\Mee$ be the set of observables whose angle margin $F_{\mathbb{T}}$ is $\veps_1$-close to the ideal observable, $d(F_{\mathbb{T}},\Theta)\leq\veps_1$, and whose [number ]{}margin satisfies $d(F_{\mathbb{Z}},N)\leq\veps_2$; then this set is closed under phase space shifts, i.e., is unchanged when we replace $F$ by $F'$ where $$\label{Fshift}
F'(A)=T_\omega(F(A+\omega))$$ Note that the fixed points of all these transformations are precisely the covariant observables. The second point to note is that the set $\Mee$ is convex, because the worst case error of an average in a convex combination is smaller than the average of the worst case errors. It is also a compact set in a suitable weak topology. This is in contrast to the set of all observables: Since there are arbitrarily large shifts in ${\mathbb{Z}}$, we can shift an observable to infinity such that the probabilities for all fixed finite regions go to zero. The weak limit of such observables would be zero or, in an alternative formulation, would acquire some weight on points at infinity (a compactification of ${\mathbb{Z}}$). For such a sequence, however, the errors would also diverge. It is shown in Ref. that this suffices to ensure the compactness of $\Mee$. Then the Markov-Kakutani Fixed Point Theorem (Ref. ) ensures that $\Mee$ contains a common fixed point of all the transformations .
In summary:
For every joint observable $F$ on the phase space $\Omega$ there is a covariant one for which the errors are at least as good.
Therefore for determining $\MU^+$ we can just assume the observable in question to be covariant, implying the very simple form of the margins described above. The argument for $\CU^+$ is the same.
The post-processing Lemma: $\CU^+=\MU^+$
----------------------------------------
We have noted that, in general, $\Delta_\alpha^c(A',A)\leq d_\alpha(A',A)$, because for calibration the worst case analysis is done over a much smaller set of states. Indeed, it is easy to construct examples of observable pairs where the inequality is strict. There is a general result, Ref. , however, which implies equality. The condition is that $A'$ arises from $A$ by classical, possibly stochastic post-processing. That is, we can simulate $A'$ by first measuring $A$, and then adding noise or, in other words, generating a random output by a process which may depend on the measured $A$-value. The noise is described then by a transition probability kernel $P(x,dy)$ for turning the measured value $x$ into somewhere in the set $dy\subset X$. $P$ is thus a description of the noise, and its relevant size is given by the formula $$\begin{aligned}
\label{postpro}
\Delta_\alpha^c(A',A)&=& d_\alpha(A',A)
= \left(\essential_A\text{-}\sup_{x\in X}\int P(x,dy)\ d(x,y)^\alpha \right)^{1/\alpha} .
$$ Here the $A$-essential supremum of a measurable function is the supremum of all $\lambda$ such that the level set $\{x|f(x)\geq\lambda\}$ has non-zero measure with respect to $A$. This is needed to ensure that $P(x,dy)$ enters this formula only for values $x$ that can actually occur as outputs of $A$.
The covariant case: $\CU^+=\MU^+=\PU^+$
---------------------------------------
In the case at hand, the noise is independent of $x$, i.e., $P$ is translation invariant and so is the metric. Therefore, the integral in is simply independent of $x$. Moreover, we know the distribution of the noise on each margin to be the respective margin of $\sigma$, so that the integral is just the $\alpha$-power deviation of the margin from zero. So we get, for any choice of exponents and translation invariant metrics on ${\mathbb{T}}$ and ${\mathbb{Z}}$: $$\label{3delta}
\Delta_\alpha^c(\Theta',\Theta)= d_\alpha(\Theta',\Theta) = d_\alpha(\sigma\mby\Theta,0),$$ and similarly for $N$. Note that the last term here is [*not*]{} the variance $d_\alpha(\sigma\mby\Theta,*)$, because the minimization over $x$ in is missing. Indeed, if $\sigma\mby\Theta$ just had zero variance, i.e, it were a point measure at some point $x\neq0$, we would get a constant shift of size $d(x)$ between the distributions $\rho\mby{\Theta'}$ and $\rho\mby\Theta$, and this would be the errors on the left hand side. So for a fixed $\sigma$ we can only say that the terms in are $\geq d_\alpha(\sigma\mby\Theta,*)$. On the other hand, we are looking for [*optimal*]{} $\sigma$ and these will be obtained by shifting $\sigma$ in such a way that is minimized. Hence, as far as uncertainty diagrams are concerned, we can replace the last term by the $\alpha$-deviation. This concludes the proof that the three uncertainty diagrams coincide.
Minimizing the variances {#sec:minvariance}
------------------------
We now describe the general method to find the tradeoff curve. The idea is to fix some negative slope $-t$ in the diagram, and ask for the lowest straight line with that slope intersecting $\PU^+$. That is, we look at the optimal lower bound $c()$ such that $$\label{leglower}
y+t x\geq c(t)\qquad \text{for all}\ (x,y)\in\PU.$$ Now both coordinates $x=d_\alpha(\sigma\mby\Theta,0)^\alpha$ and $y=d_\beta(\sigma\mby N,0)^\beta$ are linear functions of $\sigma$, so that the left hand side of is just the $\sigma$-expectation of some operator, namely $$\begin{aligned}
y+t x&=& \tr(\sigma H(t))\nonumber\\
H(t)&=& d_{\mathbb{Z}}(N)^\beta + t d_{\mathbb{T}}(\Theta)^\alpha .\label{Ht}\end{aligned}$$ Here $d_{\mathbb{T}}$ and $d_{\mathbb{Z}}$ are the metrics chosen for these spaces, and we used the notation of writing $f(\Theta)$ for the multiplication operator by $f(\theta)$, and its Fourier transformed counterpart, and also the convention that $d(x)=d(x,0)$ for a translation invariant metric. The optimal constant $c(t)$ is thus the lowest expectation $\inf H=\inf\tr(\sigma H)$, i.e., its [*ground state energy*]{}. Note that for standard position and momentum phase space and $\alpha=\beta=2$ we get here $H=P^2+t Q^2$, a harmonic oscillator, and the well-known connection between its ground state and minimum uncertainty.
We will look into these ground state problems later and for now note some general features.
1. The variable $t$ is positive because we are looking for lower bounds on $x$ and $y$ only. This corresponds in part to taking the monotone closure, and is the reason why we replace $\PU^+$ by $\PU$ in .
2. The best bound on $\PU$ obtained in this way is achieved by optimizing over $t$, i.e., $$y\geq\sup_t\{-tx+c(t)\},$$ the Legendre transform of $c$. This is automatically convex. In other words, the method does not describe $\PU^+$ in general, but its convex hull (the intersection of all half spaces with positive normal containing $\PU$).
3. There may be points on the tradeoff curve for the convex hull which do not really correspond to a realizable pair of uncertainties. However, if we take the collection of ($t$-dependent) ground states, and their variance pairs trace out a continuous curve, we know that the tradeoff curves are the same and the set $\PU^+$ is actually convex and fully characterized by the ground state method.
![Applying the estimate in an uncertainty diagram. The lower curve is the tradeoff curve obtained from the ground state problem. The upper curve is formed by the uncertainty pairs of the respective first excited states. Suppose the uncertainty pair [A]{} has been found. We want to show that the corresponding state must be close to the minimal uncertainty state corresponding the point [G]{}. Drawing the tangent to the tradeoff curve at [G]{} and the parallel tangent to the upper curve we find from the diagram that the fidelity of the given state to [G]{} must be at least $1-\varepsilon$.[]{data-label="fig:nearmin"}](gapest.pdf){width="40.00000%"}
When the ground state problem for $H(t)$ has a gap, it is known that any state with expectations close to the ground state energy must actually be close to the ground state. More precisely, suppose that $H$ has a unique ground state vector, $H\psi=E_0\psi$, and that the next largest eigenvalue is $E_1>E_0$. Then $H\geq E_1\idty- (E_1-E_0)\kettbra\psi$. Now let $E_0<\braket{\phi}{H\phi}=E_\phi<E_1$ for some unit vector $\phi$. Then by taking the $\phi$-expectation of the operator inequality, we get $$\label{nearmin}
\abs{\braket\phi\psi}^2\geq \frac{E_1-E_\phi}{E_1-E_0}.$$ In particular, when $E_\phi\approx E_0$, $\phi$ must be close in norm to $\psi$. We can directly apply this principle to the above ground state problems. The basic geometry is described in Fig. \[fig:nearmin\]. This shows that the curve of minimizers is continuous. It will also be useful in showing explicitly that the minimizers for different choices of metrics are sometimes quite close to each other, or that some simple ansatz for the minimizer is quantitatively good.
Numerics in truncated Fourier basis {#sec:numerics}
===================================
Here we only consider the case $d_{\mathbb{Z}}=\dst$ because for the discrete metric on ${\mathbb{Z}}$ the ground state problem has an elementary explicit solution (see Section \[sec:discrete\]). The numerical treatment is easiest in the Fourier basis, or rather in the even eigenspace of the [number ]{}operator $N$, for $\abs N\leq\nmax$. The matrix elements of the relevant Hamiltonians for basis vectors in this range can be written down as simple explicit expressions. From these the numerical version of the Hamiltonian is determined as floating point matrix of the desired precision, for which the ground state and first excited state are determined by standard algorithms. All these steps were carried out in Mathematica. The criterion for the choice of $\nmax$ was that the highest-$n$ components of the eigenvectors found should be negligible at the target accuracy. The target accuracy was mostly $5$ digits with computations done in machine precision with $\nmax=80$, but was chosen larger for getting a reliable estimate of the separation of the different state families.
All computations must be considered elementary and highly efficient, even at high accuracy. None of the diagrams in this paper takes computation time longer than a keystroke. It is therefore hardly of numerical advantage to implement the analytic solutions of Section \[sec:exact\], not in computation time and even less in programming and verification time.
Perhaps the only surprise in this problem is that for the two different metrics $\dar$ and $\dch$ the minimizing state families are so close. Since $V_\mch\leq V_\mar$, the ground state problems for $H_\mch$ and $H_\mar$ are related. Perturbatively one sees that the ground state energies are indeed similar, up to the expectation of $t(V_\mar(\theta)-V_\mch(\theta))$. The stability statement at the end of Section \[sec:minvariance\] then implies that the corresponding ground states are also similar. However, direct comparison gives a norm bound, which is rather better than these arguments indicate: $$\label{normdiff}
\norm{\psi_\mar(t)-\psi_\mch(t)}\leq 0.145$$ for all $t$, corresponding to a fidelity $\geq .98$. This still does not quite reflect the similarity of these two state families: When we allow the $t$-arguments to differ, we get a much better approximation. To make this precise consider the orbits $\Omega_\mar=\{e^{i\alpha}\psi_\mar(t)| t>0, \alpha\in{\mathbb{R}}\}$, and an analogously defined $\Omega_\mch$. For sets in Hilbert space we use the Hausdorff metric, so that $d_H(\Omega_1,\Omega_2)<\veps$ means that for every point in one set there is an $\veps$-close one in the other. Then one easily gets $$\label{dHOm}
d_H\bigl(\Omega_\mar,\Omega_\mch\bigr)\leq .028$$
Consequently, there is really only one diagram representing the family of minimal uncertainty states, which we show in Fig. \[fig:states\].
![Minimum uncertainty wave functions (left) and Fourier coefficients (right) for the case $(\dch,\dst)$ as a function of the family parameter $t$ from , plotted as the depth dimension. Note that the surface in the diagram on the right is only an aid for better 3D visualization — only the embedded lines have real significance.[]{data-label="fig:states"}](gswaves.pdf "fig:"){width="45.00000%"} ![Minimum uncertainty wave functions (left) and Fourier coefficients (right) for the case $(\dch,\dst)$ as a function of the family parameter $t$ from , plotted as the depth dimension. Note that the surface in the diagram on the right is only an aid for better 3D visualization — only the embedded lines have real significance.[]{data-label="fig:states"}](gscoeffs.pdf "fig:"){width="45.00000%"}
Exact ground states {#sec:exact}
===================
Schrödinger operator case
-------------------------
With $d_{\mathbb Z}=d_{\rm std}$ and $\beta=2$, the ground state problem becomes an instance of the Schrödinger operator eigenvalue problem. In fact, writing $V=d_{\mathbb Z}^\alpha(\Theta)$, the optimal constant $c(t)$ for a given $t$ is the smallest value of $\lambda$ such that the differential equation $$\label{diffeq}
-\psi''(\theta) + tV (\theta)\psi(\theta) = \lambda \psi(\theta)$$ has a solution on $[-\pi, \pi]$ satisfying the boundary conditions $\psi(-\pi) = \psi(\pi) = \psi'(-\pi) = \psi(\pi) = 0$. By the general theory, we know that the (unique) solution $\psi = \psi_\alpha$ has no zeros, can be chosen to depend smoothly on $t$ (by perturbation theory), and can be chosen to be positive and even (by parity invariance).
Hence, we are in fact looking for an even solution $\psi$ of with $\psi'(\pi)=0$. At $t = 0$ this vector is just $\psi_\alpha(\theta) = 1/\sqrt{2\pi}$, i.e. a constant. For the two choices $\dar$ and $\dch$, the solutions $\psi_\alpha$ are known special functions; we now proceed to describe them in some detail.
### $V_{\mathbb Z}(\theta)=\dar(\theta)^2=\theta^2$
The general even solution of is given by a hypergeometric function [@abramowitz]: $$\label{arcpsi}
\psi(\theta)=N^{-1}e^{-\frac{1}{2}\sqrt{t }\,\theta^2}
\hypoo\left(\frac{1}{4} \left(1-\frac{\lambda}{\sqrt{t }}\right);\frac{1}{2};\sqrt{t } \theta^2\right),$$ where $N$ is the normalisation factor. The boundary condition $\psi'(\pi)=0$ now picks out the eigenvalues $\lambda$ for every $t$ (see Fig. \[fig:arcpsiprime\]), of which the smallest is the desired constant $c(t)$. The condition can be expressed by using the standard differentiation formulas for the hypergeometric functions: $$\begin{aligned}
\left(1-\frac{\lambda}{\sqrt{t }}\right) &\, \hypoo\left(1+\frac{1}{4} \left(1-\frac{\lambda}{\sqrt{t }}\right);\frac{3}{2};\pi ^2 \sqrt{t }\right)
=\hypoo\left(\frac{1}{4} \left(1-\frac{\lambda}{\sqrt{t }}\right);\frac{1}{2};\pi ^2 \sqrt{t }\right)\end{aligned}$$ However, as far as we could see, the theory of hypergeometric functions seems to offer little help for solving it, or for evaluating the normalization constant $N$ or the Fourier coefficients. Perhaps an elementary expression for $c(t)$ is too much to hope for, since already in much simpler problems, e.g., a particle in a box, where the pertinent transcendental equations involve only trigonometric and linear functions, no “explicit” solution can be given either.
![The derivative $\psi'(\pi)$ after , for $t=1$, as a function of the eigenvalue parameter $\lambda$. The zeros of this function determine the eigenvalues, of which the lowest gives the constant $c(1)$.[]{data-label="fig:arcpsiprime"}](psiarcprime.pdf){width="40.00000%"}
### $V(\theta)=\dch(\theta)^2=2(1-\cos\theta)$
In the case of $\dch$ (distance through the circle), the equation is just the Mathieu equation up to scaling $\theta\mapsto \theta/2$. The even periodic solutions correspond to $-4(2t-\lambda) ={\mathrm a}_r(-4t)$, where the ${\mathrm a}_r(q)$, $r=0,1,\ldots$ are called Mathieu characteristic values [@abramowitz]. Our ground state eigenvalues are therefore $$c(t) =2t +a_t/4,$$ where we have used the shorthand $a_t={\mathrm a}_0(-4t)$. Since ${\mathrm a}_0$ is implemented in e.g. Mathematica, we can easily determine the values numerically. The corresponding solutions are given by $$\psi_t(\theta)=\frac{1}{N} \matze\left(a_t;-4 t ;\frac{\theta}{2}\right)$$ where $\matze$ denotes the lowest order first kind solution of the ordinary Mathieu equation, and $N$ is again the normalisation factor. We note that the Fourier coefficients $\widehat\psi_t$ are explicit functions of $a_t$ and $t$, determined by the recurrence relations. Up to second order, we have $$\begin{aligned}
\psi_t(\theta)&=\frac 1N \Big(1-\frac {a_t}{4t}\cos\theta +\left[\frac{(a_t-4)a_t}{16t^2}+8t\right]\cos 2\theta
+O(\cos 3\theta)\, \Big)\end{aligned}$$ The relevance of the Mathieu functions in the context of circular uncertainty relations has been noted e.g. in Ref. .
Discrete metric case {#sec:discrete}
--------------------
With $d_{\mathbb Z}=\ddi$, we have the eigenvalue equation $$\label{discrev}
\bigl({\mathbb{I}}-|\phi_0\rangle\langle \phi_0| +t V\bigr)\psi =\lambda \psi,$$ with $V$ as in the previous section, and $\phi_0(\theta)=1/\sqrt{2\pi}$ the constant function. This allows us to solve for $\psi$: $$\label{psidisc}
\psi_t(\theta) = \frac A{1-\lambda+t V(\theta)}\ ,$$ where $A>0$ is the normalization constant. Inserting this into gives the consistency condition $$\label{disconcis}
\frac{1}{2\pi}\int_{-\pi}^\pi \frac{d\theta}{1-\lambda +t V(\theta)}=1.$$ The smallest positive solution $\lambda=c(t)$ of this equation will give us the desired bound.
However, we can also proceed more directly by using just the functional form , which we can further simplify to the one-parameter family $\psi(\theta)=A/(\mu+ V(\theta))$, with a single parameter $\mu$. We then have to solve three integrals: $$\begin{aligned}
I_1(\mu)&=\displaystyle\int_{-\pi}^\pi \frac{d\theta}{\mu + V(\theta)} \\
I_2(\mu)&=\displaystyle\int_{-\pi}^\pi \frac{d\theta}{\bigl(\mu + V(\theta)\bigr)^2} =\displaystyle-\frac{dI_1(\mu)}{d\mu}\\
I_3(\mu)&=\displaystyle\int_{-\pi}^\pi \frac{d\theta\ V(\theta)}{\bigl(\mu + V(\theta)\bigr)^2} =\displaystyle I_1(\mu)-\mu I_2(\mu).\end{aligned}$$ Then the pair of variances $$\begin{aligned}
\label{gendis}
\Delta_\mdi(N)&=& 1-\frac{I_1(\mu)^2}{2\pi I_2(\mu)} \\
\Delta^2(\Theta)&=& \frac{I_3(\mu)}{I_2(\mu)}\end{aligned}$$ lies on the tradeoff curve. One can check that this is consistent with the Legendre transform picture, i.e., condition in the form $$\label{disconcis1}
I_1\left(\frac{1-c(t)}t\right)=2\pi t$$ and its derivative and the parameter identification $\mu=(1-c)/t$.
Now for the arc metric we have $V(\theta)=\theta^2$ and $$\label{I1arc}
I_1(\mu)=\frac2{\sqrt\mu}\arctan\bigl(\frac\pi{\sqrt\mu}\bigr).$$ Trying to eliminate $\mu$ from leads to a transcendental equation, so one cannot give a closed inequality involving just the variances.
For the chord metric we have $V(\theta)=2(1-\cos\theta)$ and $$\label{I1cho}
I_1(\mu)=\frac{2\pi}{\sqrt{\mu(\mu+4)}}.$$ In this case one can easily eliminate $\mu$ from , and the tradeoff is explicitly described by equation .
Analytic lower bounds {#sec:lower}
=====================
In this section we establish a variational method for proving uncertainty relations by applying such bounds for the ground state problem. Of course, variational methods for the ground state problem are well-known. Basically they amount to choosing some good trial state, and evaluating the energy expectation: This will be an upper bound on the ground state energy, and it will be a good one if we have guessed well. However, it is notoriously difficult to find lower bounds on the ground state energy. The idea for finding such bounds is via the following Lemma:
\[boundlemma\]Let $V$ be a $2\pi$-periodic real valued potential, and $H$ the Schrödinger operator $H\psi=-\psi''+V\psi$ with ground state energy $E_0(H)$. Consider a twice differentiable periodic function $\phi$, which is everywhere $>0$. Then the ground state energy of $H$ is larger or equal to $$\label{elow}
E_\phi=\min_\theta\left\{V(\theta)-\frac{\phi''(\theta)}{\phi(\theta)}\right\}.$$
Let $$\label{vv}
\wt V(x)=\phi''(x)/\phi(x)+ E_\phi$$ and $\wt H$ the Schrödinger operator with this potential. Then $\phi$ is an eigenfunction of $\wt H$ with eigenvalue $E_\phi$, and since it was assumed to be positive, it has no nodes and must hence be the ground state eigenfunction. On the other hand, because $E_\phi\leq V(x)-(\wt V(x)-E_\phi)$, we have $\wt V\leq V$ and hence $\wt H\leq H$. By the Rayleigh-Ritz variational principle [@ReedS] this implies the ordering of the ground state energies, i.e., $\wt E=E_0(\wt H)\leq E_0(H)$.
Finding a $\phi$ which gives a good bound is usually more demanding than finding a good $L^2$-approximant for the ground state, because of the highly discontinuous expression $\phi''/\phi$ and the infimum being taken over the whole interval. In particular, the approximate eigenvectors obtained by other methods may perform poorly, even give negative lower bounds on a manifestly positive operator.
The positivity of $\phi$ may be ensured by setting $\phi(\theta)=\exp f(\theta)$; then one has to minimize $V-(f''+(f')^2))$.
We now consider the combinations of metrics $(\dst,\dar)$ and $(\dst, \dch)$, also comparing the results with existing uncertainty relations found in the literature. This demonstrates how our systematic approach relates to many existing (seemingly ad hoc) uncertainty relations.
One remark should be made concerning the comparison with the literature: The uncertainty measure used for the [number ]{}operator is practically always taken to be the usual standard deviation, which can be different from $\Delta_{\mst}(N)$ since in the latter case the infimum is taken only over the set of integers. In general we have $$\Delta_{\mst}(N)^2 \geq \sum_{n\in \mathbb Z} (n-{\rm tr}(\rho{N}))^2 \rho\mby{N}(n),$$ where the right-hand side is the usual standard deviation, which takes a distribution on the integers as a distribution on the reals, which is supported by the integers. Due to the above inequality any uncertainty tradeoff involving the usual standard deviation also implies the same relation for $\Delta_{\mst}(N)$.
Case $(\dst,\dar)$
------------------
The literature on this case begins with the observation that the standard uncertainty relation does not hold and needs to be modified; Judge [@judge63; @judge64] showed in 1963 that the following tradeoff relation $$\label{jbound}
\Delta_\mst(N)\Delta_\mar(\Theta)\geq c \,\left(1-(3/\pi^2) \Delta_\mar(\Theta)^2\right)$$ holds with $c=0.16$, and conjectured the same with $c=1/2$. The conjecture was quickly proved in Ref. using the Lagrange multiplier method where $\Delta_\mdi(N)$ is minimised under the constraint of fixed $\Delta_\mar(\Theta)^2$, and in Ref. by showing that the admissible pairs $(\Delta_\mst(N)^2,\Delta_\mar(\Theta)^2)$ lie above the tangent lines of the curve determined by the equality in . Both methods are essentially equivalent to our approach, and explicitly involve the same eigenvalue problem. In Ref. the bound leading to with the optimal constant was obtained using special properties of the confluent hypergeometric function.
We first show how the above Lemma can easily be applied to derive with the optimal constant $c=1/2$. An essentially identical procedure works also in other cases below. The potential is $V(\theta)=\theta^2$, and we have label the uncertainty pairs as $(x,y)=(\Delta_\mar(\Theta)^2,\Delta_\mst(N)^2)$. As the simplest ansatz we take $f(\theta)=\log\phi(\theta)$ even, hence a polynomial in $\theta^2$, which take as quadratic. The boundary condition $f'(\pi)=0$ then leaves the one-parameter family $$\label{farc}
f(\theta) = -\frac a2\, \theta^2\Bigl(1-\frac{\theta^2}{2\pi^2}\Bigr),$$ where $a\in{\mathbb R}$ is to be optimised later. The bound given by the Lemma on the uncertainty pair $(x,y)$ is then $$\begin{aligned}
y+tx&\geq& E_0\geq E_\phi=\inf_\theta\Bigl\{t\theta^2-f''(\theta)-f'(\theta)^2\Bigr\} \nonumber\\
&=&a+\inf_\theta \Bigl\{\bigl(t-a^2-\frac{3 a}{\pi ^2}\bigr)\theta^2+\frac{2 a^2}{\pi ^2}\theta^4-\frac{a^2}{\pi ^4} \theta^6 \Bigr\}.\nonumber\end{aligned}$$ This inequality is valid for any $a>0$ and $t>0$. We choose $t$ so that the linear term in $\xi$ vanishes, i.e., $t=a^2+3a\pi^{-2}$. The remaining polynomial is then positive because of $\theta\leq\pi$, and hence takes its minimum at $\xi=0$. Therefore, $$\label{lowbdarc}
y+\left(a^2+\frac{3a}{\pi^2}\right)x\geq a.$$ The optimal value here is $a=(\pi^2-3x)/(2\pi^2 x)$. Note that this is alwas positive, because the equidistribution has the largest variance, namely $x=\pi^2/3$. Substituting the optimal $a$ in we get exactly .
Case $(\dst, \dch)$
-------------------
We first recall from that $|\langle e^{i\theta}\rangle_{\rho\mby\Theta}| = 1-\Delta_{\mch}(\Theta)^2/2$. Hence, the von Mises “circular variance" is associated with the sine and cosine operators $\sin \Theta$ and $\cos \Theta$, introduced by Carruthers, Nieto, Louisell, Susskind, Glogower, and others to study the “quantum phase problem" [@nietorev; @carruthers; @breitenberger]. The idea was to replace the singular commutator $[N,\Theta]$ by the well-defined relations $$\begin{aligned}
\label{sincom}
[N, \sin\Theta] & =i\cos\Theta, & [N,\cos\Theta]=-i\sin\Theta.\end{aligned}$$ Combining the usual Robertson type inequalities associated with these commutators, they obtained (Ref. ) the tradeoff relation $$\label{CN}
\Delta_{\mst}(N) \Delta_{\mch}(\Theta)\geq \frac{1-\tfrac 12\Delta_\mch(\Theta)^2}{2\bigl[1-
\tfrac14\Delta_\mch(\Theta)^2\bigr]^{1/2}},
$$ expressed here in quantities relevant for our discussion. It was shown by Jackiw [@jackiw] that there are no states for which this inequality is saturated, i.e., this bound is not sharp. It is interesting to note that by replacing the square root term with its trivial upper bound $1$, we get $$\label{judgechord}
\Delta_{\mst}(N) \Delta_{\mch}(\Theta) \geq \frac {1-\tfrac 12\Delta_\mch(\Theta)^2}{2},$$ which is just the version of Judge’s bound for this metric. Other lower bounds were studied relatively recently [@rehacek] by using approximations of the Mathieu functions associated with the exact tradeoff curve.
We first show how can be obtained using Lemma \[boundlemma\] by applying the same procedure as above. Interestingly, the relevant trial states are exactly the ones saturating the Robertson inequality for the first commutator in , that is, we take $f(\theta)=a\cos\theta$. Then the resulting variational expression is a function of the variable $\cos\theta$, and hence of the potential $V$: $$\label{lowbdcho}
y+tx\geq\inf_V\Bigl\{a-\bigl(t-\frac a2-a^2\bigr)V+\frac{a^2}4 V^2 \Bigr\}$$ Again it is a good choice to take $t$ so that the first order term in $V$ vanishes, so that the remaining infimum is attained at $V=0$. This gives $t=a/2+a^2$ and $$\label{lowbdcho1}
y\geq a-\bigl(\frac a2+a^2\bigr)x =\frac{(2-x)^2}{16 x}.$$ where at the last equality we have substituted the optimal value $a=(2-x)/(4x)$. On taking the square root this is .
In order to obtain analytic bounds better than the Carruthers-Nieto tradeoff , we apply our method with a trial function which is second order in $\cos\theta$: We take $f(\theta)=a\cos\theta+b (\cos\theta)^2$. The expression to be minimized over $\theta$ can still be written as a polynomial in the potential, and numerical inspection suggests once again that it is a good idea to choose the parameters $t$ and $b$ so that coefficients of $V$ and $V^2$ vanish. This gives linear equations for $t$ and $b$, and the resulting polynomial has its unique minimum, namely $a$, at $V=0$. The analogue of is then $$\label{lowbdcho2}
y+\frac{a \left(8 a^2+5 a+2\right)}{8 a+4}\,x\geq a.$$ Optimizing $a$ now leads to a third order algebraic equation for which the Cardano solution gives a useless expression in terms of roots. If one just wants the tradeoff curve, the solution is actually not necessary. Defining the coefficient of $x$ as a function $g(a)$, so that $y+g(a)x\geq y$. Optimality requires $xg'(a)=1$, so we get the tradeoff curve in parametrized form $a\mapsto (1/g'(a),a-g(a)/g'(a))$.
Outlook {#sec:outlook}
=======
The methods employed in this paper for obtaining preparation uncertainty bounds can be applied to a large variety of similar problems. However, the derivation of measurement uncertainty bounds relied entirely on the theorem that phase space symmetry makes the two coincide. It is therefore no surprise that the case of positive number and phase seems much harder to tackle, and the exact uncertainty region is yet to be determined although (non-strict) uncertainty bounds have recently been proven[@LPS17]. Independent efficient methods for obtaining sharp bounds for measurement uncertainty so far have not been found, and it would be highly desirable to find such methods. A possible substitute might be a proof of the conjecture that measurement uncertainty is always larger than preparation uncertainty. Although this inequality must be strict in general, in that way the easily computed preparation uncertainty bounds would automatically be valid (but usually suboptimal) measurement uncertainty bounds. However, the only evidence for supporting such a conjecture is the comparison of cases where either kind of uncertainty vanishes, so such a result is perhaps too much to hope for.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Joe Renes for suggesting also the discrete metric on ${\mathbb{Z}}$, and Rainer Hempel for helpful communications concerning the variational principle in Section \[sec:lower\].
RFW acknowledges funding by the DFG through the research training group RTG 1991. JK acknowledges funding from the EPSRC projects EP/J009776/1 and EP/M01634X/1.
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http://dx.doi.org/10.12743/quanta.v4i1.35) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, ) [****, ()](http://stacks.iop.org/1367-2630/17/i=9/a=093046) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [**]{} (, ) @noop [**]{} (, ) @noop [****, ()]{} @noop [**]{} (, ) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} [****, ()](http://iopscience.iop.org/10.1088/1751-8121/aa83bc)
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---
abstract: |
It has been recently noted that the diffeomorphism covariance of a Chiral Conformal QFT in the vacuum sector automatically ensures Möbius covariance in all charged sectors. In this article it is shown that the diffeomorphism covariance and the positivity of the energy in the vacuum sector even ensure the positivity of the energy in the charged sectors.
The main observation of this paper is that the positivity of the energy — at least in case of a Chiral Conformal QFT — is a local concept: it is related to the fact that the energy density, when smeared with some local nonnegative test functions, remains bounded from below (with the bound depending on the test function).
The presented proof relies in an essential way on recently developed methods concerning the smearing of the stress-energy tensor on nonsmooth functions.
author:
- |
MIHÁLY WEINER\
Dipartimento di Matematica\
Università di Roma “Tor Vergata”\
Via della Ricerca Scientifica, 1, I-00133, Roma, ITALY\
E-mail: weiner@mat.uniroma2.it
---
Introduction
============
The positivity of the energy is one of the most important selection criteria for a model to be “physical”. In almost all treatments of Quantum Field Theory, it appears as one of the fundamental axioms. In the vacuum sector it is usually formulated by requiring the positivity of the selfadjoint generator of every one-parameter group of future-like spacetime translations in the representation corresponding to the model.
As an axiom, one may say that it is “automatically” true, but in a concrete model it is something to be checked. In particular, to see what are the charged sectors with positive energy for a model given in the vacuum sector may be difficult (as calculating the charged sectors can already be a hard problem).
The present paper concerns chiral components of 2-dimensional Conformal QFTs in the setting of Algebraic Quantum Field Theory (see the book [@Haag] of Haag). In this framework a Chiral Conformal QFT is commonly described by means of a Möbius covariant local net of von Neumann algebras on $S^{1}$. The net is said to be conformal (or diffeomorphism) covariant if the Möbius symmetry of the net extends to a full diffeomorphism symmetry (see the next section for precise definitions). Charged sectors are described as irreducible representations of the net; their general theory was developed by Doplicher, Haag and Roberts [@DHR1; @DHR2]. They proved, among many other things, that if a covariant sector has a finite statistical dimension then it is automatically of positive energy. In [@GL] Guido and Longo showed that under some regularity condition the finiteness of the statistics even implies the covariance property of the sector.
In the particular case of chiral theories, there are many beautiful known relations concerning charged sectors; e.g. the formula [@KLM Theorem 33] of Kawahigashi, Longo e Müger, linking the statistical dimensions of the sectors to the so-called $\mu$-index of the net. In relation with the positivity of energy we may say that the case of finite statistics is more or less completely understood [@GL; @GuLo96; @BCL]. In particular, taking in account the above mentioned formula, if a theory (in its vacuum sector) is split, conformal and has a finite $\mu$-index — which means that it is [*completely rational*]{} cf. [@KLM; @LoXu] — then every sector of it is automatically of finite statistics and covariant under a positive energy representation of the Möbius group. Yet, although these conditions cover many of the interesting cases (for example all SU$(N)_k$ models [@Xu] and all models with central charge $c<1$, see [@KL]), there are interesting (not pathological!) models in which it does not hold and, what is more important in this context, indeed possessing sectors with infinite statistical dimension (and yet with positivity of energy). This is clearly in contrast with the experience coming from massive QFTs (by a theorem of Buchholz and Fredenhagen [@BuFre], a massive sector with positive energy is always localizable in a spacelike cone and has finite statistics).
The first example of a sector with infinite statistics was constructed by Fredenhagen [@Fre]. Rehren gave arguments [@Re] that even the Virasoro model, which is in some sense the most natural model, should admit sectors with infinite statistical dimensions when its central charge $c\geq 1$ and that in fact in this case “most” of its sectors should be of infinite statistics. This was then actually proved [@Carpi1] by Carpi first for the case $c=1$ and then [@Carpi2] for many other values of the central charge, leaving open the question only for some values of $c$ between $1$ and $2$. Moreover Longo e Xu proved [@LoXu] that if $\A$ is a split conformal net with $\mu =
\infty$ then $(\A \otimes \A)^{\rm flip}$ has at least one sector with infinite statistical dimension, showing that the case of infinite statistics is indeed quite general.
Recently D’Antoni, Fredenhagen and Köster published a letter [@D'AnFreKos] with a proof that diffeomorphism covariance itself (in the vacuum sector) is already enough to ensure Möbius covariance in every (not necessary irreducible) representation: there always exists a unique (projective, strongly continuous) inner implementation of the Möbius symmetry. (In Prop. \[n-cover\] we shall generalize this statement to the [*$n$-Möbius*]{} group, which is a natural realization of the $n^{\rm th}$ cover of the Möbius group in the group of all diffeomorphisms.) Thus the concept of the conformal energy, as the selfadjoint generator of rotations in a given charged sector, is at least well-defined. (Without the assumption of diffeomorphism covariance it is in general not true: there are Möbius covariant nets — see the examples in [@GLW; @koester03a] — possessing charged sectors in which the Möbius symmetry is not even implementable.) What remained an open question until now, whether this energy is automatically positive or not. The present article shall settle this question by providing a proof for the positivity (Theorem \[mainresult\]).
The idea behind the proof is simple. The total conformal energy $L_0$ is the integral of the energy-density; i.e. the stress-energy tensor $T$ evaluated on the constant $1$ function. So if we take a finite partition of the unity $\{f_n\}_{n=1}^N$ on the circle, we may write $T(1)$ as the sum $\sum T(f_n)$ where each element is [*local*]{}. Thus each term in itself (although not bounded) can be considered in a given charged sector. Moreover, it has been recently proved by Fewster and Holland [@FewHoll] that the stress-energy tensor evaluated on a nonnegative function is bounded from below. These operators then, being local elements, remain bounded from below also in the charged sector. So their sum in the charged sector, which we may expect to be the generator of rotations in that sector, should still be bounded from below.
There are several problems with this idea. For example, as the supports of the functions $\{f_n\}$ must unavoidably “overlap”, the corresponding operators will in general not commute. To deal with sums of non-commuting unbounded operators is not easy. In particular, while in the vacuum representation — due to the well known energy bounds — we have the natural common domain of the finite energy vectors, in a charged sector (unless we assume positivity of energy, which is exactly what we want to prove) we have no such domain.
To overcome the difficulties we shall modify this idea in two points. First of all, instead of $L_0=T(1)$, that is, the generator of the rotations, we may work with the generator of the translations — the positivity of any of them implies the positivity of the other one. In fact we shall go one step further by replacing the generator of translations with the generator of $2$[*-translations*]{}. (This is why, as it has been already mentioned, we shall consider the $n$-Möbius group; particularly in the case $n=2$.) This has the advantage that the function representing the corresponding vector field can be written as $f_1+f_2$, where the two local nonnegative functions $f_1,f_2$ [*do not*]{} “overlap”. These functions, at the endpoint of their support are not smooth (such decomposition is not possible with smooth functions); they are only once differentiable. However, as it was recently proved [@CaWe] by the present author and Carpi, the stress-energy tensor can be evaluated even on nonsmooth functions, given that they are “sufficiently regular”, which is exactly the case of $f_1$ and $f_2$ (see Lemma \[finite1.5norm\] and the argument before Prop. \[affiliation\]). As they are nonnegative but not smooth, to conclude that $T(f_1)$ and $T(f_2)$ are bounded from below we cannot use the result stated in [@FewHoll]. However, it turns out to be (Prop. \[affiliation\]) a rather direct and simple consequence of the construction, thus it will be deduced independently from the mentioned result, of which we shall make no explicit use. In fact the author considered this construction as an argument indicating that if $f \geq 0$ then $T(f)$ is bounded from below (which by now is of course proven, as it was already mentioned, in [@FewHoll]); see more on this in this paper at the remark after Prop \[affiliation\] and in the mentioned article of Fewster and Holland at the footnote in the proof of [@FewHoll Theorem 4.1].
Before we shall proceed to the proof, in the next section we briefly recall some definitions and basic facts regarding local nets of von Neumann algebras on the circle.
Preliminaries
=============
Möbius covariant nets and their representations
-----------------------------------------------
Let $\I$ be the set of open, nonempty and nondense arcs (intervals) of the unit circle $S^1 =\{z\in \CC : \,|z|=1 \}$. A [**Möbius covariant net on $S^1$**]{} is a map $\A$ which assigns to every $I \in \I$ a von Neumann algebra $\A(I)$ acting on a fixed complex, Hilbert space $\H_\A$ (“the vacuum Hilbert space of the theory”), together with a given strongly continuous representation $U$ of $\mob \simeq
{\rm PSL}(2,\RR)$, the group of Möbius transformations[^1] of the unit circle $S^1$ satisfying for all $I_1,I_2,I
\in \I$ and $\varphi \in \mob$ the following properties.
- [*Isotony.*]{} $I_1 \subset I_2 \,\Rightarrow\,
\A(I_1) \subset \A (I_2).$
- [*Locality.*]{} $I_1 \cap I_2 = \emptyset \,\Rightarrow\,
[\A(I_1),\A(I_2)]=0.$
- [*Covariance.*]{} $U(\varphi){\A}(I)U(\varphi)^{-1}={\A}(\varphi(I)).$
- [*Positivity of the energy.*]{} The representation $U$ is of positive energy type: the conformal Hamiltonian $L_0$, defined by $U(R_\alpha)=e^{i\alpha L_0}$ where $R_\alpha
\in \mob$ is the anticlockwise rotation by an angle of $\alpha$, is positive.
- [*Existence and uniqueness of the vacuum.*]{} Up to phase there exists a unique unit vector $\Omega \in \H_\A$ called the “vacuum vector” which is invariant under the action of $U$.
- [*Cyclicity of the vacuum.*]{} ${\Omega}$ is cyclic for the von Neumann algebra $\A(S^1)\equiv \{\A(I):I\in \I\}''$.
There are many known consequences of the above listed axioms. We shall recall some of the most important ones referring to [@FrG; @GuLo96] and [@FJ] for proves. 1. $\Omega$ is a cyclic and separating vector of the algebra $\A(I)$ for every $I \in \I$. 2. $U(\delta^I_{2\pi t})=\Delta_I^{it}$ where ${\Delta}_{I}$ is the modular operator associated to $\A(I)$ and $\Omega$, and $\delta^I$ is the one-parameter group of Möbius transformations preserving the interval $I$ (the dilations associated to $I$) with parametrization fixed in the beginning of the next section. 3. $\A(I)'= \A(I^c)$ for every $I \in \I$, where $I^c$ denotes the interior of the complement set of $I$ in $S^1$. 4.: $\A(S^1)={\rm B}(\H_\A)$, where ${\rm B}(\H_\A)$ denotes the algebra of all bounded linear operators on $\H_\A$. 5.: for an $I\in\I$ the algebra $\A(I)$ is either just the trivial algebra $\CC \mathbbm 1$ (in which case dim$(\H_\A)=1$ and the whole net is trivial) or it is a type ${\rm I\!I\!I}_1$ factor for every $I \in \I$. 6.: if $\S\subset \I$ is a covering of the interval $I$ then $\A(I)\subset
\{\A(J):J\in \S\}''$. Note that by the [*Bisognano-Wichmann property*]{}, since $\mob$ is generated by the dilations (associated to different intervals), the representation $U$ is completely determined by the local algebras and the vacuum vector via modular structure.
A [**locally normal representation**]{} $\pi$ (or for shortness, just simply representation) of a Möbius covariant local net $(\A,U)$ consits of a Hilbert space $\H_\pi$ and a normal representation $\pi_I$ of the von Neumann algebra $\A(I)$ on $\H_\pi$ for each $I \subset \I$ such that the collection of representations $\{\pi_I : I \in \I \}$ is consistent with the [*isotony*]{}: $I \subset K \Rightarrow \pi_K
|_{\A(I)} = \pi_I$. It follows easily from the axioms and the known properties of local nets listed above that if $I \cap K =
\emptyset$ then $[\pi_I(\A(I)),\pi_K(\A(K))]=0$, if $\S \subset \I$ is covering of $K \in \I$ then $\{\pi_I(\A(I)):I\in\I\}'' \supset
\pi_K(\A(K))$, if $\S \subset \I$ is a covering of $S^1$ then $\{\pi_I(\A(I)):I\in\S\}''= \{\pi_I(\A(I)):I\in\I\}''\equiv\pi(\A)$ and finally, that for each $I \in \I$ the representation $\pi_I$ is faithful. The representation $\pi$ is called [*irreducible*]{}, if $\pi(\A)' = \CC \mathbbm 1$.
Diffeomorphism covariance
-------------------------
Let $\diff$ be the group of orientation preserving (smooth) diffeomorphisms of the circle. It is an infinite dimensional Lie group whose Lie algebra is identified with the real topological vector space Vect$(S^1)$ of smooth real vector fields on $S^1$ with the usual $C^\infty$ topology [@Milnor Sect. 6] with the negative[^2] of the usual bracket of vector fields. We shall think of a the vector field symbolically written as $f(e^{i\vartheta})\frac{d}{d\vartheta}\in $ Vect$(S^1)$ as the corresponding real function $f$. We shall use the notation $f'$ (calling it simply the derivative) for the function on the circle obtained by derivating with respect to the angle: $f'(e^{i\theta})=\frac{d}{d\alpha}f(e^{i\alpha})|_{\alpha=\theta}$.
A strongly continuous projective unitary representation $V$ of $\diff$ on a Hilbert space $\H$ is a strongly continuous $\diff \rightarrow \U(\H)/\mathbb T$ homomorphism. The restriction of $V$ to $\mob \subset \diff$ always lifts to a unique strongly continuous unitary representation of the universal covering group $\widetilde{\mob}$ of $\mob$. $V$ is said to be of positive energy type, if its conformal Hamiltonian $L_0$, defined by the above representation of $\widetilde{\mob}$ (similarly as in case of a representation of the group $\mob$) has nonnegative spectrum.
Sometimes for a $\gamma \in \diff$ we shall think of $V(\gamma)$ as a unitary operator. Although there are more than one way to fix phases, note that expressions like Ad$(V(\gamma))$ or $V(\gamma) \in \M$ for a von Neumann algebra $\M \subset {\rm B}(\H)$ are unambiguous. Note also that the selfadjoint generator of a one-parameter group of strongly continuous [*projective*]{} unitaries $t \mapsto Z(t)$ is well defined up to a real additive constant: there exists a selfadjoint operator $A$ such that Ad$(Z(t))=$ Ad$(e^{iAt})$ for all $t \in \RR$, and if $A'$ is another selfadjoint with the same property then $A'=A+r\mathbbm 1$ for some $r \in \RR$.
We shall say that $V$ is an extension of the unitary representation $U$ of $\mob$ if we can arrange the phases in such a way that $V(\varphi)=U(\varphi)$, or without mentioning phases: Ad$(V(\varphi))=$ Ad$(U(\varphi))$, for all $\varphi \in \mob$. Note that such an extension of a positive energy representation of $\mob$ is of positive energy.
\[diffcov:def\] A Möbius covariant net $(\A,$U$)$ is said to be [**conformal (or diffeomorphism) covariant**]{} if there is a strongly continuous projective unitary representation of $\diff$ on $\H_\A$ which extends $U$ (and by an abuse of notation we shall denote this extension, too, by $U$), and for all $\gamma \in \diff$ and $I \in \I$ satisfies
- $U(\gamma)\A(I)U(\gamma)^* = \A(\gamma(J)),$
- $\gamma|_I={\rm{id}}_I \Rightarrow
\rm{Ad}(U(\gamma))|_{\A(I)}=\rm{id}_{\A(I)}$.
Note that as a consequence of [*Haag duality*]{}, if a diffeomorphism is localized in the interval $I$ — i.e. it acts trivially (identically) elsewhere — then, by the second listed property the corresponding unitary is also localized in $I$ in the sense that it belongs to $\A(I)$. Thus by setting $$\A_U(I)\equiv \{U(\gamma):\gamma\in\diff,
\gamma|_{I^c} = {\rm id}_{I^c}\}'' \;\;\; (I\in\I)$$ we obtain a [*conformal subnet*]{}: for all $\gamma\in\diff$ and $I\in\I$ we have that $\A_U(I)\subset
\A(I)$ and $U(\gamma) \A_U(I) U(\gamma)^*=\A_U(\gamma(I))$. The restriction of the subnet $\A_U$ onto the Hilbert space $\H_{\A_U}\equiv \overline{(\bigvee_{I\in\I} \A_U(I))\Omega}$ is again a conformal net, which — unless $\A$ is trivial — by [@Carpi2 Theorem A.1] is isomorphic to a so-called [*Virasoro net*]{}. For a representation $\pi$ of $\A$ we set $\pi(\A_U)\equiv\{\pi_I(\A_U(I)) :I\in\I\}''$.
The smooth function $f: S^1 \rightarrow \RR$, as a vector field on $S^1$, gives rise to the one-parameter group of diffeomorphisms $t\mapsto$ Exp$(tf)$. Hence, up to an additive real constant the selfadjoint generator $T(f)$ of $t \mapsto U({\rm Exp(tf)})$ is well defined. For any real smooth function $f$ on the circle $T(f)$ is essentially selfadjoint on the dense set of [*finite-energy vectors*]{}, i.e. on the algebraic span of the eigenvectors of $L_0$. By the condition $<\Omega, T(\cdot)\Omega> = 0$ fixing the additive constant in its definition, $T$ is called the [**stress-energy tensor**]{} associated to $U$. It is an operator valued linear functional in the sense that on the set of finite energy vectors $T(f + \lambda g) = T(f) + \lambda T(g)$ for all $f,g$ real smooth functions on the circle and $\lambda \in \RR$. Note that by the second listed condition in Def. \[diffcov:def\] if Supp$(f) \subset I$ for a certain $I \in \I$ then $T(f)$ is affiliated to $\A(I)$.
For a more detailed introduction on the stress-energy tensor see for example [@CaWe; @Carpi2]. The proof of the statements made in defining $T$ relies on the so-called Virasoro operators, which can always be introduced (see the remarks in the beginning of [@CaWe Sect. 4] and before [@Carpi2 Theorem A.1], all using the results [@loke] of Loke), and on the existence of some “energy bounds” (see [@GoWa; @BS-M]).
In this paper we shall often use nonsmooth functions. For a function $f\in C(S^1,\RR)$ with Fourier coefficients $\hat{f}_n=\frac{1}{2\pi}\int_0^{2\pi} e^{-in\alpha}f(e^{i\alpha})
\,d\alpha$ $(n \in {\mathbb Z})$ we set $$\|f\|_{\frac{3}{2}}=
\mathop{\sum}_{n \in {\mathbb Z}}|\hat{f}_n|(1+|n|^{\frac{3}{2}})
\in \RR^+_0 \cup \{+\infty\}.$$ In [@CaWe Sect. 4] the present author with Carpi proved that $T$ can be continuously extended to functions with finite $\|\cdot\|_{\frac{3}{2}}$ norm as
- if $f,f_n$ $(n \in \NN)$ are real smooth functions on the circle and $f_n\rightarrow f$ in the $\|\cdot\|_{\frac{3}{2}}$ sense then $T(f_n)$ converges to $T(f)$ in the strong resolvent sense,
- if $f_n$ $(n \in \NN)$ is a Cauchy sequence of real smooth functions with respect to the $\|\cdot\|_{\frac{3}{2}}$ norm then $T(f_n)$ converges to a selfadjoint operator in the strong resolvent sense, which is essentially selfadjoint on the finite energy vectors,
- the real smooth functions form a dense set among the real continuous functions with finite $\|\cdot\|_{\frac{3}{2}}$ norm.
Thus one can consider $T(f)$ even when $f$ is not smooth but its $\|\cdot\|_{\frac{3}{2}}$ norm is finite. The following lemma, which was essentially demonstrated in the proof of [@CaWe Lemma 5.3] but was not stated there can be useful in some cases to establish the finiteness of this norm.
\[finite1.5norm\] Let $f$ be a (once) differentiable function on the circle. Suppose that there exists a finite set of intervals $I_k\in\I$ and smooth functions $g_k$ on the circle $(k=1,..,N)$ such that $\overline{\cup_{k=1}^N I_k}=S^1$ and $f|_{I_k}=g_k|_{I_k}$. Then $\|f\|_{\frac{3}{2}} < \infty$.
The conditions mean that $f''$, which is everywhere defined apart from a finite set of points, has Fourier coefficients $\hat{(f'')}_n=-n^2 \hat{f}_n$ and is of bounded variation. Therefore $|n^2\hat{f}_n| \leq |\frac{{\rm Var}(f'')}{n}|$ (see [@katznelson Sect. I.4]), from which the claim follows easily.
In relation with the net $(\A,U)$ the extension to nonsmooth functions is still [*covariant*]{} and [*local*]{} in the sense of the following statement (which again was essentially proved in [@CaWe], but was not explicitly stated there).
\[aff\] Let $\gamma \in \diff$ and $f$ be a real continuous function on the circle with both $\|f\|_{\frac{3}{2}} < +\infty$ and $\|\gamma_* f\|_{\frac{3}{2}} < +\infty$ where $\gamma_*$ stand for the action of $\gamma$ on vector fields. Then up to phase factors $$U(\gamma)\, e^{iT(f)}\, U(\gamma)^*=
e^{iT(\gamma_* f)}.$$ Moreover, if Supp$(f) \subset
\overline{I}$ where $I \in \I$, then $T(f)$ is affiliated to $\A(I)$.
For the second part of the statement, by the continuity [@jors] of the net we may assume that Supp$(f)$ is already contained in $I$ (and not only in its closure). Then according to [@CaWe Lemma 4.6], there exists a sequence of smooth functions $f_n$ $(n \in \NN)$ converging to $f$ in the $\|\cdot\|_{\frac{3}{2}}$ norm whose support is contained in $I$. Then, by [@CaWe Prop. 4.5] $T(f_n)$ converges to $T(f)$ in the strong resolvent sense, and thus $T(f)$ is affiliated to $\A(I)$ as $T(f_n)$ is affiliated to $\A(I)$ for each $n \in \NN$.
The first part of the statement is again obviously true if $f$ is smooth, as then $e^{iT(f)}=U({\rm Exp}(f))$ and $e^{iT(\gamma_* f)}=$ $$U({\rm Exp}(\gamma_* f)=U(\gamma\circ{\rm
Exp}(f)\circ\gamma^{-1})=U(\gamma)\, U({\rm Exp}(f))\, U(\gamma)^*.$$ Then similarly to the first part, by approximating $f$ with smooth functions and taking limits one can easily finish the proof.
Proof of the Positivity
=======================
Apart from the subgroup $\mob \subset \diff$, for our argument we shall need to use some other important subgroups. For each positive integer $n$ the group $\mob^{(n)}$ is defined to be the subgroup of $\diff$ containing all elements $\gamma \in \diff$ for which there exists a Möbius transformation $\phi
\in \mob$ satisfying $$\gamma(z)^n = \phi(z^n) \;\; (\forall z \in S^1).$$ Thus the group $\mob^{(n)}$ gives a natural $n$-covering of $\mob$. This group has been already considered and successfully used for the analyses of conformal nets, see e.g. [@LoXu].
In $\mob$, beside the rotations one often considers the translations $a\mapsto \tau_a$ and the dilations $s \mapsto \delta_s$, that are the one-parameter groups generated by the vector fields $t(z) = \frac{1}{2}-\frac{1}{4}(z+z^{-1})$ and $d(z) = \frac{i}{2}(z+z^{-1})$, respectively. For an $I\in\I$ one may choose a transformation $\phi\in\mob$ such that $\phi(S^1_+)=I$ where $S^1_\pm = \{z \in S^1: \pm {\rm Im}(z) > 0\}$. The one-parameter group $s \mapsto \phi\circ \delta_s
\circ\phi^{-1}$ is independent of $\phi$ (see e.g. [@GLW]) and is called the dilations associated to the interval $I$. When no interval is specified, $\delta$ always stands for the one associated to $S^1_+$.
By direct calculation $[d,t] = t$ (remember that the bracket is the negative of the usual bracket of vector fields) and thus at the group level we find $$\label{scaling}
\delta_s \tau_a \delta_{-s} = \tau_{e^s a}$$ i.e. the dilations “scale” the translations.
In $\mob^{(n)}$, just like in $\mob$, one introduces the one-parameter subgroup of translations $a \mapsto \tau^{(n)}_a$, which is defined by the usual procedure of lifting: it is the unique continuous one-parameter subgroup satisfying $\tau^{(n)}_a(z)^n = \tau_a(z^n)$. Alternatively, one may define it directly with its generating vector field $t^{(n)}(z)=\frac{1}{2n} - \frac{1}{4n}(z^n+z^{-n})$. Similarly one introduces the notion of rotations $\alpha \mapsto R^{(n)}_\alpha$ and of dilations $s \mapsto \delta^{(n)}_s$. Of course the “$n$-rotations”, apart from a rescaling of the parameter, will simply coincide with the “true” rotations: $$\label{n-rotation}
R^{(n)}_\alpha=R_{\alpha/n}.$$
Let us now consider a strongly continuous projective unitary representation $V^{(n)}$ of $\mob^{(n)}$. The group $\mob^{(n)}$ is connected and its Lie algebra is isomorphic to ${\mathfrak{sl}}(2,\RR)$ which is in particular semisimple (in fact even simple, but for what follows semisimplicity is enough). Therefore, as it is well known, the representation $V^{(n)}$ has a unique strongly continuous lift $\tilde{V}$ to the universal cover of $\mob^{(n)}$ which is a [*true*]{} representation. As $\mob^{(n)}$ covers $\mob$ in a natural way, its universal cover is canonically identified with $\widetilde{\mob}$ which is isomorphic to $\widetilde{{\rm SL}(2,\RR)}$.
The following lemma, although contains some well known facts, is hereby included for the convenience of the reader. The presented proof is an adopted (and slightly modified) version of the proof of [@koester02 Prop. 1].
\[mob\^n\] Let $\tilde{V}$ be a strongly continuous unitary representation of $\widetilde{\mob}$ with $H$ and $P$ being the selfadjoint generator of rotations and translations in $\tilde{V}$, respectively. Then the following four conditions are equivalent:\
$1.\;\,H$ is bounded from below,\
$2.\;\,P$ is bounded from below,\
$3.\;\,H \geq 0$,\
$4.\;\,P \geq 0$.
Let $\tilde{R}$ be the lift of $R$ and set $P_\pi=\tilde{V}(\tilde{R}_\pi) P \tilde{V}(\tilde{R}_\pi)^*$; it is then the selfadjoint generator associated to the one-parameter group generated by the vector field $t_\pi$ which we get by rotating $t$ by $\pi$ radian i.e. $t_\pi(z)=\frac{1}{2}+\frac{1}{4}(z+z^{-1})$. As $P_\pi$ is unitary conjugate to $P$ their spectra coincide. Moreover, as $t+t_\pi =1$ on the G[å]{}rding-domain we have that $P+P_\pi=H$ which immediately proves that if $P$ is bounded from below or positive then so is $H$.
As for the rest of the statement, apart from the trivial indications there remain only to show that if $H$ is bounded from below then $P$ is positive. Consider the lifted dilations $s \mapsto \tilde{\delta}_s$. By equation (\[scaling\]) one has that $\tilde{V}(\tilde{\delta}_s)
P \tilde{V}(\tilde{\delta}_s)^* = e^s P$. Moreover, by direct calculation $[d,t_\pi]=-t_\pi$ so similarly to the case of translations the dilations also “scale” $t_\pi$, but in the converse direction. Thus $\tilde{V}(\tilde{\delta}_s)
P_\pi \tilde{V}(\tilde{\delta}_s)^* = e^{-s} P_\pi$. So if $H\geq r \mathbbm 1$ for some $r$ real (but not necessarily nonnegative) number then for any vector $\xi$ in the G[å]{}rding-domain, setting $\eta
= \tilde{V}(\tilde{\delta}_s)^*\xi$ we have that $$\begin{aligned}
\label{ineq}
r \|\xi\|^2 = r \|\eta\|^2 &\leq& <\eta,H\eta>\,
= e^s <\xi,P\xi> + \, e^{-s} <\xi,P_\pi \xi>\end{aligned}$$ from which, letting $s\rightarrow \infty$ we find that $P \geq 0$.
If any of the conditions of the above lemma is satisfied, $\tilde{V}$ is called a positive energy representation. A projective representation $V^{(n)}$ of $\mob^{(n)}$ is said to be of positive energy if its lift to $\widetilde{\mob}$ is of positive energy.
Let us now consider a conformal local net of on the circle $(\A,U)$. By equation (\[n-rotation\]), $U^{(n)}$, the restriction of the positive energy representation of $U$ of $\diff$ with stress-energy tensor $T$, is a positive energy projective representation of $\mob^{(n)}$. In particular, as $U^{(2)}$ is of positive energy, the selfadjoint operator $T(t^{(2)})$ must be bounded from below, since it generates the translations for the representation $U^{(2)}$. (Note that $T(t^{(2)})$ is bounded from below but not necessary positive: it is not [*the*]{} generator — it still generates the same projective one-parameter group of unitaries if you add a real constant to it.) The function $t^{(2)}(z)=
\frac{1}{4}-\frac{1}{8}(z^2+z^{-2})$ is a nonnegative function with two points of zero: $t^{(2)}(\pm 1)=0$. By direct calculation of the first derivative: $(t^{(2)})'(\pm 1)=0$, hence the decomposition $$\label{decomp}
t^{(2)} = t^{(2)}_+ + t^{(2)}_-$$ with the functions $t^{(2)}_\pm$ defined by the condition Supp$(t^{(2)}_\pm) = (S^1_\mp)^c$ is a decomposition of $t^{(2)}$ into the sum of two (once) differentiable nonnegative functions that satisfy the conditions of Lemma \[finite1.5norm\]. Therefore, as it was explained in the Preliminaries, we can consider the selfadjoint operators $T(t^{(2)}_\pm)$.
\[affiliation\] Let $(\A,U)$ be a conformal net of local algebras on the circle with stress-energy tensor $T$. Then $T(t^{(2)}_+)$ is affiliated to $\A(S^1_+)$ and $T(t^{(2)}_-)$ is affiliated to $\A(S^1_-)$ and so in particular they strongly commute. Moreover, the operators $T(t^{(2)}_\pm)$ are bounded from below.
Supp$(t^{(2)}_\pm) \subset \overline{S^1_\pm}$ and so by Prop. \[aff\] $T(t^{(2)}_\pm)$ is affiliated to $\A(S^1_\pm)$. So if $P_{[a,b]}$ is a nonzero spectral projection of $T(t^{(2)}_+)$ and $Q_{[c,d]}$ is a nonzero spectral projection of $T(t^{(2)}_-)$, then $P_{[a,b]} \in \A(S^1_+),\; Q_{[a,b]} \in \A(S^1_-)$ and by the algebraic independence of two commuting factors (see for example [@kadison Theorem 5.5.4]) $R=P_{[a,b]}Q_{[c,d]} \neq 0$. Of course the range of $R$ is invariant for (and included in the domain of) $T(t^{(2)}_+) + T(t^{(2)}_-)$ and the restriction of that operator for this closed subspace is clearly bigger than $a+c$ and smaller than $b+d$. Thus $${\rm Sp}\left(T(t^{(2)}_+) + T(t^{(2)}_-)\right) \supset
{\rm Sp}(T(t^{(2)}_+)) + {\rm Sp}(T(t^{(2)}_-)).$$ To conclude we only need to observe that by equation (\[decomp\]) on the common core of the finite energy vectors $T(t^{(2)}_+) + T(t^{(2)}_-) = T(t^{(2)})$, and as it was said, the latter selfadjoint operator is bounded from below.
[*Remark.*]{} The author considered this construction to indicate that if $f \geq 0$ then $T(f)$ is bounded from below, which — as it was already mentioned — by now is a proven fact (cf.[@FewHoll]). The point is the following. If $f$ is [*strictly*]{} positive then, as a vector field on $S^1$, it is conjugate to the constant vector field $r$ for some $r>0$. Thus, using the transformation rule of $T$ under diffeomorphisms, $T(f)$ is conjugate to $T(r)$ plus a constant, and so it is bounded from below by the positivity of $T(1)=L_0$. The real question is whether the statement remains true even when $f$ is nonnegative, but not strictly positive because for example it is [*local*]{} (there is an entire interval on which it is zero). One can of course consider a nonnegative function as a limit of positive functions, but then one needs to control that the lowest point of the spectrum does not go to $-\infty$ while taking this limit (which — in a slightly different manner — has been successfully carried out in the mentioned article). However, even without considering limits, by the above proposition we find nontrivial examples of local nonnegative functions $g$ such that $T(g)$ can easily be checked to be bounded from below. (Take for example $g=t^{(2)}_\pm$ but of course we may consider conjugates, sums and multiples by positive constants to generate even more examples.)
Let us now investigate what we can say about a representation $\pi$ of the conformal net $(\A,U)$. In [@D'AnFreKos] it was proved that the Möbius symmetry is continuously implementible in any (locally normal) representation $\pi$ by a unique inner projective way. By their construction the implementing operators are elements of $\pi(\A_U)$. Looking at the article, we see that the only structural properties of the Möbius subgroup of $\diff$ that the proof uses are the following.
- There exist three continuous one-parameter groups $\Gamma_1, \Gamma_2$ and $\Gamma_3$ in $\mob$, so that every element $\gamma \in \mob$ can be uniquely written as a product $\gamma=
\Gamma_1(s_1)\Gamma_2(s_2)\Gamma_3(s_3)$ where the parameters $(s_1,s_2,s_3)$ depend continuously on $\gamma$. (In the article $\Gamma_1$ is the translational, $\Gamma_2$ is the dilational and $\Gamma_3$ is the rotational subgroup; which is the so-called Iwasawa decomposition, see [@FrG].)
- The Lie algebra of $\mob$ is simple.
These two properties hold not only for the subgroup $\mob$, but also for $\mob^{(n)}$ where $n$ is any positive integer: for all $n$ the Lie algebra of $\mob^{(n)}$ is isomorphic to ${\mathfrak{sl}}(2,\RR)$, and with the rotations, dilations and translations replaced by $n$-rotations, $n$-dilations and $n$-translations we still have the required decomposition. Let us collect into a proposition what we have thus concluded.
\[n-cover\] Let $\pi$ be a locally normal representation of the conformal local net of von Neumann algebras on the circle $(\A,U)$. Then for all positive integer $n$ there exists a unique strongly continuos projective representation $U^{(n)}_\pi$ of $\mob^{(n)}$ such that $U^{(n)}_\pi(\mob^{(n)})\subset \pi(\A)$ and for all $\gamma \in \mob^{(n)}$ and $I \in \I$ $${\rm Ad}(U^{(n)}(\gamma)) \circ \pi_I =
{\pi}_{\gamma(I)} \circ {\rm Ad}(U^{(n)}(\gamma)).$$ Moreover, this unique representation satisfies $U^{(n)}_\pi(\mob^{(n)})\subset \pi(\A_U)$.
We shall now return to the particular case $n=2$. On one hand, the action of the $2$-translation $\tau^{(2)}_a$ in the representation $\pi$ can be implemented by $U^{(2)}_\pi(\tau^{(2)}_a)$. On the other hand, as $$U(\tau^{(2)}_a)= e^{iaT(t^{(2)})}=e^{iaT(t^{(2)}_+)} e^{iaT(t^{(2)}_-)}$$ we may try to implement the same action by $\pi_{S^1_+}(W_+(a)) \pi_{S^1_-}(W_-(a))$, where $$W_\pm(a)=e^{iaT(t^{(2)}_\pm)} \in \A_U(S^1_\pm).$$
\[localimplementation\] The unitary operator in $\pi(\A_U)$ $$W_\pi(a):=\pi_{S^1_+}(W_+(a)) \,\pi_{S^1_-}(W_-(a))
=\pi_{S^1_-}(W_-(a))\, \pi_{S^1_+}(W_+(a))$$ up to phase coincides with $U^{(2)}_\pi(\tau^{(2)}_a)$.
It is more or less trivial that the adjoint action of the two unitaries coincide on both $\pi_{S^1_+}(\A(S^1_+))$ and $\pi_{S^1_-}(\A(S^1_-))$. There remain two problems to overcome:
- the algebra generated by these two algebras do not necessarily contain $\pi(\A_U)$, so it is not clear why the adjoint action of these two unitaries should coincide on the mentioned algebra,
- but even if we knew that the actions coincide, the two unitaries, although both belonging to $\pi(\A_U)$, for what we know could still “differ” in an inner element.
As for the first problem, consider an open interval $I \subset S^1$ such that it contains the point $-1$ and has $1$ in the interior of its complement. Note that due to the conditions imposed on $I$, the sets $K_\pm:=I \cup S^1_\pm$ are still elements of $\I$.
If $a \geq 0$ then $W_+(a) \A(I) W_+(a)^* \subset \A(I)$.
Let us take a sequence of nonnegative smooth functions $\phi_n$ $(n=1,2,..)$ on the real line, such that the support of $\phi_n$ is contained in the interval $(-1/n,1/n)$, and its integral is $1$. Then, exactly as in [@CaWe Prop. 4.5, Lemma 4.6], we have that $T(\rho_n)$, with $\rho_n$ being the convolution $$\rho_n(e^{i\theta})\equiv (t^{(2)}_+ * \phi_n)(e^{i\theta})
\equiv \int t^{(2)}_+(e^{i(\theta + \alpha)}) \phi_n(\alpha)\,d\alpha,$$ converges to $T(t^{(2)}_+)$ in the strong resolvent sense.
The flow of a vector field given by a nonnegative function on the circle, moves all points forward (i.e. anticlockwise). Moreover, the flow cannot move points from the support of the vector field to outside, and leaves invariant all points outside.
The function $\rho_n$ — being the convolution of two nonnegative function — is nonnegative, and its support is $S^1_+$ “plus $1/n$ radius in both direction”. Taking in consideration what was said before it is easy to see that for $n$ large enough Exp$(a \rho_n)(I) \subset I$ and consequently $${\rm Ad}\left(e^{i a T(\rho_n)}\right)(\A(I))\subset \A(I).$$ Then by the convergence in the strong resolvent sense we obtain what we have claimed.
It follows that if $A \in \A(I)$ and $a \geq 0$ then $$\begin{aligned}
\nonumber
&&\pi_{S^1_+}(W_+(a))\,\pi_I(A)\,\pi_{S^1_+}(W_+(a))^* = \\
&&\pi_{K_+} (W_+(a)\,A\,W_+(a)^*) = \pi_I (W_+(a)\,A\,W_+(a)^*)\end{aligned}$$ and thus $\rm{Ad}\left(W_\pi(a)\right)(\pi_I(A))=
\rm{Ad}\left(\pi_{S^1_-}(W_-(a))\,\pi_{S^1_+}(W_+(a)) \right)
(\pi_I(A))=$ $$\begin{aligned}
\nonumber
&=&\rm{Ad}\left(\pi_{S^1_-}(W_-(a))\right)(\pi_I(W_+(a)\,A\,W_+(a)^*))
\\ \nonumber
&=&\pi_{K_-}(W_-\,(W_+(a)\,A\,W_+(a)^*)\,W_-(a)^*)
\\
&=&\pi_{K_-}U(\tau^{(2)}_a)\,A\,U(\tau^{(2)}_a)^*)
=\rm{Ad}\left(U^{(2)}_\pi(\tau^{(2)}_a)\right)(\pi_I(A))\end{aligned}$$ where in the last equality we have used the fact that for $a \geq 0$ the image of $I$ under the diffeomorphism $\tau^{(2)}_{a}$ is contained in $K_-$.
We have thus seen that for $a \geq 0$ the adjoint action of $W_\pi(a)$ and of $U^{(2)}_\pi(\tau^{(2)}_a)$ coincide on $\pi_I(A(I))$. Actually, looking at our argument we can realize that everything remains true if instead of $I$ we begin with an open interval $L$ that contains the point $1$ and has $-1$ in the interior of its complement and we exchange the “+” and “-” subindices. So in fact we have proved that for $a \geq 0$ these adjoint actions coincide on both $\pi_I(A(I))$ and $\pi_{L}(A(L))$ and therefore on the whole algebra $\pi(\A)$, since we may assume that the union of $I$ and $L$ is the whole circle. (The choice of the intervals, apart from the conditions listed, was arbitrary.) Of course the equality of the actions, as they are obviously one-parameter automorphism groups of $\pi(\A)$, is true also in case the parameter $a$ is negative. We can now also confirm that the unitary $$Z_\pi(a) \equiv W_\pi(a)^* \, U^{(2)}_\pi(\tau^{(2)}_a)$$ lies in $\Z(\pi(\A))\cap \pi(\A_U) \subset \Z(\pi(\A_U))$ where “$\Z$” stands for the word “center”. Thus $a \mapsto Z_\pi(a)$ is a strongly continuous (projective) one-parameter group. (It is easy to see that as $Z_\pi$ commutes with both $W_\pi$ and $U^{(2)}_\pi$ it is actually a one-parameter group.)
We shall now deal with the second mentioned problem. The $2$-dilations $s \mapsto \delta^{(2)}_s$ scale the $2$-translations and preserve the intervals $S^1_\pm$. Thus they also scale the functions $t^{(2)}_\pm$ and so we get some relations — both in the vacuum and in the representation $\pi$ — regarding the unitaries implementing the dilations and translations and the unitaries that were denoted by $W$ with different subindices (see Prop. \[aff\]). More concretely, with everything meant in the projective sense, in the vacuum Hilbert space we have that the adjoint action of $U^{(2)}(\delta^{(2)}_s)$ scales the parameter $a$ into $e^sa$ in $U^{(2)}(\tau^{(2)}_a)$ and in $W_\pm(a)$ while in $\H_\pi$ we have exactly the same scaling of $U^{(2)}_\pi(\tau^{(2)}_a)$ and of $\pi_{S^1_\pm}(W_\pm(a))$ by the adjoint action of $U^{(2)}_\pi(\delta^{(2)}_s)$. Thus we find that $$Ad\left(U^{(2)}_\pi(\delta^{(2)}_s)\right)(Z_\pi(a)) = Z_\pi(e^s a),$$ but on the other hand of course, as $Z_\pi$ is in the center, the left hand side should be simply equal to $Z_\pi(a)$. So $Z_\pi(a)=
Z_\pi(e^s a)$ for all values of the parameters $a$ and $s$ which means that $Z_\pi$ is trivial and hence in the projective sense $W_\pi(a)$ equals to $U^{(2)}_\pi(\tau_a)$.
\[pos:2\] The projective representation $U^{(2)}_\pi$ is of positive energy.
As the spectrum of the generator of a one-parameter unitary group remains unchanged in any normal representation, by Prop. \[affiliation\] the selfadjoint generator of the one-parameter group $$a \mapsto \pi_{S^1_+}\left(e^{iaT(t^{(2)}_+)}\right)
\pi_{S^1_-}\left(e^{iaT(t^{(2)}_-)}\right)$$ is bounded from below and by Prop. \[localimplementation\] this one-parameter group of unitaries equals to the one-parameter group $a \mapsto U^{(2)}_\pi(\tau^{(2)}_a)$ in the projective sense. So by Lemma \[mob\^n\] the representation $U^{(2)}_\pi$ is of positive energy.
Let us now take an arbitrary positive integer $n$. By equation (\[n-rotation\]) $R_\alpha \in \mob^{(n)}$ for all $\alpha \in \RR$, and by definition both $U^{(n)}_\pi(R_\alpha)$ and $U^{(2)}_\pi(R_\alpha)$ implement the same automorphism of $\pi(\A)$. Since both unitaries are actually elements of $\pi(\A_U)\subset \pi(\A)$, they must commute and $$\label{propotion}
C_{\pi}^{(n)}(\alpha)=(U^{(n)}_\pi(R_\alpha))^* \,\,
U^{(2)}_\pi(R_\alpha)$$ is a strongly continuous one-parameter group in $\Z(\pi(\A))\cap \pi(\A_U) \subset \Z(\pi(\A_U))$.
As it was mentioned, by [@Carpi2 Theorem A.1] the restriction of the subnet $\A_U$ onto $\H_{\A_U}$ — unless $\A$ is trivial, in which case dim$(\H_{\A_U})={\rm dim}(\H_\A)=1$ — is isomorphic to a Virasoro net. Thus $\H_{\A_U}$ must be separable (even if the full Hilbert space $\H_\A$ is not so; recall that we did not assume separability) as the Hilbert space of a Virasoro net is separable.
Every von Neumann algebra on a separable Hilbert space has a strongly dense separable $C^*$ subalgebra. A von Neumann algebra generated by a finite number of von Neumann algebras with strongly dense separable $C^*$ subalgebras has a strongly dense $C^*$ subalgebra. Thus considering that for an $I\in\I$ the restriction map from $\A_U(I)$ to $\A_U(I)|_{\H_{\A_U}}$ is an isomorphism, one can easily verify that the von Neumann algebra $\pi(\A_U)$ has a strongly dense $C^*$ subalgebra.
We can thus safely consider the direct integral decomposition of $\pi(\A_U)$ along its center $$\label{directint}
\pi(\A) = {\int}_{\!\!\! X}^\oplus \pi(\A)(x) d\mu(x).$$ (Even if $\H_\pi$ is not separable, by the mentioned property of the algebra $\pi(\A_U)$, it can be decomposed into the direct sum of invariant separable subspaces for $\pi(\A_U)$. Then writing the direct integral decomposition in each of those subspaces, the rest of the argument can be carried out without further changes.) For an introduction on the topic of the direct integrals see for example [@kadison Chapter 14.].
As it was mentioned the representations $U^{(n)}_\pi$ $(n=1,2,..)$ have a unique strongly continuous lift $\tilde{U}^{(n)}_\pi$ to $\widetilde{\mob}$ where $\tilde{U}^{(n)}_\pi$ is a true representation. Since the group in question is in particular second countable and locally compact, and all these representations are in $\pi(\A_U)$, the decomposition (\[directint\]) also decomposes these representations (cf. [@dixmier Lemma 8.3.1 and Remark 18.7.6]): $$\tilde{U}^{(n)}_\pi(\cdot) = {\int}_{\!\!\! X}^\oplus
\tilde{U}^{(n)}_{\pi}(\cdot)(x) d\mu(x)$$ where $\tilde{U}^{(n)}_{\pi}(\widetilde{\mob})(x) \subset \pi(\A)(x)$ and $\tilde{U}^{(n)}_{\pi}(\cdot)(x)$ is a strongly continuous representation for almost every $x \in X$.
\[posdirectint\] The representation $\tilde{U}^{(n)}_\pi$ is of positive energy if and only if $\tilde{U}^{(n)}_{\pi}(\cdot)(x)$ is of positive energy for almost every $x \in X$.
For a $t\mapsto V(t)$ strongly continuous one-parameter group of unitaries the positivity of the selfadjoint generator is for example equivalent with the fact that $\hat{V}(f)\equiv\int V(t)f(t)dt
= 0$ for a certain smooth, fast decreasing function $f$ whose Fourier transform is positive on $\RR^-$ and zero on $\RR^+$. If $V$ is a direct integral of a measurable family of strongly continuous one-parameter groups, $V(\cdot)=\int^\oplus_{\! X}
V(\cdot)(x) d\mu(x)$, then $\hat{V}(f) =\int^\oplus_{\! X}
\hat{V}(f)(x) d\mu(x)$. As $\hat{V}(f)(x) \geq 0$ for almost every $x\in X$, the operator $\hat{V}(f)$ is zero if and only if $\hat{V}(f)(x)=0$ for almost every $x\in X$.
As $C_{\pi}^{(n)}$ is a strongly continuous one-parameter group in the center, for almost all $x \in X:\;\tilde{U}^{(n)}_{\pi}
(R_{(\cdot)})(x) = \tilde{U}^{(2)}_{\pi}(R_{(\cdot)})(x)$ in the projective sense. Therefore, since by Lemma \[posdirectint\] and Corollary \[pos:2\] in $\tilde{U}^{(2)}_{\pi}(\cdot)(x)$ the selfadjoint generator of rotations is positive, also in $\tilde{U}^{(n)}_{\pi}(\cdot)(x)$ it must be at least bounded from below and hence by Lemma \[n-cover\] it is actually positive. Thus, by using again Lemma \[posdirectint\] we arrive to the following result.
\[mainresult\] Let $\pi$ be a locally normal representation of the conformal local net of von Neumann algebras on the circle $(\A,U)$. Then the strongly continuous projective representation $U^{(n)}_\pi$ of $\mob^{(n)}$, defined by Proposition \[n-cover\], is of positive energy for all positive integers $n$. In particular, the unique continuous inner implementation of the Möbius symmetry in the representation $\pi$ is of positive energy.
Carpi proved [@Carpi2 Prop. 2.1] that an irreducible representation of a Virasoro net $\A_{{\rm Vir},c}$ must be one of those that we get by integrating a positive energy unitary representation of the Virasoro algebra (corresponding to the same central charge) under the condition that the representation is of positive energy. Thus by the above theorem we may draw the following conclusion.
An irreducible representation of the local net $\A_{{\rm Vir},c}$ must be one of those that we get by integrating a positive energy unitary representation of the Virasoro algebra corresponding to the same central charge.
[**Acknowledgements.**]{} The author would like to thank Sebastiano Carpi and Roberto Longo for useful discussions, for finding some mistakes and for calling his attention to the need for more rigor at certain points (e.g. the need for considerations about separability in respect to the direct integral decomposition).
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[^1]: diffeomorphisms of $S^{1}$ of the form $z \mapsto \frac{az+b}{\overline{b}z+\overline{a}}$ with $a,b\in \CC$, $|a|^2-|b|^2=1$.
[^2]: The negative sign is “compulsory” if we want the “abstract” exponential — defined for Lie algebras of Lie groups — to be the same as the exponential of vector fields, i.e. the diffeomorphism which is the generated flow at time equal $1$.
|
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abstract: 'The axial modes for non-barotropic relativistic rotating neutron stars with uniform angular velocity are studied, using the slow-rotation formalism together with the low-frequency approximation, first investigated by Kojima. The time independent form of the equations leads to a singular eigenvalue problem, which admits a continuous spectrum. We show that for $l=2$, it is nevertheless also possible to find discrete mode solutions (the $r$-modes). However, under certain conditions related to the equation of state and the compactness of the stellar model, the eigenfrequency lies inside the continuous band and the associated velocity perturbation is divergent; hence these solutions have to be discarded as being unphysical. We corroborate our results by explicitly integrating the time dependent equations. For stellar models admitting a physical $r$-mode solution, it can indeed be excited by arbitrary initial data. For models admitting only an unphysical mode solution, the evolutions do not show any tendency to oscillate with the respective frequency. For higher values of $l$ it seems that in certain cases there are no mode solutions at all.'
author:
- |
Johannes Ruoff and Kostas D. Kokkotas\
Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54006, Greece
date: 'Accepted ???? Month ??. Received 2001 Month ??; in original form 2001 Month ??'
title: 'On the $r$-mode spectrum of relativistic stars in the low-frequency approximation'
---
== == == ==
\#1[[ \#1]{}]{} \#1[[ \#1]{}]{} @mathgroup@group @mathgroup@normal@group[eur]{}[m]{}[n]{} @mathgroup@bold@group[eur]{}[b]{}[n]{} @mathgroup@group @mathgroup@normal@group[msa]{}[m]{}[n]{} @mathgroup@bold@group[msa]{}[m]{}[n]{} =“019 =”016 =“040 =”336 ="33E == == == ==
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\[firstpage\]
relativity – methods: numerical – stars: neutron – stars: oscillations – stars: rotation
Introduction
============
Immediately after the discovery of the $r$-modes being generically unstable with respect to gravitational-wave emission (Andersson 1998a; Friedman & Morsink 1998), it was suggested that they may cause the newly born neutron stars to spin down via the emission of gravitational waves (Lindblom, Owen & Morsink 1998; Andersson, Kokkotas & Schutz 1999). Because of their surprisingly fast growth times, $r$-modes should be able to slow down a hot and rapidly spinning newly born neutron star during the first months of its existence. There is also work suggesting that the $r$-mode instability might be relevant for old neutron stars in binary systems. This potential relevance for astrophysics has attracted the interest of both the relativity and the astrophysical community on various aspects of this subject. For an exhaustive upto-date review, see for instance Andersson & Kokkotas (2001) and Friedman & Lockitch (2001).
Most of the recent work on $r$-modes is based on Newtonian calculations under the assumption of slow rotation, and the effects of gravitational radiation are incorporated through the quadrupole formula. However, it is clear that for a complete and quantitatively correct understanding, one has to use the framework of general relativity. Still the slow-rotation approximation is well justified, since the angular velocity of even the fastest spinning known pulsar corresponds to a rotational expansion parameter of $\veps =
\Omega/\sqrt{M/R^3}\approx 0.3$. The full set of equations in the slow-rotation limit was first given by Chandrasekhar & Ferrari (1991) for the axisymmetric case, and by Kojima (1992) for the general case.
In the non-rotating case, the perturbation equations are decoupled with respect to the harmonic index $l$ and degenerate with respect to the azimuthal index $m$. Furthermore the oscillation modes can be split into two independent sets, which are characterized by their behaviour under parity transformation. The [*polar*]{} (or [ *spheroidal*]{}) modes transform according to $(-1)^l$, whereas the [ *axial*]{} (or [*toroidal*]{}) modes according to $(-1)^{l+1}$. For a non-rotating perfect fluid star, the only possible fluid oscillations are the spheroidal $f$- and $p$-modes. For non-barotropic stars, i.e. stars with a temperature gradient or a composition gradient, there exists another family of modes, the $g$-modes, where the main restoring force is gravity. All axial perturbations of non-rotating perfect fluid stars have zero frequency, i.e. they represent stationary currents. In the non-barotropic case, this zero-frequency space consists only of the axial $r$-modes, while for barotropic stars [^1], it also includes the polar $g$-modes, since they require a temperature gradient for their existence.
As the star is set into rotation, the picture changes. In the slow rotation approximation, the $m$-degeneracy is removed and the polar modes with index $l$ are coupled to the axial modes with indices $l\pm
1$ and vice versa. Furthermore, the rotation imparts a finite frequency to the zero-frequency perturbations of the non-rotating stars. In non-barotropic stars, those modes, whose restoring force is the Coriolis force, all have axial parity. However, as has been first pointed out in the Newtonian framework by Lockitch & Friedman (1999), for barotropic stars the rotationally restored (inertial) modes are generically hybrids, whose limit in the non-rotating case are mixtures of axial and polar perturbations.
If one focuses only on the $r$-mode, whose frequency is proportional to the star’s angular velocity $\Omega$, one can order the perturbation variables in powers of $\Omega$. Kojima (1997, 1998) used this low-frequency approximation, sometimes also called slow-motion approximation (Schumaker & Thorne, 1983), to show that the purely axial modes of (non-barotropic) stellar models can be described by a single second-order ODE. This eigenvalue problem, however, proves to be singular, since it is possible for the highest derivate to vanish at some value of the radial coordinate inside the star. Kojima (1997,1998) then argued that this singular structure should give rise to a continuous spectrum. This has been put on a rigorous mathematical footing by Beyer & Kokkotas (1999). The appearance of a continuous spectrum can be explained as follows. It is well known that in the Newtonian limit, the eigenfrequency of the $r$-mode for an inertial observer is given by $$\label{freqNewt}
\sigma_N = -m\Omega{\left(}1 - \frac{2}{l(l+1)}{\right)}\;.$$ A first relativistic correction can be obtained by using the relativistic Cowling approximation, which consists in neglecting all metric perturbations. In this case, the only correction comes from the frame dragging $\omega$, which is a function of the radial coordinate $r$, thus leading to an $r$-dependent oscillation frequency of each fluid layer: $$\label{freqCowl}
\sigma_C(r) = -m\Omega\left[1 - \frac{2}{l(l+1)}
{\left(}1 - \frac{\omega(r)}{\Omega}{\right)}\right]\;.$$ Instead of a single frequency, there is now a continuous band of frequencies, whose boundaries are determined by the values of the frame dragging $\omega(r)$ at the centre and the surface of the star.
However, it has been pointed out (Beyer & Kokkotas 1999) that the existence of the continuous spectrum might be just an artefact of the too restricted low-frequency approximation. With the inclusion of gravitational radiation effects, the frequencies become complex-valued, thus potentially removing the singular structure of Kojima’s equation. But even in the case of real-valued frequencies, it has been recently shown by Lockitch, Andersson & Friedman (2001) that for a non-barotropic uniform density model, in addition to the continuous spectrum there also exists a single mode solution with the eigenfrequency lying outside the continuous band. It is this mode that represents the relativistic $r$-mode for non-barotropic stars.
In this paper, we extend the search of $r$-mode solutions to stars with various polytropic and realistic equations of state (EOS). We shall show that in addition to the continuous part, the eigenvalue equation always admits a single mode solution, at least for $l=2$. However, for some stellar models, depending on the polytropic index $n$ and on the compactness, the frequency of this solution lies inside the continuous band and is associated with a divergence in the fluid perturbation at the singular point. This is clearly not acceptable, and therefore we have to discard such solutions as being unphysical. As a logical consequence, we conclude that in those cases, there do not exist any $r$-modes, at least within the low-frequency approximation. In an independent work, Yoshida (2001) has come to a similar conclusion. He showed that even when studied in the post-Newtonian approximation, some polytropic models do not admit any $r$-modes.
For realistic EOS, the existence of $r$-modes depends on the average polytropic index of the high-density regime. For very stiff EOS, the neutron star models can exhibit $r$-modes throughout the complete physically acceptable mass range, whereas for the very soft EOS, none of the neutron star models does. For EOS in the intermediate range, the existence of $r$-modes depends on the compactness of the stellar model. In addition to the mode calculations, we also use the time depend form of the equations. For those cases where we can find a physical $r$-mode solution, the Fourier spectrum of the time evolution does indeed show a peak at the appropriate frequency, whereas for those cases where we only have the unphysical mode, it does not.
Mathematical formulation
========================
Assuming that the star is slowly rotating with a uniform angular velocity $\Omega$, we neglect all terms of order higher than $\Omega$. In this approximation, the star remains spherical, because the deformation due to centrifugal forces is of order $\Omega^2$. Thus, the metric can be written in the form $$ds^2_0 = -e^{2\nu} dt^2 + e^{2\lam} dr^2
+ r^2{\left(}d\theta^2 + \sin^2\theta d\phi^2{\right)}- 2\omega r^2\sin^2\theta dtd\phi\;,$$ where $\nu$, $\lam$ and the “frame dragging” $\omega$ are functions of the radial coordinate $r$ only. With the neutron star matter described by a perfect fluid with pressure $p$, energy density $\eps$, and four-velocity $${\left(}u^t, u^r, u^\theta, u^\phi{\right)}= {\left(}e^{-\nu}, 0, 0, \Omega e^{-\nu}{\right)}\;,$$ the Einstein equations, together with a one-parameter equation of state $p=p(\eps)$, yield the well-known TOV equations plus an extra equation for the function $\varpi$, defined as $$\varpi := \Omega - \omega\;.$$ To linear order, this equation is $$\label{drag}
\varpi'' - {\left(}4\pi re^{2\lam}(p + \eps) - \frac{4}{r}{\right)}\varpi'
- 16\pi e^{2\lam}{\left(}p + \eps{\right)}\varpi = 0\;.$$ In the exterior, it reduces to $$\varpi'' + \frac{4}{r}\varpi' = 0\;,$$ for which we have the solution (Hartle 1967) $$\label{oex}
\varpi = \Omega - \frac{2J}{r^3}\;,$$ with $J$ being the total angular momentum of the neutron star. Equation (\[drag\]) has to be integrated from the centre of the star to its surface $R$, where it has to match smoothly the exterior solution (\[oex\]). With the angular momentum given by (Hartle 1967, Glendenning 1997) $$J = \frac{8\pi}{3}\int_0^R{r^4e^{\lam-\nu}{\left(}p + \eps{\right)}\varpi}\,dr\;,$$ the matching condition becomes $$R^4\varpi'(R) = 6J\;,$$ and with Eq. (\[oex\]) $$\label{mcond}
\varpi'(R) = \frac{3}{R}(\Omega - \varpi(R))\;.$$
If we focus on pure axial perturbations, the perturbed metric can be written in the following form: $$\label{metric}
ds^2 = ds^2_0 + 2\sum_{l,m}{\left(}h_0^{lm}(t,r)dt + h_1^{lm}(t,r)dr{\right)}{\left(}-\frac{\d_\phi Y_{lm}}{\sin\theta}d\theta
+ \sin\theta\d_\theta Y_{lm}d\phi{\right)}\;,$$ where $Y_{lm} = Y_{lm}(\theta,\phi)$ denote the scalar spherical harmonics. In addition, the axial component of the fluid velocity perturbation can be expanded as $$\label{fluid}
4\pi(p + \eps){\left(}\delta u^\theta, \delta u^\phi{\right)}= e^{\nu}\sum_{l,m}U^{lm}(t,r)
{\left(}-\frac{\d_\phi Y_{lm}}{\sin\theta}, \sin\theta\d_\theta Y_{lm}{\right)}\;.$$ Einstein’s field equations then reduce to four equations for the three variables $h_0^{lm}$, $h_1^{lm}$ and $U^{lm}$ (Kojima 1992).
As an alternative, we can use the ADM-formalism (Arnowitt, Deser & Misner 1962) to derive the evolution equations for the axial perturbations. The usefulness of this formalism for the perturbation equations of non-rotating neutron stars has been showed in Ruoff (2001). This formalism can be taken over to rotating stars and we can deduce equations describing the evolution of purely axial oscillations of slowly rotating neutron stars in terms of metric and extrinsic curvature variables. We should mention that our derivation starts with the complete set of perturbations, including both polar and axial perturbations. Only at the end do we neglect the coupling between the two parities and focus only on the axial equations. In the Regge-Wheeler gauge, there are just 2 non-vanishing axial metric perturbations and 2 axial extrinsic curvature components. In the notation of Ruoff (2001), they metric components are given by $$\begin{aligned}
{\left(}\beta_\theta,\,\beta_\phi{\right)}&=& e^{\nu-\lam}\sum_{l,m}K_6^{lm}
{\left(}-\frac{\d_\phi Y_{lm}}{\sin\theta}, \sin\theta\d_\theta Y_{lm}{\right)}\;,\\
{\left(}h_{r\theta},\,h_{r\phi}{\right)}&=& e^{\lam-\nu}\sum_{l,m}V_4^{lm}
{\left(}-\frac{\d_\phi Y_{lm}}{\sin\theta}, \sin\theta\d_\theta Y_{lm}{\right)}\;,\end{aligned}$$ and the extrinsic curvature components read $$\begin{aligned}
{\left(}k_{r\theta},\,k_{r\phi}{\right)}&=& \half e^{\lam}\sum_{l,m}K_3^{lm}
{\left(}-\frac{\d_\phi Y_{lm}}{\sin\theta}, \sin\theta\d_\theta Y_{lm}{\right)}\;,\\
{\left(}\begin{array}{cc}
k_{\theta\theta} & k_{\theta\phi}\\
k_{\phi\theta} & k_{\phi\phi}\end{array}{\right)}&=&
\half e^{-\lam}\sum_{l,m}K_6^{lm}\sin\theta
{\left(}\begin{array}{cc}
-\sin\theta^{-2}X_{lm} & W_{lm}\\
W_{lm} & X_{lm}\end{array}{\right)}\;,\end{aligned}$$ where $W_{lm}$ and $X_{lm}$ are abbreviations for $$\begin{aligned}
W_{lm} &=& {\left(}\d^2_\theta + l(l+1){\right)}Y_{lm}\;,\\
X_{lm} &=& 2{\left(}\d_\theta - \cot\theta{\right)}\d_\phi Y_{lm}\;.\end{aligned}$$ For the fluid velocity perturbation, we use the covariant form $${\left(}\delta u_\theta, \delta u_\phi{\right)}= e^{-\lam}\sum_{l,m}u_3^{lm}(t,r)
{\left(}-\frac{\d_\phi Y_{lm}}{\sin\theta}, \sin\theta\d_\theta Y_{lm}{\right)}\;.$$ The relation to the expansions (\[metric\]) and (\[fluid\]) is given by (from now on we omit the indices $l$ and $m$): $$\begin{aligned}
h_0 &=& e^{\nu-\lam} K_6\;,\\
h_1 &=& e^{\lam-\nu} V_4\;,\\
U &=& 4\pi e^{-\lam-\nu}(p + \eps){\left(}u_3 - K_6{\right)}\;.\end{aligned}$$ We obtain the following quite simple set of evolution equations for the variables $V_4$, $K_3$, $K_6$ and $u_3$: $$\begin{aligned}
\label{V4}
{\left(}{ \frac{\partial {}}{\partial {t}} } + {\mbox{\rm i}}m\omega{\right)}V_4 &=& e^{2\nu-2\lam}
\left[K_6' + {\left(}\nu' - \lam' - \frac{2}{r}{\right)}K_6 - e^{2\lam}K_3\right]\;,\\
\label{K3}
{\left(}{ \frac{\partial {}}{\partial {t}} } + {\mbox{\rm i}}m\omega{\right)}K_3 &=& \frac{l(l+1) -2}{r^2}V_4
+ \frac{2{\mbox{\rm i}}m}{l(l+1)}\omega'e^{-2\lam}K_6\;,\\
\label{K6}
{\left(}{ \frac{\partial {}}{\partial {t}} } + {\mbox{\rm i}}m\omega{\right)}K_6 &=& V_4' - \frac{{\mbox{\rm i}}mr^2}{l(l+1)}
\left[\omega'K_3 - 16\pi(\Omega - \omega)(p + \eps)u_3\right]\;,\\
\label{u3}
{\left(}{ \frac{\partial {}}{\partial {t}} } + {\mbox{\rm i}}m\Omega{\right)}u_3 &=& \frac{2{\mbox{\rm i}}m(\Omega-\omega)}{l(l+1)}
{\left(}u_3 - K_6{\right)}\;.\end{aligned}$$ Furthermore, we have one momentum constraint: $$\label{MC_odd}
16\pi(p + \eps)u_3 = K_3' + \frac{2}{r}K_3
- \frac{l(l+1) - 2}{r^2}K_6
- \frac{2{\mbox{\rm i}}m\omega'}{l(l+1)}e^{-2\nu}V_4\;.$$ These equations are completely equivalent to the axial parts of Eqs. (20), (24), (25) and (27) of Kojima (1992) when the coupling to the polar equations is neglected.
Low-frequency approximation
===========================
The above evolution equations should not only describe the $r$-modes but also the axial $w$-modes, which have much higher oscillation frequencies. If we want to focus on the $r$-modes only, we can use the fact that from Eq. (\[freqNewt\]), it follows that the $r$-mode frequency $\sigma$ is proportional to the star’s angular velocity $\Omega$. Hence, we can require that in our evolution equations (\[V4\]) – (\[u3\]), the time derivative $\d_t$ be proportional to the $r$-mode frequency $\sigma$ or, equivalently, to $\Omega$ (Kojima 1997, 1998). In this case, we can order the perturbation variables in powers of $\Omega$ (Lockitch et al. 2001) as $$\begin{aligned}
\label{eq:order}
u_3, K_3, K_6 &\sim& O(1)\;,\non\\
V_4 &\sim& O(\Omega)\;.\end{aligned}$$ Keeping only the lowest order terms, we can neglect terms proportional to $V_4$ in the evolution equation (\[V4\]) and in the constraint (\[MC\_odd\]), which then read $$\begin{aligned}
K_6' + {\left(}\nu' - \lam' - \frac{2}{r}{\right)}K_6 - e^{2\lam}K_3 &=& 0\;,\\
K_3' + \frac{2}{r}K_3 - \frac{l(l+1) - 2}{r^2}K_6 &=& 16\pi(p + \eps)u_3\;.\end{aligned}$$ These can be easily combined to give a single second-order differential equation for $K_6$. However, it is more convenient to write this equation in terms of the variable $h_0 = e^{\nu-\lam} K_6$: $$\label{eqh0}
e^{-2\lam} h_0''-4\pi r(p + \eps)h_0' + \left[8\pi(p + \eps)
+ \frac{4M}{r^3} - \frac{l(l+1)}{r^2}\right]h_0
= 16\pi e^{\nu-\lam}(p + \eps)u_3\;,$$ together with the evolution equation for $u_3$ $$\label{equ3}
{ \frac{\partial {}}{\partial {t}} }u_3 = -{\mbox{\rm i}}m\left[\Omega u_3
+ \frac{2\varpi}{l(l+1)}{\left(}e^{\lam-\nu}h_0 - u_3{\right)}\right]\;.$$ At this point, it is worth making some comments on this approximation. The full set of axial equations (\[V4\]) – (\[u3\]) is a hyperbolic system describing the propagation of gravitational waves, which are excited on the one hand by the fluid motion ($r$-modes) and on the other hand by the curvature of spacetime itself ($w$-modes). With the above approximation, we have completely suppressed the wave propagation, and the resulting equations now correspond to a Newtonian-like picture, where the fluid oscillations are acting as a source in the equation of the gravitational field. As now being described by a Poisson-like elliptic equation, implying an infinite propagation speed, the gravitational field $h_0$ reacts instantaneously on any changes in the source $u_3$. Of course, this picture is only an analogy, since the metric variable $h_0$ corresponds to a post-Newtonian correction of the gravitational field and vanishes completely in the Newtonian limit.
Furthermore our derivation of this approximation is only valid for non-barotropic stars. This is because in general we cannot start from decoupling the polar and axial equations in the first step as we did. Instead, when we apply the low-frequency approximation, we actually have to start from the full coupled system of equations, including both polar and axial perturbations. If we then do the same ordering in powers of $\Omega$, we also would have some polar variables of order $O(1)$, namely the remaining two fluid velocity components, coming from $\delta u_r$ and the polar part of the angular components $(\delta u_\theta, \delta u_\phi)$, and the $(rt)$ component of the metric, usually denoted by $H_1$. It turns out that the polar constraint equations can be combined to give a single constraint for $H_1$, which can be reduced to $$\label{H1_const}
{\left(}\Gamma - \Gamma_1{\right)}H_1 = 0\;,$$ with $$\Gamma = \frac{p + \eps}{p}\frac{dp}{d\eps}$$ the adiabatic index corresponding to the unperturbed configuration and $\Gamma_1$ the adiabatic index of the perturbed configuration, which in general differs from $\Gamma$. This is the case for non-barotropic stellar models, and Eq. (\[H1\_const\]) can only be satisfied if $H_1$ vanishes. But this automatically implies that the polar fluid perturbations vanish, too, leaving thus only the axial equations, given above. In the barotropic case, it is $\Gamma = \Gamma_1$ and the constraint for $H_1$ is trivially satisfied, even for nonzero $H_1$. But this has as a consequence that the coupling between the polar and axial mode cannot be neglected, which means that there cannot exist pure axial mode solutions, since any kind of pure axial initial data will through the coupling automatically induce polar fluid oscillations. Hence, our analysis is strictly valid only for non-barotropic stellar models.
As a further approximation, we could completely neglect all the metric perturbations. With this so-called relativistic Cowling approximation, we would be left with a single evolution equation for the fluid variable $u_3$: $${ \frac{\partial {}}{\partial {t}} }u_3 = -im{\left(}\Omega - {\frac{2\varpi}{l(l+1)}}{\right)}u_3\;.$$ From this equation we can immediately deduce that the various fluid layers are decoupled from each other, which means that each layer has its own real oscillation frequency given by $$\label{freqCowl2}
\sigma = -m{\left(}\Omega - \frac{2\varpi}{l(l+1)}{\right)}\;.$$ In the Newtonian limit $(\varpi \rightarrow \Omega)$, this reduces to the well known result for the frequency of the $r$-mode given in Eq. (\[freqNewt\]). It should be pointed out that in the relativistic case, the presence of the frame dragging $\omega$ destroys the occurrence of a single mode frequency and gives rise to a continuous spectrum, at least to this order of approximation. Of course, it has been argued that this might be a pure artefact of the approximation, and the continuous spectrum may disappear as soon as certain approximations are relaxed.
Let us therefore return to the low-frequency approximation, which is less restricted than the Cowling approximation, and see whether or not we can find real mode solutions. To this end, we assume our variables to have a harmonic time dependence $$\begin{aligned}
u_3(t,r) = u_3(r)e^{-i\sigma t}\;,\\
h_0(t,r) = h_0(r)e^{-i\sigma t}\;.\end{aligned}$$ Note that for the sake of notational simplicity, we do not explicitely distinguish between the time dependent and time independent form of the variables. With this ansatz, we assume the $r$-mode frequency $\sigma$ to be positive for positive values of $m$ in contrast to the definitions in Eqs. (\[freqCowl2\]) and (\[freqNewt\]). From Eq. (\[equ3\]) we find that $$\label{u3alg}
u_3 = \frac{2m\varpi}
{2m\varpi + l(l+1)(\sigma - m\Omega)}e^{\lam-\nu} h_0\;,$$ which can be used to eliminate $ u_3$ in Eq. (\[eqh0\]), yielding $$\label{eveqn}
{\left(}\sigma - m\Omega + \frac{2m\varpi}{l(l+1)}{\right)}\left[e^{-2\lam} h_0''-4\pi r(p + \eps)h_0'
-{\left(}8\pi(p + \eps) - \frac{4M}{r^3}
+ \frac{l(l+1)}{r^2}{\right)}h_0\right]
+ 16\pi(p + \eps)(\sigma - m\Omega)h_0 = 0\;.$$ With appropriate boundary conditions, this equation represents an eigenvalue problem which should yield one, or possibly several, distinct eigenmodes. However, as was at first pointed out by Kojima (1997,1998), it might occur that the denominator in Eq. (\[u3alg\]) can become zero at some point inside the star, and the resulting eigenvalue problem becomes singular at this point. If the zero of the denominator lies outside the star, the eigenvalue problem is regular, since outside $u_3 = 0$ and Eq. (\[eqh0\]) can be directly solved without using Eq. (\[u3alg\]). The zeroes of the denominator occur if the frequency $\sigma$ lies in an interval determined by the values of $\varpi$ at the centre and the surface, which we denote by $\varpi_c$ and $\varpi_s$, respectively: $$\label{range}
m\Omega{\left(}1 - \frac{2\varpi_s}{\Omega l(l+1)}{\right)}< \sigma
< m\Omega{\left(}1 - \frac{2\varpi_c}{\Omega l(l+1)}{\right)}\;.$$ By comparison with similar results from fluid dynamics, Kojima concluded that this equation should have a continuous spectrum with the frequency range given by (\[range\]). This was put on a rigorous mathematical footing by Beyer & Kokkotas (1999). However, they could not exclude that there might not exist additional isolated eigenvalues, which would correspond to true mode solutions.
We will now show that there actually exist such solutions, even though in some cases they are unphysical since the corresponding fluid perturbations would be divergent at the singular point. To make things look simpler, we can rescale Eq. (\[eveqn\]) and make it independent of $\Omega$ and $m$. Following Lockitch et al. (2001), we introduce a normalized frequency $$\alpha = \half l(l+1){\left(}1 - \frac{\sigma}{m\Omega}{\right)}$$ and rewrite Eq. (\[eveqn\]) as $$\label{eveqn_a}
{\left(}\alpha - \hat\varpi{\right)}\left[e^{-2\lam} h_0''
-4\pi r(p + \eps)h_0' -{\left(}8\pi(p + \eps) - \frac{4M}{r^3}
+ \frac{l(l+1)}{r^2}{\right)}h_0\right] + 16\pi(p + \eps)\alpha h_0 = 0\;,$$ where $$\hat\varpi := \varpi/\Omega\;.$$ Equation (\[u3alg\]) then reads $$\label{u3alg_a}
u_3 = \frac{\hat\varpi}
{\hat\varpi - \alpha}e^{\lam-\nu} h_0\;.$$ Equation (\[eveqn\_a\]) becomes singular if $ \alpha$ lies in the interval limited by the values of $\hat\varpi$ at the centre and at the surface of the star, i.e. if $$\hat\varpi_c < \alpha < \hat\varpi_s\;.$$ For a solution to be acceptable, it must be regular at the origin, which amounts to $h_0(0) = 0$, and it must vanish at infinity. As already mentioned above the integration of Eq. (\[eveqn\]) is straightforward, if the singular point lies outside the star. It is only when the singular point lies inside the star that some care has to be taken.
Let us now assume the singular point $r = r_0$ lie inside the star. An analysis of Eq. (\[eveqn\]) (Andersson 1998b) shows that the singular point is a regular singularity, which admits a Frobenius expansion of the form $$\begin{aligned}
\label{expan}
h_0(r - r_0) &=& A {\left(}a_1 (r - r_0) + a_2 (r - r_0)^2 + \ldots {\right)}+ \non\\
&& B \left[{\left(}a_1 (r - r_0) + a_2 (r - r_0)^2 + \ldots{\right)}\ln|r - r_0|
+ b_0 + b_1(r - r_0) + b_2(r - r_0)^2 + \ldots \right]\;.\end{aligned}$$ Even though the solution is finite and smooth at the singular point $r
= r_0$, its derivative diverges, because of the logarithmic term. Moreover, if we want to compute the associated velocity component $
u_3$, we find that unless $h_0(r = r_0) = 0$, it will blow up at the singular point $r = r_0$. Therefore, we conclude that the coefficient $B$ has to vanish in order to obtain a physical solution and we are only left with the first power series starting with the linear term $a_1(r - r_0)$. This yields vanishing $h_0$ at $r = r_0$ and therefore $u_3$ can be finite at this point. The question is whether there are solutions satisfying both $h_0(r = r_0) = 0$ and the appropriate boundary conditions at the centre and at infinity. We will now show that this cannot be the case.
Suppose that $h_0(r = r_0) = 0$ and $h_0'(r = r_0) > 0$. We can then integrate Eq. (\[eveqn\]) from $r_0$ to $r > r_0$: $$h_0'(r) = h_0'(r_0) + \int_{r_0}^r{e^{2\lam}\left[4\pi r(p + \eps)h_0'
+{\left(}8\pi(p + \eps) - \frac{4M}{r^3} + \frac{l(l+1)}{r^2}{\right)}h_0
+ 16\pi(p + \eps)\frac{\alpha}{\hat\varpi - \alpha} h_0\right]dr},$$ Since it is $\varpi - \alpha > 0$ for $r > r_0$, each coefficient in the integral is strictly positive, hence we will have $h_0'(r) > 0$ for all $r > r_0$; i.e. $h_0$ keeps increasing as $r \rightarrow
\infty$, which is clearly incompatible with our requirement that $h_0$ vanish at infinity. Of course, the same argument holds if $h_0' < 0$ at $r = r_0$, with $h_0$ keeping decreasing. Hence, it follows that $h_0(r = r_0) \ne 0$, but this means that we cannot have a vanishing coefficient $B$. Therefore, our solution will always contain the logarithmic term, which means that the associated velocity perturbation $u_3$ is divergent at this point. This is clearly unphysical. We thus conclude that it is in principle possible to find eigensolutions to Eq. (\[eveqn\_a\]), however, if the associated eigenfrequencies lie inside the continuous band, the solutions become singular and have to be discarded on physical grounds.
It has been shown by Lockitch et al. (2001) that for the existence of mode solutions, the allowed range of the eigenvalues $\alpha$ is bounded from below by $\hat\varpi_c$: $$\label{bound1}
\hat\varpi_c \le \alpha \le 1\;.$$ However, based on our above argumentation, we can further restrict this interval for the physically allowed eigenmodes to have as lower limit the value of $\hat\varpi_s$: $$\label{bound2}
\hat\varpi_s < \alpha \le 1\;.$$
Numerical results
=================
The numerical integration of Eq. (\[eveqn\_a\]) can be easily accomplished if the singular point lies outside the star, since in the exterior $u_3 = 0$ and we can therefore use the non-singular Eq. (\[eqh0\]) for the integration toward the outer boundary. If the singular point lies inside the star, we initiate our integration with a regular solution at the origin and integrate outward close to the singular point, where we match the solution to the expansion (\[expan\]), i.e. we compute the leading coefficients $b_0$ and $a_1$. This gives us the new starting values to the right of the singular point and we can continue the integration up to a finite point outside the star, where we test if the solution satisfies the correct boundary condition. We mention again that for the integration outside the star we take the non-singular Eq. (\[eqh0\]), with $u_3$ set to zero.
We have performed mode calculations for sequences of uniform density and polytropic stars. For the uniform density models, the eigenfrequency $\alpha$ always lies outside the range of the continuous spectrum and therefore the associated eigenfunctions do not exhibit any singularities. In Fig. \[fig:em\_cd\], we show the normalized eigenvalues $\alpha$ for $l = 2$ as a function of the compactness $M/R$ together with $\hat\varpi_c$ and $\hat\varpi_s$, marking the boundaries of the continuous spectrum. For larger $l$ (not shown), the eigenvalues $\alpha$ decrease and converge to $\hat\varpi_s$, but stay always above $\hat\varpi_s$. [^2] In Fig. \[fig:ef\_cd\], we show the eigenfunctions $h_0$ and $u_3$ for a uniform density model with a compactness of $M/R = 0.153$ and corresponding mode frequency $\alpha = 0.89806$. Close to the centre of the star, the fluid perturbation $u_3$ is proportional to $r^{l+1}$, but as it approaches the stellar surface, it grows much stronger, which comes from the denominator in Eq. (\[u3alg\_a\]) becoming very small.
Polytropic models
-----------------
For polytropic models, obeying an equation of state of the form $$p = \kappa\eps^{1 + 1/n}$$ with polytropic index $n$, we obtain a quite different picture. For a polytropic index $n = 1$, as it is for instance shown in Fig. \[fig:em\_poly\], it is only for the less compact stellar models that the eigenfrequencies $\alpha$ lie outside the continuous spectrum and therefore represent physical mode solutions. However, they are already that close to the upper boundary of the continuous spectrum $\hat\varpi_s$ that in Fig. \[fig:em\_poly\] they cannot be distinguished any more. For more compact models, the eigenfrequency eventually moves inside the domain of the continuous spectrum, which means that the singular point now lies inside the star. This happens for a compactness of about $M/R \approx 0.085$. As discussed above, at the singular point the mode solution for $h_0$ exhibits an infinite slope and the associated fluid perturbation $u_3$ diverges. Therefore, we have to discard them as being unphysical mode solutions. In Table \[models\], we have listed some polytropic models with their physical parameters and the eigenvalues $\alpha$ for $l=2$ and $l=3$. The frequencies which are marked by an asterisk lie inside the continuous band and therefore correspond to unphysical mode solutions. For $l=2$, only models 1 and 2 permit physical modes, whereas for $l=3$, the modes are unphysical for all the stellar models. We should also mention that all our values are in perfect agreement with those previously obtained by Andersson (1998b).
To assess how the existence of a physical mode solution depends on the polytropic index $n$, we have computed modes for stellar models with fixed compactness $M/R$ but with different values of $n$, ranging from 0 to 1.5. The results are depicted in Fig. \[fig:index\], where we show $\alpha$ as a function of $n$ for $l=2$. For small values of $n$, i.e. for stiff equations of state, the mode eigenfrequency lies outside the range of the continuous spectrum. But as $n$ is increased, what corresponds to softening the equation of state, the mode frequency eventually crosses the boundary and migrates inside the continuous spectrum. This happens at $n \approx 0.8$, but for larger values of $l$, the transition point moves to smaller values of $n$. Actually, it is not the mode frequency $\alpha$ which moves towards the boundary of the continuous spectrum $\hat\varpi_s$ as $n$ is increased, it is rather the boundaries of the continuous spectrum which start to expand, and $\hat\varpi_s$ approaches the mode frequency $\alpha$, which more or less hovers at a constant value. For $n = 0$, the uniform density models, the range of the continuous spectrum (the shaded area in Fig. \[fig:index\]) is the smallest, and probably it is only in this case that one can find eigenvalues for quite large $l$, if not for all $l$. We should also mention that for each polytropic model, there seems to exist a maximal value of $l$, beyond which there are neither physical nor unphysical mode solutions. The frequency $\alpha$ of the unphysical mode solution quickly approaches $\hat\varpi_c$ as $l$ is continuously increased. For $l$ larger than the critical value, where $\alpha(l) = \hat\varpi_c$, we could not find any mode solution at all. For the $n=1$ models of Table 1, this happens already for $l=4$.
To check and corroborate our above results, we also numerically evolved the time dependent equations (\[eqh0\]) and (\[equ3\]) and took Fourier transforms of the resulting evolution. For initial data representing the physical mode solution of Fig. \[fig:ef\_cd\], the time evolution indeed gives a single frequency signal at each point inside and outside the star. In this case, there is no sign of a continuous spectrum at all, and all the fluid layers oscillate in a uniform manner. This is shown in Fig. \[fig:ps\_cd\_mode\], where for both the fluid variable $u_3$ (left panel) and the metric variable $h_0$ (right panel), there is one single peak, which is independent of the location $r$.
If we now choose arbitrary initial data, as for instance in Fig. \[fig:arb\_id\], we expect the power spectrum at a given location $r$ to consist of two peaks: One which is independent of the location inside the star and represents the eigenmode, and another peak which varies between the boundaries determined by $\hat\varpi_c$ and $\hat\varpi_s$ as one moves throughout the star. This is how the continuous spectrum should show up in the Fourier transform. For the fluid variable $u_3$, the power spectrum of the evolution indeed confirms our expectations, as is shown in the left panel of Fig. \[fig:ps\_cd\_arb\].
However, the spectra of $h_0$ show that for locations closer to the stellar surface, the peaks corresponding to the continuous spectrum are smaller by several orders of magnitudes compared to the peak representing the eigenmode. For $u_3$, the peaks are of the same order of magnitude. And outside the star, $h_0$ shows only the eigenmode peak, and no sign of the presence of the continuous spectrum, which should reveal itself as a superposition of all the frequencies in the range between $\hat\varpi_c$ and $\hat\varpi_s$. In therefore seems to be invisible for an external observer. We should note that those spectra are taken after a certain initial time, in which the system adjusts itself. If we had taken the Fourier transform right from $t=0$, we would have obtained a clear sign of the continuous spectrum.
Let us now turn or attention to the polytropic cases, where we can have unphysical mode solutions. We will present evolution runs for the stellar models 1 and 5 from Table \[models\] with $l=2$. For model 1, the singular point lies outside the stellar surface and therefore there exists a physical mode solution. For model 5, the singular point lies inside the star, hence $\alpha < \hat\varpi_s$ and the associated eigensolution is unphysical. It should be noted that this model is also unstable with respect to radial collapse.
[ccccccccc]{}
------------------------------------------------------------------------
Model & $\eps_0\;[$g/cm$^3]$ & $M\;[M_\odot]$ & $R\;[$km$]$ & $M/R$ & $\hat\varpi_c$ & $\hat\varpi_s$ & $\alpha(l=2)$ & $\alpha(l=3)$\
------------------------------------------------------------------------
1 & $1.0\times 10^{14}$ & 0.120 & 12.32 & 0.014 & 0.96168 & 0.99237 & 0.99254 & 0.98523$^*$\
2 & $5.0\times 10^{14}$ & 0.495 & 11.58 & 0.063 & 0.83048 & 0.96431 & 0.96453 & 0.92446$^*$\
3 & $1.0\times 10^{15}$ & 0.802 & 10.81 & 0.109 & 0.70420 & 0.93407 & 0.93362$^*$ & 0.84895$^*$\
4 & $5.0\times 10^{15}$ & 1.348 & 7.787 & 0.256 & 0.28377 & 0.80236 & 0.72579$^*$ & 0.44782$^*$\
5 & $1.0\times 10^{16}$ & 1.300 & 6.466 & 0.297 & 0.14214 & 0.74960 & 0.52932$^*$ & 0.25301$^*$\
For model 1, the physical mode solution can be used as initial data. As for the uniform density case, the numerical evolution of such data yields a purely sinusoidal oscillation with the expected $r$-mode frequency $\alpha$. Therefore, the corresponding power spectrum is similar to Fig. \[fig:ps\_cd\_mode\]. For arbitrary initial data, we obtain a picture similar to Fig. \[fig:ps\_cd\_arb\]. Note, that the values of $\alpha$ and $\hat\varpi_s$ differ only by about 0.01 per cent. Still, with a high resolution run we can numerically distinguish these values, as is shown in Fig. \[fig:ps\_poly\_1e14\]. Here, we plot the power spectra of $h_0(t)$ and $u_3(t)$ taken at the stellar surface. The spectrum of $h_0$ shows one single peak at the eigenfrequency $\alpha$, whereas $u_3$ shows two peaks at $\alpha$ and $\hat\varpi_s$.
For model 5, things are quite different. Here, we cannot evolve initial data representing the unphysical mode solutions because the fluid perturbation would diverge at the singular point. Yet, if this solution still had some physical relevance, then arbitrary initial data should be able to excite this mode, and the power spectrum of the time evolution should show a peak at the corresponding frequency. However, this is clearly not the case, as can be seen in Fig. \[fig:ps\_poly\_1e16\], where we show the late time power spectra of the time evolution of $u_3$ (left panel) and $h_0$ (right panel) for model 5. For the fluid variable $u_3$, there is always one single peak, which varies for different locations $r$ between the boundaries $\hat\varpi_c$ and $\hat\varpi_s$. There is not even the slightest trace of a common peak at the expected value of $\alpha = 0.52932$.
For the metric variable $h_0$ (right panel), we essentially observe the same picture. Here, too, no common peak can be found at the expected mode frequency $\alpha$, but curiously there is nevertheless an additional common peak for all locations with its frequency given exactly by $\hat\varpi_s$. However, this peak does not show up in the power spectrum of $u_3$, except, of course, directly at the surface.
It is obvious that it cannot be a mode solution, since first of all the eigenvalue code does not give a solution for this particular frequency $\hat\varpi_s$ or even in the close vicinity. Moreover, the time evolution shows a quite different behaviour compared to the case where a physical $r$-mode exists. In Fig. \[fig:time\_evol\], we plot the time evolutions of $h_0$ outside the star for models 1 and 5 of Table \[models\]. For model 1, where we have a physical $r$-mode, after some initial time the amplitude remains constant, whereas for model 5 the amplitude keeps decreasing with time and in this case fits very well a power law with an exponent of -2. For model 1, the dominant oscillation frequency is the corresponding $r$-mode frequency $\alpha$, whereas for model 5 it is given by $\hat\varpi_s$. In both cases, the amplitude of the fluid perturbation $u_3$ remains constant after some initial time. It now becomes clear, why we cannot observe the common peak at $\hat\varpi_s$ in the fluid spectrum. The spectra are taken at late times, where the amplitude of $h_0$ and therewith its influence on $u_3$ has considerably decreased. If we had taken the spectra at earlier times, we could observe a similar peak in the fluid spectrum, as well.
We have no clear explanation what causes this additional peak, but we suppose that it comes from the behaviour of the energy density $\eps$ at the surface. The peak is much more pronounced for polytropes with $n < 1$, since there the energy density $\eps$ has an infinite slope at the surface. For $n=1$, the slope $\eps'$ is finite and for $n >
1$, it is zero. In the latter case, the peak is strongly suppressed. Even for uniform density models, one can observe this additional peak, arising because of the discontinuity of the energy density at the surface. However, this peak is several orders of magnitudes smaller than the peak corresponding to the eigenmode, which is always present for uniform density models, and therefore hard to detect.
It is instructive to compare the evolution of the same initial fluid perturbation for a uniform density model and a polytropic model, having the same compactness but without the latter admitting a mode solution. Since in the low-frequency approximation, there is no radiation which can dissipate the energy of the fluid, the total energy of the system should be conserved. However, we have observed that in the polytropic case the amplitude of $h_0$ is constantly decreasing, hence its initial energy has to be transferred to the fluid, whereas in the uniform density model, the energy should be shared between $u_3$ and $h_0$. This is indeed, what can be observed. In the uniform density case, the fluid amplitude does not change too much, but in the polytropic case, it shows a quite strong initial growth, accompanied by the strong decrease of $h_0$.
Realistic Equations of State
----------------------------
Having found that for a quite large range of polytropic stellar models, there do not exist any physical $r$-modes, an obvious question is, whether or not realistic equations of state do admit physical mode solutions. To give an answer, we have investigated the collection of realistic equations of state which have been studied by Kokkotas & Ruoff (2001) for the radial modes. The relevant notations, references and data of the stellar models can be found in there.
The results are quite unexpected and seemingly contradictory. When trying to compute the modes through Eq. (\[eveqn\_a\]), we find that for all the equations of state, the frequencies always lie inside the continuous band. Based on our above discussion, we therefore would have to discard them as being unphysical. It then would seem that none of the existing realistic EOS admits an $r$-mode, at least in the physically relevant range from about one solar mass up to the stability limit of each EOS. The surprise is now that the time evolution does show a quite different picture. Only for the EOS B (Pandharipande, 1971), G (Canuto & Chitre, 1974) and MPA (Wu et al., 1991) do the evolutions meet our expectations and show the continuous decrease in the amplitude of the metric variable $h_0$, in accordance with the polytropic cases without $r$-modes. However, for all other EOS, the amplitude remains constant after a while, indicating that there is indeed a mode present. Only when approaching their respective stability limit do some EOS show a decay of the amplitude of $h_0$. When obtaining the frequency through Fourier transformation, we find that it always lies [*inside*]{} the continuous spectrum, however, it does [*not*]{} coincide with the frequency found from the mode calculation. Instead, the frequency is in all cases given by the value of $\hat\varpi$ close to the neutron drip point.
How can we explain this discrepancy with our previous considerations? First, we would like to stress that it is not a numerical artefact of the time evolution, for convergence tests corroborate the presence of this mode. When examining the different EOS, we find that the EOS B, G and MPA are the softest ones, with a maximal polytropic index in the high density regime around $n=0.8$. All others have indices less than 0.8, going down to $n \approx 0.5$ for the EOS I (Cohen et al. 1970) and L (Pandharipande et al. 1976). From Fig. \[fig:index\], it becomes clear that it is just for polytropic models with $n \ge 0.8$ that the eigenvalue migrates inside the continuous band, and the $r$-mode therefore ceases to exist. For models with $n < 0.8$, we usually can find a physical $r$-mode, but this depends on the compactness of the model under consideration.
At a density of $10^{14}$g/cm$^3$, the effective polytropic index of any EOS is given by $n \approx 2$. As the EOS is approaching the neutron drip point at a density of $\eps \approx 4\times
10^{11}$g/cm$^3$, the EOS becomes softer and softer, i.e. $n$ increases even further. Only for densities below the neutron drip point does the EOS stiffen again. This structure of the EOS is responsible of putting a low density layer (the crust) around the high density core. However, because of its low density compared to the core, this layer practically does not contribute to the total mass, its only effect is to slightly increase the radius of the neutron star. Thus, whether or not the EOS admits a mode, should be determined solely by the core. To assess this proposition, we can do the following. We replace the whole part of the EOS below $10^{14}$g/cm$^3$ by a smooth polytropic EOS with polytropic index of $n = 2$. By doing so, we obtain a model with practically the same mass, but a somewhat smaller radius. Computing the modes of the thus modified model, we find that the $r$-mode frequencies $\alpha$ actually do lie outside the continuous spectrum, if the average polytropic index of the core is less than 0.8, and the model is not too compact. However, $\alpha$ lies extremely close to the value of $\hat\varpi_s$. For softer EOS with an average index of $n \approx
0.8$, we still would not be able to find any physical modes. If we now go back and restore the outer layer, the $r$-mode frequency should not significantly change, because of the negligible gravitational influence of this outer layer. The only effect is the slight increase of the stellar radius $R$. But with $R$ becoming larger, the value of $\hat\varpi_s$ also increases and actually becomes larger than the $r$-mode frequency $\alpha$, which then lies inside the continuous band. This is shown in Fig. \[fig:real\_poly\], where we plot the two density profiles for a stellar model based on the EOS WFF (Wiringa, Ficks & Fabrocini, 1988) and the same model with the low-density regime replaced by a $n=2$ polytropic fit. For the polytropic fit, the zero of $\hat\varpi - \alpha$ lies right outside the star, whereas for the complete realistic model, it is inside the star. In the latter case, however, the mode still exists, but it cannot have a purely harmonic time dependence any more. If this were the case, it had to be a physical solution of Eq. (\[eveqn\_a\]), but it is clearly not since the frequency lies inside the continuous spectrum. We therefore conclude that this stable oscillation that can be seen in the time evolution is always a mixture of a mode and the continuous spectrum. We should mention that in our treatment of realistic EOS, we have assumed that the neutron stars consists entirely of a perfect fluid, even in the outer layer. This is certainly not true, instead a neutron star should have a solid crust, which clearly will modify the above results. However, this is beyond the scope of this work.
Summary
=======
We have performed both mode calculations and time evolutions of the pure axial perturbation equations for slowly rotating stars in the low-frequency approximation. Although the time independent equation (\[eveqn\_a\]) represents a singular eigenvalue problem, admitting a continuous spectrum, it is nevertheless possible to find discrete mode solutions, representing the relativistic $r$-modes. If the mode frequency lies outside the continuous spectrum, the eigenvalue problem becomes regular, and the associated solution represents a physically valid $r$-mode solution. If the eigenvalue lies inside the continuous band, the eigenfunction exhibits an infinite slope at the singular point, which is due to the presence of a $r\log|r|$ term in the series expansion. Moreover, the corresponding fluid perturbation $u_3$ diverges at the singular point. Therefore, we conclude that these mode solutions are unphysical, and the only physically valid mode solutions are the ones where the associated frequencies $\alpha$ lie outside the range of the continuous spectrum.
We have performed mode calculations for uniform density models, for various polytropic models and also for a set of realistic equations of state. In agreement with the results of Lockitch et al. (2001), we find that uniform density models generally admit $r$-modes for any compactness. For polytropic equations of state, however, the existence of physical $r$-mode solutions depends strongly on both the polytropic index $n$ and the compactness $M/R$ of the stellar model. The general picture is that the smaller $n$ is, which corresponds to a stiffer equation of state, the larger is the compactness range where one can find physical mode solutions. For a given polytropic index $n$, one usually finds physical mode solutions for models with a small $M/R$ ratio. As the compactness is increased, i.e. as the models become more relativistic, the mode frequency $\alpha$ decreases and starts approaching $\hat\varpi_s$. Eventually it crosses this point and migrates inside the range of the continuous spectrum, thus becoming unphysical, and no $r$-mode exists any more.
When considered as a function of $l$, the $r$-mode frequency $\alpha$ is monotonically decreasing. For a uniform density model, $\alpha$ approaches $\hat\varpi_s$ as $l$ is increased. In polytropic models, this has the effect that it is even harder to find physical mode solutions for higher values of $l$, since $\alpha$ will much sooner cross the border $\hat\varpi_s$ of the continuous spectrum. If $l$ is large enough, it seems that the eigenvalue equation (\[eveqn\_a\]) does not admit any mode solution at all, not even a singular one.
We have verified our results by explicitly integrating the time dependent equations. The time evolutions for the models admitting an $r$-mode can clearly be distinguished from those without a discrete mode. In the former case, both the fluid perturbation $u_3$ and the metric perturbation $h_0$ oscillate with a constant amplitude after some initial time. In the latter case, the amplitude of $h_0$ constantly decreases. The fluid amplitude, however, still remains at a constant level. This can be explained by a decoherence effect in the fluid oscillations, since the frame dragging causes each fluid layer to oscillate with a different frequency. Thus the initially uniform fluid profile becomes more and more disturbed because the fluid layers get out of phase, resulting in a continuously weakening of the strength of the fluid source term in Eq. (\[eqh0\]). When a physical $r$-mode exists, the system can oscillate in a coherent manner.
When turning to realistic equations of state, the mode calculations yielded only frequencies lying inside the continuous band, therefore being apparently unphysical. However, for most EOS the numerical time evolutions revealed the presence of a mode, but with the frequency still lying inside the continuous band and corresponding approximately to the value of $\hat\varpi$ at the neutron drip point. We explained this seemingly contradictory behaviour by making the core responsible for the existence of the $r$-mode. For, if we remove the outer layer, which does not have any significant gravitational contribution, we can indeed find eigenvalues $\alpha$ which lie outside the continuous band. However, they are very close to the upper limit of the continuous band $\hat\varpi_s$. By adding the additional layer, we increase $\hat\varpi_s$ such that it now becomes larger than $\alpha$, which remains basically unaffected. Although now lying inside the continuous spectrum, the mode still exists, but it will be always associated with an excitation of the continuous spectrum. Most of the realistic EOS do admit $r$-modes in a certain mass range, but only the stiffest ones admit modes throughout the whole mass range up to the stability limit. The less stiff ones have a maximal mass model above which there are no $r$-modes any more, and for the softest EOS, namely EOS B, G and MPA, there are no $r$-mode for the whole physically relevant mass range.
It should be kept in mind that all our results concerning the $r$-modes are obtained within the low-frequency approximation. It would be clearly much too early to infer any statements about the existence or non-existence of the $r$-modes in rapidly rotating neutron stars. And if the true EOS of neutron stars is rather stiff, and therefore would already admit $r$-modes within the low-frequency approximation, then the whole discussion about the singular structure would be irrelevant. But as the true EOS is not known yet, we cannot exclude it to be rather soft, and the appearance of the singular points has to be taken much more seriously. Still, it could still be seen as a mere artefact of the low-frequency approximation. But the work of Kojima & Hosonuma (1999) indicates that the inclusion of second-order terms in $\Omega$ even increases the range of the continuous spectrum, which is responsible for the disappearance of the $r$-mode. They worked only in the Cowling approximation, but whether or not the inclusion of more higher order terms and the complete radiation reaction can restore the existence of the $r$-modes in all cases is still an open question and deserves further investigation. As a next step in this direction, we will investigate in a subsequent paper the full set of axial equations (Eqs. (\[V4\]) – (\[u3\])), which contains the radiation reaction.
acknowledgments {#acknowledgments .unnumbered}
===============
We thank Nils Andersson, Horst Beyer, John Friedman, Luciano Rezzolla, Adamantios Stavridis, Nikolaos Stergioulas and Shin Yoshida for helpful discussions. J.R. is supported by the Marie Curie Fellowship No. HPMF-CT-1999-00364. This work has been supported by the EU Programme ’Improving the Human Research Potential and the Socio-Economic Knowledge Base’ (Research Training Network Contract HPRN-CT-2000-00137).
\[lastpage\]
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[^1]: Following Lockitch et al. (2001) we call a stellar model [*barotropic*]{} if the unperturbed configuration obeys the same one-parameter equation of state as the perturbed configuration.
[^2]: Note that in Table 1 of Lockitch et al. (2001), there is a systematic error in their given values of $\alpha$, which are too large by about 5 per cent. This might be a consequence of a misprint in their Eqs. (5.2), (5.4) and (5.7), where the terms ${\left(}1 -
2M_0/R{\right)}^{1/2}$ and ${\left(}1 - 2M_0/R {\left(}r/R{\right)}^2{\right)}^{1/2}$ got confused.
|
---
abstract: 'We consider the entanglement entropy for a line segment in the system of noninteracting one-dimensional fermions at zero temperature. In the limit of a large segment length $L$, the leading asymptotic behavior of this entropy is known to be logarithmic in $L$. We study finite-size corrections to this asymptotic behavior. Based on an earlier conjecture of the asymptotic expansion for full counting statistics in the same system, we derive a full asymptotic expansion for the von Neumann entropy and obtain first several corrections for the Rényi entropies. Our corrections for the Rényi entropies reproduce earlier results. We also discuss the entanglement spectrum in this problem in terms of single-particle occupation numbers.'
author:
- Roman Süsstrunk
- 'Dmitri A. Ivanov'
title: 'Free fermions on a line: asymptotics of the entanglement entropy and entanglement spectrum from full counting statistics'
---
#### 1. Introduction.—
Entanglement is one of the central concepts of modern quantum mechanics and quantum information theory. It characterizes the amount of correlations between parts of a quantum system. In recent years, a progress has been achieved in studying entanglement for a variety of models, with the most detailed results available for one-dimensional systems, see e.g. the review [@latorre:09].
Entanglement can be introduced in a particularly simple way in the case of a many-body system in a pure state, e.g., in the zero-temperature ground state, which will always be assumed in this paper. Let such a system be divided into two subsystems $A$ and $B$. Then the entanglement may be characterized by the properties of the reduced density matrix $\rho_A$ of the subsystem $A$, which is obtained by tracing out the remaining degrees of freedom $$\rho_A=\operatorname{tr}_B\rho$$ (here $\rho$ denotes the density matrix of the pure state of the total system). The (von Neumann) entanglement entropy is then defined as the von Neumann entropy of $\rho_A$, $$\mathcal{S}^{(A)}=-\operatorname{tr}\rho_A\ln\rho_A\,.$$ Though characterizing entanglement by a single number is appealing, it falls short in representing its full complexity. A more complete description of entanglement may be given by the set of Rényi entropies $$\mathcal{S}^{(A)}_{\alpha}=\frac{1}{1-\alpha}\ln\operatorname{tr}\rho_A^{\;\alpha}
\, , \quad \alpha \geq 0
\, , \quad \alpha \neq 1$$ (the von Neumann entropy can then be expressed as the limit $\mathcal{S}^{(A)}=\lim_{\alpha\to 1} \mathcal{S}^{(A)}_{\alpha}$).
Since the total system is assumed to be in a pure state, these definitions can be shown to be symmetric with respect to the interchange of the subsystems $A$ and $B$: $\mathcal{S}^{(A)}=\mathcal{S}^{(B)}$ for both von Neumann and Rényi entropies [@latorre:09], so we shall drop the superscript $(A)$ or $(B)$ in our notation below.
Equivalently, entanglement may be characterized by the spectrum of the reduced density matrix $\rho_A$ (which coincides with the spectrum of $\rho_B$ for a pure state) [@li:08:calabrese:08:fidkowski:10:pollmann:10]. Like the full knowledge of the Rényi entropies, the entanglement spectrum allows to determine the state of the system up to unitary transformations in the subsystems $A$ and $B$. In this sense, the Rényi entropies and the entanglement spectrum contain the full information about entanglement.
The problem of calculating the entropies or the entanglement spectrum simplifies in the case of [*noninteracting particles*]{} (bosons or fermions). In this case, the reduced density matrix ($\rho_A$ or $\rho_B$) can be factorized into density matrices of individual single-particle levels[@d-matrix], and both the entanglement spectrum and the entropies may be expressed in terms of [*single-particle*]{} occupation numbers $p_i$. In the case of noninteracting [*fermions*]{}, the entropies are given by $$\mathcal{S}=-\sum_i \left[ p_i \ln p_i + (1-p_i) \ln (1-p_i) \right]
\label{vN-noninteracting}$$ for the von Neumann entropy and $$\mathcal{S}_\alpha=\frac{1}{1-\alpha} \sum_i \ln \left[ p_i^\alpha + (1-p_i)^\alpha \right]
\label{Renyi-noninteracting}$$ for the Rényi entropies. The sums over $i$ can be converted into integrals over $p_i$ \[Eqs. (\[eq:VNInt\]) and (\[eq:RenInt\]) below\] by introducing the spectral density of the occupation number $$\mu(p) = \sum_i \delta(p - p_i)\, .$$ This spectral density, together with the entanglement entropies (\[vN-noninteracting\]) and (\[Renyi-noninteracting\]), in the model of noninteracting one-dimensional fermions, will be the main object of our study.
Note that, in the case of noninteracting particles, the same spectrum of occupation numbers $p_i$ defines the full counting statistics (FCS) of the number of particles in each of the two subsystems. This observation was used in Refs. to establish an exact relation between the FCS and the entanglement spectrum. In the case of noninteracting fermions, both the FCS and the entanglement spectrum can be expressed in terms of the spectrum of a single-particle correlation matrix (in the context of FCS, such a decomposition was done in Ref. on the basis of the Levitov-Lesovik determinant formula [@levitov:93:96]).
Moreover, for noninteracting fermionic systems with translational invariance, the corresponding spectral problem involves matrices of Toeplitz type. Therefore, the asymptotic behavior of FCS and entanglement spectrum in the limit of a large subsystem size may be obtained with the help of the theory of Toeplitz determinants. A prominent example is the spin-1/2 $XX$ chain [@jin:04; @its:09], which can be mapped to a system of noninteracting fermions via a Jordan-Wigner transformation. In many interesting one-dimensional situations (including free fermions), the relevant Toeplitz matrix has Fisher-Hartwig singularities, and the asymptotic behavior of its determinant can be found using the Fisher-Hartwig conjecture [@basor:91:deift:11]. While the leading asymptotic behavior of entanglement and FCS can be obtained by choosing the main Fisher-Hartwig branch, finding subleading contributions requires more work. Recently, corrections to the entanglement entropies accurate to order $\mathcal{O}(L^{-3})$ (for a block of size $L$) have been computed for the spin-1/2 $XX$ chain [@calabrese:10] and in the continuous limit [@calabrese:11].
Furthermore, in the continuous limit, a full asymptotic expansion of the corresponding Toeplitz determinant was conjectured in Ref. in the context of FCS. Based on the matrix Riemann-Hilbert problem [@cheianov:04] and, independently, on the Painlevé V equation in the Jimbo-Miwa form [@jimbo:80:tracy:93], an expansion was constructed for the FCS generating function of the particle number on a line interval for one-dimensional free fermions in the zero-temperature ground state. Using the periodicity conjecture for the expansion (not proven, but verified up to high orders in $1/L$), the asymptotic expansion was written in an explicitly periodic Fisher-Hartwig form [@ivanov:11]. Instead of selecting the leading Fisher-Hartwig branch, all the branches were combined to obtain a full expansion to all orders in $1/L$, taking into account the switching of branches intrinsically.
We use the relation between FCS and entanglement entropies to carry over the full expansion conjectured in Ref. of the FCS generating function to the problem of finding the entanglement entropies and the entanglement spectrum for free fermions on a line. In particular, we find the power-law asymptotic expansion for the von Neumann entropy $\mathcal{S}$, compute first several coefficients, and present an algorithm for calculating the coefficients to an arbitrary order. A similar approach to the Rényi entropies $\mathcal{S}_\alpha$ produces an expansion with oscillating terms. For the Rényi entropies, we only compute the lowest-order terms, which agree with the previously available results [@calabrese:10; @calabrese:11]. We also find finite-size corrections to the spectral density of single-particle occupation numbers $\mu(p)$.
The physical motivation for studying finite-size corrections to the entanglement entropies is twofold. First, in critical one-dimensional systems, the form of those corrections is related to the scaling dimensions of operators in the corresponding conformal field theory (CFT) [@cardy:10]. Second, knowing the structure of finite-size corrections is helpful for extracting the central charge of the CFT from numerical computations of the entropies [@finite-size-numerics].
The remaining parts of the paper are structured as follows. The next section embodies our main results. Then we review the asymptotic expansion of the FCS for one-dimensional free fermions. Subsequently we present the calculations of the spectral density $\mu(p)$ and of the von Neumann and Rényi entropies. Finally we conclude by a discussion of our results. The appendix includes details of the analysis of oscillating terms in the asymptotic expansions of the entanglement entropies.
![The function $\tau(p)$ in Eq. (\[DOS-result\]). At the end points of the interval $[0,1]$, $\tau(p)$ tends to infinity (logarithmically).[]{data-label="fig:tau"}](tau-plot-3.pdf){width="40.00000%"}
#### 2. Results.—
Based on the conjecture for the FCS in Ref. , we derive the asymptotic power series for the entanglement entropy of free fermions on a line in the ground state: $$\mathcal{S}(x)=\frac{1}{3}\ln(2x)+\Upsilon + \sum_{n=1}^{\infty} s_{2n} x^{-2n}\, .
\label{eq:S-series}$$ Here $x=k_F L$ ($L$ is the length of the line segment for which the entanglement is computed and $k_F$ is the Fermi wavevector) and the constant $\Upsilon$ is given by Eq. (\[eq:Upsilon\]). Note that this series contains only even powers of $1/x$. All the coefficients $s_{2n}$ are rational numbers which can be computed to any given order in $n$ using the methods of Ref. . The first several coefficients are: $$s_2=-\frac{1}{12}\, , \quad s_4=-\frac{31}{96}\, , \quad s_6=-\frac{7057}{1440}\, .
\label{eq:S-coefficients}$$ The leading term $(1/3)\ln(2x)$, the constant term $\Upsilon$, and the coefficient $s_2$ are known from earlier works [@jin:04; @calabrese:10; @calabrese:11]. In contrast to the Rényi entropies, there are no oscillating contributions to the von Neumann entropy at any order in $1/x$.
The calculation involves an expansion for the spectral density $\mu(p)$ based on the conjecture in Ref. . Away from the end points $p=0$ and $p=1$ (a precise condition is formulated below), the spectral density has a quasiclassical structure with locally nearly equidistant levels. The smooth (nonoscillating) part of the spectral density is given by $$\bar\mu(p,x)=\frac{1}{\pi^2 \, p(1-p)}\left[\ln(2x)-\tau(p)\right] + \mathcal{O}(x^{-1})\, ,
\label{DOS-result}$$ where $$\tau(p)=\varphi'\left[\frac{1}{2\pi}\ln\left(\frac{1-p}{p}\right)\right]\, ,
\label{eq:tau}$$ $$\varphi(\xi)=\arg\left[
\Gamma \left(\frac{1}{2} + i\xi \right) \right]\, ,
\label{eq:varphi}$$ and prime denotes the derivative of $\varphi(\xi)$ with respect to its argument. The function $\tau(p)$ is plotted in Fig. \[fig:tau\].
#### 3. FCS of free one-dimensional fermions.—
We consider free spinless fermions on a continuous line. The temperature is assumed to be zero, i.e. the system is in the ground state characterized by the Fermi wavevector $k_F$. We will study the entanglement between two subsystems: an interval of length $L$ and the remainder of the line. Both FCS and the entanglement in this setup depend only on the dimensionless parameter $x=k_FL$. For example, the average number of particles on the line segment is given by $\left< N \right>=x/\pi$. The FCS generating function for the probability distribution of the particle number $N$, $$\chi(\kappa,x)=\left<e^{i\, (2\pi\kappa)\, N}\right>,$$ was conjectured in Ref. to be given by $$\chi(\kappa, x) = \sum_{j=-\infty}^{\infty} \chi_*(\kappa+j, x) \, ,
\label{eq:fcsExpansion1}$$ $$\begin{gathered}
\chi_*(\kappa,x) =\exp\biggl[2i\kappa x - 2\kappa^2 \ln x + C(\kappa) \\
+\sum_{n=1}^{\infty} f_n(\kappa)\, (ix)^{-n} \biggr]\, ,
\label{eq:fcsExpansion2}\end{gathered}$$ $$C(\kappa) =\ln \left[ G(1+\kappa)^2 G(1-\kappa)^2 \right] - 2\kappa^2 \ln 2\, ,
\label{eq:defnC}$$ where $G(z)$ denotes the Barnes G-function and $f_n(\kappa)$ are polynomials in $\kappa$, computable order by order. For our purpose, we will use the logarithm of this expansion, which for $-1/2<\kappa <1/2$ takes the form $$\begin{gathered}
\ln\chi(\kappa,x)=2i\kappa x - 2\kappa^2 \ln x + C(\kappa) \\
+ \sum_{n=1}^\infty \; \sum_{m=-\left\lfloor \frac{n}{2}\right\rfloor}^{\left\lfloor \frac{n}{2}\right\rfloor}
C_{n,m}(\kappa)\, x^{-n-4m\kappa}e^{2imx}\, ,
\label{eq:fcsLog}\end{gathered}$$ where $\left\lfloor \cdot \right\rfloor$ denotes the integer part of the argument.
The coefficients $C_{n,m}(\kappa)$ can be expressed in terms of the polynomials $f_n(\kappa)$ order by order. They are also linearly related to the coefficients $R_{n,m}(\kappa)$ used in Ref. for the expansion of the derivative (in $x$) of Eq. (\[eq:fcsLog\]). In particular, $C_{n,0}(\kappa)=-(1/n)\, R_{n+1,0}(\kappa)$.
#### 4. Entanglement spectrum.—
The spectral density $\mu(p)$ can be obtained from the jump of $\ln\chi(\kappa,x)$ across the line $\kappa=\pm 1/2$ (see, e.g., Ref. ): $$\mu(p)=-\frac{1}{4\pi^2\, p(1-p)} \frac{\partial}{\partial \kappa} \ln \chi(\kappa,x)
\biggr|_{\kappa=-(\frac{1}{2}-\varepsilon)-i\xi}^{\kappa=+(\frac{1}{2}-\varepsilon)-i\xi}\, ,
\label{eq:entSpecY}$$ where $\varepsilon$ is an infinitesimally small positive parameter and we introduced the parameterization $$\xi=\frac{1}{2\pi}\ln\left(\frac{1-p}{p}\right)\, .
\label{eq:changeOfVariables}$$ Inserting Eq. (\[eq:fcsLog\]) into Eq. (\[eq:entSpecY\]) and using the symmetry of the generating function $\chi(-\kappa,x) = \chi^* (\kappa,x)$, we arrive at $$\begin{gathered}
\mu(p) = - \frac{1}{2\pi^2\, p(1-p)} \operatorname{Re}\frac{\partial}{\partial \kappa}\Biggl[2i\kappa x - 2 \kappa^2\ln x + C(\kappa) \\
+\sum_{n=1}^\infty \;
\sum_{m=-\left\lfloor \frac{n}{2}\right\rfloor}^{\left\lfloor \frac{n}{2}\right\rfloor}
C_{n,m}(\kappa)\, x^{-n-4m\kappa}e^{2imx}
\Biggl]_{\kappa=\frac{1}{2}-i\xi}\, .
\label{eq:muExp}\end{gathered}$$
Note that the $x$ dependence of each term in Eq. (\[eq:muExp\]) is known. The coefficients at nonoscillating terms are determined by $C_{n,0}$, so that the smooth (nonoscillating) part of $\mu(p)$ can be calculated as $$\begin{gathered}
\bar\mu(p) = \frac{1}{\pi^2\, p(1-p)}
\Big(\ln(2x) - \tau(p) \\
+ \sum_{n=1}^{\infty} \operatorname{Re}C_{n,0}^\prime \left(\frac{1}{2}-i\xi\right)\, x^{-n} \Big) \, ,
\label{eq:mubar} \end{gathered}$$ where $\tau(p)$ is given by Eqs. (\[eq:tau\])–(\[eq:varphi\]) and the prime denotes the derivative of $C_{n,0}(\kappa)$ with respect to $\kappa$. The first two terms in this expansion give the announced result (\[DOS-result\]).
Oscillating terms may, in turn, be collected by the “diagonals” $C_{2n,-n+l}(\kappa)$ and $C_{2n+1,-n+l}(\kappa)$ with $l=0,1,2,\dots$, contributing terms of the orders $x^{-2l}$ and $x^{-2l-1}$, respectively. The first two diagonals (with $l=0$) are easy to sum. By calculating the logarithm of the series (\[eq:fcsExpansion1\])–(\[eq:fcsExpansion2\]) and using $f_1(\kappa)=2\kappa^3$ (see Ref. ), we find $$\begin{aligned}
C_{2n,-n}(\kappa) &=\frac{(-1)^{n+1}}{n}\, e^{n[C(\kappa-1) - C(\kappa)]}\, , \label{eq:diag1}\\
C_{2n+1,-n}(\kappa) &= 2i (-1)^{n+1} (3\kappa^2 - 3\kappa + 1)\, e^{n[C(\kappa-1) - C(\kappa)]}\, .
\label{eq:diag2}\end{aligned}$$ Adding those contributions converts the continuous spectrum (\[eq:mubar\]) into a sum of delta functions. For example, taking into account the diagonals (\[eq:diag1\]) and (\[eq:diag2\]) results in $$\mu(p)=\sum_n \delta\left[\Phi(p,x) - \pi \left(n+\frac{1}{2}\right)\right]\,
\frac{\partial\Phi}{\partial p} \, ,
\label{eq:muPhase}$$ where $$\Phi(p,x)=x + 2\xi\ln(2x) - 2\varphi(\xi) + \left(3\xi^2-\frac{1}{4}\right) x^{-1}
+ \mathcal{O} \left(x^{-2}\right)
\label{eq:Phi}$$ and $\varphi(\xi)$ is given by Eq. (\[eq:varphi\]).
Note that the spectrum (\[eq:muPhase\]) has a quasiclassical nature: the positions of quantum levels are determined by a quantization rule of Bohr-Sommerfeld type. The resulting spectrum is regularly spaced with the average density given by $(1/\pi)\, \partial\Phi/\partial p$. This can be explained by the fact that the diagonals (\[eq:diag1\]) and (\[eq:diag2\]) stem, in fact, only from the two leading Fisher-Hartwig branches in Eq. (\[eq:fcsExpansion1\]) at $\kappa=1/2$ (those with $j=0$ and $j=-1$). The spectrum is thus determined from the condition that these two branches cancel each other, which naturally leads to an expression of the form (\[eq:muPhase\]). Including higher-order Fisher-Hartwig branches produces modulations in the level spacing, but this effect appears only at higher orders in $1/x$.
Note also that this expansion breaks down close to the end points of the spectrum $p=1$ and $p=0$. In those regions $\tau(p)$ is large and therefore the density of states given by Eq. (\[eq:mubar\]) becomes formally negative: in fact, the expansion (\[eq:mubar\]) is not applicable in those regions of $p$. Indeed, the expansion parameter in Eq. (\[eq:mubar\]) is $\xi/x$: this can be seen from the (unproven) fact observed in Ref. that the polynomial $C_{n,0}$ has degree $n+2$ in $\kappa$ (and therefore in $\xi$). Thus the expansion (\[eq:mubar\]) is only applicable at $|\xi |\ll x$. Remarkably, this condition also guarantees the positivity of $\bar\mu(p)$.
Our results (\[DOS-result\])–(\[eq:varphi\]) and the quasiclassical structure of the spectrum are consistent with the numerical studies of Refs. . In particular, the smooth part of the density of states in the middle of the spectrum is $$\begin{aligned}
& \bar\mu(p=1/2) = \frac{4}{\pi^2} \left( \ln x + b\right) + \mathcal{O} (x^{-1})\, ,
\notag \\
& b = \ln2 - \varphi'(0) \approx 2.657\, ,\end{aligned}$$ in agreement with the findings of those works.
#### 5. Von Neumann and Rényi entropies.—
Once the spectral density $\mu(p)$ is known, the entropies can be calculated using the integral forms of Eqs. (\[vN-noninteracting\]) and (\[Renyi-noninteracting\]): $$\mathcal{S} =-\int_0^1 dp\, \mu(p)\, \left[ p\ln p+(1-p)\ln(1-p) \right] \label{eq:VNInt}$$ for the von Neumann entropy and $$\mathcal{S_{\alpha}} =\frac{1}{1-\alpha}\int_0^1 dp\, \mu(p)\,
\ln \left[ p^{\alpha}+(1-p)^{\alpha}\right]
\label{eq:RenInt}$$ for the Rényi entropies. Note that, even though the expansion (\[eq:muExp\]) applies only at $|\xi|\ll x$, we may integrate in Eqs. (\[eq:VNInt\]) and (\[eq:RenInt\]) from $\xi=-\infty$ to $\xi=+\infty$ (corresponding to $0<p<1$): the contributions from large $\xi$ are exponentially smaller than all the terms of the resulting series and may be neglected.
For the von Neumann entanglement entropy, oscillating contributions vanish at all orders in $1/x$ \[the integral (\[eq:VNInt\]) may be closed in the upper or lower half plane of the variable $\xi$, see Appendix\]. Only nonoscillating contributions survive and may be found by replacing $\mu(p)$ in the integral (\[eq:VNInt\]) by its nonoscillating part (\[eq:mubar\]). As a result, we find the power series $$\mathcal{S}(x)=\frac{1}{3}\ln(2x)+\Upsilon + \sum_{n=1}^{\infty} s_n x^{-n}\, ,
\label{eq:VNFullExpansion}$$ where the coefficients are given by $$s_n = \int_{-\infty}^{\infty} d\xi\, \frac{\pi \xi}{\cosh^2(\pi \xi)}\,
\operatorname{Im}C_{n,0} \left( \frac{1}{2} - i \xi \right)\, .$$
The functions $C_{n,0}(\kappa)$ may be found from the results reported in Ref. or calculated order by order using the methods developed in that work. In particular, it follows from the results of Ref. that $C_{2n+1,0}(\kappa)$ are polynomials odd in $\kappa$ with purely imaginary coefficients. Therefore, all the odd terms in the expansion (\[eq:VNFullExpansion\]) vanish, and we arrive at the result (\[eq:S-series\]). Furthermore, since $C_{n,0}(\kappa)$ are polynomials with rational coefficients, all the coefficients $s_{2n}$ are rational numbers. The first three nonzero coefficients can be obtained from $$\begin{aligned}
C_{2,0}(\kappa)&=& -\frac{5}{2} \kappa^4\, , \notag \\
C_{4,0}(\kappa)&=& \frac{25}{16}\kappa^4+\frac{63}{4}\kappa^6\, ,\\
C_{6,0}(\kappa)&=& -\frac{35}{8}\kappa^4-\frac{889}{12}\kappa^6-\frac{3129}{16}\kappa^8\, , \notag\end{aligned}$$ which gives the result (\[eq:S-coefficients\]). Following this procedure \[with $C_{2n,0}(\kappa)$ calculated using the method of Ref. \], the coefficients $s_{2n}$ may be computed to any order, one by one, in a straightforward way.
The constant $\Upsilon$ is found to be $$\begin{gathered}
\Upsilon = -\frac{2}{\pi} \int_{-\infty}^{+\infty} d\xi \, \varphi'(\xi)
\left(\ln\left[2\cosh(\pi \xi)\right] - \pi \xi \tanh[\pi \xi] \right) \\
\approx 0.4950179081 \, ,
\label{eq:Upsilon}\end{gathered}$$ where the function $\varphi(\xi)$ is defined by Eq. (\[eq:varphi\]). This expression for $\Upsilon$ can be shown to agree with that found in Ref. .
In contrast, for the Rényi entropies, the oscillating parts do not vanish and can be classified in terms of the poles of the integrand of Eq. (\[eq:RenInt\]), see Appendix. The first orders \[calculated using Eqs. (\[eq:diag1\]) and (\[eq:diag2\])\] are given by Eqs. (\[eq:RenSeries\]) and (\[eq:UpsilonAlpha\]). Calculating higher-order oscillating terms in the Rényi entropies would require knowing higher order diagonals $C_{2n,-n+l}(\kappa)$ and $C_{2n+1,-n+l}(\kappa)$ with $l\ge 1$. Although each of those coefficient can be separately calculated (using the methods of Ref. ), deriving general formulas (valid for all $n$) is a tedious task, and we do not attempt it here.
#### 6. Numercial illustration.—
To illustrate our main result (\[eq:S-series\])–(\[eq:S-coefficients\]) and to perform an additional check of the expansion conjectured in Ref. , we have also computed the von Neumann entropies $\mathcal{S}(x)$ numerically and compared them to our analytical expansion (\[eq:S-series\])–(\[eq:S-coefficients\]). The numerical computation was performed in the lattice model (considered, e.g., in Ref. ) for blocks containing up to 1000 sites and then extrapolated to the continuous limit. This allowed us to calculate $\mathcal{S}(x)$ for $x\in [5,20]$ with the error bars not exceeding $10^{-9}$. In Fig. \[fig:deviations\] we plot the remainder of the asymptotic series (\[eq:S-series\]) $\Delta_{2n}=(1/3)\ln(2x)+\Upsilon + \sum_{m=1}^{n} s_{2m} x^{-2m} - \mathcal{S}(x)$ as a function of $x$. One can see that the remainders indeed decay as powers of $x$: in particular, $\Delta_6$ decays as $x^{-8}$, in agreement with our analytical prediction.
![The remainders of the asymptotic series (\[eq:S-series\]) as functions of $x$ (in the log-log scale). The dashed line indicates the slope of $x^{-8}$.[]{data-label="fig:deviations"}](deviations.pdf){width="45.00000%"}
#### 7. Summary and discussion.—
In this paper, we have used the asymptotic expansion of the FCS generating function for a line segment of one-dimensional free fermions to determine the asymptotic expansion of the entanglement entropy and the entanglement spectrum in the same system. The main result is the asymptotic power series in $1/x$ for the von Neumann entropy. Our method also allows to construct finite-size corrections for the Rényi entropies (we only do it to the lowest order, where we reproduce the known results) and gives an expansion for the spectrum of the single-particle correlation matrix.
Our results are based on the expansion conjectured (not rigorously proven) in Ref. , and therefore also have the status of conjecture. Two elements of the proof were missing in Ref. . First, the periodicity relations on the expansion coefficients \[which allows to convert the expansion into an explicitly periodic form (\[eq:fcsExpansion1\])\] were not proven but only checked analytically up to the 15th order in $1/x$. Second, the expansion (\[eq:fcsExpansion1\])–(\[eq:fcsExpansion2\]) was not extended to the line $\operatorname{Re}(\kappa)=1/2$: the point where the switching of the Fisher-Hartwig branches takes place and where we need the expansion for calculating the entropies. An extension of the expansion to this line is however a very plausible conjecture, since the expansion itself is regular at this line; it is also supported by a numerical study on the more general lattice model [@abanov:11] and by our numerical computations of the von Neumann entropy (Fig. \[fig:deviations\]). We thus conjecture that our results are in fact exact expressions for the model considered.
We thank P. Calabrese, V. Eisler, and I. Peschel for helpful comments on the manuscript.
#### Appendix: Oscillating contributions to the entropies.—
In this appendix, we treat the oscillating (in $x$) terms in the expansions of the von Neumann and Rényi entanglement entropies. For the von Neumann entropy (\[eq:VNInt\]), all the oscillating terms vanish, provided the expansion conjectured in Ref. is correct. For the Rényi entropies (\[eq:RenInt\]), there are oscillating terms decaying as $\alpha$-dependent powers of $x$.
Oscillating terms in the entropies are obtained by substituting the terms of the expansion (\[eq:muExp\]) with a given oscillation frequency $m$ into the integrals (\[eq:VNInt\]) and (\[eq:RenInt\]). The integrals are further calculated by using $\xi$ as the integration variable, integrating by parts and closing the integration contour in the upper (lower) half plane for $m>0$ ($m<0$, respectively).
In the case of the von Neumann entropy, this produces terms of the form $$\begin{gathered}
\operatorname{Im}\Big[ e^{2imx} x^{-n-2m} \int_{-\infty}^{\infty} d\xi \,
\frac{\pi \xi}{\cosh^2 (\pi \xi)} \\
\times C_{n,m} \left(\frac{1}{2} - i \xi \right)
e^{4 i m \xi \ln x} \Big]\, .
\label{eq:oscIntVN}\end{gathered}$$ Now the crucial ingredient of our discussion is the structure of the coefficients $C_{n,m}(\kappa)$. It can be seen from the explicit calculation in Ref. (using the Riemann-Hilbert method) that these coefficients have the following form (assuming $m>0$): $$\begin{aligned}
C_{n,m}(\kappa) & = & \tilde{c}_{n,m}(\kappa) e^{m[C(\kappa+1) - C(\kappa)]} \, , \\
C_{n,-m}(\kappa) & = & \tilde{c}_{n,-m}(\kappa) e^{m[C(\kappa-1) - C(\kappa)]} \, ,\end{aligned}$$ where $C(\kappa)$ is defined in Eq. (\[eq:defnC\]) and $\tilde{c}_{n,m}(\kappa)$ are some polynomials in $\kappa$.
From this property, it follows that, at $m>0$, the coefficient $C_{n,m}(1/2 - i \xi)$ has zeroes of degree $2m$ at all points $\xi= i(1/2 + r)$ for $r=0, 1, \ldots$, which compensate the poles of degree two of the factor $\cosh^{-2} (\pi \xi)$ in the integral (\[eq:oscIntVN\]). Therefore the integrand is analytic in the upper half plane where the contour is closed, and the integral vanishes. Similarly, at $m<0$, the coefficient $C_{n,m}(1/2 - i \xi)$ has zeroes of degree $2m$ at all points $\xi= - i (1/2 + r)$ for $r=0, 1, \ldots$, the integrand is analytic in the lower half plane, and the integral vanishes again. We therefore conclude that the asymptotic expansion of the von Neumann entanglement entropy has the form of a power series in $1/x$ (apart from the leading logarithm), without any oscillating terms.
In the case of the Rényi entropies, the oscillating terms have the form $$\begin{gathered}
\operatorname{Im}\Big[ e^{2imx} x^{-n-2m} \int_{-\infty}^{\infty} d\xi\,
\frac{\alpha \left[ \tanh (\pi \xi) - \tanh (\alpha \pi \xi) \right]}{1-\alpha} \\
\times C_{n,m} \left(\frac{1}{2} - i \xi \right) e^{4 i m \xi \ln x} \Big ]\, .
\label{eq:oscIntRen}\end{gathered}$$ They contain additional poles at $\xi=\pm (i/\alpha)(1/2+n)$. These poles are not compensated by zeroes of $C_{n,m}(1/2-i \xi)$ and produce oscillating contributions to the entropy decaying as fractional ($\alpha$-dependent) powers of $x$. A calculation of the first few terms \[based on the explicit expressions (\[eq:diag1\]) and (\[eq:diag2\])\] produces the result $$\begin{gathered}
\mathcal{S}_{\alpha}(x) =\frac{1}{6}\left[1+\frac{1}{\alpha}\right]\ln(2x)+\Upsilon_\alpha +\frac{(\alpha+1)(3\alpha^2-7)}{96\alpha^3}x^{-2} \\
+\sum_{n,j=1}^\infty\frac{(-1)^n}{\alpha-1}\,(2x)^{-\frac{2n(2j-1)}{\alpha}}
\left[\frac{\Gamma\left(\frac{1}{2}+\frac{2j-1}{2\alpha}\right)}{\Gamma\left(\frac{1}{2}-
\frac{2j-1}{2\alpha}\right)}\right]^{2n} \\
\times \left[\frac{2}{n}\cos(2nx)+x^{-1}\left[1+3\frac{(2j-1)^2}{\alpha^2}\right]\sin(2nx)\right] \\
+ o (x^{-2})\, ,
\label{eq:RenSeries}\end{gathered}$$ where $$\begin{gathered}
\Upsilon_\alpha = -\frac{2}{\pi} \int_{-\infty}^{+\infty} d\xi\, \varphi'(\xi) \\
\times \frac{\ln\left[2\cosh(\pi \xi \alpha)\right] - \alpha \ln [ 2 \cosh(\pi \xi) ]}{1-\alpha} \, .
\label{eq:UpsilonAlpha}\end{gathered}$$ These corrections reproduce the results of Ref. (in the corresponding continuous limit of the spin-1/2 $XX$ chain), , and .
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|
---
address: |
Department of Mathematics\
University of Oregon\
Eugene, OR 97403
author:
- Daniel Dugger
title: 'Bigraded cohomology of ${{\mathbb Z}}/2$-equivariant Grassmannians'
---
Introduction
============
Let ${{\mathbb R}}$ and ${{\mathbb R}}_-$ denote the two representations of ${{\mathbb Z}}/2$ on the real line: the first has the trivial action, the second has the sign action. Let ${{{\EuScript}U}}$ denote the infinite direct sum $${{{\EuScript}U}}={{\mathbb R}}\oplus {{\mathbb R}}_{-} \oplus {{\mathbb R}}\oplus {{\mathbb R}}_- \oplus \cdots$$ The subjects of this paper are the infinite Grassmannians $\operatorname{Gr}_k({{{\EuScript}U}})$, regarded as spaces with a ${{\mathbb Z}}/2$-action. Our goal is to compute the $RO({{\mathbb Z}}/2)$-graded cohomology rings $H^*(\operatorname{Gr}_k({{{\EuScript}U}});{({{\mathbb Z}}/2)_m})$, where ${({{\mathbb Z}}/2)_m}$ denotes the constant-coefficient Mackey functor. These cohomology rings are a notion of equivariant cohomology that is finer than the classical Borel theory.
Of course our results may be interpreted as giving a calculation of all characteristic classes, with values in the theory $H^*({-};{({{\mathbb Z}}/2)_m})$, for rank $k$ equivariant bundles. Previous work on related problems has been done by Ferland and Lewis [@FL] and by Kronholm [@K1; @K2], but the present paper provides the first complete computation for any single value of $k$ larger than $1$.
The rest of this introduction aims to describe the results of the computation. The context throughout the paper is the category of ${{\mathbb Z}}/2$-spaces, with equivariant maps. Unless stated otherwise all spaces and maps are in this category.
The theory $H^*({-};{({{\mathbb Z}}/2)_m})$ is graded by the representation ring $RO({{\mathbb Z}}/2)$. That is to say, if $V$ is a virtual representation then the theory yields groups $H^V({-};{({{\mathbb Z}}/2)_m})$. For the group ${{\mathbb Z}}/2$ every representation has the form ${{\mathbb R}}^p\oplus ({{\mathbb R}}_-)^q$ for some $p$ and $q$, and this implies that we may regard our cohomology theory as being bigraded. Different authors use different indexing conventions, but we will use the “motivic” indexing described as follows. The representation $V={{\mathbb R}}^p\oplus ({{\mathbb R}}_-)^q$ is denoted ${{\mathbb R}}^{p+q,q}$, and the corresponding cohomology groups $H^V({-};{({{\mathbb Z}}/2)_m})$ will be denoted $H^{p+q,q}({-};{({{\mathbb Z}}/2)_m})$. In this indexing system the first index is called the [topological degree]{} and the second is called the [weight]{}. One appeal of this system is that dropping the second index will always give statements that seem familiar from non-equivariant topology.
Before continuing, for ease of reading we will just write ${{\mathbb Z}}/2$ instead of ${({{\mathbb Z}}/2)_m}$ in coefficients of cohomology groups. In the presence of the bigrading this will never lead to any confusion.
Let ${{\mathbb M}}_2$ be the bigraded ring $H^{*,*}({pt};{{\mathbb Z}}/2)$, the cohomology ring of a point. This is the ground ring of our theory; for any ${{\mathbb Z}}/2$-space $X$, the ring $H^{*,*}(X;{{\mathbb Z}}/2)$ is an algebra over ${{\mathbb M}}_2$. A complete description of ${{\mathbb M}}_2$ is given in the next section, but for now one only needs to know that there are special elements $\tau\in {{\mathbb M}}_2^{0,1}$ and $\rho\in {{\mathbb M}}_2^{1,1}$.
The cohomology ring of the projective space $\operatorname{Gr}_1({{{\EuScript}U}})$ has been known for a while; the motivic analog was computed by Voevodsky, and the same proof works in the ${{\mathbb Z}}/2$-equivariant setting. A careful proof is written down in [@K2 Theorem 4.2]. There is an isomorphism of algebras $H^{*,*}(\operatorname{Gr}_1({{{\EuScript}U}});{{\mathbb Z}}/2){\cong}{{\mathbb M}}_2[a,b]/(a^2=\rho a+\tau b)$ where $a$ has bidegree $(1,1)$ and $b$ has bidegree $(2,1)$. In non-equivariant topology one has $\rho=0$ and $\tau=1$, so that the above relation becomes $a^2=b$ and we simply have a polynomial algebra in a variable of degree $1$—the familiar answer for the mod $2$ cohomology of real projective space.
Note that additively, $H^{*,*}(\operatorname{Gr}_1({{{\EuScript}U}});{{\mathbb Z}}/2)$ is a free module over ${{\mathbb M}}_2$ on generators of the following bidegrees: $$(0,0),(1,1),(2,1),(3,2),(4,2),(5,3),(6,3),(7,4),\ldots$$ corresponding to the monomials $1,a,b,ab,b^2,ab^2,b^3,ab^3,\ldots$ If one forgets the weights, then one gets the degrees for elements in an additive basis for the singular cohomology $H^*({{{\mathbb R}}P}^\infty;{{\mathbb Z}}/2)$. So in this case one can obtain the equivariant cohomology groups by taking a basis for the singular cohomology groups, adding appropriate weights, and changing every ${{\mathbb Z}}/2$ into a copy of ${{\mathbb M}}_2$. We mention this because it is a theorem of Kronholm [@K1] that the same is true in the case of $\operatorname{Gr}_k({{{\EuScript}U}})$ (and for many other spaces as well, though not all spaces). Because we know the singular cohomology groups $H^*(\operatorname{Gr}_k({{\mathbb R}}^\infty);{{\mathbb Z}}/2)$, computing the equivariant version becomes only a question of knowing what weights to attach to the generators. While it might seem that it should be simple to resolve this, the question has been very resistant until now; the present paper provides an answer.
To state our main results, begin by considering the map $$\eta\colon \operatorname{Gr}_1({{{\EuScript}U}})\times \cdots \times \operatorname{Gr}_1({{{\EuScript}U}}) {\longrightarrow}\operatorname{Gr}_k({{{\EuScript}U}})$$ that classifies the $k$-fold direct sum of line bundles. Using the Künneth Theorem, the induced map on cohomology gives $$\eta^*\colon H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2) {\rightarrow}H^{*,*}(\operatorname{Gr}_1({{{\EuScript}U}});{{\mathbb Z}}/2)^{{\otimes}k}.$$ Since permuting the factors in a $k$-fold sum yields an isomorphic bundle, the image of $\eta^*$ lies in the ring of invariants under the action of the symmetric group $\Sigma_k$. That is to say, we may regard $\eta^*$ as a map
\[eq:main\] \^\*H\^[\*,\*]{}(\_k([[U]{}]{});[[Z]{}]{}/2) \^[\_k]{}.
The first of our results is the following:
\[th:main1\] The map in (\[eq:main\]) is an isomorphism of bigraded rings.
This is the direct analog of what happens in the nonequivariant case. Let us note, however, that until now neither injectivity nor surjectivity has been known in the present context. It must be admitted up front that in some ways our proof of Theorem \[th:main1\] is not very satisfying: it does not give any reason, based on first principles, why $\eta^*$ should be an isomorphism. Rather, the proof proceeds by computing the codomain of $\eta^*$ explicitly and then running a complicated spectral sequence for computing the domain of $\eta^*$. By comparing what is happening on the two sides, and appealing to the nonequivariant result at key moments, one can see that there is no choice but for the map to be an isomorphism—even without resolving all the differentials in the spectral sequence (of which there are infinitely many). The argument is somewhat sneaky, but not terribly difficult in the end. However, it depends on a key result proven by Kronholm [@K1] that describes the kind of phenomena that take place inside the spectral sequence.
The proof of Theorem \[th:main1\] is the main component of this paper. It is completed in Section \[se:proof\]. Subsequent sections explore some auxilliary issues, that we describe next.
In non-equivariant topology there are several familiar techniques for proving Theorem \[th:main1\], perhaps the most familiar being use of the Serre spectral sequence. Since the theorem is really about the identification of characteristic classes, another method that comes to mind is the Grothendieck approach to characteristic classes via the cohomology of projective bundles. The equivariant analogs of both these approaches have been partially explored by Kronholm [@K2], but one runs into a fundamental problem: such calculations require the use of local coefficient systems, because the fixed sets of Grassmannians are disconnected. So they involve a level of diffculty that is far beyond what happens in the non-equivariant case, and to date no one has gotten these approaches to work. Cohomology with local coefficients has been little-explored in the equivariant setting, but see [@Sh] for work in this direction.
To access the full power of Theorem \[th:main1\] one should compute the ring of invariants $[H^{*,*}(\operatorname{Gr}_1({{{\EuScript}U}});{{\mathbb Z}}/2)^{{\otimes}k }\bigr
]^{\Sigma_k}$, which is a purely algebraic problem. The proof of Theorem \[th:main1\] only requires understanding an additive basis for this ring. The second part of the paper examines the multiplicative structure.
In regards to the additive basis, we can state one form of our results as follows. Recall that a basis for $H^n(\operatorname{Gr}_k({{\mathbb R}}^\infty);{{\mathbb Z}}/2)$ is provided by the Schubert cells of dimension $n$, and these are in bijective correspondence with partitions of $n$ into $\leq k$ pieces. For example, a basis for $H^6(\operatorname{Gr}_3({{\mathbb R}}^\infty);{{\mathbb Z}}/2)$ is in bijective correspondence with the set of partitions $$[6],\quad [51],\quad [42],\quad [411], \quad [33], \quad [321],\quad [222].$$ For any such partition $\sigma=[j_1j_2\ldots j_k]$, define its [*weight*]{} to be $$w(\sigma)=\sum \lceil \tfrac{j_i}{2}\rceil.$$ So the list of the above seven partitions have corresponding weights $3,4,3,4,4,4,3$. Using this notion, the following result shows how to write down an ${{\mathbb M}}_2$-basis for $H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$:
$H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ is a free module over ${{\mathbb M}}_2$ with a basis $S$, where the elements of $S$ in topological degree $n$ are in bijective correspondence with partitions of $n$ into at most $k$ pieces. This bijection sends a partition $\sigma$ to a basis element of bidegree $(n,w(\sigma))$ where $w(\sigma)$ is the weight of $\sigma$.
It is easy to see that for a partition $\sigma$ of $n$ the weight is also equal to $$w(\sigma)=\tfrac{1}{2}\bigl (n+(\text{\# of odd pieces in
$\sigma$})\bigr ).$$ Using this description we can reinterpret the theorem as follows:
The number of free generators for $H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ having bidegree $(p,q)$ coincides with the number of partitions of $p$ into at most $k$ pieces where exactly $2q-p$ of the pieces are odd.
For example, in $H^{7,*}(\operatorname{Gr}_5({{{\EuScript}U}});{{\mathbb Z}}/2)$ we have basis elements in weights $4$, $5$, and $6$, corresponding to the partitions $$\begin{aligned}
[7], [61], [52], [43], [421], [322], [2221] \quad &\text{(weight
$4$/one odd piece)}
\\
[511], [4111], [331], [3211], [22111] \quad &\text{(weight $5$/three odd
pieces)} \\
[31111] \quad &\text{(weight $6$/five odd pieces)}.\end{aligned}$$
We next describe a little about the ring structure. Unlike what happens in nonequivariant topology, it is not easy to write down a simple description of the ring of invariants in terms of generators and relations—except for small values of $k$. In essense, the innocuous-looking relation “$a^2=\rho a+\tau b$” propogates itself viciously into the ring of invariants, leading to some unpleasant bookkeeping. However, we are able to give a minimal set of generators for the algebra, and we investigate the relations in low dimensions.
First, for $1\leq i\leq k$ there are special classes $w_i\in
H^{i,i}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ that we call [Stiefel-Whitney classes]{}; they correspond to the usual Stiefel-Whitney classes in singular cohomology. There are also special classes $c_i\in
H^{2i,i}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ that we call [Chern classes]{}; their images in non-equivariant cohomology correspond to the mod $2$ reductions of the usual Chern classes of the complexification of a bundle. In some sense these constitute the “obvious” characteristic classes that one might expect. It is not true, however, that these generate $H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ as an algebra. This is easy to explain in terms of the ring of invariants. There are two sets of variables $a_1,\ldots,a_k$ and $b_1,\ldots,b_k$, with $\Sigma_k$ acting on each as permuation of the indices. The class $w_i$ is the $i$th elementary symmetric function in the $a$’s, and likewise $c_i$ is the elementary symmetric function in the $b$’s. But there are many other invariants, for example $a_1b_1+\cdots+a_kb_k$.
We let $w_j^{(e)}$ be the characteristic class corresponding to the invariant element $\sum a_{i_1}\ldots a_{i_j}b_{i_1}^e\ldots b_{i_j}^e$. Note that $w_j^{(0)}=w_j$. This particular choice of invariants is not the only natural one, but it seems to be convenient in a number of ways. Among other things, these characteristic classes satisfy a Whitney formula $$w_j^{(e)}(E\oplus F)=\sum_r w_{r}^{(e)}(E)\cdot w_{j-r}^{(e)}(F).$$ Using the classes $w_j^{(e)}$ we can write down a minimal set of algebra generators for $H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$:
The indecomposables of $H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ are represented by $c_1,\ldots,c_k$ together with the classes $w_{2^i}^{(e)}$ for $1\leq
2^i \leq k$ and $0\leq e \leq \frac{k}{2^i}-1$.
Note that the above result gives a slight surprise when $e=0$. The equivariant Stiefel-Whitney classes $w_i$ are indecomposable only when $i$ is a power of $2$. This phenomenon is familiar in a slightly different (but related) context—see [@M Remark 3.4].
In practice it is unwieldy to write down a complete set of relations for $H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$. To give a sense of this, however, we do it here for $k=2$:
\[pr:k=2\] The algebra $H^{*,*}(\operatorname{Gr}_2({{{\EuScript}U}});{{\mathbb Z}}/2)$ is the quotient of the ring ${{\mathbb M}}_2[c_1,c_2,w_1,w_2,w_1^{(1)}]$ by the following relations:
- $w_1^2=\rho w_1+\tau c_1$
- $w_2^2=\rho^2 w_2 + \rho \tau \bigl (w_1c_1+w_1^{(1)}\bigr ) +
\tau^2 c_2$
- $\bigl [w_1^{(1)}\bigr ]^2 = \rho\bigl (w_1^{(1)}c_1 +
w_1c_2\bigr )+\tau(c_1^3+c_1c_2)$
- $w_1w_2=\rho w_2 + \tau\bigl (w_1c_1+w_1^{(1)}\bigr )$
- $w_1w_1^{(1)}=\rho w_1^{(1)}+\tau c_1^2 +w_2c_1$
- $w_2 w_1^{(1)}=\rho w_2 c_1 + \tau (w_1c_1^2+w_1^{(1)}c_1+w_1c_2)$.
The classes $1$, $w_1$, $w_2$, and $w_1^{(1)}$ give a free basis for $H^{*,*}(\operatorname{Gr}_2({{{\EuScript}U}});{{\mathbb Z}}/2)$ as a module over the subring ${{\mathbb M}}_2[c_1,c_2]$.
The forgetful map $H^{*,*}(\operatorname{Gr}_2({{{\EuScript}U}});{{\mathbb Z}}/2) {\rightarrow}H^*(\operatorname{Gr}_2({{\mathbb R}}^\infty);{{\mathbb Z}}/2)={{\mathbb Z}}/2[w_1,w_2]$ from equivariant to non-equivariant cohomology sends
- $\rho\mapsto 0$, $\tau\mapsto 1$
- $w_1\mapsto w_1$, $w_2\mapsto w_2$
- $c_1\mapsto w_1^2$, $c_2\mapsto w_2^2$, $w_1^{(1)}\mapsto w_1w_2+w_1^3$.
(Note that the final line can be read off from the above relations and the first two lines).
The complexity of the above description is discouraging, but the main point is really that (a) it can be done, and (b) it is tedious but mostly mechanical. We discuss both the cases $k=2$ and $k=3$ in detail in Section \[se:examples\].
In nonequivariant topology there is the relation $c_i(E{\otimes}{{\mathbb C}})=w_i^2(E)$. The first two relations in Proposition \[pr:k=2\] should be thought of as deformations of this nonequivariant relation.
One might expect the problem of describing the rings $H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ to become more tractable as $k\mapsto \infty$. In some ways it does, but even in this case we have not found a convenient way to write down a complete set of relations. See Proposition \[pr:stable\] for more information.
Open questions
--------------
(1) Our computations produce the full set of characteristic classes for equivariant real vector bundles, taking values in $H^{*,*}({-};{{\mathbb Z}}/2)$. It remains to investigate possible uses for such classes, and in particular their ties to geometry.
(2) In the classical case another way to describe the ring structure on the cohomology of Grassmannians is combinatorially, via Littlewood-Richardson rules. It might be useful to work out equivariant versions of these rules, and to describe the ring structure that way instead of by generators and relations.
(3) There is an interesting duality that appears in our description of the cohomology ring for $H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$. See Corollary \[co:duality\] and the charts preceding it. Is there some geometry underlying this duality?
(4) Although we have computed the bigraded cohomology of the infinite Grassmannians $\operatorname{Gr}_k({{{\EuScript}U}})$, our techniques do not yield the cohomology of the finite Grassmannians (in which ${{{\EuScript}U}}$ is replaced by a finite-dimensional subspace). The reason is tied to our inability to resolve all the differentials in the cellular spectral sequence. So computing the cohomology in these cases remains an open problem.
(5) We have not developed any understanding of how to analyze differentials in cellular spectral sequences, since the approach of this paper essentially amounts to a sneaky way of avoiding this. Developing a method for computing such differentials, and connecting them to geometry, is an important area for exploration.
(6) If ${{\mathbb C}}^\infty$ is given the conjugation action, then the space of complex $k$-planes $\operatorname{Gr}_k({{\mathbb C}}^\infty)$ has simple cohomology, even integrally: $H^{*,*}(\operatorname{Gr}_k({{\mathbb C}}^\infty);{{\mathbb Z}})={{\mathbb Z}}[c_1,c_2,\ldots]$ where the Chern classes $c_i$ have bidegree $(2i,i)$. These are the characteristic classes for Real vector bundles (where ‘Real’ is in the sense of Atiyah [@A2]). One can attempt a similar computation but replacing ${{\mathbb C}}^\infty$ with ${{\mathbb C}}{\otimes}{{{\EuScript}U}}$: non-equivariantly this is still ${{\mathbb C}}^\infty$, but the action is different—it is ${{\mathbb C}}$-linear rather than conjugate-linear. The computation of $H^{*,*}(\operatorname{Gr}_k({{\mathbb C}}{\otimes}{{{\EuScript}U}});{{\mathbb Z}})$ seems to be an open problem, that could perhaps be tackled by the methods of this paper. See [@FL] for some relevant, early computations.
(7) The initial motivation of this work was an interest in motivic characteristic classes for quadratic bundles, generalizing the Stiefel-Whitney classes of Delzant [@De] and Milnor [@M]; see Section \[se:motivic\] for the connection with the present paper. The original motivic question remains unsolved.
Organization of the paper
-------------------------
Section \[se:background\] gives some brief background about the theory $H^{*,*}({-};{{\mathbb Z}}/2)$. Section \[se:add-basis\] gives a first look at the ring of invariants $[H^{*,*}(\operatorname{Gr}_1({{{\EuScript}U}});{{\mathbb Z}}/2)^{{\otimes}k}\Bigr ]^{\Sigma k}$, and we provide an additive basis over the ground ring ${{\mathbb M}}_2$. We also measure the size of this ring by counting the elements of this free basis that appear in each bidegree.
In Section \[se:cells\] we describe the equivariant Schubert-cell decomposition of $\operatorname{Gr}_k({{{\EuScript}U}})$. A key point here is counting the number of Schubert cells in each bidegree. We also introduce the associated spectral sequence for computing $H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$, and in Section \[se:differentials\] we discuss Kronholm’s theorems about this spectral sequence.
Section \[se:proof\] contains the main topological part of the paper. Using the results of Sections \[se:add-basis\]–\[se:differentials\] we prove that $H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ is isomorphic to the expected ring of invariants (Theorem \[th:main1\]).
In Section \[se:mult\] we turn to the multiplicative structure of our ring of invariants. We calculate some relations here, and we identify a minimal set of generators. This section is entirely algebraic. Section \[se:examples\] then gives a presentation for the ring of invariants in the cases $k=2$ and $k=3$.
Finally, Section \[se:motivic\] describes the connection between the present work and a certain motivic problem about characteristic classes of quadratic bundles. The results of this section are not needed elsewhere in the paper. An appendix is enclosed which calculates the ring of invariants for $\Sigma_n$ acting on $\Lambda_{{\mathbb{F}}_2}(a_1,\ldots,a_n){\otimes}_{{\mathbb{F}}_2} {\mathbb{F}}_2[b_1,\ldots,b_n]$ by permutation of the indices. This purely algebraic result is needed in the body of the text, and we were unable to find a suitable reference.
Throughout this paper, if $X$ is a ${{\mathbb Z}}/2$-space then we write $\sigma\colon X{\rightarrow}X$ for the involution. For general background on $RO(G)$-graded equivariant cohomology theories we refer the reader to [@Ma].
Acknowledgments
---------------
I am grateful to Mike Hopkins for a useful conversation about this subject, and to John Greenlees for expressing some early interest.
Background on equivariant cohomology {#se:background}
====================================
Recall that ${{\mathbb M}}_2$ denotes the cohomology ring $H^{*,*}({pt};{{\mathbb Z}}/2)$. This ring is best depicted via the following diagram:
(300,220)(-80,-30) (-15,-15)(0,15)[13]{}[(1,0)[180]{}]{} (-15,-15)(15,0)[13]{}[(0,1)[180]{}]{} (82,82)(0,15)[6]{} (97,97)(0,15)[5]{} (112,112)(0,15)[4]{} (127,127)(0,15)[3]{} (142,142)(0,15)[2]{} (157,157)(0,15)[1]{} (82,52)(0,-15)[5]{} (67,37)(0,-15)[4]{} (52,22)(0,-15)[3]{} (37,7)(0,-15)[2]{} (22,-8)(0,-15)[1]{} (-10,75)[(1,0)[180]{}]{} (-10,74)[(1,0)[176]{}]{} (160,75)[(-1,0)[180]{}]{} (160,74)[(-1,0)[176]{}]{} (75,-10)[(0,1)[180]{}]{} (75,20)[(0,-1)[40]{}]{} (74,-10)[(0,1)[176]{}]{} (74,20)[(0,-1)[36]{}]{} (172,71)[$p$]{} (75,174)[$q$]{} (100,95)[$\scriptstyle{\rho}$]{} (76.5,95)[$\scriptstyle{\tau}$]{} (76.5,80)[$\scriptstyle{1}$]{} (84,50)[$\scriptstyle{\theta}$]{} (82,97)[(1,1)[60]{}]{} (82,112)[(1,1)[45]{}]{} (82,127)[(1,1)[30]{}]{} (82,142)[(1,1)[15]{}]{} (97,97)[(0,1)[62]{}]{} (112,112)[(0,1)[47]{}]{} (127,127)[(0,1)[32]{}]{} (142,142)[(0,1)[17]{}]{} (82,82)[(1,1)[80]{}]{} (82,82)[(0,1)[80]{}]{} (82,52)[(0,-1)[65]{}]{} (82,52)[(-1,-1)[65]{}]{} (82,37)[(-1,-1)[50]{}]{} (82,22)[(-1,-1)[35]{}]{} (82,7)[(-1,-1)[20]{}]{} (67,37)[(0,-1)[47]{}]{} (52,22)[(0,-1)[32]{}]{} (37,7)[(0,-1)[17]{}]{} (22,-8)[(0,-1)[2]{}]{}
Each dot represents a ${{\mathbb Z}}/2$, each vertical line represents a multiplication by $\tau$, and each diagonal line represents multiplication by $\rho$. In the “positive” range $p,q \geq 0$, the ring is therefore just ${{\mathbb Z}}/2[\tau,\rho]$. In the negative range there is an element $\theta\in {{\mathbb M}}_2^{0,-2}$ together with elements that one can formally denote $\frac{\theta}{\tau^k\rho^l} \in
{{\mathbb M}}_2^{-l,-2-k-l}$. After specifying $\theta^2=0$ this gives a complete description of the ring ${{\mathbb M}}_2$. We will refer to the subalgebra ${{\mathbb Z}}/2[\tau,\rho] \subseteq
{{\mathbb M}}_2$ as the [positive cone]{}, and the direct sum of all ${{\mathbb M}}_2^{p,q}$ for $q<0$ will be called the [negative cone]{}. See [@C] and [@D] for more background on this coefficient ring.
There are natural transformations $H^{p,q}(X;{{\mathbb Z}}/2){\rightarrow}H^p_{sing}(X;{{\mathbb Z}}/2)$ from our bigraded cohomology to ordinary singular cohomology. These are compatible with the ring structure, and when $X$ is a point they send $\tau\mapsto 1$ and $\rho\mapsto 0$. Since everything in the negative cone is a multiple of $\rho$, it follows that the entire negative cone of ${{\mathbb M}}_2$ is sent to $0$.
The graded rank functor
-----------------------
Let $I{\hookrightarrow}{{\mathbb M}}_2$ denote the kernel of the projection map ${{\mathbb M}}_2 {\rightarrow}{{\mathbb Z}}/2$. Let $M$ be a bigraded, finitely-generated free module over ${{\mathbb M}}_2$. Define the [bigraded rank]{} of $M$ by the formula $$\operatorname{rank}^{p,q} M = \dim_{{{\mathbb Z}}/2} (M/IM)^{p,q}.$$ So $\operatorname{rank}M$ should be regarded as a function ${{\mathbb Z}}^2 {\rightarrow}{{\mathbb Z}}_{\geq 0}$. Clearly $M$ is determined, up to isomorphism, by its bigraded rank.
It is usually easiest to depict the bigraded rank as a chart. For example, the bigraded rank of $H^{*,*}(\operatorname{Gr}_1({{{\EuScript}U}});{{\mathbb Z}}/2)$ is
(300,120)(-50,0) (0,0)(0,15)[8]{}[(1,0)[165]{}]{} (0,0)(15,0)[12]{}[(0,1)[105]{}]{} (5,5)[$1$]{} (20,20)[$1$]{} (35,20)[$1$]{} (50,35)[$1$]{} (65,35)[$1$]{} (80,50)[$1$]{} (95,50)[$1$]{} (110,65)[$1$]{} (125,65)[$1$]{} (140,80)[$1$]{} (155,80)[$1$]{} (168,-2)[$p$]{} (-7,102)[$q$]{} (0,0)
where the lower left corner is the $(0,0)$ spot and all unmarked boxes are regarded as having a $0$ in them.
An additive basis for the ring of invariants {#se:add-basis}
============================================
Let $R={{\mathbb M}}_2[a,b]/(a^2=\rho a+\tau b)$ where $a$ has degree $(1,1)$ and $b$ has degree $(2,1)$. Fix $k\geq 1$ and let $T_k=R^{{\otimes}k}$. Let $\Sigma_k$ act on $T_k$ in the evident way, as permutation of the tensor factors. Define ${{\mathcal{I}}nv}_k = [T_k]^{\Sigma_k}$. Our goal in this section is to investigate an additive basis for the algebra ${{\mathcal{I}}nv}_k$, regarded as a module over ${{\mathbb M}}_2$. The multiplicative structure of this ring will be discussed in Section \[se:mult\].
It will be convenient to rename the variables in the $i$th copy of $R$ as $a_i$ and $b_i$. So $T_k$ is the quotient of ${{\mathbb M}}_2[a_1,\ldots,a_k,b_1,\ldots,b_k]$ by the relations $a_i^2=\rho a_i+\tau b_i$, for $1\leq i\leq k$. Let $$w_i=\sigma_i(a_1,\ldots,a_k)$$ be the $i$th elementary symmetric function in the $a$’s, and let $$c_i=\sigma_i(b_1,\ldots,b_k)$$ be the $i$th elementary symmetric function in the $b$’s. These are the most obvious elements of ${{\mathcal{I}}nv}_k$, but there are others as well. For example, the element $a_1b_1+a_2b_2+\cdots+a_kb_k$ is invariant under the action of $\Sigma_k$. We will need some notation to help us describe these other elements of ${{\mathcal{I}}nv}_k$.
If $m$ is a monomial in the $a$’s and $b$’s, write $[m]$ for the smallest homogeneous polynomial in $T_k$ which contains $m$ as one of its terms. By ‘smallest’ we mean the smallest number of monomial summands. If $H\leq \Sigma_k$ is the stabilizer of $m$, then $[m]$ is the sum $\sum_{gH\in \Sigma_k/H} gm$. Here are some examples:
(i) $[a_1b_1]=a_1b_1+a_2b_2+\ldots+a_kb_k$
(ii) $[a_1b_1b_2]=\sum\limits_{i \neq j} a_ib_ib_j$
(iii) $[a_1b_2b_3]=\sum\limits_{\text{$i,j,k$ distinct}}a_ib_jb_k$
(iv) $[a_1a_2]=w_2$.
Notice that $$[a_1^2b_2]=\sum_{i\neq j} a_i^2b_j = \sum_{i\neq j} (\rho
a_i +\tau b_i)b_j= \rho \sum_{i\neq j} a_ib_j + \tau \sum_{i\neq
j} b_ib_j = \rho[a_1b_2]+\tau[b_1b_2].$$ A similar computation shows that if $m$ is any monomial with an $a_i^2$ then $[m]$ is an ${{\mathbb M}}_2$-linear combination of monomials $[m_j]$ with $\deg m_j < \deg m$.
The following proposition is fairly clear:
As an ${{\mathbb M}}_2$-module, ${{\mathcal{I}}nv}_k$ is free with basis consisting of all elements $[a_1^{\epsilon_1}\ldots a_k^{\epsilon_k}b_1^{d_1}\ldots b_k^{d_k}]$, where each $d_i \geq 0$ and each $\epsilon_i \in \{0,1\}$.
Our next task is to count how many of the above basis elements appear in any given bidegree. Write $\operatorname{Mon}_k$ for the set of monomials in the variables $a_1,\ldots,a_k,b_1,\ldots,b_k$ having the property that the exponent on each $a_i$ is at most $1$. The above proposition implies that ${{\mathcal{I}}nv}_k$ has a free basis over ${{\mathbb M}}_2$ that is in bijective correspondence with the set of orbits $\operatorname{Mon}_k\!/\Sigma_k$; this correspondence preserves the bigraded degree. From now on we will refer to this basis as THE free basis for ${{\mathcal{I}}nv}_k$. We can easily write down a list of these basis elements in any given bidegree. For instance, here is the list in low dimensions, assuming $k$ is large (with the degrees of the elements given to the left): $$\xymatrixrowsep{0.2pc}\xymatrix{
(1,1): & [a_1] && (4,3): & [a_1a_2b_1], [a_1a_2b_3] \\
(2,1): & [b_1] && (4,4): & [a_1a_2a_3a_4] \\
(2,2): & [a_1a_2] && (5,3): & [a_1b_1^2], [a_1b_2^2], [a_1b_1b_2], [a_1b_2b_3] \\
(3,2): & [a_1b_1], [a_1b_2] && (5,4): & [a_1a_2a_3b_1], [a_1a_2a_3b_4] \\
(3,3): & [a_1a_2a_3] && (5,5): & [a_1a_2a_3a_4a_5]\\
(4,2): & [b_1^2], [b_1b_2] && (6,3): & [b_1b_2b_3],[b_1^3],[b_1^2b_2] \\
}$$
We can count the number of generators in each bidegree in terms of certain kinds of partitions. Given $n$, $k$, and $j$, let ${\operatorname{part}_{n,\leq k}[j]}$ denote the number of partitions of $n$ into $k$ nonnegative integers such that exactly $j$ of the integers are odd. For example, ${\operatorname{part}_{8,\leq 5}[4]}= 4$ because it counts the following partitions: $01133$, $01115$, $11114$, and $11123$.
\[pr:rank-Inv\] For any $p$, $q$, and $k$ one has $\operatorname{rank}^{p,q}({{\mathcal{I}}nv}_k)={\operatorname{part}_{p,\leq k}[2q-p]}$.
Let $w$ be a monomial in the variables $a_1,\ldots,a_k,b_1,\ldots,b_k$ where each $a_i$ appears at most once. We’ll say that $w$ is [pure]{} if all the symbols in $w$ have the same subscript: e.g., $a_1b_1^3$ is pure, but $a_1a_2b_1^2$ is not. The monomial $w$ can be written in a unique way as $w=w(1)w(2)\cdots w(k)$ where each $w(i)$ is pure and only contains the subscript $i$.
Regard $a_i$ as having degree $1$ and $b_i$ as having degree $2$. If $v$ is a pure monomial, let $d(v)$ be its total degree. Finally, if $w$ is any monomial then let $\eta(w)$ be the partition $$\eta(w)=[d(w(1)),\,d(w(2)),\,\ldots,\,d(w(k))].$$ For example, if $w=a_1a_2a_3b_1^2b_2b_4$ then $\eta(w)=[5312]$.
It is clear that all the $\Sigma_k$-cognates of $w$ give rise to the same partition, and so we have a function $$\operatorname{Mon}_k\!/\Sigma_k {\stackrel{\eta}{{\longrightarrow}}} \{\text{partitions with $\leq k$
pieces}\}.$$ Moreover, this is a bijection because the partition is enough to recover the invariant element $[w]$: if the $i$th number in our partition is $2r$ then we write $b_i^r$, and if it is $2r+1$ we write $a_ib_i^r$, and then we multiply these terms together. For example, given the partition $[34678]$ we would write $[a_1b_1b_2^2b_3^3a_4b_4^3b_5^4]$. This apparently depends on the order in which we listed the numbers in the partition, but this dependence goes away when we take the $\Sigma_k$-orbit.
Clearly the topological degree of the monomial $w$ equals the sum of the elements in the partition $\eta(w)$. Also, the number of odd elements of the partition is equal to the number of $a_i$’s in $w$. But one readily checks that $$\begin{aligned}
\text{weight of $w$} = \#b_i\text{'s} + \# a_i\text{'s} &=
\frac{\text{(topl. degree of $w$)}-\#a_i\text{'s}}{2} \, \,+ \, \# a_i\text{'s}\\
& =
\frac{\text{(topl. degree of $w$)}
+ \#a_i\text{'s}}{2}.\end{aligned}$$ So the number of odd elements in the partition $\eta(w)$ is $2q-p$, where $q$ is the weight of $w$ and $p$ is the topological degree of $w$.
As an example of the above proposition, here is a portion of the bigraded rank function for ${{\mathcal{I}}nv}_4$:
(300,140)(-50,0) (0,0)(0,15)[10]{}[(1,0)[225]{}]{} (0,0)(15,0)[16]{}[(0,1)[135]{}]{} (5,5)(15,15)[5]{}[$1$]{} (35,20)[$1$]{} (50,35)(15,15)[3]{}[$2$]{} (80,50)[$4$]{} (65,35)[$2$]{} (95,50)[$3$]{} (95,65)[$5$]{} (95,80)[$1$]{} (110,65)[$7$]{} (110,80)[$4$]{} (125,65)[$5$]{} (125,80)[$8$]{} (125,95)[$2$]{} (138,80)[$11$]{} (140,95)[$7$]{} (155,80)[$6$]{} (152,95)[$14$]{} (155,110)[$3$]{} (168,95)[$16$]{} (168,110)[$11$]{} (185,95)[$9$]{} (182,110)[$20$]{} (185,125)[$5$]{} (198,110)[$23$]{} (198,125)[$16$]{} (213,110)[$11$]{} (213,125)[$30$]{} (228,-2)[$p$]{} (-7,132)[$q$]{} (0,0)
And here is a similar chart for ${{\mathcal{I}}nv}_5$:
(300,150)(-50,-5) (0,0)(0,10)[15]{}[(1,0)[220]{}]{} (0,0)(10,0)[23]{}[(0,1)[140]{}]{} (2.5,2.5)(10,10)[6]{}[$\scriptstyle{1}$]{} (22.5,12.5)[$\scriptstyle{1}$]{} (32.5,22.5)(10,10)[4]{}[$\scriptstyle{2}$]{} (72.5,62.5)[$\scriptstyle{1}$]{} (42.5,22.5)(50,50)[2]{}[$\scriptstyle{2}$]{} (52.5,32.5)(30,30)[2]{}[$\scriptstyle{4}$]{} (62.5,42.5)(10,10)[2]{}[$\scriptstyle{5}$]{} (62.5,32.5)(50,50)[2]{}[$\scriptstyle{3}$]{} (72.5,42.5)(30,30)[2]{}[$\scriptstyle{7}$]{} (82.5,52.5)(10,10)[2]{}[$\scriptstyle{9}$]{} (82.5,42.5)(50,50)[2]{}[$\scriptstyle{5}$]{} (91.5,52)(30,30)[2]{}[$\scriptstyle{12}$]{} (101,62)(10,10)[2]{}[$\scriptstyle{16}$]{} (102.5,52.5)(50,50)[2]{}[$\scriptstyle{7}$]{} (111.5,62)(30,30)[2]{}[$\scriptstyle{18}$]{} (122,72)(10,10)[2]{}[$\scriptstyle{25}$]{} (121.5,62.5)(50,50)[2]{}[$\scriptstyle{10}$]{} (131.5,72.5)(30,30)[2]{}[$\scriptstyle{27}$]{} (141,82.5)(10,10)[2]{}[$\scriptstyle{39}$]{} (141.5,72.5)(50,50)[2]{}[$\scriptstyle{13}$]{} (151.5,82)(30,30)[2]{}[$\scriptstyle{38}$]{} (161,92)(10,10)[2]{}[$\scriptstyle{56}$]{} (161.5,82.5)(50,50)[2]{}[$\scriptstyle{18}$]{} (171.5,92)(30,30)[2]{}[$\scriptstyle{53}$]{} (181,102)(10,10)[2]{}[$\scriptstyle{80}$]{} (181.5,92.5)(50,50)[1]{}[$\scriptstyle{23}$]{} (191.5,102)(30,30)[1]{}[$\scriptstyle{71}$]{} (200,112)(10,10)[2]{}[$\scriptscriptstyle{109}$]{} (201.5,102.5)(50,50)[1]{}[$\scriptstyle{30}$]{} (211.5,112)(30,30)[1]{}[$\scriptstyle{94}$]{} (225,-2)[$p$]{} (-7,135)[$q$]{} (0,0)
There are some evident patterns in these charts. For example, if one starts at spot $(2p,p)$ and reads diagonally upwards along a line of slope $1$ then the resulting numbers have an evident symmetry. This comes from a symmetry of the ${\operatorname{part}_{n,\leq k}[j]}$ numbers:
\[le:part-duality\] For any $n$, $k$, and $j$, one has ${\operatorname{part}_{n,\leq k}[j]}={\operatorname{part}_{n+(k-2j),\leq k}[k-j]}$.
Suppose $u_1,\ldots,u_k$ is a partition of $n$ in which there are exactly $j$ odd numbers—we can arrange the indices so that these are $u_1,\ldots,u_j$. Subtract $1$ from all the odd numbers and add $1$ to all the even numbers: this yields the collection of numbers $u_1-1,\ldots,u_j-1,u_{j+1}+1,\ldots,u_k+1$. This is a partition of $n+k-2j$ in which there are exactly $k-j$ odd numbers. One readily checks that this gives a bijection between the two kinds of partitions.
The diagonal symmetries in our rank charts are as follows:
\[co:duality\] For any $p$, $r$, and $k$, one has $$\operatorname{rank}^{2p+r,p+r}({{\mathcal{I}}nv}_k)=\operatorname{rank}^{2p+k-r,p+k-r}({{\mathcal{I}}nv}_k).$$
This is immediate from Proposition \[pr:rank-Inv\] and Lemma \[le:part-duality\].
The numbers in the rank chart for ${{\mathcal{I}}nv}_k$ organize themselves naturally into lines of slope $\frac{1}{2}$. To explain this (and because it will be needed later) we introduce the following terminology. A [successor]{} of a partition $\alpha$ is any partition obtained by adding $2$ to exactly one of the numbers in $\alpha$. For example, $011$ has two successors: $013$ and $112$. If a partition $\beta$ is obtained from $\alpha$ by a sequence of successors, we say that $\beta$ is a [descendent]{} of $\alpha$. Finally, a partition $\alpha$ will be called [minimal]{} if it is not a successor of any other partition.
For the set of all partitions consisting of $k$ nonnegative numbers, the following facts are immediate:
(1) There are exactly $k+1$ minimal partitions: $00\ldots 0$, $00\ldots 01$, $00\ldots 011$, $\ldots$, and $11\ldots 1$.
(2) Every partition $\alpha$ is a descendent of a unique minimal partition, namely the one obtained by replacing each $\alpha_i$ with either $0$ or $1$ depending on whether $\alpha_i$ is even or odd.
The partitions consisting of $k$ nonnegative numbers, with exactly $j$ odd numbers, form a tree under the successor operation: and the numbers of such partitions forms the line of slope $\frac{1}{2}$ ascending from spot $(j,j)$ in our rank charts.
The following corollary records the evident bounds on the nonzero numbers in our rank charts. The proof is immediate from the things we have already said, or it could be proven directly from Proposition \[pr:rank-Inv\].
\[co:bounds\] The bigraded rank function of ${{\mathcal{I}}nv}_k$ is nonzero only in the region bounded by the three lines $y=x$, $y=\frac{1}{2}x$, and $y=\frac{1}{2}x+\frac{k}{2}$. That is to say, the elements of our free basis for ${{\mathcal{I}}nv}_k$ appear only in bidegrees $(a,b)$ where $\frac{a}{2}\leq b\leq a$ if $a\leq k$, and $\frac{a}{2}
\leq b \leq \frac{1}{2}a+\frac{k}{2}$ if $a\geq k$.
Schubert cells and a spectral sequence {#se:cells}
======================================
Given a sequence of integers $1\leq a_1 < a_2 <
\cdots < a_k$, define the associated Schubert cell in $\operatorname{Gr}_k({{{\EuScript}U}})$ by $$\Omega_a = \{ V \in \operatorname{Gr}_k({{{\EuScript}U}})\,|\, \dim(V\cap {{{\EuScript}U}}^{a_i}) \geq i \}.$$ Here ${{{\EuScript}U}}^{n}\subseteq {{{\EuScript}U}}$ is simply the subspace of vectors whose $r$th coordinates all vanish for $r>n$, which we note is closed under the ${{\mathbb Z}}/2$-action. It will be convenient for us to regard the $a$-sequence as giving a “$*$-pattern”, in which one takes an infinite sequence of empty boxes and places a single $*$ in each box corresponding to an $a_i$. If the boxes represent the standard basis elements of ${{{\EuScript}U}}$, then the $*$’s represent where the jumps in dimension occur for subspaces $V$ lying in the interior of $\Omega_a$. These $*$-patterns will be used several times in our discussion below.
It is somewhat more typical to use a different indexing convention here. Define $\sigma_i=a_i-i$, so that we have $0\leq \sigma_1 \leq \sigma_2\leq \cdots \leq \sigma_k$. Write $\Omega(\sigma)$ for the same Schubert cell as $\Omega_a$, which has dimension equal to $\sum_i \sigma_i$. Define a [[$k$-Schubert symbol]{}]{} to be an increasing sequence $\sigma_1\leq \sigma_2\leq \cdots \leq \sigma_k$. To get the associated $*$-pattern, skip over $\sigma_1$ empty boxes and then place a $*$; then skip over $\sigma_2-\sigma_1$ empty boxes and place another $*$; then skip over $\sigma_3-\sigma_2$ empty boxes, and so forth. For example, the Schubert symbol $[0235]$ corresponds to the $*$-pattern $[* {{\underline}{\ \ }\,}{{\underline}{\ \ }\,}* {{\underline}{\ \ }\,}* {{\underline}{\ \ }\,}{{\underline}{\ \ }\,}*]$, or the $a$-sequence $(1,4,6,9)$.
Let $F_r\subseteq \operatorname{Gr}_k({{{\EuScript}U}})$ be the union of all the Schubert cells of dimension less than or equal to $r$. This filtration gives rise to a spectral sequence on cohomology in the usual way, where the $E_1$-term is the direct sum $\oplus_{\sigma}
\tilde{H}^{*,*}(S^{a_\sigma,b_\sigma})$ where $\sigma$ ranges over all $k$-Schubert symbols and $(a_\sigma,b_\sigma)$ is the bidegree of the associated cell. We will next describe an algorithm for producing this bidegree.
Picture the row of symbols $+-+-+-\cdots$ going on forever, with the initial symbol regarded as the first (rather than the zeroth). These symbols represent the ${{\mathbb Z}}/2$-action on the standard basis elements of ${{{\EuScript}U}}$. For each $i$ in the range $1\leq i\leq k$, change the $a_i$th symbol to an asterisk $*$. Then for each $i$, define $u_i$ to be $$u_i=\begin{cases}
\text{the total number of $+$ signs to the left
of the $i$th asterisk} & \text{if $a_i$ is even} \\
\text{the total number of $-$ signs to the left
of the $i$th asterisk} & \text{if $a_i$ is odd.}
\end{cases}$$ Finally, define the [cell-weight]{} of the Schubert symbol to be $\sum_i u_i$. We claim that the open Schubert cell corresponding to $\sigma$ is isomorphic to ${{\mathbb R}}^{n,k}$ where $n=\sum \sigma_i$ and $k$ is the cell-weight of $\sigma$.
Let us say the above in a slightly different way. We think in terms of $*$-patterns, but where the boxes contain alternating $+$ and $-$ signs and the $*$’s eradicate whatever sign was in their box. For the topological dimension of a cell, we count the number of empty boxes to the left of each $*$ and add these up. For the weight we do a fancier kind of counting: if the $*$ replaced a $+$ sign then we count the number of $-$ signs to the left of it, whereas if it replaced a $-$ we count the number of $+$ signs to the left. And again, we add up our answers for each $*$ in the pattern to get the total weight. For example, consider the Schubert symbol $\sigma=[135]$ which has topological dimension $9$. The corresponding $a$-sequence is $(2,5,8)$, and this gives the $*$-pattern $+*+-*-+*+-+-\cdots$ So $u_1=1$, $u_2=1$, $u_3=3$, and therefore the bidegree of $\Omega(\sigma)$ is $(9,5)$.
Consider the Grassmannian $\operatorname{Gr}_2({{{\EuScript}U}}^6)$. There are $\tbinom{6}{2}=15$ Schubert cells. We list all the $*$-patterns and the bidegrees of the associated cells: $$\begin{aligned}
&**+-+- \quad (0,0) \qquad +**-+- \quad (2,1) \qquad +-*-*- \quad (5,3) \\
&*-*-+- \quad (1,1) \qquad +*+*+- \quad (3,3) \qquad +-*-+* \quad (6,3) \\
&*-+*+- \quad (2,1) \qquad +*+-*- \quad (4,2) \qquad +-+**- \quad (6,3) \\
&*-+-*- \quad (3,2) \qquad +*+-+* \quad (5,4) \qquad +-+*+* \quad (7,5) \\
&*-+-+* \quad (4,2) \qquad +-**+- \quad (4,2) \qquad +-+-** \quad (8,4) \\\end{aligned}$$
We need to justify our procedure for determining the weight of a Schubert cell. Given an $a$-sequence, points in the interior of the associated Schubert cell $\Omega_a$ are in bijective correspondence with matrices of a form such as $$\begin{bmatrix}
? & ? & 1 & 0 & 0 & 0 & 0 & 0\\
? & ? & 0 & ? & 1 & 0 & 0 & 0 \\
? & ? & 0 & ? & 0 & ? & ? & 1
\end{bmatrix}.$$ (The matrix given is for the case of $\operatorname{Gr}_3({{{\EuScript}U}})$ and the $a$-sequence $(3,5,8)$). The matrix in question has $1$’s in the columns given by the $a$-sequence, each $1$ is followed by only zeros in its row, and each $1$ is the only nonzero entry in its column. The set of such matrices is a Euclidean space of dimension equal to the number of “?” symbols. Such a matrix determines a point in $\operatorname{Gr}_k({{{\EuScript}U}})$ by taking the span of its rows, and any $k$-plane in the interior of $\Omega_a$ has a unique basis of the above form. This is all standard from non-equivariant Schubert calculus. In the equivariant case, we have a ${{\mathbb Z}}/2$-action on the set of such matrices induced by the ${{\mathbb Z}}/2$-action on ${{{\EuScript}U}}$. In our above example, the action is $$\begin{bmatrix}
b & c & 1 & 0 & 0 & 0 & 0 & 0\\
d & e & 0 & f & 1 & 0 & 0 & 0 \\
g & h & 0 & i & 0 & j & k & 1
\end{bmatrix}\mapsto
\begin{bmatrix}
b & -c & 1 & 0 & 0 & 0 & 0 & 0\\
d & -e & 0 & -f & 1 & 0 & 0 & 0 \\
g & -h & 0 & -i & 0 & -j & k & -1
\end{bmatrix}.$$ Notice that the matrix on the right is not in our standard form. To convert it to standard form we multiply the third row by $-1$ to get $$\begin{bmatrix}
b & c & 1 & 0 & 0 & 0 & 0 & 0\\
d & e & 0 & f & 1 & 0 & 0 & 0 \\
g & h & 0 & i & 0 & j & k & 1
\end{bmatrix}\mapsto
\begin{bmatrix}
b & -c & 1 & 0 & 0 & 0 & 0 & 0\\
d & -e & 0 & -f & 1 & 0 & 0 & 0 \\
-g & h & 0 & i & 0 & j & -k & 1
\end{bmatrix}.$$ So as a ${{\mathbb Z}}/2$-representation we have ${{\mathbb R}}^{10}$ with five sign changes, and this is ${{\mathbb R}}^{10,5}$. It is now easy to go from this overall picture to the specific formula for the cell-weight that was given above.
We now know how to compute the bigraded Schubert cell decomposition for any Grassmannian. It is useful to look at a specific example, so here is the Schubert cell picture for $\operatorname{Gr}_5({{{\EuScript}U}})$. Each box gives the number of Schubert cells of the given bidegree.
(300,200)(-50,0) (0,0)(0,10)[20]{}[(1,0)[220]{}]{} (0,0)(10,0)[23]{}[(0,1)[190]{}]{}
(2.5,2.5)[$\scriptstyle{1}$]{} (22.5,12.5)[$\scriptstyle{2}$]{} (42.5,22.5)[$\scriptstyle{5}$]{} (62.5,32.5)[$\scriptstyle{9}$]{} (81.5,42.5)[$\scriptstyle{16}$]{} (101.5,52.5)[$\scriptstyle{25}$]{} (121.5,62.5)[$\scriptstyle{39}$]{} (141.5,72.5)[$\scriptstyle{56}$]{} (161.5,82.5)[$\scriptstyle{80}$]{} (180.5,92.5)[$\scriptscriptstyle{109}$]{} (200.5,102.5)[$\scriptscriptstyle{147}$]{} (12.5,12.5)[$\scriptstyle{1}$]{} (32.5,22.5)[$\scriptstyle{2}$]{} (52.5,32.5)[$\scriptstyle{5}$]{} (72.5,42.5)[$\scriptstyle{9}$]{} (91.5,52.5)[$\scriptstyle{16}$]{} (111.5,62.5)[$\scriptstyle{25}$]{} (131.5,72.5)[$\scriptstyle{39}$]{} (151.5,82.5)[$\scriptstyle{56}$]{} (171.5,92.5)[$\scriptstyle{80}$]{} (190.5,102.5)[$\scriptscriptstyle{109}$]{} (210.5,112.5)[$\scriptscriptstyle{147}$]{} (32.5,32.5)[$\scriptstyle{1}$]{} (52.5,42.5)[$\scriptstyle{2}$]{} (72.5,52.5)[$\scriptstyle{4}$]{} (92.5,62.5)[$\scriptstyle{7}$]{} (111.5,72.5)[$\scriptstyle{12}$]{} (131.5,82.5)[$\scriptstyle{18}$]{} (151.5,92.5)[$\scriptstyle{27}$]{} (171.5,102.5)[$\scriptstyle{38}$]{} (191.5,112.5)[$\scriptstyle{53}$]{} (210.5,122.5)[$\scriptscriptstyle{71}$]{} (62.5,62.5)[$\scriptstyle{1}$]{} (82.5,72.5)[$\scriptstyle{2}$]{} (102.5,82.5)[$\scriptstyle{4}$]{} (122.5,92.5)[$\scriptstyle{7}$]{} (141.5,102.5)[$\scriptstyle{12}$]{} (161.5,112.5)[$\scriptstyle{18}$]{} (181.5,122.5)[$\scriptstyle{27}$]{} (201.5,132.5)[$\scriptstyle{38}$]{} (102.5,102.5)[$\scriptstyle{1}$]{} (122.5,112.5)[$\scriptstyle{1}$]{} (142.5,122.5)[$\scriptstyle{2}$]{} (162.5,132.5)[$\scriptstyle{3}$]{} (182.5,142.5)[$\scriptstyle{5}$]{} (202.5,152.5)[$\scriptstyle{7}$]{} (152.5,152.5)[$\scriptstyle{1}$]{} (172.5,162.5)[$\scriptstyle{1}$]{} (192.5,172.5)[$\scriptstyle{2}$]{} (212.5,182.5)[$\scriptstyle{3}$]{} (225,-2)[$p$]{} (-7,185)[$q$]{} (0,0)
Note that the numbers appearing along lines of slope $\frac{1}{2}$ are the same as the numbers we saw in the rank chart for ${{\mathcal{I}}nv}_5$, except that the lines are arranged differently in the plane. We will need a precise statement:
\[pr:cell-bound\] Let $X$ be the $E_1$-term of the cellular spectral sequence for $\operatorname{Gr}_k({{{\EuScript}U}})$ based on the Schubert cell filtration. Then the nonzero entries in the rank chart for $X$ are bordered by the lines $y=x$, $y=\frac{1}{2}x$, and $y=\frac{1}{2} (x+\binom{k+1}{2})$.
Moreover, for any $j,r$ one has $\operatorname{rank}^{j+2r,j+r}X=0$ unless $j=\binom{i}{2}$ for some $i$ in the range $1\leq i \leq k+1$. And finally, if $j=\binom{i}{2}$ then $$\operatorname{rank}^{j+2r,j+r}X = \operatorname{part}_{2r+\gamma_i,\leq k}[\gamma_i]=
\operatorname{rank}^{\gamma_i+2r,\gamma_i+r}({{\mathcal{I}}nv}_k)$$ where $\gamma$ is the function defined by $$\gamma_i=\begin{cases}
\tfrac{k+i}{2} & \text{if $k-i$ is even} \\
\tfrac{k+1-i}{2} & \text{if $k-i$ is odd}.
\end{cases}$$
The mathematical phrasing of the above proposition is somewhat awkward, but it says something very concrete. Namely, the nonzero entries in the rank chart for $X$ are divided into rays of slope $\frac{1}{2}$ emanating from the points $\bigl (\binom{i}{2},\binom{i}{2}\bigr )$ for $1\leq i \leq k+1$. Starting from $\bigl (\binom{k+1}{2},\binom{k+1}{2}\bigr )$ and working towards the origin along the $y=x$ line, mentally label each vertex with the numbers in the sequence $$0,\ k,\ 1,\ k-1, \ 2,\ k-2,\ 3,\ k-3,\ \ldots$$ These are the numbers $\gamma_{k+1}, \gamma_{k},\ldots$. Then in $\operatorname{rank}(X)$, the $r$th term from $\bigl (\binom{i}{2},\binom{i}{2}\bigr )$ along the ray of slope $\frac{1}{2}$ is equal to $\operatorname{part}_{2r+\gamma_i,\leq
k}[\gamma_i]$.
In order to prove Proposition \[pr:cell-bound\] we need to introduce some language for bookkeeping. Define a [successor]{} of a $*$-pattern to be a pattern made by moving one of the $*$’s two spots to the right (note that one can only do this if the new spot for the $*$ started out empty). In terms of $a$-sequences, a successor is an $a$-sequence obtained by adding $2$ to one of the $a_i$’s. For example, the $a$-sequence $123$ has exactly two successors, namely $125$ and $145$. A $*$-pattern (or $a$-sequence) is said to be [minimal]{} if it is not the successor of another pattern (or sequence); said differently, a $*$-pattern is minimal if one cannot move any $*$ two places to the left. The sequences $123$ and $124$ are both minimal, but $125$ is not; these correspond to the $*$-patterns $[***]$, $[**+*]$, and $[**+-*]$.
Observe that taking the successors of a $*$-pattern increases the bidegree of the associated Schubert cell by $(2,1)$. This is easy to explain in terms of the following picture, showing an arbitrary $*$-pattern and a successor obtained by moving one of the $*$’s:
$$\xymatrixcolsep{1pc}\xymatrix{
\cdots & \square & {*} \ar@/^4ex/[rr]& \square & \square & \square & {*}
& {*} & \square & \cdots
}$$
The count of empty boxes to the left of each $*$ is the same for the two patterns, except for the $*$ that got moved: and for that $*$ the count has increased by $2$. Likewise, the number of $+/-$ signs in the empty boxes stays the same for each $*$ in the two patterns, except again for the $*$ that got moved: and for that $*$ the number of $+$ and $-$ signs to the left of it each got increased by $1$.
The fact that the successor relation increases the bidegree by $(2,1)$ explains why our Schubert cell chart breaks up into rays of slope $\frac{1}{2}$. The number of such rays will be governed by the number of minimal $*$-patterns, so we investigate this next.
It is clear that for a $*$-pattern to be minimal it must be true that any two successive $*$’s have at most one empty space between them. Moreover, as soon as one has an empty space in the $*$-pattern then all successive $*$’s must be separated by one empty space. So for patterns with $k$ asterisks, there are exactly $k+1$ minimal patterns; they are completely described by saying which $*$ has the first blank space after it (the count is $k+1$ because the first blank might appear after the [*zeroth*]{} star, which doesn’t actually exist). One thing that is easy to verify about these minimal patterns is that the corresponding Schubert cells each have bidegree $(p,p)$, for some values of $p$; that is, the topological dimension and weight coincide. Recall that computing both the topological dimension and the weight from the $*$-patterns amounts to counting empty boxes to the left of each $*$, with the weight computation involving some restrictions on which boxes get counted. For the minimal $*$-patterns, the placement of the $*$’s results in these restrictions all being vacuous: that is, all empty boxes are counted.
The minimal $*$-patterns correspond to the following $a$-sequences: $$(1,2,3,\ldots,k),(1,2,\ldots,k-1,k+1),(1,2,\ldots,k-2,k,k+2),\ldots,(1,3,5,\ldots,2k-1)$$ and $(2,4,6,\ldots,2k)$ (the first $k$ of these follow a common pattern, the final one does not). The associated Schubert symbols are $$[00\ldots 0], \ [00\ldots 01], \ [00\ldots 12],\ \ldots,\ [012\ldots (k-1)],\quad\text{and}\quad
[123\ldots k].$$ The topological dimensions are therefore $\binom{i}{2}$ for $1\leq
i\leq k+1$, so the minimal $*$-patterns correspond to Schubert cells of bidegree $(\binom{i}{2},\binom{i}{2})$ for $i$ in this range.
We have determined in the preceding discussion that the successor relation breaks the Schubert-cell chart into $k+1$ rays of slope $\frac{1}{2}$, each ray starting at a point $(\binom{i}{2},\binom{i}{2})$ for $1\leq i\leq k+1$. The starting points are the minimal $*$-patterns determined above. What remains to be shown is that the number of cells counted along these rays matches similar rays in the count of partitions we saw in our study of $\operatorname{rank}({{\mathcal{I}}nv}_k)$. This is where the awkward rearrangement of the rays must be accounted for.
We have the classical bijection between Schubert cells and partitions, which associates to any $*$-pattern the corresponding Schubert symbol. For the rest of this proof we completely discard this bijection, and instead use a [*different*]{} bijection, to be described next. This is the crux of the argument. See Remark \[re:strange-bijection\] below for more information about where this new bijection comes from.
Given a partition $\sigma$ with $k$ nonnegative parts, regard this as two partitions $\sigma^{ev}$ and $\sigma^{odd}$ by simply separating the even and odd numbers. For example, if $\sigma=[00123]$ then $\sigma^{ev}=[002]$ and $\sigma^{odd}=[13]$. Note that in both $\sigma^{ev}$ and $\sigma^{odd}$ the difference of consecutive pieces (when ordered from least to greatest) will always be even.
Consider a string of empty boxes labelled $1,2,3,\ldots$ Take $\sigma^{ev}$ and convert this to a $*$-pattern in what is essentially the usual way, but placing the $*$’s only in the [*even*]{} boxes of the pattern. If $\sigma^{ev}=[u_1,\ldots,u_r]$ then skip over $\frac{u_1}{2}$ even boxes and place a $*$, then skip over $\frac{u_2-u_1}{2}$ even boxes and place a $*$, and so on. Likewise, convert $\sigma^{odd}$ to a $*$-pattern in the usual way but placing the $*$’s only in the [*odd boxes*]{}. If $\sigma^{odd}=[v_1,\ldots,v_r]$ then skip over $\frac{v_1-1}{2}$ odd boxes and place a $*$, then skip over $\frac{v_2-v_1}{2}$ odd boxes and place a $*$, and so on. This awkward procedure is best demonstrated by an example, so return to $\sigma=[00123]$. Then $\sigma^{ev}=[002]$, which corresponds to the $*$-pattern $[\,{{\underline}{\ \ }\,}* {{\underline}{\ \ }\,}* {{\underline}{\ \ }\,}{{\underline}{\ \ }\,}{{\underline}{\ \ }\,}*]$, and $\sigma^{odd}=[13]$ which corresponds to the $*$-pattern $[*{{\underline}{\ \ }\,}{{\underline}{\ \ }\,}{{\underline}{\ \ }\,}*]$. So the combined pattern is $[** {{\underline}{\ \ }\,}** {{\underline}{\ \ }\,}{{\underline}{\ \ }\,}*]$.
We have given a function from partitions with $k$ pieces to $*$-patterns with $k$ asterisks. It is easy to see that this is a bijection; an example of the inverse should suffice. For the $*$-pattern $$[* {{\underline}{\ \ }\,}{{\underline}{\ \ }\,}* {{\underline}{\ \ }\,}* * * {{\underline}{\ \ }\,}{{\underline}{\ \ }\,}{{\underline}{\ \ }\,}* {{\underline}{\ \ }\,}* {{\underline}{\ \ }\,}*]$$ the only odd boxes occupied are $1$ and $7$. The associated $\sigma^{odd}$ is $[15]$, because $\frac{5-1}{2}$ accounts for the two skipped odd boxes between them. The occupied even boxes are $4,6,8,12,14,16$ and so $\sigma^{ev}=[222444]$. The partition associated to the above $*$-pattern is therefore $\sigma=[12224445]$.
The point of this strange bijection is the following: it carries the successor relation for $*$-patterns to the successor relation for partitions (the latter defined back in Section \[se:add-basis\]). This is easy to see, and we leave it to the reader—but also see Remark \[re:strange-bijection\] below for a strong hint.
Using the above bijection, the minimal $*$-patterns of $k$ asterisks correspond to the partitions $[00\ldots 0]$, $[00\ldots 01]$, $[00\ldots
001]$,$\ldots$, and $[11\ldots 1]$ (each with $k$ pieces). For example, if $k$ is even then the $*$-pattern with $a$-sequence $(1,2,3,\ldots,k)$ corresponds to the partition $[00\ldots 011\ldots 1]$ where there are $\frac{k}{2}$ zeros and $\frac{k}{2}$ ones. It is somewhat better to order the partitions as
\[eq:min-parts\] , \[11…1\], \[00…01\], \[011…1\], \[00…001\], \[0011…1\], …
because in this order the topological degrees of the associated Schubert cells are $$\tbinom{k+1}{2}, \tbinom{k}{2}, \tbinom{k-1}{2} ,\ldots,
\tbinom{2}{2}, \tbinom{1}{2}.$$
For later use, let $\mu(i)$ be the number of $1$’s in the $i$th partition from the list (\[eq:min-parts\]), with $1\leq i \leq k+1$. This sequence is $\mu(1)=0$, $\mu(2)=k$, $\mu(3)=1$, $\mu(4)=k-1$, and so forth. Note that $\mu(i)=\gamma_{k+2-i}$, for the $\gamma$-function defined in Proposition \[pr:cell-bound\].
We can now wrap up the argument. We have a bijection between $*$-patterns and partitions, and it preserves the successor relation; it therefore also preserves the trees of descendants. In both settings (of $*$-patterns and partitions) one finds exactly $k+1$ minimal elements—and therefore $k+1$ trees. The minimal partitions are the ones in which each piece is either $0$ or $1$. For the $*$-patterns we have computed that the minimal elements correspond to cells of bidegree $\bigl (\binom{i}{2},\binom{i}{2}
\bigr )$ for $1\leq i\leq k+1$, and that the partition associated to this $*$-pattern has exactly $\gamma_i$ pieces equal to $1$ (and the rest zeros). We also found that an $r$th successor of such a $*$-pattern has bidegree $\bigl (\binom{i}{2}+2r,\binom{i}{2}+r\bigr )$.
Let $\sigma_i$ be the partition associated to the minimal $*$-pattern of bidegree $\bigl (\binom{i}{2},\binom{i}{2}\bigr )$. Then $\operatorname{rank}^{\tbinom{i}{2}+2r,\tbinom{i}{2}+r}(X)$ is the number of $r$th successors of this $*$-pattern, which is equal to the number of $r$th successors of the partition $\sigma_i$. But $\sigma_i$ contains exactly $\gamma_i$ odd numbers, so the successors of $\sigma_i$ are the partitions with exactly $\gamma_i$ odd numbers. The sum of the numbers in $\sigma_i$ is equal to $\gamma_i$ (note that $\sigma_i$ only contains $0$s and $1$s), and so the sum of the numbers in an $r$th successor of $\sigma_i$ will be $\gamma_i+2r$. One sees in this way that the number of $r$th successors of $\sigma_i$ is equal to $\operatorname{part}_{2r+\gamma_i,\leq k}[\gamma_i]$. This completes the proof.
\[re:strange-bijection\] Let us return to the classical bijection between $*$-patterns and partitions, via Schubert symbols. We claim that moving an asterisk one spot to the right corresponds to adding $1$ to an element of the associated Schubert symbol. Suppose that the $a$-sequence for the $*$-pattern is $\ldots x,y,z,\ldots$ and that we are promoting $y$ to $y+1$. Clearly this does not effect the beginning or the end of the Schubert symbol. If the original Schubert symbol was $\ldots,u,v,w\ldots$ then $v-u=y-x-1$ and $w-v=z-y-1$. The new Schubert symbol will be $\ldots,u,v',w',\ldots$ where $v'-u=(y+1)-x-1$ and $w'-v'=z-(y+1)-1$. Clearly this requires $v'=v+1$ and $w'=w$.
It is not true, however, that moving an asterisk [*two*]{} spots to the right corresponds to adding $2$ to an element of the associated Schubert symbol. The whole point of the strange bijection from the above proof was to create a situation where this does work, and the previous paragraph suggests why treating the even and odd spots separately accomplishes this.
Differentials in the cellular spectral sequence {#se:differentials}
===============================================
The main goal of this section is Kronholm’s theorem (Theorem \[th:Kronholm\] below), which to date is our best tool for governing what happens inside the cellular spectral sequence.
To begin, we give two examples demonstrating the kinds of differentials that can appear in the cellular spectral sequence for $\operatorname{Gr}_k({{{\EuScript}U}})$. The first example consists of the row of three pictures below. In the leftmost picture we have a page of the spectral sequence in which there are two copies of ${{\mathbb M}}_2$, with generators in bidegrees $(a,b)$ and $(a+3,b+4)$. (Note that one will typically have many more than two copies of ${{\mathbb M}}_2$, but we focus on this simple situation for pedagogical purposes). There is a differential (shown) that must be a $d_3$, since it maps a class from filtration degree $a$ into one from filtration degree $a+3$. The differential is only drawn on the [*generator*]{} of the first copy of ${{\mathbb M}}_2$, but the differentials in the cellular spectral sequence are ${{\mathbb M}}_2$-linear: so the one that is drawn implies several other evident differentials.
(300,145)(5,0) (0,0)(0,10)[14]{}[(1,0)[100]{}]{} (0,0)(10,0)[11]{}[(0,1)[130]{}]{} (102,-2)[$p$]{} (-7,125)[$q$]{} (0,0) (34.5,54.5)[(0,1)[30]{}]{} (34.5,54.5)[(1,1)[30]{}]{} (34.5,34.5)[(-1,-1)[30]{}]{} (34.5,34.5)[(0,-1)[30]{}]{} (64.5,94.5)[(0,1)[30]{}]{} (64.5,94.5)[(1,1)[30]{}]{} (64.5,74.5)[(-1,-1)[30]{}]{} (64.5,74.5)[(0,-1)[30]{}]{} (34.5,54.5)[(1,0)[8]{}]{} (122,0)(0,10)[14]{}[(1,0)[100]{}]{} (122,0)(10,0)[11]{}[(0,1)[130]{}]{} (224,-2)[$p$]{} (115,125)[$q$]{} (122,0) (156.5,64.5)[(0,1)[30]{}]{} (156.5,64.5)[(1,1)[25]{}]{} (156.5,34.5)[(-1,-1)[30]{}]{} (156.5,34.5)[(0,-1)[30]{}]{} (186.5,84.5)[(1,1)[30]{}]{} (186.5,84.5) (186.5,94.5)[(0,1)[30]{}]{} (186.5,94.5)[(1,1)[30]{}]{} (186.5,64.5)[(-1,-1)[25]{}]{} (186.5,64.5)[(0,-1)[30]{}]{} (156.5,44.5)[(-1,-1)[30]{}]{} (156.5,44.5) (246,0)(0,10)[14]{}[(1,0)[100]{}]{} (246,0)(10,0)[11]{}[(0,1)[130]{}]{} (349,-2)[$p$]{} (240,125)[$q$]{} (247,0) (280.5,64.5)[(0,1)[30]{}]{} (280.5,64.5)[(1,1)[30]{}]{} (280.5,44.5)[(-1,-1)[30]{}]{} (280.5,44.5)[(0,-1)[30]{}]{} (310.5,84.5)[(0,1)[30]{}]{} (310.5,84.5)[(1,1)[30]{}]{} (310.5,64.5)[(-1,-1)[30]{}]{} (310.5,64.5)[(0,-1)[30]{}]{}
In the middle panel we show the $E_{4}$-term of the spectral sequence, obtained by taking homology with respect to our differential (warning: not all $\tau$-multiplications are shown here). In our simple example this is the same as $E_\infty$, but note that there are extension problems in deducing the ${{\mathbb M}}_2$-structure. By a theorem of Kronholm [@K1 Theorem 3.2] it turns out that the cohomology we are converging to must be free over ${{\mathbb M}}_2$, and hence the extensions are resolved as shown in the third panel.
Note the net effect as one passes from the first panel to the third: the two copies of ${{\mathbb M}}_2$ remain, but their bidegrees have been shifted. The first copy has moved up one weight, and the second copy has moved down one weight.
Our next example shows a very similar phenomenon. Interpreting the pictures requires a little more imagination, though: remember that the pictures only explicitly show the edges of the cones, whereas there are an entire lattice of classes within the cones. The leftmost chart shows a situation where the differential takes the black generator to a class in the interior of the negative cone for the second copy of ${{\mathbb M}}_2$:
(300,145)(5,0) (0,0)(0,10)[14]{}[(1,0)[100]{}]{} (0,0)(10,0)[11]{}[(0,1)[130]{}]{} (102,-2)[$p$]{} (-7,125)[$q$]{} (0,0) (34.5,44.5)[(0,1)[30]{}]{} (34.5,44.5)[(1,1)[30]{}]{} (34.5,24.5)[(-1,-1)[20]{}]{} (34.5,24.5)[(0,-1)[20]{}]{} (64.5,104.5)[(0,1)[20]{}]{} (64.5,104.5)[(1,1)[20]{}]{} (64.5,84.5)[(-1,-1)[40]{}]{} (64.5,84.5)[(0,-1)[40]{}]{} (34.5,44.5)[(1,0)[8]{}]{} (122,0)(0,10)[14]{}[(1,0)[100]{}]{} (122,0)(10,0)[11]{}[(0,1)[130]{}]{} (224,-2)[$p$]{} (115,125)[$q$]{} (122,0) (156.5,74.5)[(0,1)[30]{}]{} (156.5,74.5)[(1,1)[25]{}]{} (156.5,24.5)[(-1,-1)[20]{}]{} (156.5,24.5)[(0,-1)[20]{}]{} (186.5,94.5)(0,-10)[3]{} (186.5,94.5)(10,10)[4]{}[(0,-1)[20]{}]{} (186.5,94.5)(0,-10)[3]{}[(1,1)[35]{}]{} (186.5,104.5)[(0,1)[20]{}]{} (186.5,104.5)[(1,1)[20]{}]{} (186.5,54.5)[(-1,-1)[25]{}]{} (186.5,54.5)[(0,-1)[25]{}]{} (156.5,34.5)(0,10)[3]{} (156.5,34.5)(0,10)[3]{}[(-1,-1)[34]{}]{} (156.5,34.5)(-10,-10)[4]{}[(0,1)[20]{}]{} (246,0)(0,10)[14]{}[(1,0)[100]{}]{} (246,0)(10,0)[11]{}[(0,1)[130]{}]{} (349,-2)[$p$]{} (240,125)[$q$]{} (247,0) (280.5,74.5)[(0,1)[35]{}]{} (280.5,74.5)[(1,1)[35]{}]{} (280.5,54.5)[(-1,-1)[35]{}]{} (280.5,54.5)[(0,-1)[35]{}]{} (310.5,74.5)[(0,1)[35]{}]{} (310.5,74.5)[(1,1)[35]{}]{} (310.5,54.5)[(-1,-1)[35]{}]{} (310.5,54.5)[(0,-1)[35]{}]{}
The leftmost chart is again an $E_3$-term, as the differential maps a class in filtration $a$ to a class in filtration $a+3$. The $E_4$-page is shown in the second chart. Kronholm’s theorem tells us that the cohomology our spectral sequence is converging to is free over ${{\mathbb M}}_2$, and so the relevant extension problems work out to be as shown in the third chart.
Once again, notice the difference between the first chart and the last chart: the left copy of ${{\mathbb M}}_2$ has increased its weight by three, whereas the right copy has decreased its weight by three.
Kronholm’s theorem generalizes these two examples. It says that the cohomology that the spectral sequence is converging to will be related to the $E_1$-term by a sequence of “trades” in which two copies of ${{\mathbb M}}_2$ shift up/down by the same number. The following is a rigorous statement along these lines, which covers all the applications we will need in the present paper:
\[th:Kronholm\] Let $X$ denote the $E_1$-term of the cellular spectral sequence for $\operatorname{Gr}_k({{{\EuScript}U}})$, and let $Y=H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$. Both $X$ and $Y$ are free as ${{\mathbb M}}_2$-modules, and for each $p\in {{\mathbb Z}}$ one has $$\sum_{q} \operatorname{rank}^{p,q}(X) = \sum_q \operatorname{rank}^{p,q}(Y)
\quad\text{and}\quad
\sum_c \operatorname{rank}^{c,p+c}(X) = \sum_c \operatorname{rank}^{c,p+c}(Y).$$
Note that the first equality from part (a) says that the number of basis elements in topological dimension $p$ is the same in both $E_1$ and $H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}}))$. Relative to our rank charts, the second equality from (a) says that the number of basis elements along any given diagonal is the same in both $E_1$ and $H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$.
In actuality, Theorem \[th:Kronholm\] as we have stated it is not quite found in [@K1]. However, the result is implicit in the proof of [@K1 Theorem 3.2].
The forgetful map to singular cohomology
----------------------------------------
There are natural maps $\Phi\colon H^{p,q}(X;{{\mathbb Z}}/2) {\rightarrow}H^p(X;{{\mathbb Z}}/2)$ from equivariant cohomology to singular cohomology. These fit together to give a ring map $\Phi\colon H^{*,*}(X;{{\mathbb Z}}/2){\rightarrow}H^*(X;{{\mathbb Z}}/2)$. When $X=*$ this map is completely determined by the formulas $\Phi(\tau)=1$, $\Phi(\rho)=0$. Consequently, $\Phi$ induces natural maps $$H^{*,*}(X;{{\mathbb Z}}/2)/(\rho) {\rightarrow}H^*(X;{{\mathbb Z}}/2) \quad\text{and}\quad
H^{*,*}(X;{{\mathbb Z}}/2)[\tau^{-1}] {\rightarrow}H^*(X;{{\mathbb Z}}/2).$$
If $J$ is a free ${{\mathbb M}}_2$-module, then $J/\rho J$ is a free ${{\mathbb M}}_2/\rho={{\mathbb Z}}/2[\tau]$-module. Note that $\tau$ has topological dimension zero, and so $J/\rho J$ will decompose as a ${{\mathbb Z}}/2[\tau]$-module into a direct sum over all topological dimensions: $$J/\rho J = \oplus_p \bigl [J/\rho J\bigr ]^{p,*}.$$ Note that the submodule $[J/\rho J]^{p,*}$ is only “influenced” by basis elements of $J$ in topological degree $p$: more precisely, any element of $[J/\rho J]^{p,*}$ is the image under $J{\rightarrow}J/\rho J$ of a ${{\mathbb Z}}/2[\tau]$-linear combination of basis elements of $J$ in topological degree $p$. Also, if $J$ has a finite number of free generators in each topological degree then the ${{\mathbb Z}}/2$-dimension of $[J/\rho J]^{p,q}$ is independent of $q$ for $q\gg 0$ (once $q$ is larger than the weights of all the generators in this topological degree).
Let us apply these ideas when $J=H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$. Then $[J/\rho J]^{p,*}$ is a free ${{\mathbb Z}}/2[\tau]$-module with a basis corresponding to the equivariant Schubert cells of topological dimension $p$. While these cells likely have different weights, if we look in $[J/\rho J]^{p,N}$ for $N$ large enough then we will see all of them (more precisely, $\tau$-multiples of all of them). The forgetful map $\Phi$ will send these elements to the corresponding non-equivariant Schubert classes in $H^p(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ (recall that $\Phi(\tau)=1$). This shows that in large enough weights $N$ the map $\Phi\colon [J/\rho J]^{p,N} {\rightarrow}H^p(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ is an isomorphism. This proves part (a) of the following:
\[pr:inject\] For $p,q\in {{\mathbb Z}}$ consider the map $$\Phi_{p,q}\colon\Bigl
[H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)/(\rho)\Bigr ]^{p,q} {\rightarrow}H^p(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2).$$
(a) Given $p$, there exists an $N\in {{\mathbb Z}}$ such that the map $\Phi_{p,q}$ is an isomorphism for all $q\geq N$.
(b) For any $p$ and $q$ the map $\Phi_{p,q}$ is an injection.
The proof of part (a) preceded the statement of the proposition. For (b), fix $p$ and $q$ and write $J=H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ for simplicity. By (a) we know that for large enough $N$ the map $\Phi_{p,N}$ is an isomorphism. Now just consider the diagram $$\xymatrix{
[J/\rho J]^{p,q} \ar[r]^{\cdot \tau^{N-q}}\ar[dr]_{\Phi} & [J/\rho
J]^{p,N}\ar[d]_{\cong}^{\Phi} \\
& H^{p}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2),
}$$ which commutes because $\Phi(\tau)=1$. Multiplication by $\tau$ is an injection on ${{\mathbb M}}_2/(\rho)$, and hence also on $J/\rho J$. So the diagonal map in the diagram is also injective.
Proof of the main theorem {#se:proof}
=========================
Throughout this section we let $X$ be the $E_1$-term of the cellular spectral sequence for $\operatorname{Gr}_k({{{\EuScript}U}})$, we let $Y=H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$, and we let $Z={{\mathcal{I}}nv}_k$. It will be convenient to keep in mind the diagram $$\xymatrix{
X \ar@{~>}[r] & Y \ar[r] & Z,
}$$ indicating that $Y$ maps to $Z$ and that there is a spectral sequence that starts from $X$ and converges to $Y$. Our aim is to prove Theorem \[th:main1\], stating that $Y{\rightarrow}Z$ is an isomorphism.
Each of $X$, $Y$, and $Z$ is a free module over ${{\mathbb M}}_2$, and the proof will involve a study of the bigraded rank functions for each. The following lemma collects the key results we will need:
\[le:main\]
(a) For every $p\in {{\mathbb Z}}$, $$\sum_q \operatorname{rank}^{p,q}(Y)=
\sum_q \operatorname{rank}^{p,q}(X)=\sum_q \operatorname{rank}^{p,q}(Z).$$
(b) For every $c\in {{\mathbb Z}}$, $$\sum_p \operatorname{rank}^{p,p+c}(Y)=
\sum_p \operatorname{rank}^{p,p+c}(X)=\sum_p \operatorname{rank}^{p,p+c}(Z).$$
(c) For every $p,q\in {{\mathbb Z}}$, $$\sum_{c\leq q} \operatorname{rank}^{p,c}(Y) \leq \sum_{c\leq q} \operatorname{rank}^{p,c}(Z)
\quad\text{and}\quad
\sum_{c\geq q} \operatorname{rank}^{p,c}(Y) \geq \sum_{c\geq q} \operatorname{rank}^{p,c}(Z).$$
We have written the equalities in the first two parts in the order that they will be proven: $Y$ is related to $X$, and $X$ is related to $Z$. Phrase in terms of our rank charts, the above results say:
(i) The sum of the numbers in any column is the same for $X$, $Y$, and $Z$.
(ii) The sum of the numbers along any diagonal is the same for $X$, $Y$, and $Z$.
(iii) If one fixes a particular box and adds together the numbers in all boxes directly above it, the sum for $Y$ is always at least the sum for $Z$. (This is the second inequality in (c)).
We defer the proof of the lemma for just a moment, in order to highlight the structure of the main argument. However, let us point out that the left equalities in (a) and (b) are by Kronholm’s Theorem, and the second equalities come from our combinatorial analyses of $\operatorname{rank}^{*,*}(X)$ and $\operatorname{rank}^{*,*}(Z)$. In light of (a), the two inequalities in part (c) are equivalent. The proof of these final inequalities uses some topology, namely the non-equivariant version of Theorem \[th:main1\].
Before proving the next result, we introduce a useful piece of notation. If $M$ is a free ${{\mathbb M}}_2$-module, then for each $c\in {{\mathbb Z}}$ let $d_c(M)$ denote the function ${{\mathbb Z}}{\rightarrow}{{\mathbb Z}}$ given by $p\mapsto \operatorname{rank}^{p,p-c}(M)$. These are the entries in the rank chart of $M$ along the diagonal line of slope $1$ passing through the point $(0,-c)$.
\[pr:Y->Z\] For all $p,q\in {{\mathbb Z}}$, $\operatorname{rank}^{p,q}(Y)=\operatorname{rank}^{p,q}(Z)$.
We will prove the proposition by establishing that $d_c(Y)=d_c(Z)$ for all $c\in {{\mathbb Z}}$. First note that this is easy for $c<0$. In this case we know by direct computation that $\operatorname{rank}^{p,p-c}(Z)=0$ for all $p\in {{\mathbb Z}}$ (Corollary \[co:bounds\]). So $\sum_p
\operatorname{rank}^{p,p-c}(Z)=0$, which implies by Lemma \[le:main\](b) that $\sum_p \operatorname{rank}^{p,p-c}(Y)=0$. Since the ranks are all non-negative, this means $\operatorname{rank}^{p,p-c}(Y)=0$ for all $p\in {{\mathbb Z}}$.
Next we proceed by induction on $c$. Assume $c\geq 0$ and that $d_n(Y)
=d_n(Z)$ for all $n< c$. Let $p\in {{\mathbb Z}}$, and consider the inequality $$\sum_{q\geq p-c} \operatorname{rank}^{p,q}(Y) \geq \sum_{q\geq p-c}
\operatorname{rank}^{p,q}(Z)$$ from Lemma \[le:main\](c). By induction we know that $\operatorname{rank}^{p,q}(Y)=\operatorname{rank}^{p,q}(Z)$ for $q>p-c$, and so we conclude that
\[eq:eq1\] \^[p,p-c]{}(Y) \^[p,p-c]{}(Z).
This holds for all $p\in {{\mathbb Z}}$. But we also know, by Lemma \[le:main\](b), that
\[eq:eq2\] \_p \^[p,p-c]{}(Y) = \_p \^[p,p-c]{}(Z).
Equations (\[eq:eq1\]) and (\[eq:eq2\]) can both be true only if $\operatorname{rank}^{p,p-c}(Y)=\operatorname{rank}^{p,p-c}(Z)$ for all $p \in {{\mathbb Z}}$. That is, $d_c(Y)=d_c(Z)$.
It is worth remarking that Proposition \[pr:Y->Z\] has solved one of our main questions. It completely identifies the weights of the free generators for $H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ by showing that they agree with the ranks of the generators for the combinatorially-computable ring of invariants ${{\mathcal{I}}nv}_k$.
Next we give the
The first equality in (a) is by Kronholm’s Theorem (Theorem \[th:Kronholm\]). For the second equality observe that $\sum_q
\operatorname{rank}^{p,q}(X)$ is just the number of classical Schubert cells of dimension $p$ inside $\operatorname{Gr}_k({{\mathbb R}}^\infty)$. This is the same as the number of partitions of $p$ into at most $k$ pieces, which is the same as $\sum_q \operatorname{part}_{p,\leq k}[q]$. The latter equals $\sum_q
\operatorname{rank}^{p,q}(Z)$ by Proposition \[pr:rank-Inv\].
For (b), the first equality is again by Kronholm’s Theorem. The equality $\sum_p \operatorname{rank}^{p,p+c}(X)=\sum_p \operatorname{rank}^{p,p+c}(Z)$ follows from the combinatorial identities in Proposition \[pr:rank-Inv\] and Proposition \[pr:cell-bound\]; to see why, it is best to think pictorially. Proposition \[pr:cell-bound\] says that the rank chart for $X$ is concentrated along $k+1$ rays of slope $\frac{1}{2}$, emanating from certain points on the $y=x$ line. Proposition \[pr:rank-Inv\] says that the rank chart of $Z$ also consists of $k+1$ rays of slope $\frac{1}{2}$—containing the same entries as the ones in $X$—but which emanate from different points on the $y=x$ line (in other words, the order of the rays in the two charts are both permuted and shifted along the $y=x$ line). From this it follows at once that the diagonals of the two rank charts contain the same entries, only permuted. In particular, the sum of the entries is the same in the two situations.
For (c), it will suffice to prove that $\sum_{c\leq q} \operatorname{rank}^{p,c}(Y)
\leq \sum_{c\leq q} \operatorname{rank}^{p,c}(Z)$, since the second inequality follows from this one together with part (a). Consider the diagram $$\xymatrixcolsep{1pc}\xymatrix{
Y^{p,q} \ar[rr]\ar[dr]\ar[dd]_\Phi && Z^{p,q}\ar[dr]\ar[dd]^-<<<<<<\Phi \\
& [Y/\rho Y]^{p,q}\ar@{.>}[dl]\ar[rr] && [Z/\rho Z]^{p,q}\ar@{.>}[dl]\\
H^p(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2) \ar[rr]_{\cong}&& \Bigl [ \bigl [H^*({{{\mathbb R}}P}^\infty;{{\mathbb Z}}/2)^{{\otimes}k}\bigr ]^{\Sigma_k} \Bigr ]^{p}
}$$ where the dotted arrows exist because $\rho$ is sent to zero by $\Phi$. The bottom horizontal map is an isomorphism by the classical theory, and the map $[Y/\rho Y]^{p,q} {\rightarrow}H^p(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ is an injection by Proposition \[pr:inject\](b). It follows that $[Y/\rho
Y]^{p,q} {\rightarrow}[Z/\rho Z]^{p,q}$ is an injection. However, it is easy to see that if $J$ is a free ${{\mathbb M}}_2$-module then $\dim_{{{\mathbb Z}}/2} [J/\rho
J]^{p,q}=\sum_{c\leq q}
\operatorname{rank}^{p,c}(J)$. Applying this to $Y$ and $Z$, we have completed the proof.
At this point we have only proven that $Y$ and $Z$ are free ${{\mathbb M}}_2$-modules with the same bigraded rank functions. But we have a specific map $Y{\rightarrow}Z$, and our goal is to prove that it is an isomorphism. Since both $Y^{p,q}$ and $Z^{p,q}$ are finite-dimensional over ${{\mathbb Z}}/2$ for every $p,q\in {{\mathbb Z}}$, it will be sufficient to prove that $Y {\rightarrow}Z$ is surjective. We begin with the following observation:
\[le:iso-mod-rho\] The map $Y/\rho Y {\rightarrow}Z/\rho Z$ is an isomorphism.
As in the proof of Lemma \[le:main\](c), we know that $[Y/\rho Y]^{p,q}{\rightarrow}[Z/\rho Z]^{p,q}$ is an injection. We also know that the ${{\mathbb Z}}/2$-dimensions of these two spaces are $\sum_{c\leq q} \operatorname{rank}^{p,c}(Y)$ and $\sum_{c\leq q}
\operatorname{rank}^{p,c}(Z)$, which are equal by Proposition \[pr:Y->Z\]. This proves the lemma.
The desired result will now follow from the purely algebraic lemma below:
\[le:algebra\] Let $M$ and $N$ be free ${{\mathbb M}}_2$-modules, and let $f\colon M{\rightarrow}N$ be a map such that $M/\rho M {\rightarrow}N/\rho N$ is an isomorphism. Assume that
(i) $\operatorname{rank}^{p,q}(M)=\operatorname{rank}^{p,q}(N)$ for all $p,q\in {{\mathbb Z}}$.
(ii) $\dim_{{{\mathbb Z}}/2} M^{p,q}$ is finite for all $p,q\in {{\mathbb Z}}$.
(iii) There exists an $r\in {{\mathbb Z}}$ such that $d_c(M)=d_c(N)=0$ for all $c< r$.
(iv) There exists a number $u$ such that $\operatorname{rank}^{p,q}(M)=0$ for all $p< u$.
Then $f$ is an isomorphism.
Pick a free basis $\{e_\alpha\}$ for $N$ consisting of homogeneous elements. For each $s\in {{\mathbb Z}}$ let $N_s\subseteq N$ be the submodule spanned by all $e_\alpha$ for which the bidegree $(p_\alpha,q_\alpha)$ satisfies $p_\alpha-q_\alpha \leq
s$ (these are the basis elements on all diagonals ‘higher than’ the $p-q=s$ diagonal). Note that $N_s=0$ for $s<r$, where $r$ is the number specified in condition (iii).
Condition (iv) readily implies the following fact: for every $p,q\in {{\mathbb Z}}$ there exists an $m\geq 0$ such that $[\rho^m
N]^{p,q}\subseteq N_{p-q-1}$. In other words, every element of $N^{p,q}$ that is a multiple of $\rho^m$ is in the ${{\mathbb M}}_2$-span of basis elements from higher diagonals. (One need only take $m=p-u+1$ here, where $u$ is from condition (iv)).
We will prove by induction that each $N_s$ is contained in the image of $f$. We know this for $s<r$ since in that case $N_s=0$. So assume $s\in
{{\mathbb Z}}$ and $N_{s-1}\subseteq \operatorname{im}f$. Since $M/\rho M {\rightarrow}N/\rho N$ is an isomorphism it follows that $N=(\operatorname{im}f) + \rho N$. Substituting this equation for $N$ into itself, we then find $$N = (\operatorname{im}f)+\rho N = (\operatorname{im}f)+\rho^2 N =(\operatorname{im}f)+\rho^3 N = \cdots$$ So $N=(\operatorname{im}f)+\rho^n N$ for any $n\geq 1$.
Now let $e_\alpha$ be a basis element lying in $N_s$, of bidegree $(p,q)$ (so that $p-q\leq s$). We may assume $p-q=s$, for otherwise $e_\alpha \in N_{s-1}$ and so is in the image of $f$ by induction. By the second paragraph of this proof, there exists $m\geq
1$ such that $[\rho^m N]^{p,q} \subseteq N_{s-1}$. But then we have $$N^{p,q}=(\operatorname{im}f)^{p,q} + [\rho^m
N]^{p,q} \subseteq (\operatorname{im}f) + N_{s-1} =\operatorname{im}f$$ where the last equality uses our inductive assumption that $N_{s-1}\subseteq \operatorname{im}f$. We have therefore shown that $e_\alpha \in \operatorname{im}f$, and since this holds for every basis element we have $N_s\subseteq \operatorname{im}f$.
At this point we have shown that $f$ is surjective. The finiteness condition (ii) then implies that $f$ is indeed an isomorphism.
We now restate Theorem \[th:main1\] from the introduction, and tie up its proof:
The map $\eta^*\colon H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2) {\rightarrow}\bigl
[H^{*,*}(\operatorname{Gr}_1({{{\EuScript}U}});{{\mathbb Z}}/2)^{{\otimes}k }\bigr ]^{\Sigma_k}$ is an isomorphism of bigraded rings.
This is the map $Y{\rightarrow}Z$ considered throughout this section. Both $Y$ and $Z$ are free ${{\mathbb M}}_2$-modules that satisfy hypotheses (ii)–(iv) of Lemma \[le:algebra\]. Proposition \[pr:Y->Z\] verifies condition (i) of that lemma. The result therefore follows by that lemma together with Lemma \[le:iso-mod-rho\].
The multiplicative structure of the ring of invariants {#se:mult}
======================================================
At this point in the paper we have proven that our map $$\eta^*\colon
H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}})) {\rightarrow}[H^{*,*}(\operatorname{Gr}_1({{{\EuScript}U}}))^{{\otimes}k}]^{\Sigma_k}$$ is an isomorphism of rings. We also have a combinatorial description of the bigraded rank function—that is, we understand the additive structure of these rings, or their structure as ${{\mathbb M}}_2$-module. In this section we further investigate the ring of invariants, concentrating on the multiplicative structure. Recall that this ring of invariants is denoted ${{\mathcal{I}}nv}_k$ for short.
First observations
------------------
Recall from Section \[se:add-basis\] that we use the notation $w_i=[a_1\ldots a_i]$ and $c_i=[b_1\ldots b_i]$. These are the $i$th elementary symmetric functions in the $a$’s and $b$’s, respectively. More generally, define the invariant element ${wc}_{i,j}$ by $${wc}_{i,j}=[a_1\ldots a_i b_{i+1}\cdots b_{i+j}].$$ Note that this only makes sense when $i+j\leq k$. Note also that ${wc}_{i,0}=w_i$ and ${wc}_{0,j}=c_j$. Finally, let us observe that the bidegree of ${wc}_{i,j}$ is $(i,i)+(2j,j)=(i+2j,i+j)$.
As a warm-up for our investigation let us consider some basic relations. The easiest relation one encounters is $$w_1^2=(a_1+\cdots+a_n)^2=a_1^2+\cdots+a_n^2 =
(\rho a_1+\tau b_1)+\cdots + (\rho a_n+\tau b_n) = \rho w_1 +\tau c_1.$$ Analogously, $$\begin{aligned}
w_2^2=[a_1a_2]^2 &=\sum_{i< j} (\rho a_i+\tau b_i)(\rho a_j+\tau
b_j)\\
&=\rho^2 \sum_{i<j}a_ia_j + \rho\tau \sum_{i<j}(a_ib_j+a_jb_i) + \tau^2
\sum_{i<j} b_ib_j
\\
& =\rho^2 w_2 + \rho\tau\sum_{i\neq j} a_ib_j + \tau^2 c_2 \\
&= \rho^2 w_2 + \rho\tau \cdot {wc}_{1,1} + \tau^2 c_2.\end{aligned}$$ More generally we have the following (the proof is left as an exercise):
In ${{\mathcal{I}}nv}_k$ there is the relation $$w_j^2=\tau^j c_j + \tau^{j-1}\rho {wc}_{1,j-1} +
\tau^{j-2}\rho^2 {wc}_{2,j-2} +\cdots + \tau\rho^{j-1} {wc}_{j-1,1} + \rho^j
w_j$$ for any $j\leq k$.
Next let us consider the products $w_1w_i$ for various $i$. For instance, $w_1w_2 = (a_1+\ldots+a_n)(a_1a_2+\ldots +
a_{n-1}a_n)$. When we distribute, we will get terms that look like $a_1^2a_2$, and also terms that look like $a_1a_2a_3$. Note that the former term only appears once, whereas the latter appears $\binom{3}{2}=3$ times (which is equivalent to once, since we are in characteristic two). So we can write $$w_1w_2=[a_1^2a_2] + [a_1a_2a_3]=[a_1^2a_2]+w_3.$$ We must be careful when identifying $[a_1^2a_2]$. We have $$[a_1^2a_2]=\sum_{i\neq j} a_i^2a_j =\sum_{i\neq j} (\rho a_i+\tau
b_i) a_j = \rho \sum_{i\neq j} a_ia_j + \tau \sum_{i\neq j} b_ia_j =
0 + \tau {wc}_{1,1}.$$ Note that $\sum_{i\neq j} a_ia_j =0$ only because we are in characteristic $2$.
As one more example, let’s compute $w_1w_3$. We are looking at the product $(a_1+\ldots+a_n)(a_1a_2a_3+\ldots )$, and so we have terms that look like $a_1^2a_2a_3$ and $a_1a_2a_3a_4$. The former occurs exactly once, the latter $\binom{4}{1}=4$ times (equivalent to zero times, mod $2$). So $$w_1w_3=[a_1^2a_2a_3]=\sum_{{j <k} \atop {i\notin \{j,k\}}}
a_i^2a_ja_k
=\sum_{{j <k} \atop {i\notin \{j,k\}}}
(\rho a_i + \tau b_i) a_ja_k
= \rho[a_1a_2a_3]+\tau[a_1a_2b_3].$$ The last equality takes a little thought: we must ask ourselves how many times a typical term $a_1a_2a_3$ appears in the sum $\sum\limits_{{j <k} \atop {i\notin \{j,k\}}}
a_ia_ja_k$, and the answer is that it occurs exactly three times (equivalent to once, mod $2$).
The following proposition is easily proven by the above techniques:
\[pr:w-decomp\] In ${{\mathcal{I}}nv}_k$ one has the relations $w_1w_{2i}=\tau \cdot {wc}_{2i-1,1} +
w_{2i+1}$ and $w_1w_{2i+1}=\tau {wc}_{2i,1}+\rho w_{2i+1}$.
Note that the first relation from Proposition \[pr:w-decomp\] shows that $w_{2i+1}$ is decomposable in ${{\mathcal{I}}nv}_k$. Without much trouble this generalizes to the following result. Compare [@M Remark 3.4].
\[pr:w-indecomposable\] Let $1\leq j\leq k$. Then $w_j$ is indecomposable in ${{\mathcal{I}}nv}_k$ if and only if $j$ is a power of $2$.
If $j$ is not a power of $2$, then $\binom{j}{i}$ is odd for some $i$. Consider the product $$w_i w_{j-i}=(a_1a_2\ldots a_i + \cdots)(a_1a_2\ldots a_{j-i}+
\cdots).$$ When we distribute, we have some terms which contain one or more squares—these belong to the ideal $(\rho,\tau)$ of ${{\mathcal{I}}nv}_k$ because of the relation $a_i^2=\rho a_i + \tau b_i$. A typical term which doesn’t involve squares is $a_1a_2\ldots a_j$, and this appears exactly $\binom{j}{i}$ times in the big sum. So we can write $$w_i w_{j-i} \in (\rho,\tau) + w_{j}.$$ But the elements of $(\rho,\tau)$ are by nature decomposable, and so we have that $w_j$ is decomposable.
For the proof that $w_{2^r}$ is indecomposable, we map our ring ${{\mathcal{I}}nv}_k$ to a simpler ring where it is easier to prove this. Specifically, consider the map $${{\mathbb M}}_2[a_1,\ldots,a_k,b_1,\ldots,b_k]/(a_i^2=\rho a_i+\tau b_i) {\rightarrow}\Lambda_{{{\mathbb Z}}/2}(a_1,\ldots,a_k)$$ that sends $\rho$, $\tau$, and all the $b_i$’s to zero. Upon taking invariants this gives a map $${{\mathcal{I}}nv}_k {\rightarrow}\Lambda_{{{\mathbb Z}}/2}(a_1,\ldots,a_k)^{\Sigma_k}$$ that sends each $w_i$ to the $i$th symmetric function $\sigma_i$ in the $a_j$’s. But in $\Lambda_{{{\mathbb Z}}/2}(a_1,\ldots,a_k)^{\Sigma_k}$ it is well-known that $\sigma_i$ is indecomposable when $i$ is a power of $2$ (see Proposition \[pr:exterior-inv\] below for a proof).
Generalized Stiefel-Whitney classes
-----------------------------------
One of the difficulties in studying the ring ${{\mathcal{I}}nv}_k$ is that there does not seem to be a clear choice of which algebra generators to use; every choice seems to have drawbacks. The ${wc}$ classes defined above represent one extreme: they result from making the indices on the $a$’s and $b$’s disjoint. The opposite approach is to make the indices overlap as much as possible, and that leads to the following definition: $$w_i^{(e)}=[a_1\ldots a_ib_1^e\ldots b_i^e].$$ Note that this defines an element of ${{\mathcal{I}}nv}_k$ for $1\leq i\leq k$ and $0\leq e$. It has bidegree $(i,i)+ei(2,1)=(i(2e+1),i(e+1))$, and in terms of our rank charts it lies on the same line of slope $\frac{1}{2}$ as the class $w_i$. Notice that $w_i^{(0)}=w_i$.
Indecomposables
---------------
Let $\epsilon \colon {{\mathbb M}}_2[a_1,\ldots,a_k,b_1,\ldots,b_k]/(a_i^2=\rho
a_i+\tau b_i) {\rightarrow}{{\mathbb M}}_2$ be defined by sending each $a_i$ and $b_i$ to zero. We will also write $\epsilon$ for the restriction to ${{\mathcal{I}}nv}_k$. Let $I_k\subseteq {{\mathcal{I}}nv}_k$ be the kernel of $\epsilon \colon {{\mathcal{I}}nv}_k {\rightarrow}{{\mathbb M}}_2$. Then $I_k/I_k^2$ is a bigraded ${{\mathbb M}}_2$-module that is readily checked to be free; it is called the [module of indecomposables]{} for ${{\mathcal{I}}nv}_k$ relative to ${{\mathbb M}}_2$. Our goal is to determine the bigraded rank function for $I_k/I_k^2$, as well as a basis. In other words, we aim to write down a complete set of representatives for the indecomposables in ${{\mathcal{I}}nv}_k$.
It is worth stressing that we have set things up so that ‘indecomposable’ means [*relative to*]{} ${{\mathbb M}}_2$. The elements $\rho$, $\tau$, and $\theta$ are of course indecomposable elements of ${{\mathcal{I}}nv}_k$ in the ‘absolute’ sense, but we do not want to keep track of them. They will not be reflected in the rank function for $I_k/I_k^2$, which by definition counts the number of basis elements over ${{\mathbb M}}_2$.
The main result is as follows:
\[th:indecomp\]
(a) The indecomposables of ${{\mathcal{I}}nv}_k$ are represented by the classes $c_1,\ldots,c_k$ together with the classes $w_{2^i}^{(e)}$ for $1\leq
2^i \leq k$ and $0\leq e \leq \frac{k}{2^i}-1$. That is to say, these classes give a free basis for $I_k/I_k^2$ as an ${{\mathbb M}}_2$-module.
(b) The number of indecomposables for ${{\mathcal{I}}nv}_k$ is $$3k-(\text{\# of ones in the binary expansion of $k$}).$$
(c) For $1\leq 2^i\leq k$ and $0\leq e\leq \frac{k}{2^i}-1$ the classes ${wc}_{2^i,e2^i}$ and $w_{2^i}^{(e)}$ are equivalent modulo decomposables.
(d) For $p,q\in {{\mathbb Z}}$, $\operatorname{rank}^{p,q}(I_k/I_k^2)=0$ unless $0\leq
p$ and $0\leq q\leq k$.
(e) When $p$ is odd and $0\leq p$, $$\operatorname{rank}^{p,q}(I_k/I_k^2)=\begin{cases} 1 & \text{if $q=\frac{p+1}{2}$},\\
0 & \text{otherwise}.
\end{cases}$$ The unique indecomposable in topological dimension $p$ is represented by $w_1^{(\frac{p-1}{2})}$, or equivalently by ${wc}_{1,\frac{p-1}{2}}$.
(f) When $p$ is even and positive, write $p=2^i(2e+1)$. Then $$\operatorname{rank}^{p,q}(I_k/I_k^2)=\begin{cases} 1 & \text{if $q=\frac{p}{2}$
or $q=\frac{p}{2}+2^{i-1}$},\\
0 & \text{otherwise}.
\end{cases}$$ When $q=\frac{p}{2}$, the unique indecomposable in bidegree $(p,q)$ is represented by the Chern class $c_{q}$. When $q=\frac{p}{2}+2^{i-1}$ the unique indecomposable is represented by $w_{2^i}^{(e)}$, or equivalently by ${wc}_{2^i,e\cdot 2^i}$.
To paraphrase the above theorem, in the limiting case $k{\rightarrow}\infty$ there is one indecomposable in every odd topological dimension and two indecomposables in every even topological dimension. The following chart shows the exact bidegrees, with different symbols for different types of indecomposables:
(300,180)(-50,-10) (0,0)(0,15)[11]{}[(1,0)[240]{}]{} (0,0)(15,0)[17]{}[(0,1)[150]{}]{} (7.5,7.5)(30,15)[8]{} (19,18.5)(30,15)[8]{}[$\square$]{} (21,20.5)(30,15)[8]{}[$\scriptscriptstyle{1}$]{} (36,35.5)(60,30)[4]{}[$\scriptscriptstyle{2}$]{} (66,65.5)(120,60)[2]{}[$\scriptscriptstyle{4}$]{} (126,125.5)[$\scriptscriptstyle{8}$]{} (34,34)[$\square$]{} (64,64)[$\square$]{} (124,124)[$\square$]{} (94,64)[$\square$]{} (154,94)[$\square$]{} (184,124)[$\square$]{} (214,124)[$\square$]{}
The circles represent the Chern classes, whereas the squares represent the $w$-classes. The squares with an $i$ inside represent $w_i^{(e)}$ classes, for $0\leq e$. The pattern here is that the $w_i^{(e)}$ classes start in bidegree $(i,i)$ and then proceed up along the line of slope $\frac{1}{2}$, occuring every $i$ steps along this line, where “step” means a $(2,1)$ move.
For ${{\mathcal{I}}nv}_k$ one cuts the chart off and only takes the classes in weights less than or equal to $k$. For example, in ${{\mathcal{I}}nv}_5$ there will be the following indecomposables (given in order of increasing topological degree): $$w_1,\ c_1,\ w_2,\ w_1^{(1)},\ c_2,\ w_4,\ w_1^{(2)},\ c_3,\
w_2^{(1)},\ w_1^{(3)},\ c_4,\ w_1^{(4)},\ c_5.$$ Note that Theorem \[th:indecomp\](b) predicts the number of indecomposables to be $15-2=13$, which agrees with the above list.
Our goal is now to prove Theorem \[th:indecomp\], proceeding by a series of reductions.
The complexities of ${{\mathbb M}}_2$ are irrelevant to the considerations at hand. To this end, define $R_k={{\mathbb Z}}/2[\tau,\rho,a_1,\ldots,a_k,b_1,\ldots,b_k]/(a_i^2=\rho
a_i+\tau b_i)$. Let $S_k=R_k^{\Sigma_k}$, where the $\Sigma_k$-action permutes the $a_i$’s and $b_i$’s but fixes $\rho$ and $\tau$. Let $\epsilon\colon R_k {\rightarrow}{{\mathbb Z}}/2[\tau,\rho]$ be the map that sends $a_i$ and $b_i$ all to zero, for $1\leq i \leq k$. Let $J_k$ be the augmentation ideal of $S_k$, defined as $$J_k=\ker (S_k {\rightarrow}R_k {\stackrel{\epsilon}{{\longrightarrow}}} {{\mathbb Z}}/2[\tau,\rho]).$$ It is easy to see that ${{\mathcal{I}}nv}_k{\cong}S_k{\otimes}_{{{\mathbb Z}}/2[\tau,\rho]} {{\mathbb M}}_2$ and $I_k/I_k^2 {\cong}(J_k/J_k^2){\otimes}_{{{\mathbb Z}}/2[\tau,\rho]} {{\mathbb M}}_2$. So the bigraded rank function for $J_k/J_k^2$ over ${{\mathbb Z}}/2[\tau,\rho]$ coincides with the bigraded rank function for $I_k/I_k^2$ over ${{\mathbb M}}_2$. It will therefore suffice for us to prove the theorem in the former case.
A free basis for $J_k/J_k^2$ over ${{\mathbb Z}}/2[\tau,\rho]$ is the same as a vector space basis for $J_k/[J_k^2+(\rho,\tau)J_k]$ over ${{\mathbb Z}}/2$. This is the form in which we will study the problem.
Let $\tilde{R}_k={{\mathbb Z}}/2[a_1,\ldots,a_k,b_1,\ldots,b_k]/(a_i^2)$ with the evident $\Sigma_k$-action, and let $\tilde{S}_k=\tilde{R}_k^{\Sigma_k}$. Consider the diagram $$\xymatrix{
J_k \ar[d]\ar@{ >->}[r] & S_k \ar[d]\ar[r]^-\epsilon & {{\mathbb Z}}/2[\tau,\rho]
\ar[d] \\
\tilde{J}_k \ar@{ >->}[r] & \tilde{S}_k \ar[r]^{\tilde{\epsilon}} & {{\mathbb Z}}/2
}$$ where the vertical maps send $\rho$ and $\tau$ to zero, and $\tilde{J}_k$ is the kernel of $\tilde{\epsilon}$. It is easy to see that $S_k{\rightarrow}\tilde{S_k}$ is surjective: a ${{\mathbb Z}}/2$-basis for the target is given by the orbit sums $[m]$ where $m$ is a monomial in the $a$’s and $b$’s, and such an orbit sum lifts into $S_k$. The same argument shows that $J_k{\rightarrow}\tilde{J}_k$ is surjective. We in fact have a surjection $$J_k/[J_k^2+(\rho,\tau)J_k] {\twoheadrightarrow}\tilde{J}_k/\tilde{J}_k^2,$$ and it is easy to see that this is actually an isomorphism.
We have therefore reduced our problem to understanding the module of indecomposables $\tilde{J}_k/\tilde{J}_k^2$ for the ring $\tilde{S}_k$. This is a fairly routine algebra problem; we give a full treatment in Appendix A for lack of a suitable reference. See Theorem \[th:En\] for the classification of the indecomposables, proving parts (a) and (b). The third statement in Lemma \[le:in3\] proves part (c), and parts (d)–(f) are really just restatements of (a) and (c).
Relations {#se:relations}
---------
In general it seems that writing down a complete set of relations for ${{\mathcal{I}}nv}_k$ is not practical or useful. See the cases of $k=2$ and $k=3$ described in the next section. The relations tend to be numerous and also fairly complicated. One general remark worth making is that there will always be a relation for the square of a $w_i^{(e)}$ class. The square of $[a_1\ldots a_ib_1^e\ldots b_i^e]$ will be $[a_1^2\ldots a_i^2b_1^{2e}\ldots b_i^{2e}]$, and each $a_j^2$ decomposes as $\rho a_j+\tau b_j$. For example,
\[eq:w-square\] \^2=\[a\_1\^2b\_1\^[2e]{}\]=+=w\_1\^[(2e)]{}+.
To express this in terms of indecomposables we need to write the power sum $[b_1^{2e+1}]$ as a polynomial in the elementary symmetric functions, via the mod $2$ Newton polynomials. This already produces an expression with lots of terms. If $2e>k-1$ then $w_1^{(2e)}$ is not an indecomposable and we also need to rewrite that term. This can be handled via the following result:
\[le:w1e\] In ${{\mathcal{I}}nv}_k$ one has the relation $$w_1^{(e)}=w_1^{(e-1)}c_1 + w_1^{(e-2)}c_2+\cdots + w_1^{(e-k)}c_k$$ for any $e\geq k$.
This follows from the identities $$\begin{aligned}
&[a_1b_1^e]=[a_1b_1^{e-1}]\cdot [b_1] + [a_1b_1^{e-1}b_2] \\
&[a_1b_1^{e-1}b_2]=[a_1b_1^{e-2}]\cdot [b_1b_2]+[a_1b_1^{e-2}b_2b_3]\\
&\vdots\end{aligned}$$ We stop when the right-hand term is $[a_1b_1^{e-(k-1)}b_2\ldots b_k]$, since in this case the monomial $b_1\cdots b_k$ is a common factor to all the summands in the $\Sigma_k$-orbit and can be taken out: $$[a_1b_1^{e-(k-1)}b_2\ldots b_k]=[a_1b_1^{e-k}]\cdot [b_1\cdots
b_k]=w_1^{(e-k)}\cdot c_k.$$ Substituting each identity into the previous one leads to the desired relation.
Let us work through one example. In ${{\mathcal{I}}nv}_3$ there is the indecomposable $w_1^{(2)}$, and according to our above analysis its square is
\[eq:newt\] \^2=w\_1\^[(4)]{} + =w\_1\^[(4)]{} + .
The latter expression comes from working out the appropriate Newton polynomial. For the $w_1^{(4)}$ term we have $$w_1^{(4)} =w_1^{(3)}c_1 + w_1^{(2)}c_2 + w_1^{(1)}c_3
= \bigl [ w_1^{(2)}c_1 + w_1^{(1)}c_2 + w_1 c_3\bigr ] c_1 +
w_1^{(2)}c_2 + w_1^{(1)}c_3$$ by two applications of Lemma \[le:w1e\]. Our final relation is $$\Bigl[ w_1^{(2)}\Bigr ]^2 = \rho \bigl [ w_1^{(2)}(c_1^2 +c_2) +
w_1^{(1)}(c_1c_2+c_3) + w_1 c_1c_3 \bigr ] + \tau
[c_1^5 + c_1c_2^2+ c_1^2c_3 + c_1^3c_2+c_2c_3].$$ This gives a fair indication of the level of awkwardness to this approach.
The stable case
---------------
The ring of invariants ${{\mathcal{I}}nv}_k$ will typically require many relations beyond just those for the squares on the $w$-classes—see the examples in Section \[se:examples\]. However, things become simpler in the stable case $k{\rightarrow}\infty$. We describe this next.
Recall that $T_k={{\mathbb M}}_2[a_1,\ldots,a_k,b_1,\ldots,b_k]/(a_i^2=\rho
a_i+\tau b_i)$. The map $T_{k+1}{\rightarrow}T_k$ that sends $a_{k+1}$ and $b_{k+1}$ to $0$ induces a surjection ${{\mathcal{I}}nv}_{k+1}{\rightarrow}{{\mathcal{I}}nv}_k$ which is an isomorphism in topological degrees less than $k+1$ (the latter is immediate from looking at the standard free bases over ${{\mathbb M}}_2$). Write ${{\mathcal{I}}nv}_\infty$ for the inverse limit of $$\cdots {\twoheadrightarrow}{{\mathcal{I}}nv}_3 {\twoheadrightarrow}{{\mathcal{I}}nv}_2 {\twoheadrightarrow}{{\mathcal{I}}nv}_1$$ From Theorem \[th:indecomp\] it follows that the indecomposables of this ring are the classes $c_j$ for $1\leq j$ and the classes $w_{2^i}^{(e)}$ for $0\leq i$ and $0\leq e$.
\[pr:stable\] There exist a collection of polynomials $R_{i,e}$ such that ${{\mathcal{I}}nv}_\infty$ is the quotient of ${{\mathbb M}}_2[c_j,w_i^{(e)}\,|\,i,j,e\in
{{\mathbb Z}}_{\geq 0}]$ by the relations $$\Bigl [ w_i^{(e)} \Bigr ]^2 = R_{i,e}.$$
Unfortunately the polynomials $R_{i,e}$ seem cumbersome to work out in general. We saw in (\[eq:newt\]) that $R_{1,e}=\rho
w_1^{(2e)}+\tau[N_{2e+1}(c_1,\ldots))]$ where $N_{2e+1}$ is the mod $2$ Newton polynomial for writing the $(2e+1)$-power sum as a polynomial in the elementary symmetric functions. The polynomial $R_{2,e}$ is more unpleasant; it has the form $$R_{2,e}=\rho^2 w_2^{(2e)} + \rho\tau \Bigl [ w_1^{(4e+1)} +
w_1^{(2e)} N_{2e+1}(c_1,\ldots) \Bigr ] + \tau^2[b_1^{2e+1}b_2^{2e+1}]$$ where the expression $[b_1^{2e+1}b_2^{2e+1}]$ must be replaced by a certain complicated, Newton-like polynomial in the Chern classes.
We let $R_{i,e}$ be the polynomials constructed as in Section \[se:relations\]—it is clear enough that they exist, it is just not clear how to write down their coefficients in a reasonable way. Consider the surjection $${{\mathbb M}}_2[c_j,w_i^{(e)}\,|\,i,j,e\in
{{\mathbb Z}}_{\geq 0}]/(R_{i,e}) {\twoheadrightarrow}{{\mathcal{I}}nv}_\infty.$$ We claim that the bigraded Poincaré series for these two algebras are identical, and from this it immediately follows that the map is an isomorphism. Both the domain and target are free ${{\mathbb M}}_2$-modules, so it suffices to instead look at the bigraded rank functions.
The domain has a free ${{\mathbb M}}_2$-basis consisting of monomials in the variables $c_j$ and $w_i^{(e)}$ that are square-free in the $w$-classes. So the bigraded rank function is the same as for the algebra $$\Lambda(w_i^{(e)}\,|\,0\leq i,0\leq e){\otimes}{\mathbb{F}}_2[c_1,c_2,\ldots].$$ Likewise, the bigraded rank function for ${{\mathcal{I}}nv}_\infty$ is the same as the Poincaré series for the algebra $L_\infty$ from Appendix A ($L_\infty$ is just the quotient of ${{\mathcal{I}}nv}_\infty$ obtained by killing $\rho$ and $\tau$). But Theorem \[th:En\](c) gives the isomorphism of graded rings $L_\infty{\cong}\Lambda(w_i^{(e)}\,|\,0\leq i,0\leq e){\otimes}{\mathbb{F}}_2[c_1,c_2,\ldots]$, so this completes the proof.
Examples {#se:examples}
========
Our purpose in this section is to take a close look at $H^{*,*}(\operatorname{Gr}_2({{{\EuScript}U}});{{\mathbb Z}}/2)$ and $H^{*,*}(\operatorname{Gr}_3({{{\EuScript}U}});{{\mathbb Z}}/2)$, to demonstrate our general results. We also make some remarks about $H^{*,*}(\operatorname{Gr}_4({{{\EuScript}U}});{{\mathbb Z}}/2)$.
Write ${{\mathbb M}}_2[{\underline}{c}]\subseteq H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ for the ${{\mathbb M}}_2$-subalgebra generated by the $c_i$’s. We have seen that the rank chart for the cohomology ring breaks up naturally into lines of slope $\frac{1}{2}$, and it will be convenient to consider a corresponding decomposition at the level of algebra. To this end, let $F_i\subseteq H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ be the ${{\mathbb M}}_2$-submodule spanned by the elements of our standard basis having degrees $(p,q)$ for $0\leq 2q-p\leq i$. Note that $F_0={{\mathbb M}}_2[{\underline}{c}]$, and in general $F_i$ is an ${{\mathbb M}}_2[{\underline}{c}]$-module. Let $Q_i=F_i/F_{i-1}$, and call this module the “$i$-line”. It is a free ${{\mathbb M}}_2$-module, and the ranks correspond to the ranks of $H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$ occuring along the line of slope $\frac{1}{2}$ that passes through $(i,i)$. The $0$-line is simply ${{\mathbb M}}_2[{\underline}{c}]$. The duality given by Corollary \[co:duality\] says that the ranks along the $i$-line and the $(k-i)$-line are the same, for every $i$.
We take the perspective that the $0$-line is completely understood, as this is just the polynomial ring over ${{\mathbb M}}_2$ on the classes $c_1,c_2,\ldots,c_k$. In some sense we then also understand the $k$-line, by duality. Our next observation is that we can also understand the $1$-line (and therefore the $(k-1)$-line along with it).
\[le:1-line\] Let $X=H^{*,*}(\operatorname{Gr}_k({{{\EuScript}U}});{{\mathbb Z}}/2)$. Then $$\operatorname{rank}^{2p+1,p+1}(X)=\operatorname{rank}^{2p,p}(X)+\operatorname{rank}^{2p-2,p-1}(X) + \cdots +
\operatorname{rank}^{2p-2(k-1),p-(k-1)}(X)$$ for any $p\in {{\mathbb Z}}$.
We change this into a statement about partitions, using Proposition \[pr:rank-Inv\]. The claim is that $$\operatorname{part}_{2p+1,\leq k}[1]=\sum_{i=0}^{k-1} \operatorname{part}_{2p-2i,\leq k}[0].$$ We sketch a bijective proof of this. Regard a partition with at most $k$ pieces as a partition having exactly $k$ pieces, but where some pieces are $0$. Given a partition of $2p$ into $k$ pieces that are all even, make a partition of $2p+1$ by adding $1$ to the smallest piece. Given a partition of $2p-1$ into $k$ pieces that are all even, make a partition of $2p+1$ by adding $3$ to the second smallest piece. And so on: given an element of $\operatorname{part}_{2p-2i,\leq k}[0]$, make a partition of $2p+1$ by adding $2i+1$ to the $i$th smallest piece. We leave it to the reader to check that this does indeed give the desired bijection.
The $1$-line $Q_1$ is a free ${{\mathbb M}}_2[{\underline}{c}]$-module generated by the classes $w_1^{(e)}$ for $0\leq e\leq k-1$.
We have the evident map
\[eq:map\] [[M]{}]{}\_2\[[c]{}\]w\_1,w\_1\^[(1)]{},…,w\_1\^[(k-1)]{}Q\_1.
Theorem \[th:indecomp\] says that $X$ is generated as an ${{\mathbb M}}_2$-algebra by products of elements $c_i$ and $w_j^{(e)}$. The only such products that can lie on the $1$-line are products of $c_i$’s with $w_1^{(e)}$’s. This shows that the map in (\[eq:map\]) is surjective. But Lemma \[le:1-line\] shows that the ranks of the domain and target of (\[eq:map\]) coincide, hence the map must be an isomorphism.
In the cohomology of $\operatorname{Gr}_2({{{\EuScript}U}})$ we only have the $0$-line, $1$-line, and $2$-line, and the outer two are dual—so we basically understand everything. In $\operatorname{Gr}_3({{{\EuScript}U}})$ we have the $0$-line/$3$-line and the $1$-line/$2$-line, and again we understand everything. This is why these two cases are fairly easy. When we get to $\operatorname{Gr}_4({{{\EuScript}U}})$ things become more complicated.
Let us now look in detail at $\operatorname{Gr}_2({{{\EuScript}U}})$. The rank calculations can be done by counting partitions using Proposition \[pr:rank-Inv\], and this is very easy. One finds $$\operatorname{rank}^{2p,p}=\operatorname{rank}^{2p+2,p+2}=\begin{cases} \frac{p}{2}+1 &
\text{if $p$ is even}, \\
\frac{p+1}{2} & \text{if $p$ is odd,}
\end{cases}$$ and $$\operatorname{rank}^{2p+1,p+1}=p+1.$$ By Theorem \[th:indecomp\] the indecomposables are the following elements: $$c_1,\ c_2,\ w_1,\ w_1^{(1)}, \ w_2.$$ The $1$-line is a free ${{\mathbb M}}_2[{\underline}{c}]$-module generated by $w_1$ and $w_1^{(1)}$, and the rank calculations suggest that the $2$-line is the free ${{\mathbb M}}_2[{\underline}{c}]$-module generated by $w_2$. So we guess that the three classes $w_1$, $w_1^{(1)}$ and $w_2$ span the cohomology as an ${{\mathbb M}}_2[{\underline}{c}]$-module. If this is true, there will be relations specifying the products of any two of the $w$-classes. A little work shows that $$\begin{aligned}
& w_1^2=\rho w_1+\tau c_1, \qquad w_2^2=\rho^2 w_2 + \rho \tau
\bigl (w_1c_1+w_1^{(1)}\bigr ) + \tau^2 c_2\\
& \Bigl [w_1^{(1)}\Bigr ]^2=\rho\bigl (w_1^{(1)}c_1 +
w_1c_2\bigr )+\tau(c_1^3+c_1c_2)\\\end{aligned}$$ and also that $$\begin{aligned}
& w_1w_2=\rho w_2 + \tau\bigl (w_1c_1+w_1^{(1)}\bigr ) \\
& w_1w_1^{(1)}=\rho w_1^{(1)}+\tau c_1^2 +w_2c_1 \\
& w_2 w_1^{(1)}=\rho w_2 c_1 + \tau (w_1c_1^2+w_1^{(1)}c_1+w_1c_2).\end{aligned}$$ We have separated the relations into two classes: the relations for the squares of the $w$-classes will always be present, but the relations amongst square-free monomials in the $w$-classes depend very much on the value of $k$.
Once these relations have been verified, we have a surjective algebra map $${{\mathbb M}}_2[c_1,c_2,w_1,w_1^{(1)},w_2]/(R) {\twoheadrightarrow}H^{*,*}(\operatorname{Gr}_2({{{\EuScript}U}});{{\mathbb Z}}/2)$$ where $R$ is the above list of relations. As an ${{\mathbb M}}_2[{\underline}{c}]$-module the domain is free with generators $1$, $w_1$, $w_1^{(1)}$, and $w_2$, and our rank calculations then show that the Poincaré series for the domain and target agree. So the above map must be an isomorphism.
It remains to verify the relations listed above. The ones for the squares of $w_1$ and $w_1^{(1)}$ follow readily from (\[eq:w-square\]) and Lemma \[le:w1e\]. For $w_2^2$ we write $$\begin{aligned}
[a_1a_2]^2=[a_1^2a_2^2]=[(\rho a_1+\tau b_1)(\rho a_2+\tau
b_2)] & =\rho^2[a_1a_2] + \rho\tau [a_1b_2] + \tau^2 [b_1^2] \\
&= \rho^2 w_2
+ \rho\tau \bigl ( [a_1][b_1]+[a_1b_1]\bigr ) +\tau^2 c_1^2.\end{aligned}$$ Of the remaining three relations, we leave the first two to the reader and only verify the last: $$\begin{aligned}
[a_1a_2]\cdot [a_1b_1]= [a_1^2a_2b_1] =
\rho[a_1a_2b_1]+\tau[a_1b_2^2]
&= \rho [a_1a_2][b_1] + \tau \bigl ( [a_1][b_1^2]+[a_1b_1^2] \bigr
)\\
&= \rho w_2c_1 + \tau \bigl ( w_1c_1^2 + w_1^{(2)}\bigr ).\end{aligned}$$ Now use Lemma \[le:w1e\] to decompose $w_1^{(2)}$.
Next let us look at the cohomology of $\operatorname{Gr}_3({{{\EuScript}U}})$. The indecomposables are $$c_1, \ c_2,\ c_3,\ w_1,\ w_1^{(1)},\ w_1^{(2)},\ w_2,$$ and the $1$-line is generated over ${{\mathbb M}}_2[{\underline}{c}]$ by $w_1$, $w_1^{(1)}$, and $w_1^{(2)}$. The evident elements of interest on the $2$-line are $$w_2, \ w_1\cdot w_1^{(1)}, \ w_1\cdot w_1^{(2)}.$$ Duality between the $1$-line and $2$-line suggests that we will have three generators as an ${{\mathbb M}}_2[{\underline}{c}]$-module, and since these are the only candidates there is not much choice for what can happen. Finally, we expect by duality that the $3$-line is the free ${{\mathbb M}}_2[{\underline}{c}]$-module generated by $w_1w_2$. This gives a conjectural description of the cohomology as a module over ${{\mathbb M}}_2[{\underline}{c}]$, which we will soon see is correct.
The guess suggests that we should have relations for the products $w_2\cdot w_1^{(1)}$, $w_2\cdot w_1^{(2)}$, and $w_1^{(1)}\cdot
w_1^{(2)}$—as well as for the squares of all the $w$-classes, of course. Some tedious work in the ring of invariants reveals the following relations: $$\begin{aligned}
& w_1^2=\rho w_1+\tau c_1 \\
& w_2^2=\rho^2 w_2 + \rho\tau(w_1c_1+w_1^{(1)})+\tau^2 c_2 \\
& \Bigl [ w_1^{(1)}\Bigr ]^2= \rho w_1^{(2)}+ \tau \bigl [ c_1^3 +
c_1c_2+c_3\bigr ]\\
& \Bigl [ w_1^{(2)}\Bigr ]^2= \rho \bigl [
w_1^{(2)}c_1^2 + w_1^{(1)}c_1c_2+w_1c_1c_3+w_1^{(2)}c_2+w_1^{(1)}c_3
\bigl ]\\
&\qquad\qquad\qquad\qquad\qquad\qquad + \tau\bigl [
c_1^5+c_1^3c_2+c_1^2c_3+c_1c_2^2+c_2c_3\bigr ]\end{aligned}$$ and $$\begin{aligned}
& w_2\cdot w_1^{(1)}=
w_1w_2c_1+(\rho,\tau)\\
& w_2\cdot w_1^{(2)}=
w_1w_2c_1^2+(\rho,\tau) \\
& w_1^{(1)}\cdot
w_1^{(2)}=w_2c_3+w_2c_1c_2+w_1w_1^{(1)}c_1^2+w_1w_1^{(2)}c_1
+(\rho,\tau).\end{aligned}$$ In the last three cases we are being somewhat lazy and not writing out the entire relations, which are long and complicated. We have instead written “$(\rho,\tau)$” as shorthand for all terms belonging to the ideal $(\rho,\tau)$.
Once again, we have now produced a surjective map $${{\mathbb M}}_2[c_1,c_2,w_1,w_1^{(1)},w_1^{(2)},w_2]/(R) {\twoheadrightarrow}H^{*,*}(\operatorname{Gr}_3({{{\EuScript}U}});{{\mathbb Z}}/2)$$ where $R$ is the set of relations above. The domain is a free ${{\mathbb M}}_2[{\underline}{c}]$-module generated by $1,w_1,w_1^{(1)},w_1^{(2)},w_2, w_1\cdot w_1^{(1)}, w_1\cdot
w_1^{(2)},w_1w_2$. One can analyze the Poincaré series for the cohomology ring in terms of partitions, and a little work shows that the Poincaré series of the domain and codomain agree. It follows that the above map is an isomorphism of algebras.
Finally, we make some brief remarks about $\operatorname{Gr}_4({{{\EuScript}U}})$. The indecomposables are $$c_1,\ c_2,\ c_3,\ c_4,\ w_1,\ w_1^{(1)}, \ w_1^{(2)}, \
w_1^{(3)},\ w_2, \ w_2^{(1)}, \ w_4.$$ The $0$-line is the polynomial algebra ${{\mathbb M}}_2[c_1,c_2,c_3,c_4]$, and the $1$-line is the free ${{\mathbb M}}_2[{\underline}{c}]$-module with basis elements $w_1^{(e)}$ for $0\leq e\leq 3$. The monomials on the $2$-line are $$w_2,\ w_1w_1^{(1)},\ w_1w_1^{(2)},\ w_1w_1^{(3)},\ w_2^{(1)},\
w_1^{(1)}\cdot w_1^{(2)},\ w_1^{(1)}\cdot w_1^{(3)},\ w_1^{(2)}\cdot
w_1^{(3)},$$ with bidegrees $$(2,2), \ (4,3),\ (6,4),\ (8,5),\ (6,4),\ (8,5),\
(10,6),\ (12,7).$$ The ranks along the $0$-line constitute the sequence $S=(1,1,2,3,5,6,9,11,\ldots)$. If the $2$-line were free on the above generators then the ranks along the $2$-line would be $P=(1,2,5,9,15,23,34,47,\ldots)$. This sequence is obtained by adding up eight copies of $S$ with appropriate shifts, according to the topological degrees of the eight monomials listed above: $P=\sum_i
(\Sigma^{(p_i-2)/2}S)$ where $p_i$ is the topological degree of the $i$th element of the list (we subtract two because our $2$-line “starts” at $w_2$). That is, $$P=S+\Sigma S + \Sigma^2 S+\Sigma^2 S + \Sigma^3 S+\Sigma^3
S+\Sigma^4 S+\Sigma^5 S.$$ Computations with partitions reveals that the actual rank sequence for the $2$-line is $(1,2,5,8,14,20,30,40,55,\ldots)$. Playing around with the numerology shows that removing a $\Sigma^3S$ and the $\Sigma^5 S$ from $P$ seems to yield the correct answer; this leads to the guess that there is a dependence relation amongst the two elements $w_1w_1^{(3)}$ and $w_1^{(1)}w_1^{(2)}$, and also that there should be a relation for $w_1^{(2)}w_1^{(3)}$. One can indeed find such relations, although the process is time-consuming. In the first case the relation is $$w_1w_1^{(3)}+w_1^{(1)}w_1^{(2)}+w_1w_1^{(2)}c_1+w_2^{(1)}c_1+w_2c_3+w_1w_1^{(1)}c_2+(\rho,\tau)=0$$ where the last term represents an element in the ideal $(\rho,\tau)$ that we have not gone to the trouble of determining.
It again appears that the cohomology of $\operatorname{Gr}_4({{{\EuScript}U}})$ is free as a module over ${{\mathbb M}}_2[{\underline}{c}]$, with basis consisting of certain products of $w$-classes. However, there does not seem to be a canonical choice for the basis: e.g., there is no preferred choice among $w_1w_1^{(3)}$ and $w_1^{(1)}w_1^{(2)}$ for which to include. Also, the relations are getting truly horrendous. We choose to stop here.
Connections to motivic phenomena {#se:motivic}
================================
Let $F$ be a field, not of characteristic $2$. For an algebraic variety $X$ over $F$, a [quadratic bundle]{} over $X$ is an algebraic vector bundle $E{\rightarrow}X$ together with a pairing $E{\otimes}_F E {\rightarrow}{{{\EuScript}O}}_X$ that is symmetric and restricts to nondegenerate bilinear forms on each fiber. For reasons that we will not explain here, such bundles play the role in motivic homotopy theory that ordinary real vector bundles play in classical algebraic topology (see Remark \[re:quadratic=real\] below for a bit more information). It is natural, therefore, to try to understand characteristic classes for quadratic bundles with values in mod $2$ motivic cohomology.
One can make a guess at a classifying space for quadratic vector bundles, as follows (this is known to be a true classifying space if one works stably, by a result of [@ST]). Equip the affine space ${{\mathbb A}}^{2n}$ with the quadratic form $$q_{2n}(x_1,y_1,x_2,y_2,\ldots,x_n,y_n)=x_1y_1+\cdots+x_ny_n$$ and equip ${{\mathbb A}}^{2n+1}$ with the quadratic form $$q_{2n+1}(x_1,y_1,x_2,y_2,\ldots,x_n,y_n,z)=x_1y_1+\cdots+x_ny_n+z^2.$$ These are called the [split]{} quadratic forms. Note that we have ${{\mathbb A}}^{2n}$ sitting inside ${{\mathbb A}}^{2n+1}$ as the $z=0$ subspace, which exhibits $q_{2n}$ as the restriction of $q_{2n+1}$. We will also regard ${{\mathbb A}}^{2n+1}$ as sitting inside ${{\mathbb A}}^{2n+2}$ as the subspace $x_{n+1}=y_{n+1}$, which exhibits $q_{2n+1}$ as the restriction of $q_{2n+2}$.
From now on we will write $({{\mathbb A}}^N,q)$ for either $({{\mathbb A}}^{2n},q_{2n})$ or $({{\mathbb A}}^{2n+1},q_{2n+1})$. Note that we have a series of inclusions $$({{\mathbb A}}^1,q) {\hookrightarrow}({{\mathbb A}}^2,q) {\hookrightarrow}({{\mathbb A}}^3,q){\hookrightarrow}\cdots$$
Define the [orthogonal Grassmannian]{} $\operatorname{OGr}_k({{\mathbb A}}^N)$ to be the Zariski open subspace of $\operatorname{Gr}_k({{\mathbb A}}^N)$ consisting of the $k$-planes where $q$ restricts to a nondegenerate form. Taking the colimit over $N$ gives a motivic space $\operatorname{OGr}_k({{\mathbb A}}^\infty)$, in the sense of [@MV]. It is an interesting (and unsolved) problem to compute the motivic cohomology groups of this space.
Now restrict to the case $F={{\mathbb R}}$. From an ${{\mathbb R}}$-variety $X$ we can consider the set $X({{\mathbb C}})$ of ${{\mathbb C}}$-valued points, regarded as a topological space via the analytic topology. This space has an evident ${{\mathbb Z}}/2$-action given by complex conjugation, and the assignment $X\mapsto X({{\mathbb C}})$ extends to a map of homotopy theories from motivic homotopy theory over ${{\mathbb R}}$ to ${{\mathbb Z}}/2$-equivariant homotopy theory. Our goal in this section is only to note the following result:
\[th:Z/2\] There is an equivariant weak homotopy equivalence $$[\operatorname{OGr}_k({{\mathbb A}}^{N})]({{\mathbb C}}) \
{\simeq}\operatorname{Gr}_k({{{\EuScript}U}}^N).$$ (Recall that ${{{\EuScript}U}}^N$ denotes the first $N$ summands of the infinite ${{\mathbb Z}}/2$-representation ${{{\EuScript}U}}={{\mathbb R}}\oplus {{\mathbb R}}_-\oplus {{\mathbb R}}\oplus {{\mathbb R}}_- \oplus \cdots$).
The above theorem shows that the main problem considered in this paper is indeed the ${{\mathbb Z}}/2$-equivariant analog of the problem of motivic characteristic classes for quadratic bundles.
We will need a few preliminary results before giving the proof of the theorem. To generalize our previous definition somewhat, if $V$ is any vector space with a quadratic form $q$ then we write $\operatorname{OGr}_k(V)$ for the subspace of $\operatorname{Gr}_k(V)$ consisting of $k$-planes $W\subseteq V$ such that $q|_W$ is nondegenerate. Sometimes $V$ will be a real vector space and sometimes $V$ will be a complex vector space, and in the latter case our orthogonal Grassmannian will be the space of complex $k$-planes on which $q$ is nondegenerate. Usually the intent will be clear from context.
Assume $V$ is real and the form $q$ is positive-definite. This form extends to give a complex quadratic form on $V{\otimes}_{{\mathbb R}}{{\mathbb C}}$ that we will also call $q$. The complexification map $c\colon \operatorname{Gr}_k(V) {\rightarrow}\operatorname{Gr}_k(V{\otimes}_{{\mathbb R}}{{\mathbb C}})$ has its image contained in $\operatorname{OGr}_k(V{\otimes}_{{\mathbb R}}{{\mathbb C}})$. To see this, just observe that if $U\subseteq V$ is any $k$-plane then there is a basis for $U$ with respect to which $q$ looks like the sum-of-squares form. Extending this basis to $U{\otimes}_{{\mathbb R}}{{\mathbb C}}$ shows that $q$ is nondegenerate here. Similar remarks apply to show that the direct-sum map in part (b) of the following result takes its image in $\operatorname{OGr}$ rather than just $\operatorname{Gr}$.
Note that the following result takes place in the non-equivariant setting:
\[pr:quad-form-nonequiv\] Let $V$ be a real vector space with a positive-definite quadratic form $q$.
(a) The complexification map $\operatorname{Gr}_k(V) {\rightarrow}\operatorname{OGr}_k^{{\mathbb C}}(V{\otimes}_{{\mathbb R}}{{\mathbb C}})$ is a weak homotopy equivalence;
(b) Let $V'$ be another real vector space with positive-definite form $q'$. Then the direct-sum map $\coprod_{a+b=k} \operatorname{Gr}_a(V)\times \operatorname{Gr}_b(V') {\rightarrow}\operatorname{OGr}_k(V\oplus V',q\oplus (-q'))$ is a weak homotopy equivalence.
Without loss of generality we may assume that $V={{\mathbb R}}^n$ and $q$ is the sum-of-squares form. Recall that the symmetry group of this form is the Lie group $O_n=\{A\in M_{n\times n}({{\mathbb R}})\,|\,
AA^T=I\}$. The symmetry group for the sum-of-squares form over ${{\mathbb C}}$ is $O_n({{\mathbb C}})=\{A\in M_{n\times n}({{\mathbb C}})\,|\,
AA^T=I\}$. Recall that $O_n$ is a maximal compact subgroup inside of $O_n({{\mathbb C}})$; it is therefore known by the Iwasawa decomposition that the inclusion $O_n{\hookrightarrow}O_n({{\mathbb C}})$ is a homotopy equivalence (see [@CSM Theorem 8.1 of Segal’s lecture] or [@H Chapter XV, Theorem 3.1]).
The space $\operatorname{Gr}_k({{\mathbb R}}^n)$ is homeomorphic to $O_n/[O_k\times O_{n-k}]$. Likewise, $\operatorname{OGr}_k({{\mathbb C}}^n)$ is homeomorphic to $O_n({{\mathbb C}})/[O_k({{\mathbb C}})\times
O_{n-k}({{\mathbb C}})]$. The map in part (a) is the evident comparison map between these homogeneous spaces. Consider the two fiber bundles $$\xymatrix{
O_k\times O_{n-k} \ar[r] \ar[d] & O_n \ar[r]\ar[d] & O_n/[O_k\times
O_{n-k}]\ar[d] \\
O_k({{\mathbb C}})\times O_{n-k}({{\mathbb C}}) \ar[r] & O_n({{\mathbb C}}) \ar[r] & O_n({{\mathbb C}})/[O_k({{\mathbb C}})\times
O_{n-k}({{\mathbb C}})]
}$$ (written horizontally). The left and middle vertical maps are weak equivalences, therefore the right map is as well. This proves (a).
For (b) recall that a nondegenerate quadratic form on an $n$-dimensional real vector space is classified by its signature: the pair of integers $(a,b)$ such that $a+b=n$, representing the number of positive and negative entries in any diagonalization of the form. Let $O(a,b)$ be the symmetry group for the quadratic form of signature $(a,b)$. This Lie group contains $O(a)\times O(b)$ in the evident way, and it is known that this is a maximal compact subgroup. Consequently, the inclusion $O(a)\times
O(b){\hookrightarrow}O(a,b)$ is a weak homotopy equivalence by the Iwasawa decomposition.
We can assume $V={{\mathbb R}}^n$ and $V'={{\mathbb R}}^{n'}$, with both $q$ and $q'$ being the sum-of-squares form. The group $O(n,n')$ acts on $\operatorname{OGr}_k(V\oplus
V')$ in the evident way. It is easy to see that the action decomposes the orthogonal Grassmannian into a disjoint union of orbits, one for every possible signature $(a,b)$ with $a+b=k$. The path component corresponding to such a signature is the homogeneous space $$O(n,n')/ [O(a,b)\times O(n-a,n'-b)].$$
The map in part (b) coincides with the disjoint union of the evident maps $$\xymatrix{
\Bigl [ O(n)/[O(a)\times O(n-a)] \Bigr ] \times \Bigl [
O(n')/[O(b)\times O(n'-b)]\Bigr ] \ar[d]^{\cong}\\
[O(n)\times O(n')]/\Bigl [[O(a)\times O(n-a)] \times [O(b)\times
O(n'-b)]\Bigr ] \ar[d]
\\ O(n,n')/ [O(a,b)\times O(n-a,n'-b)].
}$$ At this point one proceeds exactly in the proof of part (a): write down a map between two fiber bundles, where two of the three maps are already known to be weak homotopy equivalences.
We next move into the equivariant setting. By an [orthogonal representation]{} of ${{\mathbb Z}}/2$ we mean a pair $(V,q)$ where $V$ is a real vector space and $q\colon
V{\rightarrow}{{\mathbb R}}$ is a positive-definite quadratic form on $V$ such that $q(\sigma x)=q(x)$ for all $x\in V$. The main examples for us will be where $V={{\mathbb R}}^n$, $q$ is the standard sum-of-squares form, and ${{\mathbb Z}}/2$ acts on $V$ by changing signs on some subset of the standard basis elements.
Let $V_{{\mathbb C}}=V{\otimes}_{{\mathbb R}}{{\mathbb C}}$, with the ${{\mathbb Z}}/2$ action induced by that on $V$. The complexification map $\operatorname{Gr}_k(V){\rightarrow}\operatorname{OGr}_k(V_{{\mathbb C}})$ sending $U\subseteq V$ to $U_{{\mathbb C}}\subseteq V_{{\mathbb C}}$ is clearly equivariant, where the ${{\mathbb Z}}/2$-actions on domain and codomain are induced by those on $V$ and $V_{{\mathbb C}}$.
\[co:Gr=OGr\] For any orthogonal representation $V$ of ${{\mathbb Z}}/2$, the map of ${{\mathbb Z}}/2$-spaces $\operatorname{Gr}_k(V){\rightarrow}\operatorname{OGr}_k(V_{{\mathbb C}})$ is an equivariant weak equivalence.
Taking Proposition \[pr:quad-form-nonequiv\](a) under consideration, it suffices to prove that the induced map of fixed sets is a weak equivalence. Let $V^{{{\mathbb Z}}/2}$ and $V^{-{{\mathbb Z}}/2}$ denote the $+1$ and $-1$ eigenspaces for the involution on $V$. These are orthogonal with respect to the inner product on $V$. A subspace $U\subseteq V$ is fixed under the ${{\mathbb Z}}/2$ action if and only if $U$ equals the direct sum $(U\cap V^{{{\mathbb Z}}/2})\oplus
(U\cap V^{-{{\mathbb Z}}/2})$. From this we get a homeomorphism $$\operatorname{Gr}_k(V)^{{{\mathbb Z}}/2} {\cong}\coprod_i \operatorname{Gr}_i(V^{{{\mathbb Z}}/2})\times
\operatorname{Gr}_{k-i}(V^{-{{\mathbb Z}}/2}),$$ which sends $U\subseteq V$ to the pair $(U\cap V^{{{\mathbb Z}}/2}, U\cap
V^{-{{\mathbb Z}}/2})$. In the same way, one obtains a homeomorphism $$\operatorname{OGr}_k(V_{{\mathbb C}})^{{{\mathbb Z}}/2}{\cong}\coprod_i \operatorname{OGr}_i(V_{{\mathbb C}}^{{{\mathbb Z}}/2})\times
\operatorname{OGr}_{k-i}(V_{{\mathbb C}}^{-{{\mathbb Z}}/2}).$$ Since the inclusions $\operatorname{Gr}_i(V^{{{\mathbb Z}}/2}){\hookrightarrow}\operatorname{OGr}_i(V_{{\mathbb C}}^{{{\mathbb Z}}/2})$ and $\operatorname{Gr}_{j}(V^{-{{\mathbb Z}}/2}){\hookrightarrow}\operatorname{OGr}_{j}(V_{{\mathbb C}}^{-{{\mathbb Z}}/2})$ are (non-equivariant) weak equivalences by Proposition \[pr:quad-form-nonequiv\](a), this completes the proof.
The above corollary has been included for completeness, but it actually does not give us what we need. The ${{\mathbb Z}}/2$ action on $V{\otimes}_{{\mathbb R}}{{\mathbb C}}$ is complex linear, whereas we will find that we actually need to consider conjugate linear actions. We do this next.
Let $W$ be a complex vector space with a nondegenerate quadratic form $q$. Let $\sigma\colon W{\rightarrow}W$ be a conjugate-linear map such that $\sigma^2=1$. That is, $\sigma(zx)=\bar{z}\sigma(x)$ for every $z\in {{\mathbb C}}$ and $x\in W$. Also assume that $q(\sigma x)=\overline{q(x)}$ for every $x\in W$. The space $\operatorname{OGr}_k(W)$ then has a ${{\mathbb Z}}/2$-action induced by $\sigma$: if $J\subseteq W$ is a complex subspace such that $q|_J$ is nondegenerate, then $\sigma(J)$ is another complex subspace on which $q$ restricts to be nondegenerate. Our next task is to analyze the fixed space $\operatorname{OGr}_k(W)^{{{\mathbb Z}}/2}$.
Let $(V,q)$ be an orthogonal representation for ${{\mathbb Z}}/2$, and let $W$ be the vector space $V{\otimes}_{{\mathbb R}}{{\mathbb C}}$ with the action given by $\sigma(v{\otimes}z)=\sigma(v){\otimes}\bar{z}$. Then $(W,q)$ satisfies the conditions of the above paragraph. In this case we will use the notation $W=V{\otimes}_{{\mathbb R}}\overline{{{\mathbb C}}}$. The bar over the ${{\mathbb C}}$ just reminds us that ${{\mathbb Z}}/2$ acts on that factor by conjugation.
Returning to the case of a general $W$, note that as a real vector space $W$ decomposes as $W^{{{\mathbb Z}}/2}\oplus
W^{-{{\mathbb Z}}/2}$, where the summands are the subspaces on which $\sigma$ acts as the identity and as multiplication by $-1$. Moreover, multiplication by $i$ maps $W^{{{\mathbb Z}}/2}$ isomorphically onto $W^{-{{\mathbb Z}}/2}$. Finally, one easily checks that $q$ is real-valued on both $W^{{{\mathbb Z}}/2}$ and $W^{-{{\mathbb Z}}/2}$.
If $J\subseteq W$ is any complex subspace that is fixed by $\sigma$ then we have the decomposition $J=(J\cap W^{{{\mathbb Z}}/2})\oplus (J\cap W^{-{{\mathbb Z}}/2})$, and multiplication by $i$ interchanges the two summands. In this way we get a map $$\operatorname{OGr}_k(W,q)^{{{\mathbb Z}}/2}{\longrightarrow}\operatorname{Gr}_k(W^{{{\mathbb Z}}/2}), \qquad J\mapsto J\cap W^{{{\mathbb Z}}/2}$$ and the image is readily checked to land in $\operatorname{OGr}_k(W^{{{\mathbb Z}}/2},q)$. Conversely, if $M\subseteq W^{{{\mathbb Z}}/2}$ is any $k$-dimensional real subspace such that $q|_M$ is nondegenerate then $M\oplus iM\subseteq
W$ is a $k$-dimensional complex subspace with the same property. So we also get a map $\operatorname{OGr}_k(W^{{{\mathbb Z}}/2}) {\rightarrow}\operatorname{OGr}_k(W,q)^{{{\mathbb Z}}/2}$. It is routine to check that these maps are inverse isomorphisms. Thus, we have proven the following:
In the above setting, there is a homeomorphism $\operatorname{OGr}_k(W,q)^{{{\mathbb Z}}/2}{\cong}\operatorname{OGr}_k(W^{{{\mathbb Z}}/2},q)$.
We are now ready to prove the main theorem of this section:
Write $q_{sp}$ for the split quadratic form on ${{\mathbb C}}^N$, and $q_{ss}$ for the sum-of-squares quadratic form on ${{\mathbb C}}^N$. The theorem concerns the space $\operatorname{OGr}_k({{\mathbb C}}^N,q_{sp})$ where the ${{\mathbb Z}}/2$-action is induced by complex conjugation. Let $x_1,y_1,x_2,y_2,\ldots $ denote our standard coordinates on ${{\mathbb C}}^N$, with the convention that when $N$ is odd then the last of the $y_j$’s is just zero. By changing coordinates we can change $q_{sp}$ into $q_{ss}$. Precisely, define a map $\phi\colon {{\mathbb C}}^N{\rightarrow}{{\mathbb C}}^N$ by $$\phi(x_1,y_1,x_2,y_2,\ldots)=(x_1+iy_1,x_1-iy_1,x_2+iy_2,x_2-iy_2,\ldots).$$ Then we have $q_{sp}(\phi(v))=q_{ss}(v)$ for any $v\in {{\mathbb C}}^N$. This gives us an identification of non-equivariant spaces $\operatorname{OGr}_k({{\mathbb C}}^N,q_{sp}){\cong}\operatorname{OGr}_k({{\mathbb C}}^N,q_{ss})$. To extend this to an equivariant identification, note that if the target of $\phi$ is given the conjugation action then the domain of $\phi$ gets the action that both conjugates all coordinates AND changes the signs of the $y$-coordinates. In terms of previously-established notation, this is the equivariant homeomorphism $$\operatorname{OGr}_k({{\mathbb C}}^N,q_{sp}) {\cong}\operatorname{OGr}_k({{{\EuScript}U}}^N{\otimes}\overline{{{\mathbb C}}},q_{ss}).$$
Consider the complexification map $$c\colon \operatorname{Gr}_k({{{\EuScript}U}}^N) {\rightarrow}\operatorname{OGr}_k({{{\EuScript}U}}^N{\otimes}\overline{{{\mathbb C}}},q_{ss}).$$ We have seen in Proposition \[pr:quad-form-nonequiv\](a) that this is a non-equivariant weak equivalence. To analyze what is happening on fixed sets, let $W={{{\EuScript}U}}^N{\otimes}\overline{{{\mathbb C}}}$. Note that $W^{{{\mathbb Z}}/2}=\{(r_1,ir_2,r_3,ir_4,\ldots,(i)r_N)\,|\, r_1,\ldots,r_N\in
{{\mathbb R}}\}$, where the last coordinate has the $i$ in front when $N$ is even. Note as well that we can decompose $W^{{{\mathbb Z}}/2}=W^{{{\mathbb Z}}/2}_+\oplus W^{{{\mathbb Z}}/2}_-$ where $$W^{{{\mathbb Z}}/2}_+=\{(r_1,0,r_3,0,\ldots)\,|\, r_i\in {{\mathbb R}}\}, \qquad
W^{{{\mathbb Z}}/2}_-=\{(0,ir_2,0,ir_4,\ldots)\,|\, r_i\in {{\mathbb R}}\}.$$ The form $q_{ss}$ is positive definite on the first summand and negative definite on the second.
Let ${{{\EuScript}U}}^N_+$ and ${{{\EuScript}U}}^N_-$ be the subspaces spanned by the odd- and even-numbered basis elements, respectively. So ${{{\EuScript}U}}^N_+=({{{\EuScript}U}}^N)^{{{\mathbb Z}}/2}$ and ${{{\EuScript}U}}^N_-=({{{\EuScript}U}}^N)^{-{{\mathbb Z}}/2}$. Note the following maps: $$\xymatrixcolsep{0.001pc}\xymatrix{
\operatorname{Gr}_k({{{\EuScript}U}}^N)^{{{\mathbb Z}}/2} \ar[r]^-c &
\operatorname{OGr}_k(W,q_{ss})^{{{\mathbb Z}}/2} &
\operatorname{OGr}_k(W^{{{\mathbb Z}}/2},q_{ss})\ar[l]_-{\cong}\\
\coprod\limits_{a+b=k} \operatorname{Gr}_a({{{\EuScript}U}}^N_+)\times \operatorname{Gr}_b({{{\EuScript}U}}^N_-) \ar@{=}[u]\ar@{.>}@<0.5ex>[rr] &&
\coprod\limits_{a+b=k} \operatorname{Gr}_a(W^{{{\mathbb Z}}/2}_+) \times \operatorname{Gr}_b(W^{{{\mathbb Z}}/2}_-) \ar[u]_\sim
}$$ The map on the right is the evident one, and is a weak homotopy equivalence by Proposition \[pr:quad-form-nonequiv\](b). The dotted map is the obvious homeomorphism, obtained by identifying ${{{\EuScript}U}}^N_+=W^{{{\mathbb Z}}/2}_+$, $i\cdot{{{\EuScript}U}}^N_-= W^{{{\mathbb Z}}/2}_-$. One readily checks that the diagram commutes, and this verifies that $c$ induces a weak homotopy equivalence of fixed sets. Thus, $c$ is an equivariant weak equivalence.
\[re:quadratic=real\] The non-equivariant part of Theorem \[th:Z/2\] (equivalently, Proposition \[pr:quad-form-nonequiv\](a)) gives the homotopy equivalence of spaces $\operatorname{OGr}_k({{\mathbb C}}^N) {\simeq}\operatorname{Gr}_k({{\mathbb R}}^N)$. This is a classical result: for example, see [@A1 remarks in Section 1.5] and [@S discussion of real Grassmannians throughout Chapter 5]. Notice that this gives some corroboration to the idea that quadratic bundles are the motivic analogs of real vector bundles.
The deRham ring of invariants in characteristic two
===================================================
Let $K_n=\Lambda(a_1,\ldots,a_n){\otimes}{\mathbb{F}}_2[b_1,\ldots,b_n]$, and let $\Sigma_n$ act on $K_n$ by simultaneous permutation of the $a_i$’s and $b_j$’s. Let $L_n=K_n^{\Sigma_n}$. We call $L_n$ the “deRham ring of invariants”. Note that there is an augmentation $\epsilon\colon K_n
{\rightarrow}{\mathbb{F}}_2$ sending all the $a_i$’s and $b_j$’s to zero, and this restricts to an augmentation of $L_n$. Let $I\subseteq L_n$ be the augmentation ideal. Our first aim in this section is to give a vector space basis for the module of indecomposables $I/I^2$. Said differently, we give a minimal set of generators for the ring $L_n$.
Note that $K_{n+1}$ maps to $K_n$ by sending $a_{n+1}$ and $b_{n+1}$ to zero, and this homomorphism induces an algebra map $L_{n+1}{\rightarrow}L_n$. That is, if $f(a,b)$ is a polynomial expression in the $a$’s and $b$’s that is invariant under the $\Sigma_{n+1}$-action, then eliminating all monomials with an $a_{n+1}$ or $b_{n+1}$ produces a polynomial that is invariant under $\Sigma_n$. From this description it is also clear that $L_{n+1}{\rightarrow}L_n$ is surjective: if $f(a,b)$ is a $\Sigma_n$-invariant then one can make a $\Sigma_{n+1}$-invariant by adding on appropriate monomial terms that all have $a_{n+1}$ or $b_{n+1}$.
Let $L_\infty$ be the inverse limit of the system $$\cdots {\longrightarrow}L_3 {\longrightarrow}L_2 {\longrightarrow}L_1.$$ The second goal of this section is to give a complete description of the ring $L_\infty$.
These results are presumably well-known amongst algebraists. See Section 7 of [@R] for the case of ${\mathbb{F}}_2[a_1,\ldots,a_k,b_1,\ldots,b_k]$, which can be used to deduce some of our results. See also [@GSS Section 2] for some related work. Rather than use the machinery of [@R], however, we have chosen to give a ‘low-tech’ treatment which is perhaps more illuminating for our present purposes.
If $m\in K_n$ is a monomial in the $a_i$’s and $b_j$’s, write $[m]$ for the smallest polynomial that contains $m$ as one of its terms and is invariant under the $\Sigma_n$-action. Here ‘smallest’ is measured in terms of the number of monomial summands. We can also describe $[m]$ as $$[m]=\sum_{\sigma\in \Sigma_n/H} \sigma.m$$ where $H$ is the stabilizer of $m$ in $\Sigma_n$.
Using the above noation, write $\alpha_{i,e}=[a_1\ldots
a_{2^i}b_1^e\ldots b_{2^i}^e]$ for $1\leq 2^i\leq n$ and $0\leq e$. Also, write $\sigma_i(a)$ and $\sigma_i(b)$ for the elementary symmetric functions in the $a$’s and $b$’s, respectively. So $\sigma_i(a)=[a_1\ldots a_i]$, for example.
We can now state the main result:
\[th:En\]
(a) $L_n$ is minimally generated by the classes $\sigma_i(b)$ for $1\leq i \leq n$ together with the classes $\alpha_{i,e}$ for $1\leq
2^i \leq n$ and $0\leq e\leq
\frac{n}{2^i} -1$. That is to say, these classes give a vector space basis for $I/I^2$.
(b) The number of indecomposables for $L_n$ is $$3n-(\text{$\#$ of
ones in the binary expansion for $n$}).$$
(c) $L_\infty {\cong}\Lambda \bigl (\alpha_{i,e}\,|\, 0\leq i, 0\leq
e\bigr ){\otimes}{\mathbb{F}}_2[\sigma_1(b),\sigma_2(b),\ldots]$.
The Online Encyclopedia of Integer Sequences [@OE] was useful in discovering the formula in part (b).
The proof of this theorem will be given after establishing several lemmas. The first result we give is not directly needed for the proof, but is included for two reasons: it provides some context that helps explain the more complicated theorem above, and we actually need the result in the proof of Proposition \[pr:w-indecomposable\]. The result is probably well-known, but we are not aware of a reference.
\[pr:exterior-inv\] Let $\Sigma_n$ act on $\Lambda_{{\mathbb{F}}_2}(a_1,\ldots,a_n)$ by permutation of indices. Then $$\Lambda(a_1,\ldots,a_n)^{\Sigma_n}=
\Lambda(\sigma_1,\sigma_2,\sigma_4,\ldots,\sigma_{2^k})/R$$ where $k$ is the largest integer such that $2^k\leq n$ and $R$ is the ideal generated by all products $\sigma_{2^{i_1}}\sigma_{2^{i_2}}\cdots\sigma_{2^{i_s}}$ where $2^{i_1}+2^{i_2}+\cdots+2^{i_s}>n$.
It is easy to see that the classes $1,\sigma_1,\ldots,\sigma_n$ form a vector space basis for the ring of invariants over ${\mathbb{F}}_2$. Put a grading on $\Lambda(a_1,\ldots,a_n)$ by having the degree of each $a_i$ be $1$. Then the ring of invariants is also graded; the dimension of each homogeneous piece equals $1$ in degrees from $0$ through $n$, and zero in degrees larger than $n$.
It is also easy to see that $\sigma_i^2=0$ for each $i$, and so we get a map of rings $$\Lambda(\sigma_1,\ldots,\sigma_n)/R {\twoheadrightarrow}\Lambda(a_1,\ldots,a_n)^{\Sigma_n}.$$
The next thing to note is that $\sigma_r\cdot
\sigma_s=\tbinom{r+s}{r}\sigma_{r+s}$. This is an easy computation: distributing the product in $[a_1\ldots a_r]\cdot [a_1\ldots a_s]$ one finds that the products of monomials are all zero if the monomials have any variables in common. The products that are not zero have the form $a_{i_1}\ldots a_{i_{r+s}}$, and such a monomial appears exactly $\binom{r+s}{r}$ times.
If $r$ is not a power of $2$ then there exists an $i$ such that $\binom{r}{i}$ is odd, which implies that $\sigma_r=\sigma_i\cdot
\sigma_{r-i}$. So such classes are decomposable. We therefore have a map $$\Lambda(\sigma_1,\sigma_2,\sigma_4,\ldots,\sigma_{2^k})/R {\twoheadrightarrow}\Lambda(a_1,\ldots,a_n)^{\Sigma_n}.$$ This is a map of graded algebras, and the Poincaré Series for the domain and target are readily checked to coincide. Since the map is a surjection, it must be an isomorphism.
We next establish a series of lemmas directly dealing with the situation of Theorem \[th:En\]. We begin by introducing some notation and terminology. If $I=\{i_1,\ldots,i_k\}$ then write $a_I$ for $a_{i_1}a_{i_2}\cdots
a_{i_k}$. Likewise, if $d_I$ is a function $I{\rightarrow}{{\mathbb Z}}_{\geq 0}$ then write $b_I^{d_I}$ for the monomial $b_{i_1}^{d_{i_1}}b_{i_2}^{d_{i_2}}\cdots b_{i_k}^{d_{i_k}}$. If $m$ is a monomial in the $a$’s and $b$’s, then the variables $a_i$ and $b_i$ are said to be [bound]{} in $m$ if $a_ib_i$ divides $m$. If $a_i$ divides $m$ but $b_i$ does not, we will say that $a_i$ is [free]{} in $m$ (and in the opposite situation we’ll say that $b_i$ is free). Any monomial may be written uniquely in the form $$m=a_Ib_I^{d_I} a_J b_K^{e_K}$$ where the indices in $I$ represent all the bound variables: so $I\cap
J = I\cap K = J\cap K =\emptyset$. Finally, recall that $[m]$ denotes the smallest invariant polynomial containing $m$ as one of its terms.
\[le:in1\] Let $m=a_Ib_I^{d_I} a_J b_K^{e_K}$. Then $[m]$ is decomposable in $L_n$ if any of the following conditions are satisfied:
(1) $I\neq \emptyset$ and $J\neq \emptyset$ (i.e., some of the $a$’s are bound and some are free).
(2) $J=\emptyset$ and $d_{i_1}\neq d_{i_2}$ for some $i_1,i_2\in I$.
(3) $J=K=\emptyset$ and $\#I$ is not a power of $2$.
(4) $I=K=\emptyset$ and $\#J$ is not a power of $2$.
For (1) first assume that $K=\emptyset$, and consider the product $[a_Ib_I^{d_I}]\cdot [a_J]$. Distributing this into sums of products of monomials, such products vanish if $I$ and $J$ intersect. A typical term that remains is $a_Ia_Jb_I^{d_I}$, and it is clear that this term occurs exactly once. In other words, $$[a_Ib_I^{d_I}]\cdot [a_J]=[a_Ia_Jb_I^{d^I}].$$
To finish the proof of (1) we do an induction on the size of $\#K$. If $m=a_Ib_I^{d_I}a_Jb_K^{e_K}$ then consider the product $[a_Ib_I^{d_I}]\cdot [a_Jb_K^{e_K}]$. Distributing this into sums of products of monomials, we find that $$[a_Ib_I^{d_I}]\cdot [a_Jb_K^{e_K}] = [m] + \Bigl (\text{terms of the form
$[a_Ib_I^{d'_I}a_Jb_{K'}^{e_{K'}}]$ where $\#K'<\#K$}\Bigr ).$$ The latter terms come from products where the indices in $I$ match some of those in $K$. By induction these latter terms are all decomposable in $L_n$, so $[m]$ is also decomposable.
For (2), we again first assume that $K=\emptyset$ so that we are looking at $[a_1\ldots a_s b_1^{d_1}\cdots b_s^{d_s}]$. By rearranging the labels we may assume $d_1 \geq d_2 \geq \cdots \geq d_s$. Let $r$ be the smallest index for which $d_r=d_s$, and consider the product $$[a_1\ldots a_{r-1}b_1^{d_1}\ldots b_{r-1}^{d_{r-1}}]\cdot
[a_r\ldots a_s b_r^f\cdots b_s^f]$$ where $f=d_s$. Once again considering the pairwise product of monomials, all such terms vanish except for ones of the form $a_{i_1}\ldots a_{i_s} b_{i_1}^{d_1}\ldots
b_{i_{r-1}}^{d_{r-1}}b_{i_r}^f\ldots b_{i_s}^f$. The fact that $f$ is the smallest degree on the $b_i$’s guarantees that this term appears exactly once in the sum, and hence $$[a_1\ldots a_{r-1}b_1^{d_1}\ldots b_{r-1}^{d_{r-1}}]\cdot
[a_r\ldots a_s b_r^f\cdots b_s^f] = [a_1\ldots a_s b_1^{d_1}\ldots
b_s^{d_s}].$$
To complete the proof of (2) we perform an induction on $\#K$. Consider a monomial $$m=a_Ib_I^{d_I}b_K^{e_K}=a_1\ldots a_sb_1^{d_1}\ldots b_s^{d_s}
b_{s+1}^{e_1}\ldots b_{s+k}^{e_k}.$$ Again arrange things so that $d_1\geq d_2 \geq \cdots \geq d_s$ and let $r$ be the smallest index for which $d_r= d_s$. If we again write $f=d_s$, then one readily checks that $$[a_1\ldots a_{r-1}b_1^{d_1}\ldots b_{r-1}^{d_{r-1}}]\cdot
[a_r\ldots a_s b_r^f\cdots b_s^f b_{s+1}^{e_1}\ldots b_{s+k}^{e_k}] =
[m] + \sum [a_I b_I^{d'_I}b_{K'}^{e_{K'}}]$$ where for each term in the sum $K'$ is a proper subset of $K$. These terms inside the sum correspond to pairs of monomials in the product for which a $b_i$ for $1\leq i\leq r-1$ matches a $b_{s+j}$ for $1\leq j\leq k$. However, by induction on $\#K$ each $[a_Ib_I^{d'_I}b_{K'}^{e_{K'}}]$ is decomposable, hence $[m]$ is also decomposable.
To prove (3) it suffices (in light of (2)) to show that $[a_1\ldots a_k b_1^e\ldots
b_k^e]$ is decomposable whenever $k$ is not a power of $2$. This assumption guarantees that $\binom{k}{i}$ is odd for some $i$ in the range $1\leq i \leq k-1$. We claim that $$[a_1\ldots a_ib_1^e\ldots b_i^e]\cdot [a_{i+1}\ldots
a_kb_{i+1}^e\ldots b_k^e]=[a_1\ldots a_kb_1^e\ldots b_k^e].$$ To see this, note that all terms in the product vanish except for ones of the form $a_{i_1}\ldots
a_{i_k}b_{i_1}^e\ldots b_{i_k}^e$, and such a term appears exactly $\binom{k}{i}$ times. Use that $\binom{k}{i}$ is odd.
The proof of (4) is the same as for (3), it is really the special case $e=0$.
\[le:in2\] $L_n$ is generated as an algebra by the elements $\sigma_i(b)$ for $1\leq i \leq n$ together with the classes $[m]$ where $m=a_Ib_I^{d_I}a_J$ (that is, where $m$ has no free $b$’s).
Let $Q\subseteq L_n$ denote the subalgebra generated by the elements from the statement of the lemma. We will prove that if $m=a_Ib_I^{d_I}a_Jb_K^{e_K}$ is an arbitrary monomial then $[m]$ is equivalent modulo decomposables to an element of $Q$. This readily yields the result by an induction on degree.
First consider the case where $I=J=\emptyset$, so that $m=b_K^{e_K}$. Note that ${{\mathbb Z}}/2[b_1,\ldots,b_n]\subseteq L_n$, and we know ${{\mathbb Z}}/2[b_1,\ldots,b_n]^{\Sigma_n}$ is a polynomial algebra on the $\sigma_i(b)$ for $1\leq i\leq n$. It follows at once that $[m]$ is equivalent modulo decomposables to a multiple of a $\sigma_i(b)$.
The next stage of the proof is done by induction on $\#K$. The base case $K=\emptyset$ is trivial, as such monomials lie in $Q$ by definition. So assume $K\neq \emptyset$ and consider the product $[a_Ib_I^{d_I}a_J]\cdot [b_K^{e_K}]$. This product decomposes into a sum $[m]+[m_1]+[m_2]+\cdots$ where each $m_i$ has fewer free $b$’s than $m$. Therefore $[m]$ is equivalent to $\sum_i [m_i]$ modulo decomposables, and each $[m_i]$ is equivalent to an element of $Q$ by induction.
\[co:Ln-gen\] $L_n$ is generated as an algebra by the following elements:
(1) $\sigma_i(b)$ for $1\leq i\leq n$;
(2) $[a_1\ldots a_{2^i}b_1^e\ldots b_{2^i}^e]$ for $1\leq 2^i\leq n$ and $e\geq 0$.
Lemma \[le:in2\] gives the generators $\sigma_i(b)$ and $[a_Ib_I^{d_I}a_J]$. Using Lemma \[le:in1\](1) we reduce the second class to all elements $[a_Ib_I^{d_I}]$ and $[a_J]$. Finally, Lemma \[le:in1\](2,3,4) further reduces the class to the set of elements in the statement of the corollary.
We need one more lemma before completing the proof of Theorem \[th:En\]. For $x,y\in L_n$ let us write $x\equiv y$ to mean $x$ and $y$ are equivalent modulo decomposables (that is, $x-y\in
I^2$).
\[le:in3\] If $r\geq k$ and $n\geq r+k$ then $$[a_1\ldots a_k b_1^e \ldots b_k^e b_{k+1}\ldots b_{k+r}] \equiv
[a_1\ldots a_k b_1^{e+1} \ldots b_k^{e+1} b_{k+1}\ldots b_{r}].$$ Consequently, provided $ke\leq n$ one has that $$[a_1\ldots a_k b_1^e\ldots b_k^e]\equiv [a_1\ldots a_kb_1\ldots
b_{ke}].$$ If $k+ke \leq n$ we also have $$[a_1\ldots a_k b_1^e\ldots b_k^e]\equiv [a_1\ldots a_kb_{k+1}\ldots
b_{k+ke}].$$
For the first statement consider the product $$[a_1\ldots a_kb_1^e\ldots b_k^e]\cdot [b_{1}\ldots b_r].$$ The product contains $[a_1\ldots a_k b_1^e\ldots b_k^e b_{k+1}\ldots
b_{k+r}]$ and $[a_1\ldots a_k b_1^{e+1}\ldots b_k^{e+1} b_{k+1}\ldots
b_r]$, as well as other terms that look like $[a_1\ldots a_k
b_1^{d_1}\ldots b_k^{d_k}b_{k+1}\ldots b_{k+i}]$ in which the $d_i$’s are not all equal. But such terms are all decomposable by Lemma \[le:in1\](2).
The second statement follows from the first using an induction: $$\begin{aligned}
[a_1\ldots a_kb_1\ldots b_{ke}] & \equiv
[a_1\ldots a_kb_1^2\ldots b_k^2
b_{k+1}\ldots b_{ke-k}] \\
& \equiv [a_1\ldots a_kb_1^3\ldots b_k^3
b_{k+1}\ldots b_{ke-2k}] \\
& \equiv \ldots \\
&\equiv [a_1\ldots a_kb_1^e\ldots b_k^e].\end{aligned}$$
Finally, for the third statement we consider the product $$[a_1\ldots a_k]\cdot [b_1\cdots b_{ke}].$$ This is a sum of terms $[m_i]$ where $[a_1\ldots a_kb_1\ldots b_{ke}]$ appears exactly once, $[a_1\ldots a_kb_{k+1}\ldots b_{k+ke}]$ appears exactly once, and all other $m_i$’s have at least one free $a$ and one bound $a$. But Lemma \[le:in1\](1) then tells us that these other $m_i$’s are all decomposable.
\[co:Ln-gen2\] If $e> \frac{n}{k}-1$ then $[a_1\ldots a_kb_1^e\ldots b_k^e]$ is decomposable in $L_n$.
Let $N=ke+k$, which is larger than $n$ by assumption. We begin by considering the element $[a_1\ldots a_kb_1^e\ldots b_k^e]$ in $L_N$. Lemma \[le:in3\] gives that $$[a_1\ldots a_kb_1^e\ldots b_k^e]\equiv [a_1\ldots a_kb_{k+1}\ldots
b_{k+ke}].$$ Now apply the homomorphism $L_N {\rightarrow}L_n$, and note that since $ke+k>n$ the element on the right maps to zero (every monomial term has at least one index that is larger than $n$). This proves that $[a_1\ldots
a_kb_1^e\ldots b_k^e]$ is decomposable in $L_n$.
At this point we have verified that $L_n$ is generated, as an algebra, by the classes $\sigma_i(b)$ for $1\leq i\leq n$ together with the classes $[a_1\ldots a_{2^i}b_1^e\ldots b_{2^i}^e]$ for $1\leq 2^i\leq
n$ and $0\leq e \leq \frac{n}{2^i}-1$. It remains to verify that these classes are a minimal set of algebra generators—or equivalently, that they give a ${{\mathbb Z}}/2$-basis for $I/I^2$. The approach will be to first grade the algebras in a convenient way. Then we identify the indecomposables in $L_\infty$, which can be done by a counting argument. Finally, we observe that $L_\infty {\rightarrow}L_n$ is an isomorphism in degrees less than or equal to $n$, and use this to deduce the desired facts about the indecomposables in $L_n$.
Grade the algebra $K_n=\Lambda(a_1,\ldots,a_n){\otimes}{\mathbb{F}}_2[b_1,\ldots,b_n]$ by having the degree of each $a_i$ be $1$ and the degree of each $b_i$ be $2$. Then $L_n$ inherits a corresponding grading. The invariant element $\sigma_i(b)$ has degree $2i$, whereas the element $\alpha_{i,e}=[a_1\ldots a_{2^i}b_1^e\ldots b_{2^i}^e]$ has degree $2^i+2e\cdot 2^i=2^i(2e+1)$. Notice that for every positive integer $r$ the set $\{\alpha_{i,e}\,|\,
0\leq i,0\leq e\}$ has exactly one element of degree $r$.
\[pr:stable-iso\] The map $\Lambda(\alpha_{i,e}\,|\,0\leq i, 0\leq e) {\otimes}{\mathbb{F}}_2[\sigma_i\,|\, i\geq 0] {\rightarrow}L_\infty$ is an isomorphism.
We have already proven in Corollary \[co:Ln-gen\] that the map is a surjection. The injectivity will be deduced from a counting argument. For convenience, let $D$ denote the domain of the map from the statement of the proposition. Let $S={\mathbb{F}}_2[v_1,v_2,\ldots]$ where $v_i$ has degree $i$. We will prove that the Poincaré series for $D$ and $L_\infty$ both coincide with the Poincaré series for $S$. Since $D$ and $L_\infty$ will therefore have identical Poincaré series, the surjection $D{\twoheadrightarrow}L_\infty$ must in fact be an isomorphism.
Note that $S$ has a basis over ${\mathbb{F}}_2$ consisting of monomials $$v_{i_1}v_{i_2}\cdots v_{i_r} v_1^{2e_1}v_2^{2e_2}\cdots
v_k^{2e_k}$$ with each $e_j\geq 0$, where the $i_u$’s are distinct. There is an evident bijection between the elements of this basis and the basis for $D$ consisting of monomials in the $\alpha_{i,e}$’s and $\sigma_i$’s: we replace each $v_{i_r}$ with the unique $\alpha_{i,e}$ having degree $i_r$, and we replace each $v_i^{2e}$ with $\sigma_i^e$ This identifies the Poincaré series for $S$ and $D$.
Recall that $L_\infty$ has a ${{\mathbb Z}}/2$-basis consisting of the invariants $[a_{i_1}\ldots a_{i_r}b_{j_1}^{e_1}\ldots b_{j_s}^{e_s}]$ where there is allowed to be overlap between the $i$- and $j$-indices. Say that a monomial is [pure]{} if it only contains $a$’s and $b$’s of a single index. So $b_i^{e}$ and $a_ib_i^e$ are pure, but $a_1a_2b_1^2$ is not. An arbitrary monomial $m$ can be written uniquely (up to permutation of the factors) as $$m=m_1\cdot m_2 \cdots m_t$$ where each $m_i$ is pure and the indices appearing in $m_i$ and $m_j$ are different for every $i\neq j$. For example,
\[eq:monomial\] a\_1a\_2a\_3a\_4 b\_1\^4b\_2b\_4b\_5\^2 = (a\_1b\_1\^4)(a\_2b\_2) (a\_3)(a\_4b\_4) (b\_5\^2).
For a pure monomial $m$, let $d(m)$ be its degree and let $\eta(m)=v_{d(m)}$. Finally, for an arbitrary monomial $m$ as above define $\eta(m)=\eta(m_1)\cdots\eta(m_t)=v_{d(1)}\cdot v_{d(2)} \cdots v_{d(t)}$. For example, for the monomial in (\[eq:monomial\]) we have $\eta(m)=v_1 v_3^2 v_9 v_{10}$.
Note that if $\sigma$ is a permutation of the indices then $\eta(\sigma m)=\eta(m)$. One readily checks that the function $\eta$ gives a bijection between our basis for $L_\infty$ and the standard monomial basis for $S$; it should be enough to see the inverse in one example, e.g. $$v_1^3v_2^2v_3^2v_6v_{10} = \eta([a_1a_2a_3\cdot b_4b_5\cdot
a_6b_6a_7b_7\cdot b_8^3\cdot b_9^5]).$$ Clearly $\eta$ preserves the homogeneous degrees of the elements, so the Poincaré series for $L_\infty$ and $S$ coincide. This completes our proof.
\[le:iso\] The surjections $L_{n+1}{\twoheadrightarrow}L_n$ and $L_\infty {\twoheadrightarrow}L_n$ are isomorphisms in degrees less than or equal to $n$.
This is clear from our description of the additive basis for $L_n$.
We have already proven (c) in Proposition \[pr:stable-iso\], so it only remains to prove (a) and (b). For (a) we have proven in Corollaries \[co:Ln-gen\] and \[co:Ln-gen2\] that $L_n$ is generated by the given classes, so we need only show that those classes are independent modulo $I^2$. However, all of the classes in question are in degrees less than $n$. If there were a relation among them in $L_n$, this relation would lift to $L_\infty$ by Lemma \[le:iso\]. Yet in $L_\infty$ the classes are obviously independent modulo $I^2$.
Finally, we prove (b). In our list of indecomposables there are $n$ of the form $\sigma_i(b)$ ($1\leq i\leq n$). The ones of the form $[a_1\ldots a_{2^i}]$ number $\lfloor \log_2(n)\rfloor$ since we must have $2^i\leq n$. The ones of the form $[a_1b_1^e]$ number $\lfloor
n-1\rfloor$, the ones of the form $[a_1a_2b_1^eb_2^e]$ number $\lfloor
\frac{n}{2}-1\rfloor$, etc. So we have the formula $$\#(\text{indecomposables in $L_n$}) =
n + \lfloor \log_2(n)\rfloor + (n-1) + \lfloor \tfrac{n}{2}-1\rfloor + \lfloor
\tfrac{n}{4}-1\rfloor + \cdots$$ where the series stops when $\frac{n}{2^i}$ becomes smaller than $1$. Thus, excluding the first two terms we have $\lfloor \log_2(n)\rfloor$ terms, all of which have a “-1” in them. These negative ones together cancel the $\lfloor \log_2(n)\rfloor$ term, leaving $$\#(\text{indecomposables in $L_n$}) =
2n +
\lfloor \tfrac{n}{2}\rfloor + \lfloor
\tfrac{n}{4}\rfloor + \cdots$$
Let $\alpha(n)=\lfloor \tfrac{n}{2}\rfloor + \lfloor
\tfrac{n}{4}\rfloor + \cdots$. We complete the proof of (b) by showing that $$\alpha(n)=n-(\text{number of ones in the binary expansion of $n$}).$$ We do this by induction on $n$, the case $n=1$ being trivial. For the general case write $n=2^k+n'$ where $n'<2^k$. Then $$\begin{aligned}
\alpha(n)=(2^{k-1}+2^{k-2}+\cdots + 1) + \alpha(n') = 2^k-1 +
\alpha(n') &= n-n'-1 + \alpha(n') \\
&= n-(n'-\alpha(n')+1).\end{aligned}$$ By induction, $n'-\alpha(n')$ is the number of ones in the binary expansion of $n'$—which is also one less than the number in the binary expansion of $n$. This completes the proof.
[JTTW]{}
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---
abstract: 'Recent progress in pushing the sensitivity of the Imaging Atmospheric Cherenkov Technique into the 10 mCrab regime has enabled first sensitive observations of the innermost few 100 pc of the Milky Way in Very High Energy (VHE; $>100$ GeV) [$\gamma$-rays]{}. These observations are a valuable tool to understand the acceleration and propagation of energetic particles near the Galactic Centre. Remarkably, besides two compact [$\gamma$-ray]{} sources, faint diffuse [$\gamma$-ray]{} emission has been discovered with high significance. The current VHE [$\gamma$-ray]{} view of the Galactic Centre region is reviewed, and possible counterparts of the [$\gamma$-ray]{} sources and the origin of the diffuse emission are discussed. The future prospects for VHE Galactic Centre observations are discussed based on order-of-magnitude estimates for a CTA type array of telescopes.'
author:
- Christopher van Eldik
bibliography:
- 'vaneldik\_GC\_review.bib'
title: 'Very High Energy $\gamma$-ray Observations of the Galactic Centre Region'
---
[ address=[Max-Planck-Institut für Kernphysik, P.O. Box 103980, D-69029 Heidelberg, Germany]{}]{}
The inner few 100 pc of the Milky Way
=====================================
Ever since the discovery of the strong compact radio source [Sgr A$^*$]{}[@Balick:1974aa], the Galactic Centre (GC) has been subject to intense astrophysics and astronomy research. In the last decade, precise data from this peculiar region, that evades observations at optical wavelengths due to obscuration by dust along the line-of-sight, have been obtained at radio, infrared (IR), X-ray, and hard X-ray/soft [$\gamma$-ray]{} energies. Because of its proximity, the GC is a unique, however complex, laboratory for investigating the astrophysics believed to be taking place in galactic nuclei in general.
In the first large-scale compilation of 90 cm radio observations provided by La Rosa, Kassim, and Lazio [@LaRosa00], the central few 100 pc region reveals a complicated morphology with various objects, mostly supernova remnants, H II regions, and Giant Molecular Clouds (see Fig. \[fig:LaRosa\]). Thread-like filaments, notably the GC radio arc, exhibit highly polarised radiation with no line emission [@Yusef84], and are therefore, amongst others, regions with populations of non-thermal electrons, emitting synchrotron radiation.
![Large-scale compilation of VLA 90 cm radio observations of the Galactic Centre region [@LaRosa00]. The Galactic Plane is oriented top-left to bottom-right in this image. The Galactic Centre is located inside the Sgr A region.](LaRosa.eps "fig:"){width="48.00000%"} \[fig:LaRosa\]
The structure of molecular clouds in the region has been mapped already in the 1970s using $^{12}\mathrm{CO}$ and $^{13}\mathrm{CO}$ lines (e.g. [@bania77; @liszt77]). These measurements, however, suffer from background and foreground contamination from molecular clouds in the Galactic disk. In the velocity range of interest for mapping the GC region, $|v|<30$ km/s, the CS (J=1-0) line is expected to be essentially free of such contaminations. Albeit being less sensitive because of its higher critical density, CS emission provides an efficient tracer of the dense molecular clouds close to the GC. The most complete CS map of the region is provided by measurements of the NRO radio telescope [@cs] and yields a total mass in molecular clouds of $(2-5)\times
10^7 M_{\astrosun}$ in the inner 150 pc region. These clouds are a potential target for cosmic rays accelerated within the region.
The radio view of the inner 50 pc region is dominated by the Sgr A complex, with [Sgr A$^*$]{} at its centre. Along the line-of-sight of [Sgr A$^*$]{} lies [Sgr A East]{}, at a projected distance of 2.5 pc from the GC, enclosing in projection Sgr A West, a three-armed structure which spirals around [Sgr A$^*$]{} and exhibits a thermal spectrum.
[Sgr A East]{} resembles a compact morphology because of the high density of interstellar material ($\approx 10^3$ cm$^{-3}$) which prevents a fast evolution of the forward shock. Furthermore, it interacts with a dense molecular cloud (density $\approx 10^{5}$ cm$^{-3}$) on its eastern side. Based on recent X-ray observations [@Maeda2002] and older radio measurements [@Jones74], [Sgr A East]{} is very likely the remnant of a massive star which exploded about 10.000 years ago (SNR 000.0+00.0). An overabundance of heavy elements is found, favouring a SN type II explosion of a 13-20 $M_{\astrosun}$ star. The X-ray emitting region of the remnant appears more compact (2 pc radius) then at radio wavelengths (6-9 pc), and is caused by the reverse shock that heats plasma in the inner parts of the remnant.
Observations in the near-infrared (NIR, e.g. [@Eisenhauer:2005cv]) have been taken to precisely measure the orbits of young stars in the direct (as close as 0.1”) vicinity of [Sgr A$^*$]{}. From these, the distance of the solar system to the Galactic Centre, $d_{\mathrm{GC}}
= (7.62\pm 0.32)$ kpc, and the mass of the central compact object, $m_{\mathrm{A^*}} = (3.61\pm 0.32)\times 10^6~M_{\astrosun}$, can be inferred [@Eisenhauer:2005cv]. The orbits are consistent with Keplarian motion around a point mass centred on [Sgr A$^*$]{}. Furthermore, VLBA measurements put constraining limits on the proper motion of [Sgr A$^*$]{}, requiring an enclosed mass of at least $~4\times 10^5~M_{\astrosun}$ [@Reid2004]. At a wavelength of 7 mm, VLBI observations have resolved the size of the radio emission region to $24\pm 2$ Schwarzschild radii [@Bower2004]. Combining these findings, there is not much doubt that [Sgr A$^*$]{} can only be a supermassive black hole (SMBH, see, e.g., [@Genzel:2007aa; @Melia07] for recent reviews). Its energy spectrum in the millimeter to IR domain is characterised by a hard power-law with spectral index $\approx 0.3$, a turn-over at about 1 GHz, followed by a cutoff at about $10^3$ GHz [@Zylka1995], explained as synchrotron radiation of relativistic electrons (see e.g. [@Duschl1994; @Melia2000]).
While being relatively bright at radio frequencies, [Sgr A$^*$]{} is only a faint X-ray source [@Skinner1987], but shows bright outbursts on time scales of a few minutes to several hours (see e.g. [@Baganoff2001; @Porquet2003]). These short flare durations limit the emission region to less than 10 Schwarzschild radii of the black hole, where non-thermal processes near the event horizon might produce relativistic electrons (e.g. [@Markoff2001; @Aharonian:2005ti; @Liu2006]). Such models to a certain extend predict flares in the NIR band. Such flares have been observed [@Genzel2003], but occur much more frequently than at X-ray energies. In contrast to this, INTEGRAL observations in the hard X-ray/soft [$\gamma$-ray]{} band show a faint, but steady emission from the direction of the GC [@Belanger2006].
In 1998, observations with the EGRET instrument onboard the Compton Gamma-Ray Observatory provided a strong excess (3EG J1746-2851) of $>30$ MeV [$\gamma$-rays]{} on top of the expected Galactic diffuse emission. Within an error circle of $0.2^\circ$, the position of this excess is compatible with the position of [Sgr A$^*$]{}. However, although not completely ruled out, the extension of the excess fits better a picture where the emission is produced by several distributed objects or diffuse interactions rather than by a single compact object like [Sgr A$^*$]{}. Moreover, the energy output of 3EG J1746 in the MeV-GeV range ($\approx 10^{37}$ erg s$^{-1}$) exceeds by at least an order of magnitude the energy released close to [Sgr A$^*$]{} at any other wavelength. In any case, due to the relatively poor angular resolution of EGRET, source confusion hampers the interpretation of the signal especially at low energies, where the EGRET point spread function is worst. A follow-up analysis of the position of 3EG J1746, using only events with energies $>1$ GeV, disfavours its association with [Sgr A$^*$]{} at the 99.9% CL [@Hooper2002].
It has been argued that 3EG J1746 may be associated to the SNR [Sgr A East]{}, since its [$\gamma$-ray]{} spectrum is similar to other SNRs detected by EGRET. One caveat is, however, that the [$\gamma$-ray]{}luminosity of 3EG J1746 is two orders of magnitude larger than what is found for other EGRET SNRs. Fatuzzo et al. [@Fatuzzo2003], however, convincingly explain the high energy ($>100$ MeV) [$\gamma$-ray]{} emission from the source as being produced in inelastic collisions of shock-accelerated protons with the ambient medium, self-constitently accounting for the $<100$ MeV and radio emission by bremsstrahlung processes and synchrotron emission, respectively, of electrons produced as secondaries in the decay of charged pions (via muon decay).
Discovery of the GC in VHE [$\gamma$-rays]{}
============================================
The recent detection of VHE [$\gamma$-rays]{} from the direction of the Galactic Centre by several Imaging Atmospheric Cherenkov Telescopes (IACTs) [@Tsuchiya:2004wv; @Kosack:2004ri; @Aharonian:2004wa; @Albert:2005kh] has firmly established the existence of particle acceleration to multi-TeV energies within the central few pc of our galaxy. Prior attempts to detect VHE [$\gamma$-rays]{} from this region with the HEGRA stereoscopic system were not successful, and a weak flux upper limit of 8.7 times the flux of the Crab nebula above 4.5 TeV was reported [@HEGRAGalacticPlane].
In 2004, the discovery of a VHE [$\gamma$-ray]{} signal from the direction of the GC was almost simultaneously reported by the CANGAROO-II [@Tsuchiya:2004wv] and Whipple [@Kosack:2004ri] collaborations, with energy thresholds of 250 GeV and 2.8 TeV, respectively. For both analyses the emission regions include the position of [Sgr A$^*$]{} within the statistical errors, and are compatible with a point-like origin. No hint for flux variability on timescales of months or years is found.
The CANGAROO-II telescope detects the source with a significance of 10 $\sigma$ above the background in 67 hours of observations. The differential energy spectrum reported is very steep, $\propto
E^{-4.6\pm 0.5}$ [@Tsuchiya:2004wv], with a flux normalisation at 1 TeV of about $2.7\times 10^{-12}$ cm$^{-2}$ s$^{-1}$ TeV$^{-1}$.
In 26 hours of large zenith angle observations, Whipple detects the GC with a marginal significance of 3.7 $\sigma$ above the background. The integral [$\gamma$-ray]{} flux reported is $(1.6\pm 0.5_{\mathrm{stat}} \pm
0.3_{\mathrm{syst}})\times 10^{-12}$ cm$^{-2}$ s$^{-1}$ above 2.8 TeV, roughly two orders of magnitude larger than the flux measured by CANGAROO-II at these energies.
The [H.E.S.S.]{} telescope array observed the GC region first during the commissioning of the partially incomplete array in June-August 2003. Only two of the four telescopes were operational. During the first phase of measurements (June/July 2003, 4.7 hours observing time) the telescopes were triggered independently, and events were combined using GPS timestamps. For the second observation campaign (July/August 2003, 11.8 hours observing time) a hardware stereo trigger was used, reducing the energy threshold to 165 GeV for this data set. Also [H.E.S.S.]{} reports the detection of a point-like VHE [$\gamma$-ray]{} source (henceforth called [HESS J1745-290]{}) coincident with [Sgr A$^*$]{}. Differential energy spectra have been produced separately for the two data sets, best described by hard power-laws up to the highest energies measured. 49 hours of observations with the completed [H.E.S.S.]{} array were carried out in 2004, yielding consistent results. From a power-law fit of the 2004 data, a photon index of $\Gamma=2.25\pm
0.04_\mathrm{stat}\pm 0.10_\mathrm{syst}$ and an integral flux above 1 TeV of $[1.87\pm 0.10_\mathrm{stat} \pm
0.30_\mathrm{syst}] \times 10^{-12}$ cm$^{-2}$ s$^{-1}$ is obtained [@Aharonian:2006wh]. Recent MAGIC observations of [HESS J1745-290]{} at large zenith angles in 2004 and 2005 verify the hard spectrum found by [H.E.S.S.]{}, with consistent flux levels, and confirm the point-like and non-variable characteristics of the source [@Albert:2005kh].
The [$\gamma$-ray]{} flux measured by [H.E.S.S.]{} and MAGIC is a factor of three lower than that provided by Whipple, and the hard spectral index is in clear contradiction with the CANGAROO-II results. This either makes [HESS J1745-290]{}a rapidly varying [$\gamma$-ray]{} source (with the caveat being that none of the experiments detected significant variability in its own data set), or points to some hidden systematics in the analysis. Indeed, in a careful reanalysis of the Whipple data [@Kosack2005] the flux level has been corrected, and a differential energy spectrum which matches the [H.E.S.S.]{} and MAGIC spectra is obtained. Moreover, observations with the CANGAROO-III array recently yielded a differential energy spectrum consistent with the [H.E.S.S.]{} and MAGIC results [@Mizukami2008].
![Compilation of spectral energy distributions ($E^2\times$ flux) of the GC source [HESS J1745-290]{}. Data points are taken from [@Tsuchiya:2004wv; @Aharonian:2004wa; @Albert:2005kh; @Aharonian:2006wh; @Kosack2005]. The grey shaded band shows a power-law fit $F(E)\propto E^{-\Gamma}$ to the [H.E.S.S.]{} 2004 data [@Aharonian:2006wh]. The recent CANGAROO-III result [@Mizukami2008] shown at this conference is not yet included, nor is the integrated flux measurement by Whipple [@Kosack:2004ri]. Note that for the [H.E.S.S.]{} 2004 result a contribution of 17% of the total flux from diffuse emission was subtracted first.](HESS_GC2004_Spectrum.eps "fig:"){width="48.00000%"} \[fig:Spectra\]
Fig. \[fig:Spectra\] shows a compilation of the at date available VHE [$\gamma$-ray]{} flux measurements from the direction of [HESS J1745-290]{}, indicating the recently achieved agreement between the experiments.
Today’s VHE [$\gamma$-ray]{} view of the Galactic Centre Region
===============================================================
{width="85.00000%"}
Whilst the first detections of the GC were based on data sets of limited statistics and/or high energy threshold, follow-up observations with the completed [H.E.S.S.]{} instrument provide much better sensitivity and deliver the at date most sensitive VHE [$\gamma$-ray]{} images of the GC region [@Aharonian:2006au]. Thanks to its large field-of-view [H.E.S.S.]{} is able to observe a region of $\approx$400 pc diameter (at an assumed distance to the GC of 8 kpc) with a single pointing of the instrument. The so-far published results are based on a deep exposure of GC region with the full [H.E.S.S.]{} array in 2004. In 49 hours of quality-selected data, the detection of [HESS J1745-290]{} is confirmed with a high significance of about $38~\sigma$ above the background. In the [$\gamma$-ray]{} count map shown in Fig. \[fig:GCDiffuse\] (top) a second discrete source is visible $1^\circ$ away from [HESS J1745-290]{}, associated with the SNR [G 0.9+0.1]{}.
While for previous VHE instruments sources like [G 0.9+0.1]{} were close to the detection limit, the [H.E.S.S.]{} data set enables the search for fainter emission. Subtracting the best-fit model for point-like emission from the positions of [HESS J1745-290]{} and G0.9+0.1 yields the sky map shown in the bottom part of Fig. \[fig:GCDiffuse\]. It reveals the presence of diffuse emission along the Galactic Plane [@Aharonian:2006au], as well as the extended [$\gamma$-ray]{} source HESS J1745-303 [@Aharonian:2006zz], located about $1.4^\circ$ south-west of [Sgr A$^*$]{}. HESS J1745-303 belongs to a rather long list of unidentified Galactic VHE [$\gamma$-ray]{} sources, for which the lack of solid couterparts at other wavelengths renders a firm identification difficult so far. A multi-wavelength analysis including VLA, XMM and recent [H.E.S.S.]{} data suggests that at least parts of the emission of HESS J1745-303 can be explained by the shock wave of the SNR 359.1-00.5 (Fig. \[fig:LaRosa\]) running into a dense molecular cloud and by a pulsar wind nebula driven by the pulsar PSR B1742-30 [@Aharonian:2008aa]. An association of HESS 1745-303 and 3EG J1744-3011 is difficult both in terms of the energetics and possibly detected variability of the EGRET source.
[G 0.9+0.1]{}: a Pulsar Wind Nebula in VHE [$\gamma$-rays]{}
------------------------------------------------------------
[G 0.9+0.1]{} is a well-known composite SNR with a clear shell-like radio morphology (see Fig. \[fig:LaRosa\]). It consists of a radio shell of 8’ diameter and a bright compact core. Given the extension of the shell and assuming a distance of 8 kpc, the supernova took place a few thousand years ago. Although no pulsed emission has been detected from the central core region, X-ray observations [@Gaensler2001; @Porquet2003a] have identified it as a pulsar wind nebula (PWN) and also found spectral softening away from the core of the nebula, suggesting an electron population which cools due to synchrotron radiation on its way outwards. No non-thermal X-ray emission has been detected from the SNR shell.
The [H.E.S.S.]{} instrument has discovered [$\gamma$-ray]{} emission from the direction of [G 0.9+0.1]{} with a significance of 13 $\sigma$ after 50 hours of observations [@Aharonian:2005br]. The fact that [G 0.9+0.1]{} is not detected by the MAGIC instrument is not in conflict with the [H.E.S.S.]{}result, given the lower sensitivity of the MAGIC data set. The morphology of the VHE source is compatible with a point-like excess, and an upper limit on the source size of 1.3’ at 95% CL is derived, excluding particle acceleration in the SNR shell as the main source of the VHE [$\gamma$-rays]{}. Instead the centroid of the [$\gamma$-ray]{} excess coincides within statistical errors with the Chandra position of the PWN, making a PWN association compelling. The VHE [$\gamma$-ray]{} spectrum of [G 0.9+0.1]{} extends from 230 GeV to 6 TeV and is best described by a straight power-law with a photon index of 2.40. The total power radiated in VHE [$\gamma$-rays]{} is $2\times 10^{34}$ erg s$^{-1}$, which can be easily accounted for in a one-zone inverse Compton (IC) model yielding a reasonable magnetic field strength ($6\mu$G) and a photon density of 5.7 eV cm$^{-3}$ [@Aharonian:2005br], somewhat smaller than the conventional value used in GALPROP.
[HESS J1745-290]{}: a Prime Example of an Unidentified [$\gamma$-ray]{} Source
------------------------------------------------------------------------------
While the nature of the emission from [G 0.9+0.1]{} seems to be fairly settled, this is certainly not true for the VHE emission from [HESS J1745-290]{}. Although all IACTs which have observed the source have found it being positionally coincident with the SMBH [Sgr A$^*$]{}, the actual mechanism that produces the emission is still not identified. Besides a couple of different [Sgr A$^*$]{}-related emission mechanisms proposed, there are at least two other objects in direct vicinity of the SMBH which are convincing candidates for producing to observed VHE [$\gamma$-ray]{} flux in parts or in combination:
- Various models predict VHE [$\gamma$-ray]{} production near the super-massive black hole (SMBH) itself [@Aharonian:2005ti] or in termination shocks driven by a wind from the SMBH [@Atoyan2004].
- Annihilation of Dark Matter (DM) particles clustering in a cusp around the SMBH could potentially produce VHE emission [@Bergstroem2000].
- The PWN [G359.95-0.04]{}, recently discovered in a deep Chandra exposure [@Wang:2005ya], and only 8.7” away from [Sgr A$^*$]{}in projection, may accelerate electrons to TeV energies.
- Finally, the SNR [Sgr A East]{} is a prime candidate counterpart, given its non-thermal radio shell and the fact that SNRs are proven sites of efficient particle acceleration to highest energies.
From the observer’s standpoint, an identification is particularly hampered by the relatively poor angular resolution of current IACT installations which gives rise to source confusion in this region. The point spread function (PSF) of the [H.E.S.S.]{} instrument is O(0.1$^\circ$), resulting in a relatively large emission region (see Fig. \[fig:GCDiffuse\]). Nevertheless can VHE [$\gamma$-ray]{} observations of [HESS J1745-290]{} put constraints on counterparts and emission models in various ways. The most sensitive data set is currently provided by the [H.E.S.S.]{}collaboration. Without being in conflict with measurements at longer wavelengths, models must explain the following properties of [HESS J1745-290]{}[@Aharonian:2006wh]:
- The centroid of [HESS J1745-290]{} is coincident within $7''\pm
14_\mathrm{stat}'' \pm 28_\mathrm{syst}''$ with the position of [Sgr A$^*$]{}, and the intrinsic size of the source amounts to less than 1.2’ (95% CL).
- The spectrum measured between 160 GeV and 30 TeV can be characterised by a power-law with photon index $2.25\pm
0.04_\mathrm{stat} \pm 0.10_\mathrm{syst}$. The integral flux above 1 TeV is $[1.87\pm 0.10_\mathrm{stat} \pm
0.30_\mathrm{syst}] \times 10^{-12}$ cm$^{-2}$ s$^{-1}$. This implies a [$\gamma$-ray]{} luminosity of $10^{35}$ erg s$^{-1}$ in the 1-10 TeV range. No hint for curvature is found. Assuming an exponential cutoff, a lower limit of 9 TeV (95% CL) on the cutoff energy is derived.
- There is no hint for significant flux variability on any timescale from minutes to years.
### The Case of [Sgr A East]{}
The existence of synchrotron radiation, i.e. the presence of relativistic electrons, and a large magnetic field ($\approx 2-4$ mG, determined from Zeeman splitting of OH masers [@Yusef96]) make [Sgr A East]{} a compelling candidate for [$\gamma$-ray]{} emission at VHE energies. In particular do the observated flux spectra at radio, X-ray, and [$\gamma$-ray]{}energies match a scenario in which protons have been shock-accelerated to at least 100 GeV of energy [@Fatuzzo2003] (see above). Adopting a 4 mG magnetic field, Crocker et al. [@Crocker2005] estimate the maximum proton energy achievable in the [Sgr A East]{} blast wave to be $10^{19}$ eV. The fact that none of the IACTs reports flux variability from [HESS J1745-290]{} certainly fits into this scenario.
The shell of [Sgr A East]{} partially surrounds [Sgr A$^*$]{} in projection; its emission maximum in 90 cm radio is 1.5’ (or about 3.5 pc) away from [Sgr A$^*$]{}. Due to the systematic error of 28” on the position of the [HESS J1745-290]{} centroid from uncertainties in the absolute pointing of the [H.E.S.S.]{} telescopes, the position of the VHE emission is marginally consistent with the [Sgr A East]{} radio maximum.
![Best fit positions of [HESS J1745-290]{} on top of a smoothed 90 cm VLA radio image of SNR [Sgr A East]{} in Galactic coordinates. The position of [Sgr A$^*$]{} and [G359.95-0.04]{} are marked with a cross and a star, respectively. The red triangle and red circle mark the preliminary best fit position and total error of the improved position measurement of [HESS J1745-290]{}. The blue triangle and circle show the results obtained in [@Aharonian:2006wh]. Figure taken from [@vanEldik:2007icrc].[]{data-label="fig:GCPosition"}](ImprovedPosition.eps){width="48.00000%"}
Recent progress (although subject to final checks, [@vanEldik:2007icrc]) in understanding and compensating the pointing systematics of the [H.E.S.S.]{} array has led to a reduced systematic error of 8.5” [@vanEldik:2007icrc]. Fig. \[fig:GCPosition\] shows the improved [H.E.S.S.]{} position measurement, based on 73 hours of observations, on top of a 90 cm VLA radio image of [Sgr A East]{}. The best fit position is coincident within $7.3'' \pm 8.7''_\mathrm{stat} \pm8.5''_\mathrm{syst}$ with [Sgr A$^*$]{} and effectively rules out [Sgr A East]{} as the dominant source of the VHE emission.
### [HESS J1745-290]{}: a Pulsar Wind Nebula?
The recent detection of the PWN [G359.95-0.04]{} in a deep Chandra exposure of the GC region [@Wang:2005ya] very much complicates the counterpart search for [HESS J1745-290]{}. [G359.95-0.04]{} is located only $8.7''$ in projection (or 0.3 pc) away from [Sgr A$^*$]{}, rendering a discrimination of the two by position measurements impossible (cf. Fig. \[fig:GCPosition\]). [G359.95-0.04]{} is rather faint at X-ray energies, with an implied luminosity of $10^{34}$ erg s$^{-1}$ in the 2-10 keV band [@Wang:2005ya], yet about four times brighter than [Sgr A$^*$]{}. It shows a cometary shape and exhibits a hard and non-thermal spectrum which gradually softens when going away from the “head” of the PWN, where the yet undiscovered pulsar is believed to be located. No radio counterpart of the PWN is found.
Numerical calculations show that, despite its faint X-ray flux, a population of non-thermal electrons can naturally explain both the X-ray emission of [G359.95-0.04]{} and the VHE [$\gamma$-ray]{} emission of [HESS J1745-290]{} [@Hinton:2006zk]. Compared to other locations in the galactic disk, where many VHE [$\gamma$-ray]{} sources have been found associated to PWNe, the Galactic Centre region is, however, special because of its dense radiation fields. The fact that the X-ray spectra steepen rather than harden the further one gets away from the pulsar position is an indication that the TeV electrons are cooled by synchrotron radiation rather than by IC processes in the Klein-Nishina regime, putting a lower limit of $\approx 100 \mathrm{\mu G}$ on the value of the magnetic field for typical Galactic Centre radiation fields [@Hinton:2006zk]. It is predominantly the far-IR component of the radiation field that TeV electrons upscatter to TeV energies, providing roughly an order of magnitude larger luminosity in the 1-10 TeV [$\gamma$-ray]{} band than in the 2-10 keV X-ray domain.
### Emission models involving [Sgr A$^*$]{}
The low bolometric luminosity ($< 10^{-8} L_{\mathrm{Edd}}$ in the range from millimeter to optical wavelenghts) renders [Sgr A$^*$]{} an unusually quiet representative of galactic nuclei. At the same time, this property makes the immediate vicinity of the SMBH transparent for VHE [$\gamma$-rays]{}. Aharonian & Neronov [@Aharonian:2005ti] show that the absence of dense IR radiation fields enables photons with an energy of up to several TeV to escape almost unabsorbed from regions as close as several Schwarzschild radii from the centre of the SMBH. Therefore, VHE [$\gamma$-ray]{} emission produced close to the event horizon of [Sgr A$^*$]{} provides a unique opportunity to study particle acceleration and radiation in the vicinity of a black hole.
There are several possibilities to produce the observed VHE [$\gamma$-ray]{} flux, depending on the type of particles accelerated, the model of acceleration, and finally the interaction of the accelerated particles with the ambient magnetic field or matter. Common scenarios, which do not contradict the emission at longer wavelenghts, include [@Aharonian:2005ti]:
- Synchrotron radiation of ultra-relativistic protons. In the strongly magnetised environment of a SMBH with magnetic field strengths as large as $10^4$ G, protons can be accelerated to energies of up to $10^{18}$ eV. However, the synchrotron spectrum extends only to roughly 300 GeV, and thus cannot account for the multi-TeV radiation seen by [H.E.S.S.]{}. Curvature radiation of protons can in principle extend the [$\gamma$-ray]{} spectrum 10 TeV, but only at the expense of very large magnetic fields ($10^6$ G), for which the source is opaque for [$\gamma$-rays]{} due to $e^+e^-$ pair production.
- Photo-Meson interactions. Despite its low bolometric luminosity, the IR radiation fields in the vicinity of [Sgr A$^*$]{} appear dense enough to produce a sizable number of VHE [$\gamma$-rays]{} in the interactions of the accelerated protons with IR photons. The required power to meet the luminosity in VHE [$\gamma$-rays]{} is $10^{38}$ erg s$^{-1}$, well below the Eddington luminosity of a $3\times 10^6 M_{\astrosun}$ BH.
- Proton-Proton interactions. VHE [$\gamma$-ray]{} production by interactions of accelerated protons with the ambient plasma requires an acceleration power of $10^{39}$ erg s$^{-1}$, but at the same time only $\geq 10^{13}$ eV protons are required to produce TeV radiation. In this case, possible acceleration sites include a strong electric field close to the event horizon or strong shocks in the accretion disk. This scenario in particular predicts correlated flux variability at VHE, X-ray, and IR wavelengths.
- Inverse Compton radiation of electrons. Compared to the aforementioned proton scenarios, electrons provide a much more efficient way to convert energy into radiation. To accelerate electrons to multi-TeV energies, however, a well-ordered magnetic or electric field is necessary to prevent radiation losses during acceleration. Such properties are e.g. provided by the rotation-induced electric fields near the black hole. [$\gamma$-rays]{} at the highest energies ($> 100$ TeV), however, cannot escape the source because of efficient interaction with the IR radiation field, but contribute to the spectrum with sub-100 TeV photons.
While some of the above mentioned scenarios suggest correlated multi-wavelength variability, non-observation of variability does not striktly rule out acceleration close to the black hole. Moreover, there are models which explain the absence of VHE variability by diffusion of protons away from the acceleration region into the neighbourhood of [Sgr A$^*$]{} and subsequent interaction with the ambient medium [@Liu2006a; @Liu2006]. On the other hand, the detection of variability in the VHE data would immediately point to particle acceleration near the SMBH. The most convincing signature would be the detection of correlated flaring in X-rays (or NIR) and VHE [$\gamma$-rays]{}. Such searches have been carried out [@vivier07; @hinton07]. No evidence of flaring or quasi-periodic oscillations has been found. In particular, in a coordinated multi-wavelength campaign both Chandra and [H.E.S.S.]{} observed the GC region, when a major (factor 9 increase) X-ray outburst was detected. During this 13-minutes flare the VHE [$\gamma$-ray]{}flux stayed constant within errors, and a 99% CL upper limit on a doubling of the VHE flux is derived [@hinton07].
### Dark Matter Annihilation near the GC?
Besides being of astrophysical origin, the observed TeV flux could potentially stem from annihilation of dark-matter particles, which are believed to cluster in a compact cusp around Sgr A\* [@Bergstroem2000]. Halo density profiles are believed to scale with the radius $r$ like $r^{-\alpha}$, with $\alpha$ between 1 [@Navarro1997] and 1.5 [@Moore1999] in the most common models. The fact that [HESS J1745-290]{} is point-like (after having accounted for the underlying diffuse emission) translates into $\alpha>1.2$, i.e. a cuspy halo is favoured by the observations.
![Spectral energy density of [HESS J1745-290]{}. The shaded band shows the power law fit $dN/dE\sim E^{-2.25}$ to the 2004 data points ([@Aharonian:2006wh], see also Fig. \[fig:Spectra\]. The curves show typical spectra of [$\gamma$-rays]{} from the annihilation of 14 TeV MSSM neutralinos (green), of 5 TeV Kaluza-Klein particles (purple), and of a 10 TeV DM particle decaying into 30% $\tau^+\tau^-$ and 70% $b\bar{b}$ (blue). Figure reproduced from [@Aharonian:2006wh].[]{data-label="fig:GCSpectrum"}](GCSpectrumDM.eps){width="46.00000%"}
Predicted energy spectra for [$\gamma$-rays]{} produced in cascade decays of DM particles such as MSSM neutralinos or Kaluza-Klein particles can be compared to the VHE observations. These spectra are usually curved both at high energies – for reasons of energy conservation –, and low energies, in clear disagreement with the observations (Fig. \[fig:GCSpectrum\], see also [@Aharonian:2006wh]). Furthermore, unusually large DM particle masses have to be assumed to account for the fact that the [$\gamma$-ray]{}spectrum extends up to 10 TeV.
The observed [$\gamma$-ray]{} emission is therefore not compatible with being dominantly produced in the framework of the most common DM scenarios. As a consequence, the bulk of the observed [$\gamma$-ray]{} excess is probably of astro- rather than of particle physics origin. However, an O(10%) admixture of [$\gamma$-rays]{} from DM annihilations in the signal from the GC cannot be ruled out. Assuming a NFW-type [@Navarro1997] halo profile, 99% CL upper limits on the velocity-weighted annihilation cross section $<\sigma
v>$ are at least two orders of magnitude above theoretical expectations [@Aharonian:2006wh], and thus are not able to put constraints on current model predictions.
Diffuse [$\gamma$-ray]{} Emission
---------------------------------
The diffuse emission (Fig. \[fig:GCDiffuse\] bottom) spans in a region of roughly $2^\circ$ in galactic longitude ($l$) with an rms width of about $0.2^\circ$ in galactic latitude ($b$). The reconstructed $\gamma$-ray spectrum integrated within $|l|\leq 0.8$ and $|b|\leq 0.3$ is well-described by a power law with photon index $\Gamma=2.29$ [@Aharonian:2006au], in agreement with the index observed for [HESS J1745-290]{}. Assuming that the emission is produced in the Galactic Centre region at 8 kpc distance from the observer, the latitude extension translates into a scale of about 30 pc. This is very similar to the extent of giant molecular clouds in this region [@cs]. Indeed, at least for $|l|\leq 1^\circ$, there is a strong correlation between the morphology of the observed [$\gamma$-rays]{} and the density of molecular clouds as traced by the CS emission observed with NRO ([@cs], Fig. \[fig:GCDiffuse\] bottom). This is the first time such correlation is seen, and is a strong indication for the presence of an accelerator of (hadronic) cosmic rays in the Galactic Centre region, since the energetic hadrons would interact with the material in the clouds, giving rise to the observed [$\gamma$-ray]{} flux via $\pi^0\to\gamma\gamma$ decays. The idea of acceleration in the GC region is further supported by the fact that the measured [$\gamma$-ray]{} flux is both larger and harder than expected in a scenario where the molecular material is bathened in a sea of galactic cosmic rays, with similar properties as measured in our solar neighbourhood, only. The energy necessary to fill the entire region with cosmic rays can be estimated from the measured [$\gamma$-ray]{} flux (extrapolated to 1 GeV) and amounts to $10^{50}$ erg. This number is close to the energy believed to be transferred into cosmic rays in a typical galactic supernova. A distribution of electron accelerators, such as PWNe, that cluster similarly to the gas distribution, has also been discussed (e.g. [@Aharonian:2006au]). Given the O(100$\mathrm{\mu}$G) magnetic fields in the region, electrons of several TeV energy would, however, rapidly cool via synchrotron radiation, such that their VHE [$\gamma$-ray]{} emission would appear point-like in the [H.E.S.S.]{} data.
In the context of identifying the accelerator, the fact that no emission is seen farther than $|l|\approx 1^\circ$ might be particularly important. In their discovery paper [@Aharonian:2006au], the [H.E.S.S.]{} collaboration came up with the rather simple, yet convincing explanation that the cosmic rays may have been accelerated in a rather young source near the very centre of the galaxy, having subsequently undergone diffusion away from the accelerator into the surrounding medium. Assuming a typical diffusion coefficient of $10^{30}$ cm$^2$ s$^{-1}$, or 3 kpc$^2$ Myr$^{-1}$, for TeV protons in the Galactic disk, a source age of about $10^4$ years can reproduce the observed [$\gamma$-ray]{} flux distribution [@Aharonian:2006au], and in particular the lack of emission beyond $1^\circ$ distance from the centre.
Büsching et al. [@Buesching2007] follow a similar idea. Starting from a source of non-thermal protons at the GC and the known distribution of molecular material, the authors model the [$\gamma$-ray]{} flux from the region in a time dependent diffusion picture. Neglecting a possible energy dependence of the diffusion process (suggested by the fact that the [H.E.S.S.]{} data are not sensitive enough yet to measure such an effect), they use $\chi^2$ minimisation to find the diffusion coefficient for which the [H.E.S.S.]{}results are matched best, for a variety of source ages and source on-times (Fig. \[fig:DiffusionCoeff\]), resulting in diffusion coefficients in the range of the one assumed in [@Aharonian:2006au]. In a similar approach, Dimitrakoudis et al. [@Dimitrakoudis2008] obtain a best-fit diffusion coefficient of 3 kpc Myr$^{-1}$, also close to [@Aharonian:2006au]. Scaling the diffusion coefficient $k$ with the cosmic ray rigidity $\zeta$, $k=k_0
(\zeta/\zeta_0)^{0.6}$, $\zeta_0 = 1$ GV$/c$, Büsching et al. find a value of $k_0$ which is significantly smaller than the local value, suggesting enhanced turbulence and larger magnetic fields than in the solar neighbourhood. Uncertainties in the derived diffusion coefficients of up to 50% arise from uncertainties in the molecular gas distribution and from the contribution of galactic cosmic rays to the overall [$\gamma$-ray]{} flux.
![Best-fit diffusion coefficients (see text) for CR diffusion away from a central source into the GC region. Diffusion coefficients are given as a function of source on-time, for three different source ages. Reproduced from [@Buesching2007].[]{data-label="fig:DiffusionCoeff"}](Buesching2.eps){width="46.00000%"}
In a separate paper [@Buesching2008], Büsching et al. explain both the diffuse emission and the point-source [HESS J1745-290]{} within a single model in which the cosmic rays responsible for the diffuse emission were accelerated in the shock wave of the SNR [Sgr A East]{} 5-10 kyr ago, but acceleration stopped well before the present time. When, however, the shock wave of [Sgr A East]{} collided with [Sgr A$^*$]{}, particle acceleration near the SMBH was initiated, leading to the observed VHE [$\gamma$-ray]{} emission from [HESS J1745-290]{}. Assuming that the diffusion coefficient found for the diffuse emission is also valid close to the SMBH, this last round of particle acceleration can only have happened in the recent past (O(100) yr) to be consistent with the point-like morphology of [HESS J1745-290]{}.
It should, however, be noted that there are other processes which can explain the emission from [HESS J1745-290]{} (see above). Furthermore, recent simulations seem to indicate that the diffuse emission might be better explained by inter-cloud acceleration of cosmic rays via the Fermi-II process [@Wommer2008]. More sensitive observations are needed to ultimately prove which of the discussed scenarios of the VHE [$\gamma$-ray]{} view of the GC is correct.
The Role of CTA
===============
Despite the exciting progress in recent years, a robust understanding of the GC VHE [$\gamma$-ray]{} sky needs a more refined data set than currently available. Significant progress in the identification of the VHE sources and the physics processes involved requires an instrument with better sensitivity, wider energy coverage, and, possibly, improved angular resolution. Probing the GC region with instruments like CTA [@Hermann2007] or AGIS will answer many of the open questions within a reasonable amount of observing time.
For the following order-of-magnitude estimations we assume a CTA-like array of IACTs with a core sensitivity of O(1 mCrab), 4 orders of magnitude energy coverage (10 GeV – 100 TeV), and an angular resolution of 0.02$^\circ$ per event.
Angular resolution
------------------
As discussed above, one way of identifying [HESS J1745-290]{} is to search for plausible candidate counterparts within the error circle of the emission centroid. Since the GC is a densely packed region, naively, a good angular resolution $\theta$ is important. Since the statistical error of the centroid position scales like $\theta/\sqrt{N_\gamma}$, and therefore linearly with sensitivity, a factor of 50 improvement in the statistical error of the centroid position over [H.E.S.S.]{} (about 6” per axis for 73 hours of exposure, [@vanEldik:2007icrc]) is expected. On the other hand, the systematic pointing error of the [H.E.S.S.]{} telescopes is about 6” per axis [@vanEldik:2007icrc], probably close to the limit of what can be achieved with future instruments. In the special case of [HESS J1745-290]{}, subtracting the underlying (asymmetric) diffuse emission imposes additional systematic uncertainties, which for [H.E.S.S.]{} are of the order of 1”. As a consequence, improved angular resolution is not of much help what regards a measurement of [HESS J1745-290]{}’s position.
On the other hand, superior angular resolution would help in understanding the properties of the diffuse emission. It would allow to probe the region with a few pc binning and test the [$\gamma$-ray]{}-cloud correlation in much more detail than currently possible. Fig. \[fig:DiffuseCTA\] sketches the improvement in angular resolution over [H.E.S.S.]{} for a CTA-like instrument with an angular resolution of $0.02^\circ$ per reconstructed [$\gamma$-ray]{}. Shown are [$\gamma$-ray]{} maps expected from cosmic rays interacing with the molecular material in the GC region. The cosmic rays diffuse away from the Galactic Centre, their assumed origin of production. The improvement in the quality of the data is clearly visible. With such an improvement one might in the not too distant future also get a handle on the possible existence of electron accelerators along the Galactic Plane, which might or might not be responsible for the observed [$\gamma$-ray]{} emission in parts or in total.
{width="\textwidth"}
{width="\textwidth"}
\[fig:DiffuseCTA\]
Sensitivity
-----------
An unambiguous proof that the VHE [$\gamma$-ray]{} emission from [HESS J1745-290]{} is associated with [Sgr A$^*$]{} would be the observation of correlated [$\gamma$-ray]{}/X-ray (or IR) variability. With the assumed CTA sensitivity similar X-ray flare events like the one discussed above (factor 9 increase over the quiescent level) would test a level as low as 10% of the quiescent state [$\gamma$-ray]{} flux.
To test models of cosmic ray/electron propagation through the central region of the galaxy and to study the penetration of molecular clouds by cosmic rays, energy spectra have to be provided of the diffuse [$\gamma$-ray]{}emission in small regions of a few $10$ pc $\times$ a few $10$ pc only. With these at hand, energy-dependent diffusion processes could be studied in great detail. For a simple power-law fit, the statistical error on the spectral index scales linearly with sensitivity. Therefore, because of its enhanced sensitivity, a spectrum measured by CTA in a $0.1^\circ\times 0.1^\circ$ portion of the sky (in the [H.E.S.S.]{} energy range) will reach comparable statistical accuracy in the spectral index as [H.E.S.S.]{} does in a $1^\circ\times 1^\circ$ sky area (e.g. $\Delta\Gamma_{stat} = 0.07$ for 50 hours of observations of the GC diffuse emission [@Aharonian:2006au]).
Energy coverage
---------------
The energy spectrum of [HESS J1745-290]{} covers an energy range of 160 GeV – 30 TeV and is well fitted by a straight powerlaw (see above). For the most likely counterparts of the VHE emission, [Sgr A$^*$]{} and [G359.95-0.04]{}, emission models fitting the combined spectral energy distributions have been presented by various authors (e.g. [@Hinton:2006zk], [@Aharonian:2005ti], among others). While most models can satisfactorily fit the [H.E.S.S.]{} data points, they do substantially differ at energies $< 100$ GeV. CTA energy coverage down to 10 GeV would constrain some of the models, and therefore help to identify the source of the [$\gamma$-rays]{} and the underlying physical acceleration and radiation processes.
Conclusions
===========
Less than five years after the discovery of VHE [$\gamma$-ray]{} emission from the direction of the GC, observations with Imaging Atmospheric Cherenkov Telescopes provide a very sensitive view of this interesting region. With the recent data from the [H.E.S.S.]{} instrument, a rich VHE [$\gamma$-ray]{}morphology becomes evident, giving strong evidence for the existence of a cosmic ray accelerator within the central 10 pc of the Milky Way.
An intense [$\gamma$-ray]{} point source is found coincident within errors with the position of [Sgr A$^*$]{}. Source confusion near the GC make a solid identification difficult, given the – compared to X-ray satellites or IR observatories – moderate angular resolution of current IACTs. Recent progress in improving on the systematic and statical errors of the centroid of HESS J1745-290 effectively excludes the SNR [Sgr A East]{} as the dominant source of the [$\gamma$-ray]{} emission. A major contribution from the annihilation of DM particles can also be excluded.
Future observations with even more sensitive instruments such as CTA will significantly advance our knowledge about the GC region at VHE energies. The recently launched Fermi satellite will extend the energy range down to about 100 MeV, such that unbroken sensitivity coverage will be provided over 6 orders of magnitude in energy.
The author would like to thank the organisers for having invited him to present this overview at the symposium.
|
---
abstract: 'A simple observation about the action for geodesics in a stationary spacetime with separable geodesic equations leads to a natural class of slicings of that spacetime whose orthogonal geodesic trajectories represent freely falling observers. The time coordinate function can then be taken to be the observer proper time, leading to a unit lapse function. This explains some of the properties of the original Painlevé-Gullstrand coordinates on the Schwarzschild spacetime and their generalization to the Kerr-Newman family of spacetimes, reproducible also locally for the Gödel spacetime. For the static spherically symmetric case the slicing can be chosen to be intrinsically flat with spherically symmetric geodesic observers, leaving all the gravitational field information in the shift vector field.'
author:
- Donato Bini
- Andrea Geralico
- 'Robert T. Jantzen'
date: 'Received: date / Accepted: date / Version: date '
title: Separable geodesic action slicing in stationary spacetimes
---
Introduction
============
Natural spacelike slicings of spacetimes characterized by special geometric properties are very helpful in elucidating the structure of those spacetimes, as advocated by Smarr and York [@smarr-york], for example, who show how various choices of lapse function and shift vector field can simplify the spatial metric. The family of orthogonal trajectories to such slicings represent the world lines of a family of test observers (‘fiducial observers’) which then experience the spacetime geometry in $3+1$ form, with the future-pointing unit normal vector field as their 4-velocity field. The most familiar and useful slicings in a stationary spacetime are often associated with nongeodesic slicings which are accelerated in order to resist the pull of gravity and link to our mental image from Newtonian physics of points fixed in space. The zero-angular-momentum observers (ZAMOs) in the stationary axisymmetric spacetimes like in the Kerr family of black holes are the standard tool for decomposing spacetime quantities in those spacetimes [@MTW], resisting the attraction towards the hole while being dragged along by its rotation relative to the Boyer-Lindquist coordinate grid. The shift vector field describes the motion of these fiducial observers with respect to the time lines anchored in the spatial coordinate grid, while the lapse function acts like a potential for the acceleration field characterizing those observers, which move orthogonally to the time coordinate hypersurfaces. However, these coordinates have a singularity at the outer event horizon where the time coordinate slices go null and then timelike as one continues inside, where the ZAMOs no longer exist.
Painlevé-Gullstrand coordinates [@doran; @cook; @hamilton], which exist in the Kerr and Kerr-Newman spacetimes generalizing those first found for the Schwarzschild spacetime [@Painleve21; @Gullstrand22], are instead associated with a unit lapse gauge slicing whose corresponding orthogonal fiducial observers are both stationary and geodesic, and represent a stationary field of freely falling observers whose adapted coordinates have desirable properties. First and foremost these new coordinates remain valid inside the outer horizon in these spacetimes, leading to the terminology ‘horizon-penetrating coordinates,’ while for example, in the Schwarzschild case the intrinsic geometry of the time slices is indeed flat. Retaining the original spatial coordinate functions (and therefore the same time coordinate lines adapted to the Killing vector associated with the stationary symmetry), passing to the new time function simply changes the fiducial observers used to interpret spacetime quantities, with an additional contribution to the shift to represent their motion with respect to the time lines. In the Kerr-Newman case, an additional change of azimuthal coordinate is then possible which drags the coordinate along by the geodesic motion in that direction. A similar situation occurs locally for the Gödel spacetime, leading to a new analogous representation of that metric.
The definition of such new time and space coordinates can be explained by examining the action for the timelike geodesics of these spacetimes, which are special in the sense that the action itself and the affinely parametrized geodesic equations are separable. In fact the geodesics are completely integrable, and in the Kerr case are determined by four first integrals related to that separability property which is due to the existence of Killing vectors associated with the stationary axisymmetric symmetry and the existence of a Killing tensor [@carter; @MTW]. Of these four constants of the motion for a given geodesic, one can be absorbed into the choice of parametrization for the affinely parametrized geodesics, leaving three constants to determine the direction of their 4-velocity in spacetime. By fixing these constants for the entire 4-parameter spacetime, one determines a geodesic congruence which is vorticity free, and admits a family of orthogonal spacelike hypersurfaces serving as the time slices of a useful coordinate system adapted to this family in a unit lapse time gauge. These choices of the constants of the motion are further limited in order to avoid limiting the range of validity of the new coordinates by energy or angular momentum barriers. A similar situation holds for the Gödel spacetime due to its stationary cylindrical symmetry when expressed in cylindrical-like coordinates.
The new time slices are closely related to the hypersurfaces of constant action for the geodesic problem because of the separability property, and their parametrization measures the proper time along the geodesics. This fact unifies the derivation of the various examples of unit lapse time gauge coordinates generalizing the original Painlevé-Gullstrand coordinates found for the Schwarzschild spacetime. For that spacetime in addition to unit lapse, the new coordinates have intrinsically flat time coordinate slicings, a property which is another route to generalize the Painlevé-Gullstrand coordinates with accelerated observers, as done for the de Sitter spacetime [@parikh]. In that case as well as the anti de Sitter case, in a region which admits a static spherically symmetric slicing, one can also introduce unit lapse gauge slicings with geodesic fiducial observers following from this separability discussion. In the static spherically symmetric case, the geodesic slicing corresponding to purely radial motion of the observer family can always be chosen to be intrinsically flat. One motivation for considering these kinds of coordinate systems comes from numerical relativity where horizon-penetrating coordinates like these can be useful [@cook; @gourgo].
Separable geodesic action slicings
==================================
Spacetimes with sufficient Killing symmetries allow coordinates to be introduced which allow the separation of variables for the geodesic equations [@carter; @woodhouse; @dietz; @collinson1; @collinson2; @demianski; @koutras; @houri2008; @houri2011]. Carter [@carter] was the first to appreciate this fact and determine a large class of exact solution stationary axisymmetric spacetimes with this property.
Let the coordinates $x^\alpha$ ($\alpha=0\ldots 3$, with $x^0=t$) be such that the geodesic equations are separable in the metric $\rmd s^2=g_{\alpha\beta}\rmd x^\alpha \rmd x^\beta$ of signature $-$$+$$+$$+$. Using the Hamilton-Jacobi formalism we can write the tangent vector $U^\alpha=\rmd x^\alpha(\lambda)/\rmd\lambda$ to the affinely parametrized timelike geodesics as the gradient of the fundamental action function $S=S(x^\alpha,\lambda)$, $U_\alpha=\partial_\alpha S$, satisfying the Hamilton-Jacobi equation -=H(x\^,\_S), with $\lambda$ an affine parameter for the integral curves of $U$ and the Hamiltonian H=12 g\^\_S \_S=-12\^2=[*const*]{}, the latter identity following from the normalization condition $U^\alpha U_\alpha =-\mu^2$ for timelike geodesics. The choice $\mu=1$ makes $\lambda$ equal to the proper time along the geodesics and $U^\alpha$ the unit 4-velocity [^1].
Assume that $S$ can be separated in its dependence on the variables $x^\alpha$ and $\lambda$, namely S=12\^2+S\_t(t)+S\_1(x\^1)+S\_2(x\^2)+S\_3(x\^3). Thus we have for the 1-form $U^\flat\equiv U_\alpha \rmd x^\alpha =\partial_\alpha S \rmd x^\alpha=\rmd (S-\frac12\mu^2 \lambda)$, where here $\rmd$ stands for the spacetime differential only, or explicitly U\^= \_t S\_t(t) t+ \_1 S\_1(x\^1) x\^1+ \_2 S\_2(x\^2) x\^2+ \_3 S\_3(x\^3) x\^3. Moreover, since in this case $U$ is a gradient it is also necessarily vorticity-free: $\rmd U^\flat=0$, and there exists a distribution of constant action hypersurfaces $T\equiv -S+\frac12\mu^2 \lambda =const$ with \[eq:T1\] -T=U\_x\^, such that $U^\alpha$ is the associated unit normal vector field. When one sets $\mu=1$, then the time function $T$ measures the proper time along the geodesics and the corresponding lapse function has the fixed value $N=1$. For a stationary spacetime in which $t$ is taken to be a Killing time coordinate, then $U_t = -E$ is a constant interpreted as a conserved energy, with $S_t(t)=-Et$, and the metric is independent of $t$. One then has \[eq:T\] -T=-Et + U\_a x\^a.
In the Schwarzschild case following this procedure starting from the usual Boyer-Lindquist coordinates, one not only simplifies the lapse function but also the spatial metric, which becomes flat for certain choices of the parameters, leaving the shift vector field to carry the information about the gravitational field. To study the intrinsic geometry one needs to evaluate the induced metric on this new slicing, whose $3$-dimensional line element will be denoted by ${}^{(3)}\rmd s^2=\gamma_{ab}\rmd x^a \rmd x^b$.
When $\partial_t$ is a timelike Killing vector the spacetime metric coefficients $g_{\alpha\beta}$ do not depend on $t$. Moreover, from Eq. (\[eq:T\]), on the $T=\hbox{\it const}$ slicings we have \[dt\_su\_T\] t= x\^a,(a=1,2,3). This represents a tilting of the original slicing tangent spaces to adapt them to the new stationary geodesic observer family. Substituting the above expression (\[dt\_su\_T\]) into the metric we then have $$\begin{aligned}
{}^{(3)}\rmd s^2&=&
(g_{tt}\rmd t^2 +2g_{ta}\rmd t \rmd x^a +g_{ab}\rmd x^a \rmd x^b)|_{dt=dx^a \,U_a/E}
\nonumber \\
&=& g_{tt}\frac{U_aU_b}{E^2} \rmd x^a\rmd x^b +2g_{ta}\frac{U_b}{E} \rmd x^a\rmd x^b +g_{ab}\rmd x^a \rmd x^b
%\nonumber \\ &\equiv&
\equiv\gamma_{ab}\rmd x^a\rmd x^b\end{aligned}$$ where the spatial metric components in the original spatial coordinates \[eq:gamma\] \_[ab]{}=g\_[tt]{} +g\_[t(a]{} U\_[b)]{} +g\_[ab]{}, are also independent of $t$ since the time coordinate lines are still Killing trajectories.
In the unit lapse time gauge $N=1$ of these new coordinates, the components of the shift vector field $N^a = N^2 g^{Ta} = g^{Ta}$ are given simply by the mixed components of the contravariant metric tensor [@mfg]. These describe the motion of the Killing time lines relative to the new fiducial observers. Although the relative velocity of these time lines with respect to these geodesic observers exceeds the speed of light within the horizon, the new coordinates remain well defined. The spacetime metric is then s\^2=- T\^2 +\_[ab]{} (x\^a+N\^aT) (x\^b+N\^b T).
Static spherically symmetric spacetimes {#stat}
=======================================
Before considering the more complicated case of stationary axisymmetric spacetimes like the Kerr spacetime, consider first the static spherically symmetric spacetimes. The metric written in standard spherical-like coordinates is $$\label{metricgen}
\rmd s^2=- e^{\nu}\rmd t^2 + e^{\lambda}\rmd r^2+r^2(\rmd \theta^2 +\sin ^2\theta \rmd \phi^2)\,,$$ where the functions $\nu$ and $\lambda$ depend only on the radial coordinate. Then $L=U_\phi$ is an additional Killing constant associated with the conserved angular momentum so U\_x\^= -Et + (\_r S\_r) r +(\_S\_) +L, and the corresponding Hamilton-Jacobi equation - e\^[-]{} E\^2 + e\^[-]{} (\_r S\_r)\^2 + =-\^2 can be easily separated in its dependence on the coordinates leading to =\_r e\^[/2]{}, =\_, where $K$ is a separation constant arising from the angular contribution to the Hamilton-Jacobi equation, and $|\epsilon_r|=1=|\epsilon_\theta|$.
As stated above, we can set $\mu=1$ to characterize a new foliation by a new temporal coordinate $T$ measuring proper time along the orthogonal geodesics. We are left to specify $E$, $L$, and $K$ to obtain a specific family of timelike geodesics covering the spacetime. The simplest choice would be a spherically symmetric 4-velocity field involving only radial motion of the geodesics relative to the original coordinates. We can achieve this in two steps. First we can require that this family of geodesics be tangent to the equatorial plane $\theta=\pi/2$, which requires $K=L^2$ to make $U_\theta=0$, resulting in U\_r\^2=e\^[-]{}. We then impose the radial condition $L=0$, so that \[eq:Ur\] U\_r = \_r e\^[(-)/2]{} , leaving finally the choice of the energy constant $E$. For spatially asymptotically flat spacetimes where $e^{\nu}<1$ approaches 1 as $r\to\infty$, to have a choice which works even at spatial infinity, we must have $E\ge 1$, in which case the value may be interpreted as the energy of the radially moving geodesics at spatial infinity. Of course one could choose $E<1$ but this would limit the slicing to the interior of a cylinder in spacetime inside the radial turning point of the geodesic motion.
The new time differential is then \[Tpglike\] T=Et -U\_r r . A new global coordinate system for static spacetimes is given by $(X^\alpha)=(T,R,\theta,\phi)$ with $R=r$ and $\theta$ and $\phi$ unchanged and $T=Et+f(r)$ given by integrating the differential equation $f'(r) =- e^\lambda U_r$. This leads to \_T = [E]{}\^[-1]{} \_t, \_R = \_r +U\_r \_t, and the transformed metric is \[newmet\] s\^2=-T\^2+\_[ab]{}(X\^a+N\^aT)(X\^b+N\^bT), with unit lapse function and the shift vector field aligned with the new radial direction, i.e., N\^a=-\^a\_R \_r e\^[-]{} U\_r = - \^a\_R \_r e\^[-(+)/2]{} . The 3-metric induced on the $T=\hbox{\it const}$ hypersurfaces is then given by \[3\_metric\] \^[(3)]{}s\^2=r \^2+r\^2(\^2+\^2\^2).
In the case of vacuum as well as in the presence of a nonzero cosmological constant one has $\lambda+\nu=0$, so that the induced metric is then \[indmetE\] \^[(3)]{}s\^2 =+ r\^2 (\^2 +\^2 \^2), whose only nonvanishing component of the spatial Riemann curvature tensor and the spatial curvature scalar are \^[(3)]{} R\^\_ = =12 \^[(3)]{} R , with positive or negative curvature respectively for $0<E<1 $ (bound geodesics) or $E>1$ (unbound geodesics). The choice $E=1$ leads to a flat $3$-geometry. The additional sign choice $\epsilon_r=-1$ corresponds to the radially infalling geodesics which start at rest at spatial infinity. This is the case for the Schwarzschild spacetime where the Painlevé-Gullstrand coordinates were originally found. For more details on flat foliations of spherically symmetric spacetimes see Refs. [@guven; @beig; @herrero].
If one does not choose $L=0$, then an angular momentum barrier where $U_r^2$ changes sign prevents the new slicing from reaching the horizon in a way complementary to the way the choice $E<1$ prevents the slicing from reaching spatial infinity. Thus modulo the choice of sign for incoming or outgoing radial geodesics, the $L=0, E\ge1$ slicings are the only ones which cover the original region exterior to the horizon in the Schwarzschild case, as well as extending through that horizon up to the singularity at $r=0$.
For other examples, one can specialize these general results to the case of the de Sitter and anti de Sitter spacetimes, which in static coordinate systems have the same metric function expressions e\^=1-r\^2=e\^[-]{}. but for the two different signs of $\Lambda$, respectively positive and negative. In the de Sitter case $\Lambda>0$, the original coordinates are limited by a coordinate singularity at the radius at which this expression goes to zero. Expressing the cosmological constant in terms of Hubble parameter $H$ in that case we have $\Lambda=3H^2$ and the metric (\[metricgen\]) in the original coordinates only covers the region $0<r<1/H$. Timelike geodesics with $E=1$ give rise to a new slicing as specified above with flat 3-metric on the slices and a purely radial shift vector N\^a=-\^a\_R\_r Hr. This new slicing extends through the above-mentioned coordinate singularity out to $r\to\infty$.
In the anti de Sitter case we have $\Lambda=-3H^2$ and the metric (\[metricgen\]) is valid out to infinity in the radial direction. However, examining Eq. (\[eq:Ur\]), one sees that $U_r^2\ge0$ only for $E>1$ and then only within a ball $0<r<\sqrt{E^2-1}/H$ centered at the origin, thus limiting the range of the new coordinates compared to the original ones. The associated spatial 3-metric is no longer flat as follows from Eq. (\[indmetE\]) and the radial shift vector is given by N\^a=-\^a\_R\_r .
Painlevé-Gullstrand coordinates in the de Sitter spacetime have been introduced by Parikh [@parikh] to study tunneling processes across the cosmological horizon. The same problem has been investigated in the Schwarzschild-anti de Sitter spacetime in Ref. [@hemming], but using a non-geodesic slicing.
The Kerr spacetime
==================
Now consider the Kerr spacetime with its metric written in Boyer-Lindquist coordinates $$\begin{aligned}
\label{eq:met}
\rmd s^2 &=& -\left(1-\frac{2Mr}{\Sigma}\right)\rmd t^2 -\frac{4aMr}{\Sigma}\sin^2\theta\rmd t\rmd\phi+ \frac{\Sigma}{\Delta}\rmd r^2 +\Sigma\rmd \theta^2\nonumber\\
&&+\frac{(r^2+a^2)^2-\Delta a^2\sin^2\theta}{\Sigma}\sin^2 \theta \rmd \phi^2\,,\end{aligned}$$ where $\Delta=r^2-2Mr+a^2$ and $\Sigma=r^2+a^2\cos^2\theta$; here $a$ and $M$ are the specific angular momentum and total mass characterizing the spacetime. The event horizons are located at $r_\pm=M\pm\sqrt{M^2-a^2}$. In this case we have for timelike geodesics [@MTW] S=12 \^2 -Et +L+S\_r(r) +S\_(), with S\_r=\_r r, S\_=\_ , where $\epsilon_r=\pm1$ and $\epsilon_\theta=\pm1$ are sign indicators, and $$\begin{aligned}
\label{defsvarie}
P&=& E(r^2+a^2)-La\,,\quad
B= L-aE \sin^2\theta\,, \quad
R= P^2-\Delta (\mu^2 r^2+K)\,,\nonumber\\
\Theta&=&Q-\cos^2 \theta\left[a^2(\mu^2-E^2)+\frac{L^2}{\sin^2 \theta} \right]\,,\quad
Q= K-(L-aE)^2\,.\end{aligned}$$ We set $\mu=1$ so that -T = U\^=-Et+\_rr+\_+Lwith $R(r)$ and $\Theta(\theta)$ now given by Eq. (\[defsvarie\]) with $\mu=1$. For completeness, we also list below the contravariant components of $U$: $$\begin{aligned}
U^t&=& \frac{1}{\Sigma}\left[aB+\frac{(r^2+a^2)}{\Delta}P\right]\,,\quad
U^r=\epsilon_r \frac{1}{\Sigma}\sqrt{R}\,,\nonumber \\
U^\theta&=&\epsilon_\theta \frac{1}{\Sigma}\sqrt{\Theta}\,,\quad
U^\phi= \frac{1}{\Sigma}\left[\frac{B}{\sin^2\theta}+\frac{a}{\Delta}P\right]\,.\end{aligned}$$
The induced metric Eq. (\[eq:gamma\]) on the hypersurfaces of the foliation $T=\hbox{\it const}$ is obtained by replacing $\rmd t$ by t=U\_a x\^a =\_rr+\_+in Eq. (\[eq:met\]), where $x^a=\{r,\theta,\phi\}$. This depends only on the spatial coordinates $x^a$ and on the three constants of the motion $E,L,K$.
One is free to pick any choice of the three parameters $E,L,K$ to determine a geodesic slicing. If we choose them so that the family includes equatorial geodesics, we must impose first the condition $Q=0$ so that on the equatorial plane $\theta=\pi/2$ one has $U_\theta=0$. However, off the equatorial plane $U_\theta^2$ will be negative if $a\neq0 $ and $E<1$, and will change sign if $E>1$, $L\neq0$, limiting the slicing to exclude a range of $\theta$ values around the polar axis that these geodesics cannot reach. For the slicing to be valid everywhere outside the horizon, one must therefore have $L=0$ and $E=1$, leaving $K=a^2$. This limits the zero angular momentum geodesics to be at rest at spatial infinity. The infalling geodesics have $\epsilon_r=-1$. These are the world lines of the Painlevé-Gullstrand observers.
The $T=\hbox{\it const}$ hypersurfaces then correspond to t=U\_r r=\_r r, so that the induced metric is given by $$\begin{aligned}
\label{eq:3met}
{}^{(3)}\rmd s^2
&=& \frac{\Sigma}{\Delta}\rmd r^2+\Sigma\rmd \theta^2+
\frac{\Delta\Sigma\sin^2\theta}{\Sigma -2Mr}\rmd \phi^2 \nonumber\\
&& -\left(1-\frac{2Mr}{\Sigma}\right)\left( U_r\rmd r +\frac{2aMr\sin^2\theta}{\Sigma -2Mr} \rmd\phi\right)^2\,.\end{aligned}$$
A direct calculation shows that the metric determinant $\gamma$ and the associated Ricci scalar ${}^{(3)}R$ evaluate to =, \^[(3)]{}R= , while the trace of the extrinsic curvature is given by (K)= . The geometry associated with Painlevé-Gullstrand observers in the Kerr spacetime is not intrinsically or extrinsically flat, nor is it conformally flat [@prigar; @kroon1; @kroon2].
Finally a new global coordinate system for Kerr spacetime is given by $(X^\alpha)=(T,R,\Theta,\Phi)$ with $R=r$ and $\Theta=\theta$ unchanged and $T=t+f(r)$ and $\Phi=\phi+h(r)$ such that $$\begin{aligned}
-\rmd T&=&-\rmd t+U_r\rmd r=U^\flat\,,
\qquad \rmd R=\rmd r\,, \qquad \rmd\Theta=\rmd\theta\,,
\nonumber\\
%\rmd R&=&\rmd r\,, \qquad \rmd\Theta=\rmd\theta\,,\nonumber\\
\rmd\Phi&=&\rmd \phi +\frac{g^{t\phi}}{g^{rr}U_r}\rmd r=\rmd \phi -\frac{a}{r^2+a^2}U_r\rmd r\,,\end{aligned}$$ with inverse relations $$\begin{aligned}
\partial_T&=&\partial_t\,,
\qquad \partial_\Theta=\partial_\theta\,, \qquad \partial_\Phi=\partial_\phi\,,
\nonumber\\
\partial_R&=&U_r\left(\partial_t+\frac{a}{r^2+a^2}\partial_\phi\right)+\partial_r\,.
%\nonumber\\
%\partial_\Theta&=&\partial_\theta\,, \qquad \partial_\Phi=\partial_\phi\,.\end{aligned}$$ The new azimuthal coordinate is of the form $\Phi=\phi+F(r)$, where $F'(r)=g^{t\phi}/({g^{rr}U_r})$ is a function only of $r$ since $U_r$ is such a function because of the separability condition while the ratio =- serendipidously happens to be independent of $\theta$. The introduction of $\Phi$ leads to a zero shift vector component along the azimuthal direction in the new coordinate system N\^= g\^[T]{}=0, thus aligning the new radial coordinate lines with the geodesic observers by incorporating their azimuthal motion into the new azimuthal coordinate. One finds that $
g^{\Phi\Phi}=({(r^2+a^2)\sin^2\theta})^{-1}
$, so the spatial 1-form |U\^= is a unit 1-form orthogonal to $U^\flat$. The Kerr metric in this new set of coordinates $X^\alpha$ has been given by Doran [@doran] (see also [@natario]). It is again Eq. (\[newmet\]), with unit lapse factor and the shift vector aligned with the radial direction, i.e., N\^a=\^a\_R=-\^a\_R\_r. The nonvanishing components of the spatial metric are instead given by $$\begin{aligned}
\gamma_{RR}&=&\frac{1+g_{tt}}{(N^R)^2}=\frac{\Sigma}{r^2+a^2}\,, \quad
\gamma_{R\Phi}=\frac{g_{t\phi}}{N^R}=\epsilon_r a\sin^2\theta\sqrt{\frac{2Mr}{r^2+a^2}}\,, \nonumber\\
\gamma_{\theta\theta}&=&g_{\theta\theta}\,, \quad
\gamma_{\Phi\Phi}=g_{\phi\phi}\,.\end{aligned}$$ The Doran form of the metric (\[newmet\]) is obtained simply by completing the square on the radial term as follows $$\begin{aligned}
\label{completesquare}
\rmd s^2&=&-\rmd T^2+\gamma_{RR}(\rmd r+N^R \rmd T)^2+2\gamma_{R\Phi}(\rmd r+N^R \rmd T)\rmd \Phi\nonumber\\
&&+\gamma_{\theta\theta}\rmd\theta^2+\gamma_{\Phi\Phi}\rmd\Phi^2\nonumber\\
&=&-\rmd T^2+\gamma_{RR}\left[\rmd r+N^R \rmd T+\frac{\gamma_{R\Phi}}{\gamma_{RR}}\rmd \Phi\right]^2\nonumber\\
&&+\gamma_{\theta\theta}\rmd\theta^2+\left(\gamma_{\Phi\Phi}-\frac{\gamma_{R\Phi}^2}{\gamma_{RR}}\right)\rmd\Phi^2\,,\end{aligned}$$ with \_-=(r\^2+a\^2)\^2, =\_r a\^2. Recalling that $U^\flat=-\rmd T$ and $\bar U^\flat=\sqrt{r^2+a^2}\, \sin \theta \rmd \Phi$ the final form of the metric is then s\^2=-(U\^)\^2+(|U\^)\^2+\_[RR]{}\^2+\_\^2, identifying in this way a natural orthonormal frame adapted to $U$ introduced by Doran [@doran]. $$\begin{aligned}
\omega^0&=-U^\flat\,,\qquad&
\omega^1=\sqrt{\gamma_{RR}}\left[\rmd r+N^R \rmd T+\frac{\gamma_{R\Phi}}{\gamma_{RR}}\rmd \Phi\right]\,,\nonumber\\
\omega^2&=\sqrt{\gamma_{\theta\theta}}\rmd\theta\,,\qquad&
\omega^3=\bar U^\flat\,.\end{aligned}$$
In the special case of the Schwarzschild spacetime, the induced metric reduces to the flat spatial metric as shown above, and the single nonzero shift component in the new coordinate system reduces to $N^R=-\epsilon_r\sqrt{{2M}/{r}}$. This entire discussion can be repeated for the more general Kerr-Newman family of spacetimes with similar results.
The Gödel spacetime
===================
The Gödel spacetime [@godel; @Hawell] is a stationary and axisymmetric solution of the Einstein’s equations whose metric expressed in cylindrical coordinates $(t,r,\phi,z)$ is $$\begin{aligned}
\rmd s^2&=&\frac{2}{\omega^2}\left[-\rmd t^2 +\rmd r^2 +\sinh^2 r(1-\sinh^2 r)\rmd \phi^2 \right.
\nonumber\\
&&\qquad \left. +2\sqrt{2}\sinh^2r \rmd t \rmd \phi +\rmd z^2\right]\,.\end{aligned}$$ Its matter source is a constant dust energy density $\rho$ (i.e., stress-energy tensor $T=\rho u\otimes u$), with unit 4-velocity $u=(\omega/\sqrt{2})\partial_t$ aligned with the time coordinate lines and cosmological constant $\Lambda=-\omega^2=-4\pi\rho$, where $\omega>0$ is the constant rotation parameter chosen to be positive to describe an intrinsic counterclockwise rotation of the spacetime around the $z$-axis.
Define the radius $r_h$ where $ g_{\phi\phi}=0$ (the $\phi$ coordinate circles are null here, then timelike for larger radii) by r\_h=(1+)0.88137, r\_h=1, r\_h=. Unlike the Kerr case, the time lines here are always timelike geodesics, so the coordinates are valid at all radii, but the spacelike time coordinate hypersurfaces used to introduce fiducial observers along their normal direction turn timelike beyond $r_h$, so this represents an observer horizon for this family. The 4-velocity of these fiducial observers is n= (\_t - \_) , r<r\_h. This observer horizon is similar to the one which occurs for uniformly rotating cylindrical coordinates in Minkowski spacetime where the angular speed of the corotating observer family grows to the speed of light at the light cylinder which terminates their existence.
The covariant representation of the matter 4-velocity $u$ is \[uflatgodel\] u\^=-(t -r ), which is easily seen to be geodesic and shear-free but which has constant nonzero vorticity (u)=\_z,(u)\^= z. Due to the existence of the three Killing vectors fields $\partial_t$, $\partial_\phi$ and $\partial_z$, the geodesic equations are separable and the covariant 4-velocity of a general timelike geodesic has the following separated form (setting already $\mu=1$ for a proper time parametrization) U\^=-Et +L+bz +\_r S\_r(r)r, where from the normalization condition $U^\alpha U_\alpha=-1$ one then finds \[eq:c2\] ()\^2=U\_r\^2=with the constants ${\mathcal A}$, ${\mathcal B}$ and ${\mathcal C}$ given by $$\begin{aligned}
{\mathcal A}&=& -(\omega^2b^2+2+\omega^2E^2)\,,\nonumber\\
{\mathcal B}&=& \omega^2b^2+2+3\omega^2E^2+2\sqrt{2}\omega^2LE\,,\nonumber\\
{\mathcal C}&=& -\omega^2(\sqrt{2}E+L)^2\,,\end{aligned}$$ with ${\mathcal A}+{\mathcal B}+{\mathcal C}=-\omega^2 L^2$. The 4-velocity vector is $$\begin{aligned}
U &=& \frac{\omega^2}{2\cosh^2 r} \left[ \sqrt{2} L + E (2-\cosh^2 r) \right] \,\partial_t
+\frac{\omega^2}{2} U_r \partial_r
\nonumber\\
&& \quad
- \frac{\omega^2}{2 \sinh^2 r \cosh^2 r} \left( E \sqrt{2} \sinh^2 r - L\right) \,\partial_\phi
+ \frac{\omega^2}{2} b \partial_z
\,.\end{aligned}$$
By setting $b=0$ we can avoid the unnecessary complication of additional translational motion along the axis of cylindrical symmetry, so that the above normalization condition simplifies to \[eq:U2\] U\_r\^2 = ( )\^2 =(E-V\_+)(E-V\_-), where the overall coefficient $\kappa (r)=({2-\cosh^2r})/{\cosh^2 r}$ is positive when $0<r<r_h$ and negative when $r>r_h$ and the effective potentials $V_\pm$ are given by V\_= . Noting that only the combinations $L\omega$ and $E\omega$ occur in these formulas for $V_\pm$ and $U_r$, we set $\omega=1$ (measuring $L,E$ in units of $1/\omega$). $V_\pm$ are real only when $0<r<r_*$, where $\sinh r_*={\sqrt{1+\sqrt{1+2L^2}}}/{\sqrt{2}}$. At $r=r_*$, the two potentials meet $V_+=V_-$ and assume the common value $\lim_{r\to r_*^-}V_\pm ={\sqrt{2}}(1+\sqrt{1+2L^2})/{L} $. Since $u=\omega/\sqrt{2} \partial_t$ is always a future-pointing timelike 4-vector, and the sign-reversed inner product of any two future-pointing unit vectors must be greater than 1, then -u\_U\^= \_t U = 1 E , so only positive values of the energy $E$ greater than $\sqrt{2}/\omega$ are allowed.
Fig. 1 shows typical profiles for the effective potentials. Increasing $L>0$ pulls the asymmetric potential well in (a) farther and farther to the right past the horizon radius $r_h$, with the maximum value $r_*$ a slowly increasing function of $L$. Thus for geodesics rotating in the same sense as the matter content of the spacetime ($L>0$), the corresponding time slices remain spacelike farther into the region $r>r_h$ containing timelike azimuthal coordinate circles as the angular momentum increases.
$\begin{array}{cc}
\includegraphics[scale=\SC]{fig1a.eps}&\quad
\includegraphics[scale=\SC]{fig1b.eps}\\[.4cm]
\quad\mbox{(a)}\quad &\quad \mbox{(b)}\\
\end{array}$\
![The behavior of the effective potentials for radial motion as functions of $r$ is shown for fixed values of $L=[2,-2,0]$ in Figs. (a) to (c) respectively, having set $\omega=1$. For $U_r^2\ge0$, $E$ must lie above or below both potentials if $0\le r< r_h$ but between them for $ r_h\le r\le r_*$, which are the unshaded regions of the plane. Changing the sign of $L$ reflects (a) into (b), while the case $L=0$ in (c) has an obvious reflection symmetry across the horizontal axis, but since $E>\sqrt{2}$, the lower half planes are forbidden regions (corresponding to past-directed 4-velocities). Thus horizontal lines representing energy levels in the white region of the upper half plane describe the allowed radial motion. ](fig1c.eps "fig:")\
\[fig:1\]
Radial turning points occur at the zeros $r_\pm$ of (\[eq:c2\]) or (\[eq:U2\]) where the energy level intersects the effective potential graph at $E=V_+$ or $E=V_-$, namely \^2 r\_=, where =[B]{}\^2-4[A]{}[C]{}=( E\^2-2)( E- E\_-)( E- E\_+), and E\_=-2 L. If $L>0$ then $E_-<0$, whereas $E_+>0$ for $ L<1/\sqrt{2}$ and is always negative otherwise. In contrast, if $L<0$ then $E_+>0$, whereas $E_-<0$ for $ L>-1/\sqrt{2}$ and is always positive otherwise. Finally, for $L=0$ we have $E_\pm=\pm\sqrt{2}$.
The radial turning points limit the radial range of this family of geodesics, and hence the range of the new time coordinate defined by their normal hypersurfaces, as in the case of the bound geodesics in the Kerr spacetime. For nonzero angular momentum $L>0$ aligned with the angular velocity of the spacetime itself, although one introduces a centrifugal barrier around the axis $r=0$ of cylindrical symmetry, one extends the range of the new time coordinate hypersurfaces orthogonal to this family of geodesics through the original coordinate horizon, thus “penetrating" this artificial horizon used in defining the fiducial observers associated with the time foliation. By increasing $L>0$, this penetration is increased, at the expense of pushing the centrifugal wall near the origin farther to the right, but as one lowers $E$, eventually the penetration radius $r_+$ is decreased to meet the increasing $r_-$ and the interval over which the family of geodesics is defined shrinks to zero width. This effect does not occur for $L<0$ where the geodesics counterrotate with respect to the angular velocity of the spacetime; the centrifugal potential barriers at $r=0$ and $r=r_h$ only shrink the zone of validity of the new coordinates in this case.
Next we can repeat the same subsequent steps as for the Kerr spacetime, adapting the slicing to a new temporal coordinate $T$ such that T=Et -U\_r r - L. The spatial metric induced on the $T=\hbox{\it const} $ hypersurfaces is then $$\begin{aligned}
\label{3_metricgodel}
{}^{(3)}\rmd s^2 &=& \gamma_{rr}\rmd r^2 + \gamma_{\phi\phi} \rmd \phi^2+2 \gamma_{r\phi} \rmd r \rmd \phi +\gamma_{zz}\rmd z^2\,,\end{aligned}$$ with the new coordinate components $$\begin{aligned}
\gamma_{rr}&=& g_{rr}+\frac{g_{tt}}{E^2}U_r^2\,,\quad
\gamma_{\phi\phi}=g_{\phi\phi}+g_{tt}\frac{L^2}{E^2}+2g_{t\phi}\frac{L}{E}\,,\nonumber\\
\gamma_{r\phi}&=&\frac{U_r}{E}\left(g_{t\phi}+g_{tt}\frac{L}{E}\right)\,, \quad
\gamma_{zz}=g_{zz}\,,\end{aligned}$$ depending only on $r$. The spatial geometry is not intrinsically flat, since $
{}^{(3)}{}R^{r\phi}{}_{ r\phi}=\omega^4 E^2
$ is the only nonvanishing coordinate component of the spatial Riemann tensor, corresponding to a constant spatial Ricci scalar ${}^{(3)}{}R=2\omega^4E^2$. However, the Cotton-York tensor [@cotton_bob; @bindef] is identically zero implying that the spatial metric is conformally flat. The extrinsic curvature is nonzero.
Finally a new coordinate system for the Gödel spacetime within the horizon radius of this family of geodesics is given by $(T,R,\Phi,Z)$ with $R=r$ and $Z=z$ unchanged and $T=T(t,r,\phi)$ and $\Phi=\Phi(\phi,r)$ such that \[trasfgod\] T =Et-U\_rr-L,R=r, =+[F]{}r, Z=z, with inverse relations $$\begin{aligned}
\partial_T&=&\frac{1}{E}\partial_t\,,\
\partial_\Phi=\frac{L}{E}\partial_t+\partial_\phi\,,\nonumber\\
\partial_Z&=&\partial_z\,,\
\partial_R=\frac{U_r-L{\mathcal F}}{E}\partial_t-{\mathcal F}\partial_\phi+\partial_r\,, \end{aligned}$$ where =. This choice corresponds to aligning the shift vector field with the new radial direction so that $
N^\Phi =g^{T\Phi}=0
$, exactly as in the Kerr case. One then finds g\^ =, reintroducing the general value of $\omega$ into the discussion. Therefore the spatial 1-form $\bar U^\flat=(g^{\Phi\Phi})^{-1/2} \rmd \Phi$ has unit length and is orthogonal to $U$. The Gödel metric in this new set of coordinates is then given by Eq. (\[newmet\]), with unit lapse factor and the shift vector aligned with the radial direction, i.e., N\^a=-\^a\_R12\^2U\_r , and nonvanishing components of the spatial metric given by $$\begin{aligned}
\gamma_{RR}&=&\frac{4(E^2\omega^2-2)}{E^2U_r^2 \omega^6}\,, \quad
\gamma_{R\Phi}=-\frac{4(E\sqrt{2}\sinh^2 r-L)}{E^2U_r\omega^4}\,, \quad
\gamma_{zz}=g_{zz}=\frac2{\omega^2}\,, \nonumber\\
\gamma_{\Phi\Phi}&=&\frac{2}{\omega^2 E^2} \sinh^2 r (2- \cosh^2 r) (E-W_+)(E-W_-)\,,\end{aligned}$$ where W\_=.
One can also easily diagonalize the new form of the spacetime metric exactly as done in Eq. (\[completesquare\]) for the Kerr case leading to a Doran-like sum of squares representation of the new form of the metric, namely $$\begin{aligned}
\rmd s^2&=&-\rmd T^2+\gamma_{RR}(\rmd r+N^R \rmd T)^2+2\gamma_{R\Phi}(\rmd r+N^R \rmd T)\rmd \Phi\nonumber\\
&&+\gamma_{\Phi\Phi}\rmd\Phi^2+\gamma_{zz}\rmd z^2\nonumber\\
&=&-\rmd T^2+\gamma_{RR}\left[\rmd r+N^R \rmd T+\frac{\gamma_{R\Phi}}{\gamma_{RR}}\rmd \Phi\right]^2\nonumber\\
&&+\left(\gamma_{\Phi\Phi}-\frac{\gamma_{R\Phi}^2}{\gamma_{RR}}\right)\rmd\Phi^2+\gamma_{zz}\rmd z^2\,,\end{aligned}$$ with coefficients $$\begin{aligned}
\gamma_{\Phi\Phi}-\frac{\gamma_{R\Phi}^2}{\gamma_{RR}}&=&\frac{2\sinh^2r\cosh^2r U_r^2}{(E^2\omega^2-2)}
= (g^{\Phi\Phi})^{-1}
\,, \nonumber\\
\frac{\gamma_{R\Phi}}{\gamma_{RR}}&=&\frac{\omega^2U_r}{E^2\omega^2-2}(L-\sqrt{2}E\sinh^2r)\,,\end{aligned}$$ while the new spatial metric determinant is $4\sinh^2 2r/(E^2 \omega^8)>0$. Recalling that $U^\flat=-\rmd T$ and $\bar U^\flat=(g^{\Phi\Phi})^{-1/2}\rmd \Phi$ we get s\^2=-(U\^)\^2+(|U\^)\^2+\_[RR]{}\^2+\_[zz]{}z\^2, identifying in this way a natural orthonormal frame adapted to $U$ $$\begin{aligned}
\omega^0&=-U^\flat\,,\qquad&
\omega^1=\sqrt{\gamma_{RR}}\left[\rmd r+N^R \rmd T+\frac{\gamma_{R\Phi}}{\gamma_{RR}}\rmd \Phi\right]\,,\nonumber\\
\omega^2&=\bar U^\flat\,,\qquad&
\omega^3=\sqrt{\gamma_{zz}}\rmd z\,.\end{aligned}$$
Concluding remarks
==================
We have shown how stationary spacetimes admitting separable geodesic equations admit a new spacetime slicing orthogonal to a particular family of timelike geodesics, corresponding to a unit lapse gauge. For stationary spherically symmetric vacuum spacetimes, an intrinsically flat slicing is possible, as for the Schwarzschild spacetime, reproducible also for the de Sitter spacetime. In the stationary axisymmetric case of the Kerr and Gödel spacetimes, the separability also explains the existence of a new azimuthal coordinate which allows the alignment of the shift vector field with the spherical radial or cylindrical radial direction respectively. In the Kerr spacetime this leads to the usual Painlevé-Gullstrand slicing and adapted coordinate system, while in the Gödel case it leads to a new analogous form of the metric.
Acknowledgments {#acknowledgments .unnumbered}
===============
All authors thank ICRANet for support. DB acknowledges O. Semerák for useful discussion.
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[^1]: The present analysis can be easily generalized to the case of charged background spacetimes and nongeodesic orbits of charged particles (with charge $e$) still allowing a separable action and Hamilton-Jacobi equation g\^(\_S -eA\_)(\_S -eA\_)=-\^2 as occurs for a Kerr-Newman black hole. We will not discuss such a generalization here.
|
---
abstract: 'A nucleation model for the breakdown phenomenon in an initial non-homogeneous free traffic flow that occurs at an on-ramp bottleneck is presented. This model is in the context of three-phase traffic theory. In this theory, the breakdown phenomenon is associated with a first-order phase transition from the $\lq\lq$free flow“ phase to the $\lq\lq$synchronized flow” phase. In contrast with many other nucleation models for phase transitions in different system of statistical physics in which random precluster emergence from fluctuations in an initial homogeneous system foregoes subsequent cluster evolution towards a critical cluster (critical nuclei), random precluster occurrence in free flow at the bottleneck is not necessary for traffic breakdown. In the model, the breakdown phenomenon can also occur if there were no fluctuations in free flow. This is because there is a permanent and motionless non-homogeneity that can be considered a deterministic vehicle cluster localized in a neighborhood of the bottleneck. The presented nucleation model and a nucleation rate of traffic breakdown that follows from the model exhibit qualitatively different features in comparison with previous results. In the nucleation model, traffic breakdown nucleation occurs through a random increase in vehicle number within the deterministic vehicle cluster, if the amplitude of the resulting random vehicle cluster exceeds some critical amplitude. The mean time delay and the associated nucleation rate of traffic breakdown at the bottleneck are found and investigated. The nucleation rate of traffic breakdown as a function of the flow rates to the on-ramp and upstream of the bottleneck is studied. Boundaries for traffic breakdown in the diagram of congested patterns at the bottleneck are found. These boundaries are qualitatively correlated with numerical results of simulation of microscopic traffic flow models in the context of three-phase traffic theory.'
address:
- ' DaimlerChrysler AG, REI/VF, HPC: G021, 71059 Sindelfingen, Germany '
- ' Moscow Institute of Physics and Technology, Department of Physics, 141700 Dolgoprudny, Moscow Region, Russia '
author:
- 'Boris S. Kerner'
- 'Sergey L. Klenov'
title: 'Probabilistic Breakdown Phenomenon at On-Ramp Bottlenecks in Three-Phase Traffic Theory'
---
and
Introduction {#Introduction}
============
Empirical observations of freeway traffic made in various countries show that the onset of congestion in an initial free flow is associated with an abrupt decrease in vehicle speed. This traffic breakdown called the $\lq\lq$breakdown phenomenon" occurs mostly at freeway bottlenecks, in particular on-ramp bottlenecks. The traffic breakdown is accompanied by a hysteresis effect (see references in the reviews [@Hall1992A; @Banks2002A], the book [@KernerBook], and the conference proceedings [@Lesort; @Ceder; @Taylor]). The breakdown phenomenon has a probabilistic nature [@Elefteriadou1995A; @Persaud1998B; @Lorenz2000A]: At the same on-ramp bottleneck, traffic breakdown is observed at different flow rates in different realizations (days). The probability of the breakdown is a strong increasing function of flow rate downstream of the bottleneck [@Persaud1998B; @Lorenz2000A].
Most microscopic, macroscopic, probabilistic, and other models of freeway traffic explain the onset of congestion in free flow by moving jam emergence (see, e.g. [@KK1994; @Bando; @Schreck; @Mahnke1997; @Mahnke1999; @Kuehne2002] and references in the reviews [@Sch; @Helbing2001; @Nagatani_R; @Nagel2003A], as well as the conference proceedings [@SW1; @SW2; @SW3; @SW4; @SW5]). In particular, in models of an on-ramp bottleneck moving jam(s) occurs spontaneously in free flow at the bottleneck when the flow rate upstream of the bottleneck is high enough and the flow rate to the on-ramp increases gradually beginning from zero [@Helbing2001; @Nagatani_R; @KKS1995; @Helbing1999A; @Lee1999A; @Helbing2000]. However, the fundamental model result that the onset of congestion in free flow on a homogeneous road and at freeway bottlenecks is associated with spontaneous moving jam emergence [@KK1994; @Mahnke1997; @Mahnke1999; @Kuehne2002; @KKS1995; @Helbing1999A; @Lee1999A; @Helbing2000; @Sch; @Helbing2001; @Nagatani_R; @Nagel2003A] is in a serious conflict with empirical evidence [@Kerner1998B; @Kerner2002B; @KernerBook].
Consequently, in 1996–1999 Kerner introduced three-phase traffic theory (see [@KernerBook] for a review). In this theory, there are three traffic phases: free flow, synchronized flow, and wide moving jams. In accordance with empirical investigations of phase transitions, in this theory moving jams do [*not*]{} emerge spontaneously in free flow. Rather than moving jam emergence, a phase transition from free flow to synchronized flow (F$\rightarrow$S transition for short) governs the onset of congestion in free flow [@Kerner1998B; @Kerner2002B]. A first-order F$\rightarrow$S transition postulated in three-phase traffic theory [@Kerner1998B] discloses the nature of the breakdown phenomenon at freeway bottlenecks found in empirical observations [@Hall1992A; @Banks2002A]. In other words, the terms $\lq\lq$F$\rightarrow$S transition“, $\lq\lq$breakdown phenomenon”, $\lq\lq$speed breakdown“, and $\lq\lq$traffic breakdown” are synonyms. The first microscopic models in the context of three-phase traffic theory are stochastic models [@KKl; @KKW]. These models exhibit phase transitions as well as all types of congested patterns found in empirical observations [@KKl; @KKW; @KKl2003A; @KKl2004AA; @KernerBook]. Recently, some new microscopic models based on three-phase traffic theory have been developed [@Davis2003B; @Lee_Sch2004A; @Jiang2004A], which can show some congested pattern features found earlier in [@KKl; @KKW].
One of the important methods for a study of phase transitions in non-linear distributed multiple-particle systems is a probabilistic theory based on an analysis of a master equation (e.g., [@Gardiner; @vanKampen; @MahnkeRev]). First probabilistic theories for traffic flow based on a master equation for a random vehicle cluster have been introduced by Mahnke et al. [@Mahnke1997; @Mahnke1999] and further developed by Kühne et al. [@Kuehne2002; @Kuehne2004] (see the recent review by Mahnke et al. [@MahnkeRev]). As usual for other metastable systems of statistical physics, for an initial metastable traffic flow a well-known two-well potential nucleation problem arises from the master equation, which analysis is made based on general well-known methods of statistical physics [@Gardiner; @vanKampen]. One of the main results of this analysis is the nucleation rate for the critical vehicle cluster (critical nuclei) whose occurence leads to a phase transition within the initial metastable state of traffic flow. As usual for other metastable systems of statistical physics [@Gardiner; @vanKampen], in the metastable traffic flow the nucleation rate for a phase transition is an exponential function of a control parameter, flow rate (or vehicle density) for traffic flow [@Mahnke1997; @Mahnke1999; @Kuehne2002; @Kuehne2004; @MahnkeRev]. Rather than these common well-known results, an interest for the field has a nucleation model for a metastable traffic flow. Both the model and associated dependencies of the nucleation rates for phase transitions on traffic variables and control parameters should be adequate with real traffic flow features.
In [@Mahnke1997; @Mahnke1999; @Kuehne2002], models for vehicle cluster nucleation in an initial spatially homogeneous traffic flow on a homogeneous circular road are introduced (see, however, Sect. \[Comparison\_S\]). One of the basic assumptions of these models is that in an initial homogeneous free flow firstly random precluster should emerge from fluctuations. In other words, this precluster foregoes subsequent cluster evolution towards a critical cluster (critical nuclei) whose growth leads to a phase transition. The rate of precluster emergence $w_{+}(0)$ in traffic, in which initially no vehicle cluster exists, can be different from the attachment rate of cluster evolution, when a random cluster with $n\geq 1$ vehicles already exists at the road [@Mahnke1997; @Mahnke1999; @Kuehne2002; @MahnkeRev].
A first nucleation model based on the master equation for traffic breakdown at an on-ramp bottleneck, i.e., in an open traffic system has been suggested by Kühne et al. [@Kuehne2004] and Mahnke et al. (Chap. 17 of Ref. [@MahnkeRev]). As in the models of homogeneous road [@Mahnke1997; @Mahnke1999; @Kuehne2002], in this model at given flow rates to the on-ramp $q_{\rm on}$ and on the main road upstream of the on-ramp $q_{\rm in}$ random vehicle cluster occurrence and evolution that can lead to traffic breakdown are realized [*only*]{} after a vehicle precluster consisting of one vehicle randomly appears at the bottleneck. The rate of precluster emergence is equal to the flow rate to the on-ramp [@Kuehne2004; @MahnkeRev]: $w_{+}(0)=q_{\rm on}$. This idea about foregoing precluster emergence is associated with a basic model assumption that at $q_{\rm on}=0$ no vehicle cluster can randomly appear, specifically no traffic breakdown is possible [@Kuehne2004; @MahnkeRev]. Probably, this basic model assumption leads to the nucleation rate for traffic breakdown that is proportional to $q_{\rm on}$ [@Kuehne2004; @MahnkeRev]. However, in real traffic flow both flow rates $q_{\rm on}$ [*and*]{} $q_{\rm in}$ exhibit random fluctuations that can cause cluster emergence.
Moreover, whereas the basic assumption about the necessity of precluster emergence is physically justified for a homogeneous road [@Mahnke1997; @Mahnke1999; @Kuehne2002], this is not the case for for traffic breakdown at the bottleneck. This is because in contrast with the model of a homogeneous road [@Mahnke1997; @Mahnke1999; @Kuehne2002], an initial state of free flow at the bottleneck at $q_{\rm on}>0$ and $q_{\rm in}>0$ is [*non-homogeneous regardless of fluctuations*]{} [@KernerBook]. This means that even there were no fluctuations in free flow at the bottleneck, nevertheless free flow is non-homogeneous in a neighborhood of the bottleneck. This is because two different flows permanent merge within the merging region of the on-ramp – the flow with the rate $q_{\rm on}$ and the flow with the rate $q_{\rm in}$.
It has been shown in three-phase traffic theory [@KernerBook] that due to the merging of these flows a steady state of free flow at the bottleneck is non-homogeneous: A permanent and motionless local perturbation (deterministic perturbation) occurs at the bottleneck. Within this perturbation that can be considered a deterministic vehicle cluster the speed is lower and density is greater than downstream of the cluster. This deterministic cluster can lead to traffic breakdown even if there were [*no*]{} random fluctuations at the bottleneck [@Kerner2000A; @KernerBook]. However, in the nucleation model [@Kuehne2004; @MahnkeRev] the master equation is formulated for probability of a vehicle cluster, which can occur due to fluctuations [*only*]{}. We see that for an initially non-homogeneous traffic flow, which occurs at the bottleneck, another physical approach should be applied. This is due to at least two reasons mentioned above: (i) Regardless of fluctuations, there is already a vehicle cluster at the bottleneck that growth can lead to traffic breakdown: No vehicle precluster emerging from fluctuations is necessary for traffic breakdown. (ii) When influence of fluctuations on traffic breakdown is considered, fluctuations caused by both flow rates $q_{\rm on}$ and $q_{\rm in}$ should be taken into account.
Thus, in contrast with the nucleation model at the bottleneck of Ref. [@Kuehne2004; @MahnkeRev] a nucleation model that is adequate with empirical results for traffic breakdown at the bottleneck should consider vehicle cluster evolution in an initially non-homogeneous free traffic. This random vehicle cluster should include [*both*]{} the total vehicle number within an initial deterministic cluster [*and*]{} addition vehicles within the cluster, which occur due to fluctuations.
However, there are no nucleation models and associated probabilistic theories for an initially spatially non-homogeneous free flow. Moreover, structure non-homogeneities play also a very important role for phase transitions in many other multiple-particle systems of statistical physics like granular flow, biological physics, reaction-diffusion systems, etc. (see references, e.g. in [@SW1; @SW2; @SW3; @SW4; @SW5; @KO; @Michailov1]). Thus, it seems important to derive a probabilistic theory for traffic breakdown in non-homogeneous traffic flow.
In this paper, a nucleation model for the probabilistic breakdown phenomenon in an initially spatially non-homogeneous traffic flow at an on-ramp bottleneck is presented. The article is organized as follows. In Sect. \[Sect\_prob\], the nucleation model is considered. Nucleation rate and the mean time delay for traffic breakdown that result from this model are studied in Sect. \[Sect\_time\]. Results of the paper as well as their comparison with earlier nucleation models for traffic breakdown of Ref. [@Mahnke1997; @Mahnke1999; @Kuehne2002; @Kuehne2004; @MahnkeRev] are discussed in Sect. \[Sect\_Dis\].
Nucleation model of traffic breakdown in initially non-homogeneous free flow at on-ramp bottleneck {#Sect_prob}
==================================================================================================
Non-homogeniety in free flow at bottleneck as reason for vehicle cluster \[Nonhom\_S\]
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In accordance with three-phase traffic theory [@Kerner2000A; @KernerBook], in a nucleation model we assume that the breakdown phenomenon at an on-ramp bottleneck is associated with an initial non-homogeneity of free flow at the bottleneck. This non-homogeneity exists at the bottleneck even in a hypothetical case in which there were no fluctuations in traffic flow. In this hypothetical case, this non-homogeneity can be considered a [*deterministic*]{} (permanent) local perturbation localized at the bottleneck. The non-homogeneity of an initial free flow at the bottleneck is caused by two flows, which merge at the bottleneck: (i) An on-ramp inflow with the rate $q_{\rm on}$. (ii) A flow on the main road upstream of the bottleneck with the rate $q_{\rm in}$. This flow merging occurs permanent and on the same freeway location (within an on-ramp merging region). For this reason, the non-homogeneity of free flow at the bottleneck is motionless and permanent (Fig. \[FS\_Intr\] (a)).
![Explanation of the basis of nucleation model: (a) Sketch of an on-ramp bottleneck. (b, c) – Z-shaped speed–flow (a) and the associated S-shaped density–flow characteristics for an F$\rightarrow$S transition. (d, e) – Simplified Z-shaped speed–flow (d) and S-shaped density–flow characteristics (e) related to (b) and (c), respectively. \[FS\_Intr\] ](FS_Intr.eps){width="7"}
To explain features of this non-homogeneity, note that at a given high enough flow rate $q_{\rm in}$ in free flow on the main road upstream of the bottleneck, vehicles that merge from the on-ramp onto the main road force the vehicles on the main road to decelerate in the vicinity of an on-ramp merging region. This deceleration leads to a local decrease in speed and consequently to a local increase in density in the vicinity of the bottleneck. As a result, a local perturbation, i.e., non-homogeneity in free flow appears.
If no fluctuations in free flow are considered, as mentioned above the non-homogeneity in free flow in the form of a deterministic local perturbation occurs at the bottleneck. The speed $v^{\rm (B)}_{\rm free}$ and density $\rho^{\rm (B)}_{\rm free}$ within this deterministic perturbation correspond to the conditions $$v^{\rm (B)}_{\rm free}<v^{\rm (free)}, \ \rho^{\rm (B)}_{\rm free}>\rho^{\rm (free)},
\label{v_free_rho_free}$$ where $v^{\rm (free)}$ and $\rho^{\rm (free)}$ are the average vehicle speed and density in homogeneous free flow on the main road downstream of the perturbation (Fig. \[FS\_Intr\] (a)). Because the deterministic local perturbation is permanent and motionless, at given $q_{\rm in}$ and $q_{\rm on}$, the total flow rate does not depend on the spatial co-ordinate, i.e., the speed and density on the main road satisfy the following condition: $$q_{\rm sum}=v^{\rm (free)}\rho^{\rm (free)}=v^{\rm (B)}_{\rm free}\rho^{\rm (B)}_{\rm free},
\label{q_sum_x}$$ where $$\begin{aligned}
q_{\rm sum}=q_{\rm in}+q_{\rm on}.
\label{inflow_sum}
\end{aligned}$$ The inhomogeneous steady state within the deterministic local perturbation can be considered $\lq\lq$deterministic vehicle cluster“ in free flow localized at the bottleneck or $\lq\lq$deterministic cluster” for short.
When $q_{\rm on}$ is a given value and the total flow rate $q_{\rm sum}$ increases, then the vehicle speed $v^{\rm (B)}_{\rm free}$ within the deterministic cluster decreases and in accordance with (\[q\_sum\_x\]) the associated density $\rho^{\rm (B)}_{\rm free}$ increases. However, this increase is limited by some critical density $\rho^{\rm (B)}_{\rm free}=\rho^{\rm (B)}_{\rm determ, \ FS}$ (Fig. \[FS\_Intr\] (c)) within the deterministic cluster associated with a critical flow rate $$q_{\rm sum}=q^{\rm (B)}_{\rm determ, \ FS}.
\label{critical_determ}$$ After this critical deterministic perturbation is reached, the further increase in $q_{\rm sum}$ leads to [*deterministic traffic breakdown*]{} at the bottleneck causing spontaneous synchronized flow emergence at the bottleneck. The critical deterministic perturbation can be considered $\lq\lq$critical deterministic vehicle cluster“ ($\lq\lq$critical deterministic cluster” for short) or $\lq\lq$deterministic nuclei for traffic breakdown". After the critical deterministic cluster is reached, deterministic traffic breakdown occurs at the bottleneck even if there were no random perturbations in traffic flow at the bottleneck.
Random perturbations within the initial deterministic cluster can cause random traffic breakdown (F$\rightarrow$S transition) at the flow rate $$q_{\rm sum}<q_{\rm determ, \ FS}^{\rm (B)},
\label{random_FS}$$ i.e., before the deterministic nuclei for traffic breakdown is reached. In this case, random traffic breakdown nucleation can occur at the bottleneck (arrows labeled F$\rightarrow$S in Fig. \[FS\_Intr\] (b, c)). This is realized, if through a random increase in vehicle number within the initial deterministic cluster, the amplitude of the resulting $\lq\lq$random vehicle cluster“ ($\lq\lq$random cluster” for short) exceeds some critical amplitude associated with a critical density within the random cluster $\rho^{\rm (B)}_{\rm cr, \ FS}$ (Fig. \[FS\_Intr\] (c)). The random cluster with the critical density $\rho^{\rm (B)}_{\rm cr, \ FS}$ can be considered $\lq\lq$critical random vehicle cluster“ at the bottleneck ($\lq\lq$critical random cluster” for short) or $\lq\lq$random nuclei for traffic breakdown". If the amplitude of a random cluster is smaller than the critical one, the random cluster decays towards the initial deterministic cluster.
Master equation
---------------
### Basic assumptions for master equation \[Assum\_Sect\]
There are the following basic assumptions for the nucleation model in an initial non-homogeneous free flow at the bottleneck:
\(i) There is no vehicle precluster, which random occurrence through a random decrease in speed of one of the vehicles in the initial free flow should be necessary for vehicle cluster emergence and subsequent cluster evolution. Traffic breakdown occurs within a deterministic vehicle cluster that is motionless and permanently exists in a neighborhood of the bottleneck.
\(ii) A master equation should describe probability $p$ for random vehicle cluster evolution in which the cluster size $N$, i.e., the total vehicle number within the motionless vehicle cluster randomly changes due to fluctuations in a neighborhood of the deterministic cluster (Fig. \[Cluster\]). The size of the deterministic cluster $N^{\rm (determ)}$ does not depend on fluctuations in traffic flow.
\(iii) The attachment rate $w_{+}$ onto this vehicle cluster is not a function of the cluster size $N$, i.e. $w_{+}(0)=w_{+}(N)=q_{\rm in}+q_{\rm on}$.
\(iv) A dependence of the detachment rate $w_{-}$ from a cluster is a non-linear function on $N$, which consists of at least two branches associated with the deterministic cluster and a critical vehicle cluster (nuclei for traffic breakdown).
\(v) The on-ramp inflow and the flow upstream of the bottleneck make a different influence on the cluster size. To take into account this assymetric behavior in the model, the detachment rate $w_{-}$ also depends on $q_{\rm on}$.
![Schematic illustration of cluster transformation in an initially non-homogeneous free flow at the bottlneck. \[Cluster\] ](Cluster.eps){width="10"}
The assumption (i) can be explained by existence of a deterministic vehicle cluster at the bottleneck. This cluster can lead to traffic breakdown at the bottleneck even there were no fluctuations in traffic flow, i.e., no precluster is necessary for vehicle cluster occurrence in an initially steady state of non-homogeneous free flow at the bottleneck. To explain the assumption (ii), note that real random fluctuations can cause either a decrease or an increase in the cluster size $N$ and, respectively, either a decrease or an increase in vehicle density within the cluster. Real random fluctuations, which decrease the cluster size $N$, are associated with an increase in speed within the cluster (Sect. \[Nonhom\_S\]). Therefore, these fluctuations can prevent the deterministic breakdown. In contrast, random fluctuations, which increase the cluster size $N$, i.e., decrease the speed within the cluster, can cause traffic breakdown before the deterministic nuclei for traffic breakdown is reached, i.e., when the condition (\[random\_FS\]) is satisfied. Regardless of the cluster size $N$, the attachment rate into the cluster is determined by fluctuations in both flow rates $q_{\rm in}$ [*and*]{} $q_{\rm on}$. This explains the assumption (iii). Grounds for the assumptions (iv) and (v) appear in Sect. \[N\_grounds\].
We consider the dynamics of the total vehicle number $N$ within a vehicle cluster localized at the bottleneck (dashed region in Fig. \[FS\_Intr\] (a)). It is assumed that within the cluster free flow is spatially non-homogeneous, specifically the region of spatial cluster localization is bounded upstream and downstream with homogeneous free flows. The total number of vehicles $N$ within the cluster can either increase or decrease over time randomly in comparison with a value $N=N^{\rm (determ)}$ for the case in which the deterministic cluster exists at the bottleneck only.
In accordance with the above basic assumptions, the probability $p(N,t)$ to find $N$ vehicles within the cluster at the bottleneck reads as follows $$\begin{aligned}
%\begin{split}
\label{Prob}
\frac{\partial p(N,t)}{\partial t}= w_{+}(N-1) p(N-1,t) +w_{-}(N+1)p(N+1,t) \nonumber
\\
-[w_{+}(N) +w_{-}(N)] p(N,t), \quad \textrm{at $N > 0$},
%\end{split} \end{aligned}$$ $$\begin{aligned}
\frac{\partial p(0,t)}{\partial t}=w_{-}(1) p(1,t) -w_{+}(0)p(0,t), \quad \textrm{at $N = 0$}, \quad
\label{Prob0}\end{aligned}$$ with the boundary condition $$\begin{aligned}
w_{-}(0)=0,
\label{Prob0_boundary}\end{aligned}$$ In accordance with the assumption (3), the vehicle attachment rate $w_{+}$ is independent of $N$, i.e., $$\begin{aligned}
w_{+}=q_{\rm sum}.
\label{inflow}
\end{aligned}$$
### Vehicle detachment rate from cluster \[N\_grounds\]
The vehicle detachment rate $w_{-}$ is obviously equal to the outflow rate from the cluster $$\begin{aligned}
w_{-}(N)=q^{\rm (bottle)}_{\rm down}(N).
\label{w_}
\end{aligned}$$ To understant a qualitative shape of the function $q^{\rm (bottle)}_{\rm down}(N)$, note that in accordance with three-phase traffic theory [@KernerBook], speed states within the deterministic cluster $v^{\rm (B)}_{\rm free}(q_{\rm sum})$, the speed within the critical random cluster $v^{\rm (B)}_{\rm cr, \ FS}(q_{\rm sum})$, along with a 2D synchronized flow speed states [@KernerBook], together form a Z-shaped speed–flow characteristic for traffic breakdown (Fig. \[FS\_Intr\] (b)). The associated density–flow characteristic, which consists of density states within the deterministic cluster $\rho^{\rm (B)}_{\rm free}(q_{\rm sum})$, the density within the critical random cluster $\rho^{\rm (B)}_{\rm cr, \ FS}(q_{\rm sum})$, along with a 2D synchronized flow states, has obviously a S-shaped form (Fig. \[FS\_Intr\] (c)) [^1] Due to an F$\rightarrow$S transition, fast cluster growth at the bottleneck occurs leading to congested pattern formation, i.e., either a synchronized flow pattern (SP) or a general pattern (GP) appears upstream of the bottleneck [@KernerBook]. However, the nucleation effect leading to traffic breakdown and its characteristics are fully independent of possible congested patterns resulting from this F$\rightarrow$S transition. For this reason, we can average the infinity of synchronized flow states (dashed regions in (b, c)) for each given flow rate $q_{\rm sum}$ to one speed $v^{\rm (B)}_{\rm syn, \ aver}(q_{\rm sum})$ and to one density $\rho^{\rm (B)}_{\rm syn, \ aver}(q_{\rm sum})$ (Fig. \[FS\_Intr\] (d, e)) [^2]. A consideration of the resulting congested patterns is beyond the scope of this article.
In the model, it is assumed that the shape of the characteristic $q^{\rm (bottle)}_{\rm down}(N)$ (Fig. \[q\_rho\]) follows from the S-shaped density–flow characteristic of three-phase traffic theory (Fig. \[FS\_Intr\] (e)): The characteristic $q^{\rm (bottle)}_{\rm down}(N)$ has at least two different branches $q^{\rm (bottle)}_{\rm down}(N)$ labeled $N^{\rm (determ)}$ and $N_{\rm c}$ in Fig. \[q\_rho\] (a). These branches are related to the vehicle number ranges, respectively, given by the conditions $$\begin{aligned}
0\leq N \leq N_{\rm d}
\label{range1}
\end{aligned}$$ and $$\begin{aligned}
N_{\rm d}< N \leq N_{\rm s}.
\label{range2}
\end{aligned}$$ The branches $N^{\rm (determ)}$ and $N_{\rm c}$ in Fig. \[q\_rho\] (a) are associated with the density branches $\rho^{\rm (B)}_{\rm free}$ and $\rho^{\rm (B)}_{\rm cr, \ FS}$ of the S-shaped density–flow characteristic in Fig. \[FS\_Intr\] (e), respectively. The branch $N^{\rm (determ)}$ is associated with the case in which at a high enough flow rate $q_{\rm sum}$ and the on-ramp flow rate $q_{\rm on}>0$ the deterministic cluster exists at the bottleneck. The branch $N_{\rm c}$ is associated with the case in which the critical random cluster whose growth leads to an F$\rightarrow$S transition occurs at the bottleneck.
![Qualitative dependencies of the outflow rate $q^{\rm (bottle)}_{\rm down}$ on the total vehicle number $N$ within the cluster localized at the bottleneck (a), and possible dependencies of the N-shaped function $q^{\rm (bottle)}_{\rm down}(N)$ on $q_{\rm on}$ for two different values $q_{\rm on}$ (b); curve 1 for $q_{\rm on}=q^{(1)}_{\rm on}$, curve 2 for $q_{\rm on}=q^{(2)}_{\rm on}>q^{(1)}_{\rm on}$. \[q\_rho\] ](q_rho.eps){width="7"}
In addition, from the S-shaped density–flow characteristic (Fig. \[FS\_Intr\] (e)) can be seen that for the case in which an LSP results from an F$\rightarrow$S transition, at $$\begin{aligned}
N>N_{\rm s},
\label{range3}
\end{aligned}$$ there is a third branch $N^{\rm (syn)}$ on the characteristic $q^{\rm (bottle)}_{\rm down}(N)$ (Fig. \[q\_rho\] (a)) associated with the branch $\rho^{\rm (B)}_{\rm syn, \ aver}$ for averaged synchronized flow states in Fig. \[FS\_Intr\] (e). In this case, $q^{\rm (bottle)}_{\rm down}(N)$ (\[w\_\]) is a N-shaped flow–vehicle-number characteristic.
At the critical point $N=N_{\rm d}$ at which the branches $N^{\rm (determ)}$ and $N_{\rm c}$ merges, the function $q^{\rm (bottle)}_{\rm down}(N)$ has its maximum point. At the threshold point $N=N_{\rm s}$ at which the branches $N_{\rm c}$ and $N^{\rm (syn)}$ merges, the function $q^{\rm (bottle)}_{\rm down}(N)$ has its minimum point.
Quantitative characteristics of the N-shaped function $q^{\rm (bottle)}_{\rm down}(N)$ (e.g., values $N_{\rm d}$ and $N_{\rm s}$) can depend on the flow rate to the on-ramp $q_{\rm on}$ (assumption (v) in Sect. \[N\_grounds\]). This is because the on-ramp inflow and the flow upstream of the bottleneck can make a considerable different influence on the cluster size and the outflow rate from the cluster $q^{\rm (bottle)}_{\rm down}$. In particlular, it can turn out that at the same $N$ the greater $q_{\rm on}$, the more difficult for vehicles to escape from the cluster, i.e., the less $q^{\rm (bottle)}_{\rm down}$ is. This is confirmed by microscopic simulations [@KKW] and reflected in Fig. \[q\_rho\] (b) in which it is assumed that the greater $q_{\rm on}$, the greater $N_{\rm d}$ and $N_{\rm s}$ are. Thus, in a general case instead of (\[w\_\]) we should use $$w_{-}(N)=q^{\rm (bottle)}_{\rm down}(N, \ q_{\rm on}).
\label{w_final}$$ A possible impact of the flow rate $q_{\rm on}$ on quantitative characteristics of the mean time delay for an F$\rightarrow$S transition is discussed in Sect. \[Sect\_Dis\].
Through the use of the basic assumptions (i)-(v) of Sect. \[Assum\_Sect\] and the chosen shape of the function (\[w\_\]) discussed above (Fig. \[q\_rho\] (a)), critical cluster occurrence describes an F$\rightarrow$S transition at the bottleneck. There are two reasons for this statement: (i) A vehicle cluster is motionless, i.e., fixed at the bottleneck. This is related to the definition of synchronized flow whose downstream front is usually fixed at the bottleneck, whereas the downstream from of a wide moving jam propagates trough any bottleneck while maintaining the mean downstream jam velocity [@KernerBook]. (ii) The shape of the chosen function (\[w\_\]) in the nucleation model associated with this motionless cluster (Fig. \[q\_rho\] (a)) follows from a Z-shaped speed–flow characteristic for traffic breakdown (Fig. \[FS\_Intr\] (b)) for an F$\rightarrow$S transition at the bottleneck found in a microscopic traffic flow theory. In this theory has been shown that if these two requirements are satisfied, then rather than an F$\rightarrow$J transition (moving jam emergence) an F$\rightarrow$S transition, i.e., synchronized flow emergence occurs at the bottleneck. As follows from this microscopic theory, after the F$\rightarrow$S transition has already occurred, moving jams can emerge in this synchronized flow. However, the nucleation model describes traffic breakdown, i.e., an F$\rightarrow$S transition, specifically the rate of traffic breakdown (synchronized flow) nucleation [*only*]{}.
In addition, it should be noted that the branch for synchronized flow $v^{\rm (B)}_{\rm syn, \ aver}$ in Figs. \[FS\_Intr\] (b-e) and the associated branch $N^{\rm (syn)}$ on the characteristic $q^{\rm (bottle)}_{\rm down}(N)$ (Fig. \[q\_rho\] (a)) follow from the microscopic traffic theory, rather than from the nucleation model. This branch, which is shown only with the aim of a qualitative illustration of a possible traffic flow state after synchronized flow nucleation, has no influence on the nucleation rate of an F$\rightarrow$S transition.
Steady states
-------------
Steady states of vehicle number $N$ at given $q_{\rm in}$ and $q_{\rm on}$ are associated with solutions of the equation $$\begin{aligned}
w_{+}=w_{-}(N).
\label{steady_states}
\end{aligned}$$ In accordance with (\[inflow\]), (\[w\_final\]), they are found from the condition $$\begin{aligned}
q_{\rm sum}=q^{\rm (bottle)}_{\rm down}(N, \ q_{\rm on}).
\label{steady_states_q}
\end{aligned}$$ As can be seen from Fig. \[q\_rho\] (a), at given flow rates $q_{\rm on}$ and $q_{\rm in}$ that satisfy the condition $$q^{\rm (B)}_{\rm th}<q_{\rm sum}<q^{\rm (B)}_{\rm determ, \ FS}
\label{q_sum_cond}$$ there can be at least two steady states: $N=N_{1}$ associated with the deterministic cluster and $N=N_{2}$ associated with the critical random cluster. These steady states are the roots of Eq. (\[steady\_states\_q\]), i.e., they are associated with the intersection points of the horizontal line $q^{\rm (bottle)}_{\rm down}=q_{\rm sum}$ with the branches $N^{\rm (determ)}$ and $N_{\rm c}$ of the characteristic $q^{\rm (bottle)}_{\rm down}(N, \ q_{\rm on})$ (Fig. \[q\_rho\] (a)), respectively. In addition, if an LSP occurs as a result of an F$\rightarrow$S transition, then there is a third root of Eq. (\[steady\_states\_q\]), $N=N_{3}$, associated with the intersection point of the horizontal line $q^{\rm (bottle)}_{\rm down}=q_{\rm sum}$ with the branch $N^{\rm (syn)}$ of the characteristic $q^{\rm (bottle)}_{\rm down}(N)$.
If the flow rate $q_{\rm sum}$ increases, then the critical vehicle number difference within the cluster $$\Delta N_{\rm c}=N_{2}-N_{1}
\label{N_2_N_1}$$ decreases. This critical vehicle number difference is associated with the vehicle number difference within the critical random cluster and within the initial deterministic cluster at the bottleneck. The growth of the critical random cluster leads to traffic breakdown at the bottleneck.
At the critical flow rate (\[critical\_determ\]), we get $\Delta N_{\rm c}=0$: The steady states $N_{1}$ and $N_{2}$ merge into one point with the critical vehicle number $N=N_{\rm d}$ at which $q^{\rm (B)}_{\rm determ, \ FS}=q^{\rm (bottle)}_{\rm down}(N_{\rm d}, \ q_{\rm on})$. At $q_{\rm sum}\geq q^{\rm (B)}_{\rm determ, \ FS}$ the deterministic traffic breakdown should occur even if there is no random increase in the vehicle number within the initial deterministic cluster at the bottleneck.
If the flow rate $q_{\rm sum}$ decreases gradually, then the threshold flow rate $$q_{\rm sum}=q^{\rm (B)}_{\rm th}
\label{critical_thresh}$$ is reached at which the steady states $N_{2}$ and $N_{3}$ merge into one threshold steady state $N=N_{\rm s}$ at which $q^{\rm (B)}_{\rm th}=q^{\rm (bottle)}_{\rm down}(N_{\rm s}, \ q_{\rm on})$.
Nucleation rate of traffic breakdown at bottleneck {#Sect_time}
==================================================
As follows from the analysis of the model (\[Prob\])-(\[inflow\]), (\[w\_final\]) (Appendix \[App\]), in the flow rate range (\[q\_sum\_cond\]) the mean time delay of an F$\rightarrow $S transition at the bottleneck is $$\begin{aligned}
T^{\rm (B, \ mean)}_{\rm FS}=C \exp{\big \{\Delta \Phi \big \}},
\label{time_FS}\end{aligned}$$ where a potential barrier $$\begin{aligned}
\Delta \Phi= \Phi(N_{2})- \Phi(N_{1}),
\label{delta_pi}\end{aligned}$$ the potential $\Phi(N)$ is $$\begin{aligned}
\label{potential_eq}
\Phi(N)=\left\{
\begin{array}{ll}
\sum^{N}_{n=1} \ln{\frac{w_{-}(n)}{w_{+}}} & \textrm{at $N >0$}, \\
0 & \textrm{at $N=0,$}
\end{array} \right.\end{aligned}$$ $$\begin{aligned}
C=2\pi \Big(w^{\prime}_{-}(N_{1})
\mid w^{\prime}_{-}(N_{2})\mid \Big)^{-\frac{1}{2}},
\label{factor}\end{aligned}$$ $w^{\prime}_{-}(N)=dw_{-}/dN$. Respectively, the nucleation rate for traffic breakdown at the bottleneck is $$G^{\rm (B)}_{\rm FS}=\frac{1}{T^{\rm (B, \ mean)}_{\rm FS}}=C^{-1} \exp{\big \{-\Delta \Phi \big \}}.
\label{Probab_FS}$$
![Qualitative shape of the potential $\Phi(N)$ (\[potential\_eq\]) for different flow rates $q_{\rm sum}$: curves 1, 2, and 3 are related to the corresponding flow rates $q^{(1)}_{\rm sum}$, $q^{(2)}_{\rm sum}$, and $q^{(3)}_{\rm sum}$ satisfying the condition $q^{(3)}_{\rm sum}<q^{(1)}_{\rm sum}<q^{(2)}_{\rm sum}$. \[potential\] ](potential.eps){width="7"}
To find a qualitative shape $\Phi(N)$ (\[potential\_eq\]) (Fig. \[potential\]), a change in $\Phi(N)$ between two neighboring points $N$ and $N-1$ that equals $$\begin{aligned}
\label{potential_change}
\delta \Phi(N)=\Phi(N)-\Phi(N-1)= {\ln \frac{w_{-}(N)}{w_{+}}}\end{aligned}$$ can be used. The value $\delta \Phi(N)$ (\[potential\_change\]) becomes zero at the maximum and minimum points of the function $\Phi(N)$, i.e., at the roots of Eq. (\[steady\_states\]) that are the points $N=N_{\rm i}, \ i=1,2,3$ discussed above (Fig. \[q\_rho\] (a)). The value $\delta \Phi(N)>0$ at $w_{-}(N)>w_{+}$, i.e., at points of the curve $w_{-}(N)$ above the horizontal line $q=q_{\rm sum}$ in (Fig. \[q\_rho\] (a)). In contrast, $\delta \Phi(N)<0$ at $w_{-}(N)<w_{+}$, i.e., at points of the curve $w_{-}(N)$ below the horizontal line $q=q_{\rm sum}$.
It can be seen from (\[time\_FS\]) that the mean time delay for traffic breakdown decreases exponentionally with increase in potential barrier $\Delta \Phi$ (\[delta\_pi\]). If in Fig. \[potential\] the total flow rate increases from $q_{\rm sum}=q^{(1)}_{\rm sum}$ to $q_{\rm sum}=q^{(2)}_{\rm sum}$, which is close to the critical flow rate (\[critical\_determ\]) for deterministic traffic breakdown, then the potential barrier $\Delta \Phi$ (\[delta\_pi\]) decreases from $\Delta \Phi_{1}$ to $\Delta \Phi_{2}$.
In contrast, if the total flow rate decreases from $q_{\rm sum}=q^{(1)}_{\rm sum}$ to $q_{\rm sum}=q^{(3)}_{\rm sum}$, which is close to the threshold flow rate $q^{(B)}_{\rm th}$ (\[critical\_thresh\]) for random traffic breakdown, then the potential barrier $\Delta \Phi$ (\[delta\_pi\]) increases from $\Delta \Phi_{1}$ to $\Delta \Phi_{3}$ (Fig. \[potential\]). At the threshold point $q_{\rm sum}=q^{\rm (B)}_{\rm th}$ (\[critical\_thresh\]), the potential barrier $\Delta \Phi(N)$ reaches the maximum value $$\begin{aligned}
\Delta \Phi= \Phi(N_{s})- \Phi(N_{\rm th}),
\label{potential_th}\end{aligned}$$ where $N_{\rm th}=N_{1}$ at $ q_{\rm sum}=q^{\rm (B)}_{\rm th}$. As a result, the mean time delay $T^{\rm (B, \ mean)}_{\rm FS}$ (\[time\_FS\]) strongly increases as $q_{\rm sum}$ approaches the threshold point $q^{\rm (B)}_{\rm th}$. Under the condition $$q_{\rm sum}<q^{\rm (B)}_{\rm th}
\label{no_speed_bre}$$ no traffic breakdown at the bottleneck regardless of a random increase in the vehicle number within the cluster is possible at the bottleneck.
If in the vicinity of the critical vehicle number $N_{\rm d}$ the function $w_{-}(N)$ (\[w\_final\]) can be approximated by a parabolic function of $N$, then the following approximate formula can be derived from (\[time\_FS\]) (Appendix \[App\]): $$T^{\rm (B, \ mean)}_{\rm FS} =
\frac{\sqrt{2} \pi N_{\rm d}}{q^{\rm (B)}_{\rm determ, \ FS}(\xi_{\rm d} \Delta_{\rm c})^{1/2} }
\bigg(\frac{1+\Delta_{\rm c}^{1/2}}{1-\Delta_{\rm c}^{1/2}}\bigg) ^{2\sqrt{2/\xi_{\rm d}}N_{\rm d}}\exp{\bigg(-\frac{4\sqrt{2} N_{\rm d}{\Delta_{\rm c}^{1/2}}}{\sqrt{\xi_{\rm d}}} \bigg)},
\label{time_FS_approx_new}$$ where $$\xi_{\rm d}=-(N^{2}d^{2}\ln{w_{-}}/dN^{2})\big| _{N=N_{\rm d}}
\label{xi_determ}$$ is a dimensionless value of the order of 1, $$\Delta_{\rm c}=\frac{q^{\rm (B)}_{\rm determ, \ FS}- q_{\rm sum}}{q^{\rm (B)}_{\rm determ, \ FS}},
\label{overcr}$$ i.e., $\Delta_{\rm c}$ is the relative difference between the critical flow rate $q^{\rm (B)}_{\rm determ, \ FS}$ for the deterministic F$\rightarrow$S transition and the total flow rate $q_{\rm sum}$ (\[inflow\_sum\]). If in (\[time\_FS\_approx\_new\]) $\Delta_{\rm c}\ll$ 1, then we get $$T^{\rm (B, \ mean)}_{\rm FS} =
\frac{\sqrt{2} \pi N_{\rm d}}{q^{\rm (B)}_{\rm determ, \ FS}(\xi_{\rm d} \Delta_{\rm c})^{1/2} }
\exp{\bigg(\frac{8 N_{\rm d}{\Delta_{\rm c}^{3/2}}}{3 \sqrt{2\xi_{\rm d}}} \bigg)}.
\label{time_FS_approx}$$ Respectively, the nucleation rate $G^{\rm (B)}_{\rm FS}=1/T^{\rm (B, \ mean)}_{\rm FS}$ for traffic breakdown at the bottleneck associated with (\[time\_FS\_approx\]) is $$G^{\rm (B)}_{\rm FS} = \frac{ q^{\rm (B)}_{\rm determ, \ FS}(\xi_{\rm d} \Delta_{\rm c})^{1/2} }{\sqrt{2} \pi N_{\rm d}}
\exp{\bigg(-\frac{8 N_{\rm d}{\Delta_{\rm c}^{3/2}}}{3 \sqrt{2\xi_{\rm d}}} \bigg)}.
\label{probability_FS_approx}$$ Note that $q^{\rm (B)}_{\rm determ, \ FS}$, $N_{\rm d}$, and $\xi_{\rm d}$ depend $q_{\rm on}$. Therefore, the mean time delay $T^{\rm (B, \ mean)}_{\rm FS}$ (\[time\_FS\_approx\]) and the nucleation rate $G^{\rm (B)}_{\rm FS}$ ( \[probability\_FS\_approx\]) are functions of $q_{\rm sum}$ and $q_{\rm on}$.
If the flow rate $q_{\rm on}$ decreases continuously up to a small enough value (however, we assume that $q_{\rm on}>0$, i.e., the deterministic cluster still exists at the bottleneck), then the values $\xi_{\rm d}$, $N_{\rm d}$, and $q^{\rm (B)}_{\rm determ, \ FS}$ in (\[overcr\]), (\[probability\_FS\_approx\]) and, therefore, the nucleation rate $G^{\rm (B)}_{\rm FS}$ ( \[probability\_FS\_approx\]) do [*not*]{} decrease proportionally to this decrease in $q_{\rm on}$. In contrast, in this limit case $\xi_{\rm d}\rightarrow \xi_{\rm d, \ lim}$, $N_{\rm d} \rightarrow N_{\rm d, \ lim}$, and $q^{\rm (B)}_{\rm determ, \ FS}\rightarrow q^{\rm (B)}_{\rm determ, \ lim}$, where $\xi_{\rm d, \ lim}$, $N_{\rm d, \ lim}$, and $q^{\rm (B)}_{\rm determ, \ lim}$ are constants. Taking into account that in this case in $\Delta_{\rm c}$ (\[overcr\]) the flow rate $q^{\rm (B)}_{\rm determ, \ FS}\approx q^{\rm (B)}_{\rm determ, \ lim}={\rm const}$, we can see that at small enough values of $q_{\rm on}$ the nucleation rate for traffic breakdown (\[probability\_FS\_approx\]) depends on the total flow rate $q_{\rm sum}$ only. In other words, in this limit case at a given $q_{\rm sum}$ within the flow rate range (\[q\_sum\_cond\]) the nucleation rate for traffic breakdown at the bottleneck (\[probability\_FS\_approx\]) tends to a finite constant value, which is greater than zero (see Sect. \[Num\_Sym\_S\]).
When $q_{\rm on}= 0$, the road can be considered homogeneous one without bottlenecks. Then there is no deterministic perturbation (deterministic cluster) at the bottleneck and, therefore, the nucleation model and results of this article cannot be applied. In three-phase traffic theory, the breakdown phenomenon can also occur in this case. However, at the same conditions, in particlular, the same flow rates downstream of the bottleneck and on a homogeneous road, the nucleation rate for traffic breakdown on the homogeneous road is considerably smaller than at the bottleneck [@KKl2003A; @KernerBook]. This is associated with empirical results in which the breakdown phenomenon has also been observed away from bottlenecks, however, this traffic breakdown is much more rare than at an on-ramp bottleneck [@KernerBook]. A consideration of a nucleation model of the breakdown phenomenon for a homogeneous road is beyond the scope of this article.
As usual for each first-order phase transition observed in many other systems in natural science [@Gardiner], the nucleation rate for traffic breakdown (\[probability\_FS\_approx\]) is an exponentional function of $\Delta_{\rm c}$ (\[overcr\]). For traffic flow, in accordance with (\[probability\_FS\_approx\]) and (\[overcr\]) the exponential growth of the nucleation rate with $\Delta_{\rm c}$ (\[overcr\]) is very sensible to the critical value for the deterministic breakdown phenomenon $q^{\rm (B)}_{\rm determ, \ FS}$. This emphasizes the important impact of the deterministic cluster, which occurs at the bottleneck at $q_{\rm on}>0$, on the nucleation rate for traffic breakdown (\[probability\_FS\_approx\]) at a given total flow rate $q_{\rm sum}$.
Discussion {#Sect_Dis}
==========
Numerical simulations of general results of nucleation model for traffic breakdown \[Num\_Sym\_S\]
--------------------------------------------------------------------------------------------------
Let us compare general results of the nucleation model presented in Sect. \[Sect\_time\] with the diagram of congested patterns at an on-ramp bottleneck postulated in [@Kerner2002B] and found in numerical simulations of microscopic traffic flow models [@KKl; @KKW], as well as with a microscopic theory of the breakdown phenomenon [@KKl2003A]. To reach this goal, we consider an example of the function $w_{-}$ (\[w\_final\]) $$\begin{aligned}
w_{-}(N)=N \bigg[ \frac{a}{1+(N/N_{0})^{4}}+b \bigg],
\label{analytical}\end{aligned}$$ where $a$, $b$, and $N_{0}$ are functions of $q_{\rm on}$: $a(q_{\rm on})=1.32q_{0}(q_{\rm on})/N_{0}(q_{\rm on})$ 1/h, $q_{0}(q_{\rm on})=2700+370(1+q_{\rm on}/300)^{-1}$ vehicles/h, $b(q_{\rm on})=33+10(1+q_{\rm on}/250)^{-1}$ 1/h, $N_{0}(q_{\rm on})=25-6.5(1+q_{\rm on}/300)^{-1}$ vehicles; the unit of $q_{\rm on}$ is vehicles/h.
The analytical function (\[analytical\]) allows us to perform a numerical analysis of the mean time delay (\[time\_FS\]) and the associated nucleation rate (\[Probab\_FS\]) for the breakdown phenomenon (F$\rightarrow$S transition). For the analysis of (\[time\_FS\]) and (\[Probab\_FS\]), only branches $N^{\rm (determ)}$ and $N_{\rm c}$ (Fig. \[q\_rho\]) of the function (\[analytical\]) associated with the deterministic and critical clusters within which $N\leq N_{\rm s}$ are relevant. This is because the maximum possible value of $N=N_{2}$ in the potential barrier $\Delta \Phi$ (\[delta\_pi\]) that determines the nucleation rate (\[Probab\_FS\]) is equal to $N_{\rm s}$.
![N-shaped function $q^{\rm (bottle)}_{\rm down}(N)$ (\[analytical\]) (a) for $q_{\rm on}=100$ vehicles/h (curve 1) and $q_{\rm on}=600$ vehicles/h (curve 2), and the associated potential $\Phi$ (\[potential\_eq\]) (b) as functions of the vehicle number $N$ for $q_{\rm on}=100$ vehicles/h and for three different total flow rates $q_{\rm sum}$: 2070 (curve 3), 2200 (curve 1), 2400 vehicles/h (curve 2). \[N\_pot\] ](N_pot.eps){width="10"}
However, for a qualitative illustration of a possible synchronized flow state resulting from an F$\rightarrow$S transition, in (\[analytical\]) the branch for the synchronized flow state is added, in which the detachment rate $w_{-}(N)$ increases with $N$ [^3]. This branch corresponds to $N>N_{\rm s}$. Respectively, this branch of the detachment rate $w_{-}(N)$ has no influence on the analysis of (\[time\_FS\]) and (\[Probab\_FS\]). For this reason, a simple mathematical approximation (\[analytical\]) of the latter branch of $w_{-}(N)$ is chosen, in which the detachment rate $w_{-}(N)$ exhibits formally unlimited growth with $N$. As mentioned, this does not impact on results discussed below. Moreover, as follows from (\[Prob\_stationary\_1\]) (see Appendix) probability of cluster emergence, which size $N$ is large, is negligible.
![Potential barrier $\Delta \Phi$ (\[delta\_pi\]) (a), critical vehicle number difference $\Delta N_{\rm c}$ (\[N\_2\_N\_1\]) (b), mean time delay for traffic breakdown $T^{\rm (B, \ mean)}_{\rm FS}$ (\[time\_FS\]) (c, d), and nucleation rate for traffic breakdown $G^{\rm (B)}_{\rm FS}$ (\[Probab\_FS\]) (e, f) as functions of the total flow rate $q_{\rm sum}$ for three different flow rates $q_{\rm on}$: 100 (curves 1), 300 (curves 2), 800 (curves 3) vehicles/h. \[Freq\] ](Freq.eps){width="10"}
A numerical study shows that the potential $\Phi$ exhibits qualitatively the same behavior at different total flow rates $q_{\rm sum}$ (Fig. \[N\_pot\] (b)) as those in Fig. \[potential\]. The potential barrier $\Delta \Phi$ in (\[time\_FS\]), (\[Probab\_FS\]) (Fig. \[Freq\] (a)) and the associated critical vehicle number difference $\Delta N_{\rm c}$ (\[N\_2\_N\_1\]) are decreasing functions of the total flow rate $q_{\rm sum}$; at a given $q_{\rm sum}$ they can also be decreasing functions of $q_{\rm on}$ (Fig. \[Freq\] (b)). For these reasons, the total flow rate dependences of the mean time delay $T^{\rm (B, \ mean)}_{\rm FS}$ (\[time\_FS\]) (Fig. \[Freq\] (c, d)) and of the associated nucleation rate for traffic breakdown at the bottleneck (Fig. \[Freq\] (e, f)) exhibit qualitative features observed in traffic flow at on-ramp bottlenecks [@Persaud1998B; @Lorenz2000A] and found in a microscopic three-phase traffic theory [@KKW; @KKl2003A]. This confirms that the breakdown phenomenon at the bottleneck is a first-order F$\rightarrow$S transition [@KernerBook]. In all these curves, the total flow rate $q_{\rm sum}$ is smaller than the critical flow rate for the deterministic traffic breakdown $q^{\rm (B)}_{\rm determ, \ FS}$ (\[critical\_determ\]). This means that $\Delta_{\rm c}$ (\[overcr\]) is not equal zero for all results in Fig. \[Freq\], i.e., traffic breakdown occurs due to a random density increase within an initial deterministic cluster at the bottleneck. The total flow rate dependences of the nucleation rate for traffic breakdown $G^{\rm (B)}_{\rm FS}$ (\[Probab\_FS\]) calculated at different flow rates $q_{\rm on}$ exhibit features of three-phase traffic theory in which the breakdown phenomenon can also occur at small values $q_{\rm on}$.
![Characteristics of the nucleation model: (a, b) - Boundaries of constant values of the nucleation rate $G^{\rm (B)}_{\rm FS}$ of traffic breakdown (curves 1–4), the critical boundary $F^{\rm (B)}_{\rm determ, \ S}$ for deterministic traffic breakdown (curves $F^{\rm (B)}_{\rm determ, \ S}$), and the threshold boundary $F^{\rm (B)}_{\rm th}$ (curve $F^{\rm (B)}_{\rm th}$) as functions of the flow rates $q_{\rm on}$ and $q_{\rm in}$. (c, d) - Dependencies of the flow rate $q^{\rm (B)}_{\rm G}$ (\[critical\_no\]) (curves 1–4), the critical flow rate $q^{\rm (B)}_{\rm determ, \ FS}$ (\[critical\_determ\]) for deterministic traffic breakdown (curves $q^{\rm (B)}_{\rm determ, \ FS}$), and the threshold flow rate $q^{(B)}_{\rm th}$ (\[critical\_thresh\]) (curve $q^{(B)}_{\rm th}$) as functions of $q_{\rm on}$. Curves 1–4 are related to different given values $\zeta$ for the nucleation rate of traffic breakdown $G^{\rm (B)}_{\rm FS}$ in (\[critical\_no\_p\]): 1/3.5 (curves 1), 0.2 (curves 2), 0.1 (curves 3), 1/60 (curves 4) ${\rm min}^{-1}$. The nucleation model cannot be applied for $q_{\rm on}= 0$, therefore, the points in all figures in the vicinity of $q_{\rm on}= 0$ show only the tendency of the boundaries in (a, b) and the flow rates in (c, d) for the limiting case of small values $q_{\rm on}$ in which, however, the on-ramp inflow rate $q_{\rm on}>0$, specifically, it is assumed that the deterministic cluster still exists at the bottleneck. \[diag\] ](diag.eps){width="10"}
The critical boundary $F^{\rm (B)}_{\rm S, \ \zeta}$ (Fig. \[diag\] (a, b)) in the diagram of congested patterns at the bottleneck (flow–flow plane with the coordinates $(q_{\rm on}, \ q_{\rm in}$)) is associated with the cases in which the nucleation rate for traffic breakdown is a given value $\zeta$. Therefore, the boundary $F^{\rm (B)}_{\rm S, \ \zeta}$ satisfies the condition $$G^{\rm (B)}_{\rm FS} (q_{\rm sum}, \ q_{\rm on})=\zeta, \quad \zeta={\rm const},
\label{critical_no_p}$$ i.e., at the boundary $F^{\rm (B)}_{\rm S, \ \zeta}$ the flow rate $$q_{\rm sum}=q^{\rm (B)}_{\rm G}(q_{\rm on})
\label{critical_no}$$ depends on $q_{\rm on}$. This boundary is qualitatively similar with the critical boundary $F^{\rm (B)}_{\rm S}$ in the diagram at which the probability for traffic breakdown for a given time $T_{\rm ob}$ for observing traffic flow is 1 [@KernerBook]. In the diagram, there is also the threshold boundary $F^{\rm (B)}_{\rm th}$ (curve $F^{\rm (B)}_{\rm th}$ in Fig. \[diag\] (b)) at which the condition (\[critical\_thresh\]) is satisfied. The threshold boundary also exhibits the same qualitative features as those found in simulation of phase transitions and spatiotemporal congested patterns in a microscopic three-phase traffic theory [@KKl2003A; @KernerBook]. In particular, in the limiting case of small values $q_{\rm on}$ (but $q_{\rm on}>0$, i.e., it is assumed that the deterministic cluster still exists at the bottleneck) the flow rate $q^{\rm (B)}_{\rm G}$ reaches the maximum (limit) value $q^{\rm (B)}_{\rm G, \ lim}$ at a given nucleation rate for traffic breakdown $\zeta$ (\[critical\_no\_p\]).
The greater the nucleation rate for traffic breakdown, the greater $q^{\rm (B)}_{\rm G, \ lim}$ should be. However, the increase in nucleation rate for traffic breakdown has a limit associated with deterministic traffic breakdown occurrence: When $\zeta$ in (\[critical\_no\_p\]) increases, the boundary $F^{\rm (B)}_{\rm S, \ \zeta}$ for random traffic breakdown tends to the boundary $F^{\rm (B)}_{\rm determ, \ S}$ (curves $F^{\rm (B)}_{\rm determ, \ S}$ in Fig. \[diag\] (a, b)) for deterministic traffic breakdown in the diagram of congested patterns. At the boundary $F^{\rm (B)}_{\rm determ, \ S}$, the deterministic breakdown phenomenon occurs within the deterministic cluster even if [*no*]{} random vehicle number increase within the deterministic cluster appears at the bottleneck. When $\zeta$ in (\[critical\_no\_p\]) decreases, the boundary $F^{\rm (B)}_{\rm S, \ \zeta}$ tends to the threshold boundary $F^{\rm (B)}_{\rm th}$ (curve $F^{\rm (B)}_{\rm th}$ in Fig. \[diag\] (b)). In accordance with a microscopic theory [@KKl2003A], in the nucleation model the flow rate $q^{\rm (B)}_{\rm G}$ (\[critical\_no\]) (curves 1–4 in Fig. \[diag\] (c, d)), the critical flow rate $q^{\rm (B)}_{\rm determ, \ FS}$ for deterministic traffic breakdown (curves $q^{\rm (B)}_{\rm determ, \ FS}$ in Fig. \[diag\] (c, d)), as well as the threshold flow rate $q^{(B)}_{\rm th}$ (\[critical\_thresh\]) (curve $q^{(B)}_{\rm th}$ in Fig. \[diag\] (d)) can be the smaller, the greater $q_{\rm on}$ is.
Comparison with earlier nucleation models for traffic breakdown \[Comparison\_S\]
---------------------------------------------------------------------------------
In [@Mahnke1997; @Mahnke1999; @Kuehne2002] nucleation models for traffic breakdown for a [*homogeneous*]{} circular road have been developed (see Fig. 11 in the review [@MahnkeRev]). However, rather than traffic breakdown (F$\rightarrow$S transition), in [@Mahnke1997; @Mahnke1999; @Kuehne2002] a nucleation theory for wide moving jam emergence in an initially homogeneous free flow (F$\rightarrow$J transition) has been derived. Indeed, in final results of this probabilistic theory the vehicle speed within the vehicle cluster is chosen to be zero and a fundamental diagram for traffic flow with the vehicle cluster derived in the probabilistic theory [@Kuehne2002] is qualitatively the same (see Fig. 48 in [@MahnkeRev]) as those first found in [@KK1994] in a macroscopic theory of free flow metastability associated with F$\rightarrow$J transition. This fundamental diagram is confirmed by empirical results associated with wide moving jam propagation (see Fig. 17 in [@MahnkeRev]). However, even on a homogeneous road, traffic breakdown is governed by an F$\rightarrow$S transition rather than by an F$\rightarrow$J transition [@KernerBook]. Thus, the nucleation theory of [@Mahnke1997; @Mahnke1999; @Kuehne2002; @MahnkeRev] does not describe traffic breakdown on a homogeneous road.
The F$\rightarrow$S transition that can occur away from freeway bottlenecks is a very rare event [@KernerBook]. This is because a freeway bottleneck introduces a spatial non-homogeneity in free flow at the road. The average speed within this non-homogeneity, which is permanently localized in a neighborhood of the bottleneck location, is lower and the vehicle density is greater than on the road away from the bottleneck [@KernerBook]. This explains why in empirical observations traffic breakdowns are mostly observed at bottlenecks [@Elefteriadou1995A; @Persaud1998B; @Lorenz2000A; @KernerBook].
The first nucleation model for traffic breakdown at an on-ramp bottleneck has been suggested by K[ü]{}hne, Mahnke et al. [@Kuehne2004; @MahnkeRev]. In this model (see Chap. 17 in [@MahnkeRev]), a hypothesis of three-phase traffic theory about the sequence of the F$\rightarrow$S$\rightarrow$J transitions that governs phase transitions at the bottleneck [@KernerBook] have been taken into account. In addition, in accordance with this theory [@KernerBook] a random vehicle cluster, whose occurrence can lead to an F$\rightarrow$S transition, is localized at the bottleneck [@Kuehne2004; @MahnkeRev].
However, in this nucleation model a random vehicle precluster that emerges from fluctuations is necessary. This precluster, which consists of one vehicle ($n=1$), should occur in an initial hypothetical unperturbed free flow at bottleneck in which no vehicle cluster exists before. The precluster, which can be associated with a random decrease in speed of one of the vehicle in a neighborhood of the bottleneck, foregoes subsequent vehicle cluster evolution towards a critical cluster (critical nuclei) for traffic breakdown [@Kuehne2004; @MahnkeRev]. The attachment rate of precluster formation $w_{+}(0)$ is equal to the flow rate to the on-ramp $q_{\rm on}$ [@Kuehne2004; @MahnkeRev]: $$w_{+}(0)=q_{\rm on}.
\label{K_M}$$ The attachment rate (\[K\_M\]) of vehicle precluster formation does not depend on the flow rate $q_{\rm in}$ upstream of the bottleneck. However, in real traffic flow both flow rates $q_{\rm on}$ [*and*]{} $q_{\rm in}$ exhibit random fluctuations. The formula (\[K\_M\]) should be associated with the basic model assumption that at $q_{\rm on}=0$ no vehicle cluster can randomly appear, specifically no traffic breakdown is possible [@Kuehne2004; @MahnkeRev]. Apparently the assumption (\[K\_M\]) leads to the nucleation rate for an F$\rightarrow$S transition at the bottleneck that is proportional to the flow rate to the on-ramp [@Kuehne2004; @MahnkeRev].
Whereas for a homogeneous road the model assumption for the necessity of precluster formation [@Mahnke1997; @Mahnke1999; @Kuehne2002] is physically justified, this is not the case for the nucleation model at the bottleneck introduced in [@Kuehne2004; @MahnkeRev]. To explain this, note that in contrast with the model of a homogeneous road, at $q_{\rm on}>0$ and $q_{\rm in}>0$ an initial state of free flow at the bottleneck is non-homogeneous regardless of fluctuations. This means that even no fluctuations would occur in free flow at the bottleneck, nevertheless free flow is non-homogeneous in a neighborhood of the bottleneck [@KernerBook]. This is because two different flows permanent merge within the merging region of the on-ramp – the flow with the rate $q_{\rm on}$ and the flow with the rate $q_{\rm in}$. Vehicles merging from the on-ramp onto the main road force vehicles on the main road to slow down. In turn, these slower moving vehicles on the main road force vehicles merging from the on-ramp onto the main road to decrease the speed too. Thus, the speed is lower and the density is greater at the bottleneck, i.e., a local cluster appears regardless of fluctuations. Thus, a permanent and motionless (deterministic) vehicle cluster in which speed is lower and the density is greater than away from bottleneck exists already on the road, even if there were no fluctuations in traffic flow. For this reason, the formula (\[K\_M\]) [@Kuehne2004; @MahnkeRev] that assumes no vehicle cluster existence without random fluctuations in free flow at the bottleneck is in serious conflict with empirical results and results a microscopic three-phase traffic theory [@KernerBook].
Moreover, the master equation of this model [@Kuehne2004; @MahnkeRev] searches the probability for a random vehicle cluster with $n$ vehicles. In this master equation, a vehicle cluster exists ($n>0$) only then, if the precluster has appeared. When there were no fluctuations at the bottleneck at all, then a vehicle cluster cannot appear ($n=0$) and no traffic breakdown is possible in the model. In contrast, in three-phase traffic theory deterministic traffic breakdown is possible regardless of fluctuations. This deterministic traffic breakdown occurs when the size of the deterministic cluster exceeds some critical value. This is the consequence of the non-homogeneity in free flow at the bottleneck mentioned above.
Thus, in contrast with the above basic assumptions of Ref. [@Kuehne2004; @MahnkeRev], a nucleation model that can be adequate with empirical results should search probability $p$ for random vehicle cluster evolution in which the cluster size $N$ randomly changes due to fluctuations in a neighborhood of the deterministic cluster (Fig. \[Cluster\]). The size of this deterministic cluster $N^{\rm (determ)}$ does not depend on fluctuations in traffic flow. Random fluctuation either increases the speed within the cluster or decreases it. Consequently, the density and the cluster size $N$ either decreases or increases. In the latter case, traffic breakdown occurs at a smaller flow rate $q_{\rm sum}=q_{\rm in}+q_{\rm on}$ than the critical flow rate $q_{\rm sum}=q^{\rm (B)}_{\rm determ \ FS}$ associated with the deterministic traffic breakdown that occurs without any fluctuations at the bottleneck.
These fundamental differences in the nucleation model of Ref. [@Kuehne2004; @MahnkeRev] and the model presented in this article can explain different results of these models. In [@Kuehne2004; @MahnkeRev], even if the on-ramp inflow rate $q_{\rm on}$ is high, in an initial steady state of traffic flow there is no deterministic cluster at the bottleneck. As a result, there is no deterministic breakdown phenomenon of three-phase traffic theory in this model. Probably for this reason, in the model [@Kuehne2004; @MahnkeRev] the nucleation rate for the breakdown phenomenon (generation rate of traffic breakdown critical nuclei) is proportional to the on-ramp inflow rate $q_{\rm on}$. As a result, if $q_{\rm on}$ decreases below a small enough value (but $q_{\rm on}>0$) and the total flow rate $q_{\rm sum}$ increases (through an increase in $q_{\rm in}$), a reasonable given nucleation rate for traffic breakdown at the bottleneck (the nucleation rate should be greater than $\approx 1/20 \ {\rm min}^{-1}$, in accordance with empirical observations [@Persaud1998B; @Lorenz2000A]) cannot be reached. This is true in the nucleation model of [@Kuehne2004; @MahnkeRev] even if the total flow rate is equal to a critical value associated with the critical nuclei for traffic breakdown consisting of [*one vehicle*]{} only (in [@Kuehne2004; @MahnkeRev] this critical value is denoted by $q_{\rm c2}$).
In contrast, in our model the nucleation rate (generation rate of critical nuclei) for the breakdown phenomenon is [*not*]{} proportional to $q_{\rm on}$ and, therefore, as mentioned in Sect. \[Sect\_time\], for the limiting case of small values $q_{\rm on}$ (however, we assume that $q_{\rm on}>0$, specifically, the deterministic cluster still exists at the bottleneck) this generation rate of critical nuclei depends on the total flow rate $q_{\rm sum}$ [*only*]{}, i.e., this generation rate does [*not*]{} depend on $q_{\rm on}$. At a given $q_{\rm sum}$, an increase in $q_{\rm on}$ can influence [*only*]{} on such characteristics of traffic breakdown as the critical flow rate $q^{\rm (B)}_{\rm determ, \ FS}$ and the threshold flow rate $q^{\rm (B)}_{\rm th}$, as well as on congested traffic states at the bottleneck that result from the breakdown phenomenon.
When $q_{\rm sum}$ increases, the nucleation rate of traffic breakdown increases in the both models. However, in our model the nucleation rate cannot exceed the nucleation rate for traffic breakdown associated with the deterministic breakdown phenomenon. In contrast with assumptions of the nucleation model of Ref. [@Kuehne2004; @MahnkeRev], in our nucleation model the deterministic traffic breakdown occurs even [*without*]{} any random vehicle number increase within the initial steady state of free flow at the bottleneck. This is because if $q_{\rm on}>0$, then in our model there is a deterministic vehicle cluster localized at the bottleneck, which exists permanent at the bottleneck due to the on-ramp inflow. In our model, random traffic breakdown nucleation can occurs through a random increase in vehicle number within this deterministic cluster. The mentioned qualitative differences in the nucleation model of Ref. [@Kuehne2004; @MahnkeRev] and our nucleation model are also responsible for different dependences of the generation rate of traffic breakdown on the total flow rate in these nucleation models.
Derivation of nucleation rate {#App}
=============================
In order to derive formula (\[time\_FS\]), we use a general formula for the mean time delay $T$ of escaping from the potential well for the master equation (\[Prob\]) [@Gardiner]: $$\begin{aligned}
\label{time_general}
T=\sum^{N_{3}}_{n=N_{1}}{ \bigg[ (w_{+}p_{\rm s}(n))^{-1}
\sum^{n}_{k=0}p_{\rm s}(k) \bigg ]},\end{aligned}$$ where $p_{\rm s}(N)$ is a steady solution of (\[Prob\]), (\[Prob0\]): $$\begin{aligned}
p_{\rm s}(N)=p_{\rm s}(0) \prod^{N}_{n=1}\frac{w_{+}}{w_{-}(n)} \quad \textrm{at $N>0$}.
\label{Prob_stationary}\end{aligned}$$ When $N_{\rm i} \gg 1, \ i=1,2$ (more rigorous conditions are given below), the distribution $p_{\rm s}(N)$ has a sharp maximum at $N=N_{\rm 1}$, and the function $p^{-1}_{\rm s}(n)$ in (\[time\_general\]) has a sharp maximum at $n=N_{\rm 2}$. Then the formula (\[time\_general\]) can be written as follows [@Gardiner]: $$\begin{aligned}
T=(w_{+})^{-1}\sum^{N_{2}}_{n=0}{p_{\rm s}(n) } \sum^{N_{3}}_{n=N_{1}}{ p^{-1}_{\rm s}(n)}.
\label{time_general_1}\end{aligned}$$ Formula (\[Prob\_stationary\]) can be written as $$\begin{aligned}
p_{\rm s}(N)=p_{\rm s}(0) \exp{[-\Phi(N)]} \quad \textrm{at $N \geq 0$},
\label{Prob_stationary_1}\end{aligned}$$ where the potential $\Phi(N)$ is given by (\[potential\_eq\]). Substituting (\[Prob\_stationary\_1\]) into (\[time\_general\_1\]), we can find the exponentially large factor in (\[time\_general\_1\]) explicitly $$\begin{aligned}
T=(w_{+})^{-1} c_{1}c_{2}\exp{[\Phi(N_{2})-\Phi(N_{1})]},
\label{time_general_2}\end{aligned}$$ where $$\label{factors}
c_{1}=\sum^{N_{2}}_{n=0}{ \exp{[-\Delta \Phi^{(1)}(n)]}}, \
c_{2}=\sum^{N_{3}}_{n=N_{1}}{ \exp{[\Delta \Phi^{(2)}(n)]}}, \\$$ $$\Delta \Phi^{(i)}(N)=\Phi(N)-\Phi(N_{i}), \quad i=1,\ 2.
\label{delta_exponent}$$ The factors $ c_{1}$, $ c_{2}$ can be estimated using the parabolic approximation of potential $\Phi(N)$ near the extremum points $N=N_{1}, \ N_{2}$ [@vanKampen]. For instance, to find the factor $ c_{1}$, we introduce a new variable $y=N/N_{1}$ and approximate the sums in (\[potential\_eq\]), (\[factors\]) by integrals: $$\begin{aligned}
\Delta \Phi^{(1)}(N) \approx \phi^{(1)}(y)=
N_{1} \int^{y}_{1} { \ln{ \frac{w_{-}(N_{1}z)} {w_{+}}}dz}, \\
c_{1} \approx N_{1} \int^{N_{2}/N_{1}}_{0}{ \exp{[- \phi^{(1)}(y)}]}dy.
\label{c1}\end{aligned}$$ Using the series expansion $$\phi^{(1)}(y) = N_{1} \eta_{1}(y-1)^{2}/2+O((y-1)^{3})
\label{potential_expansion}$$ near the point $y=1$, where $\eta_{1}=d\ln{w_{-}}/d\ln{N}|_{N=N_{1}}$, we find $ c_{1}=\sqrt{2 \pi N_{1}/ \eta_{1}}$. Similarly, $ c_{2}=\sqrt{2 \pi N_{2}/ \eta_{2}}$, where $\eta_{2}=-d\ln{w_{-}}/d\ln{N}|_{N=N_{2}}$. The substitution of $ c_{1}$ and $ c_{2}$ into (\[time\_general\_2\]) yields the formula (\[time\_FS\]).
The parabolic approximation used for estimation of factor $ c_{1}$ holds only when we can neglect in integral (\[c1\]) third-order terms in the potential expansion (\[potential\_expansion\]) [@vanKampen]. The same is true for calculation of $ c_{2}$. The conditions of the parabolic approximation are $$N_{i}\eta_{i}^{3} \gtrsim \xi_{i}^{2}, \quad i=1,2,
\label{approximation_conditions}$$ where $\xi_{i}=N^{2}_{i}d^{2}\ln{w_{-}}/dN^{2}|_{N=N_{i}}, \ i=1,2$.
To derive the formula (\[time\_FS\_approx\_new\]), we approximate the value $\Delta \Phi$ in (\[time\_FS\]) by integral $$\begin{aligned}
\Delta \Phi=N_{1} \int^{N_{2}/N_{1}}_{1} { \ln{ \frac{w_{-}(N_{1}y)} {w_{+}}}dy}.
\label{exponent}\end{aligned}$$ Under approximation of the function $w_{-}(N)$ in (\[exponent\]) near the maximum point $N=N_{\rm d}$ by parabola, we get: $$\begin{aligned}
w_{-}(N) = q^{\rm (B)}_{\rm determ, \ FS}
\left[1-\xi_{\rm d}\frac{(N-N_{\rm d})^{2}}{2N_{\rm d}^{2}}\right],
\label{w_expansion}\end{aligned}$$ where the formula $q^{\rm (B)}_{\rm determ, \ FS}=w_{-}(N_{\rm d})$ is taken into account. The roots $N=N_{1}$ and $N=N_{2}$ of equation $q_{\rm sum}=w_{-}(N)$ given by formulae (\[w\_final\]), (\[steady\_states\_q\]) are $$\begin{aligned}
N_{1, \ 2}=N_{d} \mp \Delta N_{\rm c}/2,
\label{roots_approx}\end{aligned}$$ where the critical value $\Delta N_{\rm c}$ (\[N\_2\_N\_1\]) is $$\begin{aligned}
\Delta N_{\rm c}=2\sqrt{2} \xi^{-1/2}_{\rm d}N_{d}\Delta_{\rm c}^{1/2}.
\label{delta_N_approx}\end{aligned}$$ Substituting (\[w\_expansion\])–(\[delta\_N\_approx\]) into (\[exponent\]), and calculating the derivatives $w^{\prime}_{-}(N_{i})$ in (\[factor\]): $w^{\prime}_{-}(N_{i}) \approx \pm
q^{\rm (B)}_{\rm determ, \ FS}\xi_{\rm d}\Delta N_{\rm c}/(2N_{\rm d}^{2}), \ i=1,2$, we get (\[time\_FS\_approx\_new\]).
Under the condition that $q_{\rm sum}$ is close to the critical point $q^{\rm (B)}_{\rm determ, \ FS}$, i.e., when $$\Delta_{\rm c}\ll 1,
\label{close_critical}$$ from (\[time\_FS\_approx\_new\]), we get the approximate formula (\[time\_FS\_approx\]). The latter is applicable under the condition (\[close\_critical\]) only if the conditions (\[approximation\_conditions\]) are still satisfied. Using the formula for derivatives $w^{\prime}_{-}(N_{i}), \ i=1,2$ and that $N_{i}, \ i=1,2$ is close to $N_{\rm d}$, we can estimate the values $\xi_{i} \approx \xi_{\rm d}$ and $\eta_{i}\approx \xi_{\rm d}\Delta N_{\rm c} /N_{\rm d}, \ i=1,2$ in (\[approximation\_conditions\]). Thus, the condition (\[approximation\_conditions\]) reads $\xi_{\rm d}\Delta N_{\rm c}^{3} / N^{2}_{\rm d} \gtrsim 1$. Taking into account formula (\[delta\_N\_approx\]) for $\Delta N_{\rm c}$, the condition (\[approximation\_conditions\]) can be written in terms of $\Delta_{\rm c}$: $$N_{\rm d}\Delta_{\rm c}^{3/2}\xi_{\rm d}^{-1/2} \gtrsim 1.$$ This inequality together with (\[close\_critical\]) determine the range of $\Delta_{\rm c}$ in which the approximate formula (\[time\_FS\_approx\]) for the mean time delay $T^{\rm (B, \ mean)}_{\rm FS}$ of an F$\rightarrow $S transition at the bottleneck is valid.
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[^1]: Within the deterministic cluster the speed $v^{\rm (B)}_{\rm free}$ and density $\rho^{\rm (B)}_{\rm free}$ shown in Fig. \[FS\_Intr\] (b–e) are connected by the formula (\[q\_sum\_x\]). The number of vehicles within the cluster $N=\int{\rho dx}$, where integration is performed over the region of cluster localization.
[^2]: Concerning synchronized flow states in the vicinity of the bottleneck associated with a resulting congested pattern, we should note that assumptions used in our nucleation model are satisfied only for a localized SP (LSP) [@KernerBook]. An LSP is an SP whose downstream front is fixed at the bottleneck. The upstream front of the LSP is localized on the main road at some finite distance upstream of the bottleneck, i.e., the width (in the longitudinal direction) of the LSP is always limited. Synchronized flow states at the bottleneck are drawn in Fig. \[FS\_Intr\] (b, c) and in other illustrations only for the case in which the congested pattern is an LSP.
[^3]: For more detail explanation of the approximation (\[analytical\]), note that as in the general model (Fig. \[q\_rho\]), the function (\[analytical\]) is a N-shape flow–vehicle-number characteristic (Fig. \[N\_pot\] (a)). This N-shape is chosen to satisfy those results of a microscopic three-phase traffic theory [@KernerBook] in which a Z-shaped speed–flow characteristic for an F$\rightarrow$S transition has been found (Fig. \[FS\_Intr\] (b, d)). The branch $v^{\rm (B)}_{\rm syn, \ aver}$ on this characteristic (Fig. \[FS\_Intr\] (d)) as well as the associated branch $N^{\rm (syn)}$ on the N-shape flow–vehicle-number characteristic (Fig. \[potential\] (a)) are associated with a synchronized flow state, which results from the F$\rightarrow$S transition. The greater the density, i.e., the vehicle number $N$ within the synchronized flow state, the greater the flow rate $q^{\rm (bottle)}_{\rm down}$ (Fig. \[FS\_Intr\] (c, e)), i.e., the detachment rate $w_{-}(N)$ (\[w\_final\]). In real traffic flow, the growth of $w_{-}(N)$ with $N$ has obviously a limit. This limit is related to spontaneous moving jam emergence in synchronized flow of lower speed and greater density (i.e., greater $N$). In this case, an SP transforms into an GP, which consists of two traffic phases, synchronized flow and wide moving jams [@KernerBook]. However, these effects are beyond the scope of this article. As mentioned above, the simple mathematical approximation (\[analytical\]) of the branch of $w_{-}(N)$ for synchronized flow states that are associated with $N>N_{\rm s}$ can be used, because at $N>N_{\rm s}$ the function $w_{-}(N)$ has no impact on results presented in the article.
|
---
abstract: |
In this paper, the finite volume method is developed to analyze coupled dynamic problems of nonlinear thermoelasticity. The major focus is given to the description of martensitic phase transformations essential in the modelling of shape memory alloys. Computational experiments are carried out to study the thermo-mechanical wave interactions in a shape memory alloy rod, and a patch. Both mechanically and thermally induced phase transformations, as well as hysteresis effects, in a one-dimensional structure are successfully simulated with the developed methodology. In the two-dimensional case, the main focus is given to square-to-rectangular transformations and examples of martensitic combinations under different mechanical loadings are provided.
**Key words**: Shape memory alloys, phase transformations, nonlinear thermo-elasticity, finite volume method.
author:
- |
L. X. Wang$^1$ and Roderick V.N. Melnik$^2$[^1]\
$^1$MCI, Faculty of Science and Engineering,\
University of Southern Denmark,\
Sonderborg, DK-6400, Denmark\
$^2$Mathematical Modelling and Computational Sciences,\
Wilfrid Laurier University, 75 University Ave West,\
Waterloo, ON, Canada N2L 3C5
title: |
**Finite Volume Analysis of Nonlinear Thermo-mechanical Dynamics of\
Shape Memory Alloys**
---
Introduction
============
The existing and potential applications of Shape Memory Alloys (SMA) lead to an increasing interest to the analysis of these materials by means of both experimental and theoretical approaches [@Birman1997]. These materials have unique properties thanks to their unique ability to undergo reversible phase transformations when subjected to appropriate thermal and/or mechanical loadings. Mathematical modelling tools play an important role in studying such transformations and computational experiments, based on mathematical models, can be carried out to predict the response of the material under various loadings, different types of phase transformations, and reorientations. The development of such tools is far from straightforward even in the one-dimensional case where the analysis of the dynamics is quite involved due to a strongly nonlinear pattern of interactions between mechanical and thermal fields (e.g., [@Birman1997; @Matus2004] and references therein). For a number of practical applications a better understanding of the dynamics of SMA structures with dimensions higher than one becomes critical. This makes the investigation more demanding, both theoretically and numerically.
Most results reported so far for the one-dimensional case have been obtained with the Finite Element Method (FEM) [@Bubner1996; @Bubner2000; @Niezgodka1991]. In addition to the challenges pertinent to coupling effects, we have to deal also with strong nonlinearities of the problem at hand. One of the approaches is to employ a FEM using cubic spline basis functions, in which case the nonlinear terms can be smoothed out by one of the available averaging algorithms. As an explicit time integration is typically employed in such situations, this results in a very small time step discretization. Seeking for a more efficient numerical approach, Melnik et al. [@Melnik2000; @Melnik2002] used a differential-algebraic methodology to study the dynamics of martensitic transformations in a SMA rod. An extension of that approach has been recently developed in [@Matus2003; @Matus2004; @Melnik2003] where the authors constructed a fully conservative, second-order finite-difference scheme that allowed them to carry out computations on a minimal stencil. However, a direct generalization of the scheme to a higher dimensional case appeared to be difficult.
In this paper, we approach the same problem from the Finite Volume Method (FVM) point of view. The method is based on the integral form of the governing equations, leading to inherently conservative properties of FVM numerical schemes. The methodology is well suited for treating complicated, coupled multiphysics nonlinear problems [@Berezovski2003; @Demirdzic1994; @Demirdzic1997]. It can be relatively easily generalized to higher dimensional cases. In addition to its wide-spread popularity in CFD, the method has been applied previously to linear elastic and thermoelastic problems [@Berezovski2001; @Demirdzic1994; @Demirdzic1997; @Jasak2000]. There are several recent results on the application of FVM to nonlinear thermo-mechanical problems and nonlinear elastic problems [@Berezovski2003; @Tuzel2004]. In this paper, we develop a FVM specifically in the context of studying martensitic transformations in SMAs and demonstrate its performance in simulating the dynamical behavior of SMA rods and patches.
The paper is organized as follows. The mathematical models for the dynamics of martensitic transformations in 1D and 2D SMA structures are described in Section 2. Key issues of numerical discretization of these models, including the FVM and its computational implementation via the Differential-Algebraic Equations (DAE) approach, are discussed in Section 3. Mechanically and thermally induced transformations and hysteresis effects in SMA rods are analyzed in Section 4. Section 5 is devoted to studying nonlinear thermomechanical behavior and square-to-rectangular transformations in a SMA patch. Finally, conclusions are given in Section 6.
Mathematical Model for SMA Dynamics
===================================
We start our consideration from a mathematical model for the SMA dynamics based on a coupled system of the three fundamental laws, conservation of mass, linear momentum, and energy balance, in a way we described previously in [@Melnik2003; @Matus2004; @Wang2004]. Using these laws, the system that describes coupled thermo-mechanical wave interactions for the first order martensitic phase transformations in a three dimensional SMA structure can be written as follows [@Melnik2000; @Melnik2003; @Pawlow2000] $$\label{eq2-1}
\begin{array}{l} \displaystyle
\rho\frac{\partial^{2}u_{i}}{\partial t^{2}} =
\nabla_{x}\cdot{\mbox{\boldmath$\sigma$}}+f_{i},\quad i,j=1,2,3
\\[10pt] \displaystyle
\rho\frac{\partial e}{\partial t } -
{\mbox{\boldmath$\sigma$}}^{T}:\nabla\mathbf{v}+\nabla\cdot
\mathbf{q}=g ,
\end{array}$$ where $\rho$ is the density of the material, $\textbf{u}=\{
u_{i}\}|_{i=1,2,3}$ is the displacement vector, $\textbf{v}$ is the velocity, ${\mbox{\boldmath$\sigma$}}=\{\sigma_{ij}\}$ is the stress tensor, $\textbf{q}$ is the heat flux, $e$ is the internal energy, $\textbf{f}=(f_{1},f_{2},f_{3})^{T}$ and $g$ are distributed mechanical and thermal loadings, respectively. Let $\phi$ be the free energy function of a thermo-mechanical system described by (\[eq2-1\]), then, the stress and the internal energy function are connected with $\phi$ by the following relationships:
$$\label{eq2} \displaystyle
{\mbox{\boldmath$\sigma$}}= \frac{\partial\phi}{\partial{\mbox{\boldmath$\eta$}}},
\quad{e=\phi-\theta\frac{\partial\phi}{\partial\theta}},$$
where $\theta$ is the temperature, and ${\mbox{\boldmath$\eta$}}$ the Cauchy-Lagrangian strain tensor defined as follows: $$\label{eq2-3} \displaystyle
\eta_{ij}\left(\textrm{\textbf{x}},t\right)=\left(\frac{\partial
u_{i}\left(\textrm{\textbf{x}},t\right)}{\partial x_{j}}+\frac{\partial
u_{j}\left(\textbf{x},t\right)}{\partial x_{i}}\right)/2.$$
In what follows, we employ the Landau-Ginzburg form of the free energy function for both 1D and 2D SMA dynamical models [@Bubner1996; @Falk1980; @Melnik2000]. In the 2D case, we focus our attention on the square-to-rectangular transformations that can be regarded as a 2D analog of the realistic cubic-to-tetragonal and tetragonal-to-orthorhombic transformations [@Ichitsubo2000; @Jacobs2000]. It is known that for this kind of transformations, the free energy function $\phi$ can be constructed by taking advantage of a Landau free energy function $F_{L}$. In particular, following [@Ichitsubo2000; @Jacobs2000; @Lookman2003] (see also references therein), we have: $$\label{eq2-4} \displaystyle
\phi=-c_{v} \theta \ln \theta + \frac{1}{2} a_{1}
e_{1}^{2}+ \frac{1}{2}a_{3}e_{3}^{2}+F_{L},\quad
F_{L}=\frac{1}{2}a_{2}\left(\theta-\theta_{0}\right)e_{2}^{2}-\frac{1}{4}a_{4}
e_{2}^{4}+\frac{1}{6}a_{6}e_{2}^{6},$$ where $c_{v}$ is the specific heat constant, $\theta_0$ is the reference temperature for the martensite transition, $a_{i}$, $i=1,2,3,4,6$ are the material-specific coefficients, and $e_{1}$, $e_{2}$, $e_{3}$ are dilatational, deviatoric, and shear components of strain, respectively. The latter are defined as follows: $$\label{eq2-5} \displaystyle
e_{1}=\left(\eta_{11}+\eta_{22}\right)/\sqrt{2},\quad
e_{2}=\left(\eta_{11}-\eta_{22}\right)/\sqrt{2},\quad
e_{3}=\left(\eta_{12}+\eta_{21}\right)/2.$$ This free energy function $\phi$ is a convex function of the chosen order parameters when the temperature is much higher than $\theta_0$, in which case only austenite is stable. When the temperature is much lower than $\theta_0$, $\phi$ becomes non-convex and has two local minima associated with two martensite variants, which are the only stable variants. If the temperature is around $\theta_0$, the free energy function has totally three local minima, two of which are symmetric and associated with the martensitic phases and the remaining one is associated with the austenitic phase. In this case both martensite and austenite phases could co-exist in the system. By substituting the above free energy function into the conservation laws for momentum and energy, and using Fourier’s heat flux definition $$q=-k\theta_{x}
\label{eq2-6}$$ with $k>0$ being the heat conductivity of the material, the governing equations for 2D SMA patches can be written in the following form: $$\label{eq2-7}
\begin{array}{l} \displaystyle
\rho \frac{\partial^2 u_1}{\partial t^2} = \frac{\sqrt{2}}{2} \frac{\partial}
{\partial x} \left ( a_1 e_1 + a_2 (\theta-\theta_0) e_2 - a_4 e_2^3 + a_6
e_2^5\right )+ \frac{\partial }{\partial y}
\left ( \frac{1}{2} a_3 e_3 \right ) +f_1,
\\[10pt] \displaystyle
\rho \frac{\partial^2 u_2}{\partial t^2} = \frac{\partial }{\partial x}
\left ( \frac{1}{2} a_3 e_3 \right) + \frac{\sqrt{2}}{2} \frac{\partial}
{\partial y} \left ( a_1 e_1 - a_2 (\theta - \theta_0) e_2 + e_4 e_2^3 - a_6
e_2^5\right )+f_2 ,
\\[10pt] \displaystyle
c_v \frac{\partial \theta}{\partial t} = k \left (\frac{\partial^2
\theta}{\partial x^2} + \frac{\partial^2 \theta}{\partial y^2} \right )
+ a_2 \theta e_2 \frac{\partial e_2}{\partial t} + g.
\end{array}$$ As always, we complete system (\[eq2-7\]) by appropriate initial and boundary conditions which are problem specific (see Sections 4 and 5). As discussed before in [@Matus2003; @Wang2004], the 2D model given by Eq.\[eq2-7\] can be reduced to the Falk model in the 1D case $$\label{eq2-8}
\begin{array}{l} \displaystyle
\rho\frac{\partial^{2} u}{\partial t^{2}}=\frac{\partial}{\partial
x}\left(k_{1}\left(\theta-\theta_{1}\right)\frac{\partial u}{\partial
x}-k_{2}(\frac{\partial u}{\partial x})^{3}+k_{3}(\frac{\partial u}{\partial
x})^{5}\right)+F,
\\[10pt] \displaystyle
c_{v}\frac{\partial\theta}{\partial
t}=k\frac{\partial^{2}\theta}{\partial x^{2}}+k_{1}\theta \frac{\partial
u}{\partial x}\frac{\partial v}{\partial t}+G,
\end{array}$$ where $k_{1}$, $k_{2}$, $k_{3}$, $c_{v}$ and $k$ are re-normalized material-specific constants, $\theta_1$ is the reference temperature for 1D martensitic transformations, and $F$ and $G$ are distributed mechanical and thermal loadings.
In the subsequent sections, the above models are applied to the description of the first order martensitic transformations. While such transformations are reasonably well documented for the 1D case, only few results are known for the 2D case. In what follows, we develop a FVM to simulate the dynamics described by the models (\[eq2-7\]) and (\[eq2-8\]) and apply it in both 1D and 2D cases.
Numerical Algorithm
===================
The systems (\[eq2-7\]) and (\[eq2-8\]) are analyzed numerically with the FVM implemented here with the help of the DAE approach. For the 1D case, the FVM method yields the same result as the conservative scheme already discussed in [@Matus2004]. However, the approach developed here is generalized in a straightforward manner to a higher dimensional case and we demonstrate its applicability by a numerical example in the case of two spatial dimensions. First, we note that it is convenient to replace the original model (\[eq2-7\]) by a system of equivalent differential-algebraic equations as it was proposed earlier in [@Melnik2000; @Melnik2002]: $$\label{eq3-1}
\begin{array}{l}
\displaystyle
\frac{\partial e_1}{\partial t} = \frac{\sqrt 2}{2} \left( \frac{\partial
v_1}{\partial x}+\frac{\partial v_2}{\partial y} \right),
\quad
\frac{\partial e_2}{\partial t} = \frac{\sqrt 2}{2} \left( \frac{\partial
v_1}{\partial x} - \frac{\partial v_2}{\partial y} \right),
\\[10pt] \displaystyle
\rho \frac{\partial v_1}{\partial t}= \frac{\partial\sigma_{11}}{\partial
x}+\frac{\partial\sigma_{12}}{\partial y}+f_{x},
\quad
\rho \frac{\partial v_2}{\partial t}=\frac{\partial\sigma_{12}}{\partial
x}+\frac{\partial\sigma_{22}}{\partial y}+f_{y},
\\[10pt] \displaystyle
c_{v}\frac{\partial\theta}{\partial t}=k\left(\frac{\partial^{2}\theta}{\partial
x^{2}}+\frac{\partial^{2}\theta}{\partial y^{2}}\right)+a_2\theta
e_{2}\frac{\partial e_{2}}{\partial t}+g,
\\[10pt]\displaystyle
\sigma_{11}=\frac{\sqrt{2}}{2}(a_1e_{1}+a_2\left(\theta-\theta_{0}\right)e_{2}-
a_4 e_{2}^{3}+ a_6e_{2}^{5}),
\\[10pt] \displaystyle
\sigma_{12}=\sigma_{21}=\frac{1}{2}a_3e_{3},
\\[10pt] \displaystyle
\sigma_{22}=\frac{\sqrt{2}}{2}(a_1 e_{1}-
a_2 \left(\theta-\theta_{0}\right)e_{2} + a_4 e_{2}^{3} - a_6e_{2}^{5}).
\end{array}$$ This system is solved numerically together with the compatibility relation written below in terms of strains: $$\label{eq3-2}
\frac{\partial^{2}e_{1}}{\partial x_{1}^{2}}+\frac{\partial^{2}e_{1}}
{\partial x_{2}^{2}}-\sqrt{8}\frac{\partial^{2}e_{3}}{\partial x_{1}\partial
x_{2}}-\frac{\partial^{2}e_{2}}{\partial
x_{1}^{2}}+\frac{\partial^{2}e_{2}}{\partial x_{1}^{2}}=0 .$$ There are eight variables in total that the problem needs to be solved for in this 2D case and there are eight equations. The equations for strains, velocities and temperature are all differential equations, complemented by stress-strain relationships which are treated as algebraic.
In what follows, we highlight the key elements of our numerical procedure based on the FVM implemented with the help of the DAE approach. First, all equations in the system (\[eq3-1\]) are discretized on a staggered grid represented schematically in Fig 1. Assuming that the entire computational domain is a rectangle with an area of $ L_x \times L_y
$ ${\rm cm}^2$, we define the spatial integer grid points $(x_{i},y_j)$ and the spatial flux points $ (\bar{x}_{i}, \bar y_j )$ as follows: $$\label{eq3-3}
\begin{array}{l} \displaystyle
x_{i}=ih_x,\quad, i=0,1,2,\cdots,M,\qquad\bar{x}_{i}=(i-\frac{1}{2})h_x,\quad,
i=1,2,\cdots,M \\[10pt] \displaystyle
y_{j}=jh_y,\quad, j=0,1,2,\cdots,N,\qquad\bar{y}_{j}=(j-\frac{1}{2})h_y,\quad,
j=1,2,\cdots,N
\end{array}$$ where $M$ and $N$ are the number of grid points such that $M\times h_x=L_x$ and $N\times
h_y=L_y$, respectively. The $(i,j)$th control volume for the velocities is $[\bar{x}_{i},\bar{x}_{i+1} ] \times [\bar{y}_{j},\bar{y}_{j+1} ]$, as sketched by the rectangular tiled mosaic area in Fig.1, including the upper right part overlapped with the hatched area. The variables, defined in this control volume, that will be differentiated are marked by a top bar, for instance $\bar{v}_1(i,j)$. The control volume for the strains $e_1$ and $e_2$, temperature $\theta$, and stresses $\sigma_{11}$, $\sigma_{12}$, and $\sigma_{22}$ is given by $[x_{i},x_{i+1} ] \times [y_{j},y_{j+1} ]$, represented by the rectangular hatched area in Fig. 1. We refer to these variables, defined in this control volume, without a top bar, for instance $e_1(i,j)$ for $e_1$, etc.
By integrating all the differential equations over their own control volumes and assuming that all the unknowns are linear in each single control volume while being continuous and piecewise linear in the entire computational domain, the five partial differential equations are reduced to a system of ordinary differential equations. The remaining three algebraic equations of the original system are discretized directly on the grid. The result is the following system: $$\label{eq3-4}\begin{array}{l}
\displaystyle
\frac{d e_1(i,j)}{dt}=(I_y D_x \bar v_1(i,j) +
I_x D_y \bar v_2(i,j))/ \sqrt{2},
\\[10pt] \displaystyle
\frac{d e_2(i,j)}{dt}=( I_y D_x \bar v_1(i,j) - I_x D_y
\bar v_2(i,j))/ \sqrt{2},
\\[10pt] \displaystyle
\rho \frac{d \bar v_1(i,j)}{d t} =
I_y D_x \sigma_{11}(i-1,j-1) + I_x D_y \sigma_{12}(i-1,j-1) + f_{1},
\\[10pt] \displaystyle
\rho \frac{d \bar v_2(i,j)}{d t} =
I_yD_x \sigma_{12}(i-1,j-1) +I_x D_y \sigma_{22}(i-1,j-1) + f_{2},
\\[10pt] \displaystyle
c_v \frac{ d\theta(i,j)}{dt} = k (\triangle \theta(i,j))
+ \frac{\sqrt{a_2}}{2} \theta(i,j) e_2(i,j) \frac{d e_2}{dt} + g,
\\[10pt] \displaystyle
\sigma_{11}(i,j) = \frac{\sqrt{2}}{2}(a_{1}e_{1}(i,j)+a_{2}(\theta(i,j)-
\theta_{0})e_{2}(i,j)-\frac{a_4}{4}g_{1}(i,j)+\frac{a_6}{6}g_{2}(i,j)) ,
\\[10pt] \displaystyle
\sigma_{12}(i,j)=\sigma_{21}(i,j)= \frac{1}{2}(a_{3}e_{3}(i,j) ),
\\[10pt] \displaystyle
\sigma_{22}(i,j) = \frac{\sqrt{2}}{2}( a_{1}e_{1}(i,j)- a_{2}(\theta(i,j)
-\theta_{0})e_{2}(i,j)+\frac{a_4}{4}g_{1}(i,j)-\frac{a_6}{6}g_{2}(i,j)).
\end{array}$$ where $D_x$ and $D_y$ are the discrete difference operators in the $x$ and $y$ directions, respectively, while $I_x$ and $I_y$ are the discrete interpolation operator in the $x$ and $y$ directions, and $\triangle$ is the discrete Laplace operator. For example, for the simplest case of the first order accurate scheme, the operators $D_x$ and $I_y$ could be written as follows $$D_x \bar v_1(i,j) = ( \bar v_1(i,j+1) - \bar v_1(i,j) ) / h_x , \quad
I_y \bar v_1(i,j) = ( \bar v_1(i,j) + \bar v_1(i+1,j) ) / 2,$$ with similar representations for the second order accurate schemes. Moving to the time discretization procedure, it is convenient to re-write system (\[eq3-4\]) in the following vector-matrix form: $$\begin{aligned}
\label{eq3-5}
{\bf{A}}\frac{{d{\bf{U}}}}{{dt}} + {\bf{H}}\left(
{t,{\bf{X}},{\bf{U}}} \right) = {\bf{0}}\end{aligned}$$ with matrix $A={\rm diag}(a_1, a_2,...,a_N)$ having entries “one” for differential and “zero” for algebraic equations for stress-strain relationships, and vector-function ${\bf
H}$ defined by the right hand side parts of (\[eq3-4\]). This (stiff) system is solved with respect to the vector of unknowns ${\bf U}$ that have $6 \times m_x \times m_y + 2 \times
(m_x+1) \times (m_y+1)$ components by using the second order backward differentiation formula (BDF) [@Hairer1996]: $$\label{eq3-6}
{\rm {\bf A}}\left( {\frac{3}{2}
{\rm {\bf U}}^n - 2{\rm {\bf U}}^{n - 1} +\frac{1}{2}{\rm {\bf U}}^{n - 2}}
\right) + \Delta t{\rm {\bf H}}\left( {t_n,{\rm {\bf X}},{\rm {\bf U}}^n}
\right) = 0$$ where $n$ denotes the current time layer.
This spatio-temporal discretization is applied to the analysis of phase transformations with the following modification. In order to improve convergence properties of the scheme, we employ a relaxation process connecting two consecutive time layers via a relaxation factor $\omega$ as follows: $$\label{eq3-7}
y(i,j) =(1-\omega) \times y(i,j)^n + \omega \times y(i,j) ^ {n+1},$$ where the variable $y$ could be any of the following: $e_1(i,j)$, $e_2(i,j)$, $v_1(i,j)$, $v_2(i,j)$, or $\theta(i,j)$. Note that in the general case the relaxation factors need not be the same for all the variables. In the present paper, all the numerical results have been obtained using (\[eq3-7\]) with all the relaxation factors set to $0.85$.
We note that nonlinear terms in the model are averaged in the Steklov sense [@Matus2004], so that for nonlinear function $f(e_2)$ (in particular, for $e_2^3$ and $e_2^5$), averaged in the interval $[e_2^{n}, e_2^{n+1}]$, we have $$\label{eq3-8}
g\left( e_2^{n}, e_2^{n+1}\right ) =
\frac{1}{e_2^{n+1} - e_2^{n}} \int_{e_2^{n}}^{e_2^{n+1}}
f(e_2)d e_2.$$ Applying this idea to $e_2^3$ and $e_2^5$, we have: $$\label{eq3-9}
\begin{array}{c}
\displaystyle
g_1 \left( i,j \right) =
\frac{\left(e_2^{n+1}\right)^{4}-(e_2^{n})^{4}}{e_2^{n+1}-e_2^n}=
\frac{1}{4}\sum_{k=0}^{3} (e_2^{n+1})^{3-k} (e_2^n)^{k},
\\[10pt] \displaystyle
g_2\left( i,j \right ) =
\frac{(e_2^{n+1})^6-(e_2^{n})^6}{e_2^{n+1}-e_2^n} =
\frac{1}{6} \sum_{k=0}^5 (e_2^{n+1})^{5-k} (e_2^n)^{k}
.\end{array}$$ where $e_2^{n}$ and $e_2^{n+1}$ stands for $e_2(i,j)^n$ and $e_2(i,j)^{n+1}$, respectively.
Finally, we note that in our FVM implementation the nonlinear coupling term in the energy balance equation is regarded as a time-dependent source term. In the $(i,j)$th control volume for the discretization of $\theta$, we approximate that term as follows: $$\label{eq3-10}
\int_{x_{i}}^{x_{i+1}}\int_{y_{j}}^{y_{j+1}}\left(k_{1}\theta e_2
\frac{\partial e_2}{\partial t}\right) dxdy \approx
\theta(i,j) e_2(i,j)\frac{d e_2(i,j)}{d t}.$$ As seen from (\[eq3-6\]), we use an implicit time integrator based on the BDF. At each time step we apply the bi-conjugate gradient method to solve the resultant system of algebraic equations with the Jacobian matrix updated on each iteration.
Dynamics of SMA Rods and Strips
===============================
We first consider a situation where the deformation of a 2D SMA sample in the $x_1$ direction substantially exceeds the deformation in the other direction, so that the deformation in the $x_2$ direction can be neglected and the sample can be treated as a SMA long strip or simply as a rod. Introducing formally ${\varepsilon}= \partial{u}/\partial x$ and $ v=
\partial{u}/\partial t$, system (\[eq2-8\]) can be recast in the following form: $$\label{eq4-1}
\begin{array}{c}
\displaystyle
\frac{{\partial \epsilon }}{{\partial t}} = \frac{{\partial v}}{{\partial x}},
\quad {\rho \frac{{\partial v}}{{\partial t}} = \frac{{\partial
s}}{{\partial x}} + F},
\\[10pt] \displaystyle
s = {k_1 \left( {\theta - \theta _1 } \right)\epsilon - k_2 \epsilon ^3 + k_3
\epsilon ^5 },
\\[10pt] \displaystyle
c_v \frac{{\partial \theta }}{{\partial t}} = k\frac{{\partial ^2 \theta
}}{{\partial x^2 }} + k_1 \theta \epsilon \frac{{\partial v}}{{\partial x}} + G,
\end{array}$$ where $\epsilon$ is strain and $s$ is stress.
The numerical procedure described in Section 3 is applied here to the solution of system (\[eq4-1\]). It is aimed at the analysis of martensitic transformations in the SMA rod, including hysteresis effects during the transformations. Computational experiments reported in this section were performed for a $Au_{23}Cu_{30}Zn_{47}$ rod with a length of $L=1$ ${\rm
cm}$ and all parameter values found in [@Falk1980; @Melnik2001; @Niezgodka1991], in particular: $$\label{eq4-3}
\begin{array}{c}
\displaystyle
k_{1}=480\, g/ms^{2}cmK,\qquad
k_{2}=6\times10^{6}g/ms^{2}cmK,\qquad k_{3}=4.5\times10^{8}g/ms^{2}cmK,
\\[10pt]\displaystyle
\theta_{1}=208K,\quad \rho=11.1g/cm{}^{3},\quad
C_{v}=3.1274g/ms^{2}cmK, \quad k=1.9\times10^{-2}cmg/ms^{3}K.
\nonumber
\end{array}$$
The boundary conditions for $u$ and $\theta$ for all the numerical experiments reported in this section are: $$\label{eq4-4}
\begin{array}{c} \displaystyle
u(0,t)= u_L(t), \quad u(L,t)= u_R(t),qquad \frac{\partial
\theta}{\partial x}(0,t)= \theta_L(t), \quad \frac{\partial
\theta}{\partial x}(L,t)= \theta_R(t)
\end{array}$$ with given functions $u_i(t)$ and $\theta_i(t)$, $i=L, R$ and corresponding conditions for the velocities.
In the numerical experiments reported below, we used only $9$ nodes for the velocity discretization (and 8, excluding boundaries, for the rest of variables). The time stepsize in all the experiments was set to $\tau=1. \times 10^{-4}$. All the simulations were performed for the time period $[0 , 24] $ which spans two periods of the loading cycle.
Mechanically Induced Transformations and Hysteresis
---------------------------------------------------
The first numerical experiment deals with the case of mechanical loading in the low-temperature regime. The initial conditions for this computational experiment are defined by the following configuration of martensites ([@Klein1995; @Melnik2001; @Melnik2003]): $$\label{eq4-13}
\theta(x,0)=220,\qquad u_{0}=\left\{ \begin{array}{l}
0.11869x,\qquad \qquad 0\leq x\leq0.25\\
0.11869(0.5-x), \quad 0.25\leq x\leq0.75\\
0.11869(x-1),\qquad 0.75\leq x\leq1\end{array}\right.,v^{0}=u^{1}=0$$ with the time varying distributed mechanical loading defined as $$\label{eq4-14}
F=7000\sin^{3}\left(\frac{\pi t}{2}\right)\,\, g/\left(ms^{2}cm^{2}\right),\,\,\,\,
G=0.$$
Under the given distributed mechanical loading, the SMA rod is expected to switch between different combinations of the martensite variants, and a hysteresis loop must be observed similar to those reported for ferroelastic materials at low temperature. In Fig. 2 we present simulation results for this case. The mechanical hysteresis is obtained by plotting displacement $u$ as a function of $F$ at $x=3/8$cm (the upper right plot). The time-varying mechanical loading for this case is plotted in the upper left plot. The simulated strain and the displacement distribution are also plotted as functions of time and space (lower plots). The combination of martensitic variants is changing with time-dependent mechanical loading and no stable austenite is observed at this low temperature.
Our next goal is to analyze the behavior of the same SMA rod under a medium temperature where both martensite and austenite phases may co-exist. The following initial conditions will allow us to start from the austenitic phase: $$\label{eq4-15} \theta(x,0)=250,\,\, u_{0}=0,\,\,\, v^{0}=u^{1}=0.$$ The boundary conditions as well as mechanical and thermal loadings in this case are kept identical the previous experiment. In this case, the free energy function has three minima that correspond to two martensites and one austenite. The numerical results for this case are presented in the left column of Fig.3. It is observed that when the applied loading exceeds a certain value, the austenite is transformed to a combination of martensitic variants. The reverse transformation is taken place when the loading changes its sign. In contrast to the results presented in Fig. 2, we observe that the wide hysteresis loop, typical for the low temperature case, disappears.
If we increase the initial temperature further to $\theta(x,0)=300$, the free energy function becomes convex and has only one minimum associated with the austenite phase. During the entire loading cycle, no martensite is expected under these thermal conditions. The dynamics of the SMA rod in this case exhibits nonlinear thermomechanical behaviour without phase transformations. This is confirmed by the numerical results presented in the right column of Fig.3.
Thermally induced Phase Transformations and Hysteresis
------------------------------------------------------
Thermally induced martensitic phase transformations and thermal hysteresis in SMA rods can be analyzed with the same model under time-dependent thermal loading conditions. Indeed, let us choose the initial conditions as follows: $$\theta(x,0)=230,\,\, u_{0}=\left\{ \begin{array}{l}
0.11869x,\qquad \qquad 0\leq x\leq0.5\\
0.11869(1-x),\quad 0\leq x\leq0.5\end{array}\right.,v^{0}=u^{1}=0$$
The boundary conditions remain the same as in the previous computational experiment, but the loadings conditions now become: $$G=600 \sin \left(\frac{\pi t}{6}\right)\,\, g/\left(ms^{3}cm\right),\,\,\, F=500\,
g/\left(ms^{2}cm^{2}\right).$$
Numerical results for this case are presented in Fig.4. Analyzing strain and displacement distributions, we observe that the combination of martensitic variants is transformed into the austenite phase when the temperature exceeds a certain value. The reverse process is taken place when the temperature decreases, passing the critical threshold. Note that due to the presence of thermal hysteresis, the critical temperature value for the martensite-to-austenite transformation is different from that of the austenite-to-martensite transformation. A schematic representation of the observed thermal hysteresis is given in the lower right part of Fig. 4 where we presented the temperature at $x=3/8$ as a function of strain at the same spatial point.
Dynamics of SMA Patches
=======================
The situation becomes more involved for 2D structures. Experimental, let alone numerical, results for this situation are scarce [@Wang2004]. In order to apply the FVM to the 2D model discussed in Section 2, we chose the same material as before, assuming that $a_{2}=k_{1}, a_{3}=k_{2},\, a_{4}=k_{3}, a_{1}=k_{1},\, a_{3}=2k_{1}$ and therefore effectively linking parameters in model (\[eq2-7\]) and model (\[eq2-8\]).
Nonlinear Thermomechanical Behavior
-----------------------------------
The first numerical experiment on a SMA patch is aimed at the analysis of the dynamical thermo-mechanical response of the patch to a varying distributed mechanical loading, too small to induce any phase transformations. The initial temperature of the patch is set to $250^o$ while all other variables are set initially to zero. Conditions at the boundaries are: $$\begin{array}{l} \displaystyle
\frac{\partial \theta}{\partial x}=0, \quad \frac{\partial u_2}{\partial x}=0,
\quad u_1=0, \quad \textrm{on left and right boundaries}, \\[10pt]
\displaystyle
\frac{\partial \theta}{\partial y}=0, \quad \frac{\partial u_1}{\partial
y}=0, \quad u_2=0, \quad \textrm{on top and bottom boundaries}.
\end{array}
\label{eq11}$$ Similarly, the mechanical boundary conditions are enforced in terms of velocity components. The loading conditions in this experiment are: $$f_1 = 200 \sin ( \pi t /6 ) {\textrm{g}}/(\textrm{ms}^{2}\textrm{cm}^{2}), \quad
f_2 = 200 \sin ( \pi t /40 ) {\textrm{g}}/(\textrm{ms}^{2}\textrm{cm}^{2}).$$ The time span for this simulation, $[0,24]$, covers two periods of loading. The time stepsize is set to $1\times 10^{-4}$. We take $15$ nodes used in each direction. The dimensions of the SMA patch are taken as $1 \times 0.4$ ${\rm cm}^2$.
The variations in the displacements $u_1$, $u_2$ , deviatoric strain $e_2$, and the temperature $\theta$ along the line $y=0.2\rm{cm}$ (the central horizontal line) as functions of time are presented in Fig.5. These simulations show that both thermal and mechanical fields are driven periodically by the distributed mechanical loading. Under such a small loading, the SMA patch behaves just like a conventional thermoelastic material. Observed oscillations are due to nonlinear thermomechanical coupling, but no phase transformations are observed in this case.
Phase Transformations in SMA Patches
------------------------------------
Our aim in this section is to analyze spatio-temporal patterns of martensitic transformations in a 2D SMA patch. The SMA patch, used in this computational experiment, is made of the same material as before. The patch is assumed square in shape with dimensions $1 \times 1$ ${\rm
cm}^2$. The initial temperature distribution is set to $\theta^0=240^o $, and all other variables are initially set to zero. The boundary conditions are homogeneous and $$\frac{\partial \theta}{\partial n}=0, \quad u_2 = u_1 =0,
\quad \textrm{on all the four boundaries,}$$ where n is the unit normal vector. We apply the following loading to the sample, specified below for one period: $$\begin{aligned}
f_1 = f_2 = 6000 \left \{
\begin{array}{llll}
\sin (\pi t/3), & 0 \leq t \leq 4, \\
0 ,\qquad & 4 \leq t \leq 6, \\
\sin(\pi (t-2))/3), & 6 \leq t \leq 10, \\
0 ,\qquad & 10 \leq t \leq 12. \\
\end{array}
\right.\end{aligned}$$ The numerical results for this case are presented in Fig.6 where the values of “y” are taken in the middle of the sample. The spatio-temporal plot of the order parameter $e_2$ demonstrates a periodicity pattern in the observed phase transformations due to periodicity of the loading. It is observed also that the temperature oscillates synchronously with the mechanical field variables due to the thermo-mechanical coupling.
As we mentioned earlier, there are two martensitic variants in the square-to-rectangular transformations. The following analysis proves to be useful in validating the results of computational experiments. Assuming the temperature difference $d\theta = \theta-\theta_0$, one can easily calculate the deviatoric strain that corresponds to the austenite and martensite variants by minimizing the Landau free energy functional. In particular from the condition $\partial{F_l}/\partial{e_2}=0$ we get: $$e_2 = 0 ; \quad e_2^2 = \frac{a_4\pm \sqrt{a_4^2
-4a_2d\theta a_6}}{2a_6}.$$ The value $e_2=0$ corresponds to the austenitic phase. If we denote $(a_4 +
\sqrt{a_4^2-4a_2d\theta a_6})/2a_6$ by $e_m$, then $e_{2+} = +\sqrt{e_m}$ or $e_{2-} =
-\sqrt{e_m}$ are the strains that correspond to the two martensite variants. We call them martensite plus and martensite minus, respectively. If we take $d\theta = 42^o $, then for the material considered here we can estimate that $e_{2+} = 0.12$ and $e_{2-} = 0.12$. This provides a fairly good estimate for the 1D case. However, as was pointed out in [@Jacobs2000; @Lookman2003], for the 2D case such an estimate can be adequate only in homogenous cases. Although the quality of this estimate is dependent on the boundary conditions for a specific problem, this estimate proves to be a reasonable initial approximation to the deviatoric strain.
In Fig. 7 we present two snapshots (at $t=2$ and $t=8$) of the spatial distributions of $e_2$ and $\theta$. It is observed that when the mechanical loading achieves its (positive) maximum, the SMA patch is divided into two sub-domains determined by the deviatoric strain, as seen from the $e_2$ plot at $t=2$. In the upper-left triangular-shape area, the simulated deviatoric strain corresponds to the martensite plus, while on the opposite side, the deviatoric strain corresponds to the martensite minus. At $t=8$, when the mechanical loading changes its sign to the opposite, the martensitic transformation is observed again, but now in the reverse direction. The second period of loading confirms these observations.
Conclusion
==========
In this paper, we developed a finite volume methodology for the analysis of nonlinear coupled thermomechanical problems, focusing on the dynamics of SMA rods and patches. Both mechanically and thermally induced phase transformations, as well as hysteresis effects, in one-dimensional structures are successfully simulated. While these results can be obtained with the recently developed conservative difference schemes, their generalization to higher dimensional cases is not trivial. In this paper, we also highlighted the application of the developed FVM to the 2D problems focusing on square-to-rectangular transformations in SMA materials demonstrating practical capabilities of the developed methodology.
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[^1]: Corresponding author: tel: +1-519-884-1970, fax: +1-519-884-9738, email: rmelnik@wlu.ca
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abstract: 'We consider a black hole (BH) density cusp in a nuclear star cluster (NSC) hosting a supermassive back hole (SMBH) at its center. Assuming the stars and BHs inside the SMBH sphere of influence are mass-segregated, we calculate the number of BHs that sink into this region under the influence of dynamical friction. We find that the total number of BHs increases significantly in this region due to this process for lower mass SMBHs by up to a factor of 5, but there is no increase in the vicinity of the highest mass SMBHs. Due to the high BH number density in the NSC, BH-BH binaries form during close approaches due to GW emission. We update the previous estimate of O’Leary et al. for the rate of such GW capture events by estimating the $\langle n^2\rangle/\langle n\rangle^2$ parameter where $n$ is the number density. We find a BH merger rate for this channel to be in the range $\sim0.002-0.04\,\si{Gpc^{-3}yr^{-1}}$. The total merger rate is dominated by the smallest galaxies hosting SMBHs, and the number of heaviest BHs in the NSC. It is also exponentially sensitive to the radial number density profile exponent, reaching $>\SI{100}{Gpc^{-3}yr^{-1}}$ when the BH mass function is $m^{-2.3}$ or shallower and the heaviest BH radial number density is close to $r^{-3}$. Even if the rate is much lower than the range constrained by the current LIGO detections, the GW captures around SMBHs can be distinguished by their high eccentricity in the LIGO band.'
author:
- Alexander Rasskazov and Bence Kocsis
bibliography:
- 'bib.bib'
title: The rate of stellar mass black hole scattering in galactic nuclei
---
Introduction
============
Ten stellar black hole - black hole (BH-BH) detections of binary mergers have been announced to date by Advanced LIGO and Virgo, which implies a merger rate density of $24-112\,\si{yr^{-1}Gpc^{-3}}$ in the Universe for a power law BH mass function prior [@ligo2018; @ligo2018a]. Several astrophysical channels have been proposed to explain these rates including isolated binary evolution in the galactic field [@Belczynski2016] and dynamically formed binaries in globular clusters [@Rodriguez2016; @Fragione2018]. All these events are consistent with being approximately circular, which is expected if the binaries form with a sufficiently large periapsis, since gravitational wave (GW) emission circularizes the orbit as it shrinks [@Peters1964].
However, a few channels are predicted to produce eccentric BH binaries that retain significantly nonzero eccentricity in the Advanced LIGO frequency range ($\gtrsim\SI{10}{Hz}$ for design sensitivity). First, the Kozai-Lidov effect can enhance the BH binary eccentricity in hierarchical triples. The tertiary component can be a star or another stellar mass BH. Such triples can form in the galactic field [@Antonini2017; @Silsbee2017] or in globular clusters as a result of a binary-binary interaction [@Antonini2016]. Alternatively, the tertiary component may be a supermassive BH (SMBH) in a galactic center [@Hoang2018; @Antonini2012; @Hamers2018] Furthermore, “resonant” binary-single scattering interactions may lead to highly eccentric binaries in globular clusters where the tertiary increases the eccentricity of the inner binary during close pericenter passages [@Samsing2018; @Rodriguez2018]. Eccentric BH binary GW sources can also be created from non-hierarchical triples [@Arca-Sedda2018].
The focus of this paper is on another way to produce highly eccentric BH–BH binaries, the so-called “GW captures” in which two single BHs undergo a close encounter and lose a sufficient amount of energy due to GW emission to become bound [@OLeary2009]. For a sufficiently low impact parameter, the newly-formed binary has a sufficiently small semimajor axis and high eccentricity to merge quickly before it is disrupted by an interaction with another star or BH. These events are most frequent in dense stellar clusters, e.g. galactic nuclei and globular clusters. However, the low relative velocity of BHs in globular clusters implies that most GW captures will form binaries on a wide orbit, and the binary eccentricity will typically become low due to GW emission when reaching the LIGO band [@OLeary2009 Figure 6]. In contrast, BHs in galactic nuclei sink to the inner regions close to the supermassive black hole (SMBH) due to dynamical friction, and form a mass segregated steep density cusp, where the velocity dispersion is much higher [@BW77; @FAK06; @HA06; @OLeary2009]. In these environments, GW capture binaries typically form in the LIGO band with high eccentricities [@Gondan2017]. As was shown in @Gondan2018, the aLIGO-adVirgo-KAGRA detector network will be able to measure the merging BH binary’s eccentricity with high accuracy and therefore potentially distinguish the GW capture events from other astrophysical formation channels.
The purpose of this paper is to refine the rate estimate of GW capture events in galactic nuclei, highlight the main sources of uncertainties, and to calculate the distribution of their total masses and mass ratios. Previously, @OLeary2009 has estimated the event rates by solving the Fokker-Planck equations for isotropic multimass BH distributions. The results were quite different for models with a limited mass range of stellar BHs and models with a BH mass function that extends to higher masses (e.g. $m_{\max}=15\msun$ vs. $45\msun$). Importantly, the rates were found to be proportional to a parameter $\xi$ defined as the mean squared number density over the square of the mean number density of galactic nuclei. The value of this parameter was not estimated, its fiducial value was assumed to be $\xi=30$. @Kocsis2012 extended @OLeary2009 using post-Newtonian simulations, and showed with simple analytical estimates that rare galaxies with a high $\xi$ and BH mass fractions may dominate the rates, but did not determine these quantities. @Tsang2013 estimated $\xi$, but did not calculate BH-BH merger rates, but rather focused on NS-NS binaries which do not form highly mass segregated density cusps. @Gondan2017 derived the mass and eccentricity distribution of GW capture binaries, but also did not estimate their total merger rate.
In this paper we fill in the missing pieces in the puzzle to determine the GW capture rate using simple analytical estimates and examine the dependence on various model parameters. We estimate $\xi$ based on the observed scatter of the $M-\sigma$ relation. Further, we calculate the BH number density taking into account the dynamical friction bringing BHs (predominantly heavier ones) into the galactic center [@Miralda2000]. Given the number density, the event rate does not depend on any additional parameters [@OLeary2009; @Gondan2017]. We consider various assumptions for the initial BH mass function and,[ based on the heaviest BH detected by LIGO [@ligo2018], assume it extends up to $50\msun$.]{} We also briefly consider the effect of the steepness of the BH density cusp [@Keshet2009].
The paper is organized as follows. In Section \[section:number-of-bhs\], we calculate the number of BHs around the SMBH using the results of previous papers about their mass-segregated density profile. Then in Section \[section:mergerrate\] we utilize this result to calculate the rate of GW captures in a galactic nucleus. Finally, in Section \[section:total-merger-rate\] we integrate over all galaxies and calculate the event rate per unit volume. In Section \[section:conclusions\] we summarize our conclusions and briefly discuss other eccentric BH merger mechanisms. [Several details about the calculations are given in the appendix.]{}
Number of BHs in a galactic center {#section:number-of-bhs}
==================================
In this section we calculate the increase in the number of stellar-mass BHs within the SMBH radius of influence $r_0$ due to the sinking of BHs from larger radii caused by dynamical friction.
Initial conditions
------------------
First, we assume that all stars and BHs formed in the galactic center early in the galactic lifetime ($T=\SI{12}{Gyr}$ ago) and that every star heavier than a certain mass $\mcr$ produced a BH remnant. We will also consider the case of continuous star formation in the next subsection. The initial stellar mass function is taken from [@KroupaIMF]: \[eq:KroupaIMF\] f\_ (m) = C
25()\^[-0.3]{}m<0.08,\
2()\^[-1.3]{}0.080.5,
where $C$ is a normalization parameter. The total stellar mass is then \[eq:mstellar\] M\_= \_0\^m f\_ (m) = 5.58M\_\^2 C and the total (initial) number of BHs is $$\begin{aligned}
\ninit &= \int_{\mcr}^\infty f_\mathrm{IMF} (m) \dd{m} =
k M_\ast.
\label{eq:ninit}\end{aligned}$$ where \[eq:k\] k = M\_\^[-1]{} ()\^[-1.3]{} Eq. (\[eq:ninit\]) allows us to calculate the initial number of BHs in any region where we know the total stellar mass; for example, inside the influence radius $r_0$ defined where $M_\ast=2\mSMBH$ \[eq:ninitr0\] (r\_0) &=& 2k. Thus, the initial BH mass fraction is \_ = = k m\_[BH]{}where $\langle m_{\rm BH}\rangle$ is the average BH mass. Given a power-law BH mass distribution $$\label{eq:fBH}
\fbhi\propto m_\mathrm{BH}^{-\beta}\,,~~\mmin<m_\mathrm{BH}<\mmax$$ the average BH mass is \[eq:<mbh>\] m\_[BH]{}= = . For example, $\mmin=5\msun$ and $\mmax=40\msun$ give $\kappa_\mathrm{init} \approx 0.03$ for $2<\beta<3$.
The mass distribution of BHs born inside $r_0$ is $$\begin{aligned}
\dv{\ninit}{m} &= \fbhi k M_\ast(r_0)
\nonumber\\&= \fbhi k \int_0^{r_0} \rho_\ast(r)\,4\pi r^2\dd{r},\end{aligned}$$ where $\rho_\ast(r)$ is stellar density: \[eq:rhoinside\] (r) =
\_0 ()\^[-\_1]{}rr\_0,\
\_0 ()\^[-\_2]{}r>r\_0,
where $$\rho_{0} = \frac{(3-\gamma_1)}{4\pi}\frac{2\mSMBH}{ r_0^3}.\label{eq:rho0}$$ This gives = k\_0r\_0\^3 As we consider the Milky Way (MW) NSC to be relaxed [@BW77], we assume $\gamma_1=1.5$, which is consistent with observed deep star counts[^1] [@GallegoCano2018]. For the density profile outside $r_0$, we assume $\gamma_2=3.2$ which is consistent with both star counts and diffuse light measurements [@GallegoCano2018; @Schodel2018].
The effect of dynamical friction on the BH number density
---------------------------------------------------------
The black hole mass function in the NSC is affected by dynamical friction, which delivers BHs into this region. The total number of BHs with a given mass within $r_0$ at present is defined by the maximum radius $\rdf$ from where a BH sinks to within $r_0$ in a Hubble time: \[eq:ntot\] &=& \_0\^[(m)]{} n(r)4r\^2. where $n(r)$ is the BH number density. Here $\rdf(m)$ can be defined as the initial orbital radius of a BH given the final radius $r_0$ and BH mass $m$. The evolution of a BH orbital radius can be approximated as [@BinneyTremaine] \[eq:drdt\] - = -r \_0\^4u\^2F(u), where (M\_/m) 13, \[eq:lnLambda\]$\mSMBH$ is the central SMBH mass, $\upsilon$ is the BH velocity and $F(u)$ is the velocity distribution of ambient stars. For a Maxwellian velocity distribution the value of the integral is 0.54 for $v^2 = \langle u^2\rangle$. Assuming a circular BH orbit, $$\begin{aligned}
v = \sqrt\frac{GM(r)}{r} \end{aligned}$$ where $M(r)$ is the total mass inside of radius $r$: $$\begin{aligned}
M(r) &= 3\mSMBH + \int_{r_0}^{r} \rho_0 \qty(\frac{r}{r_0})^{-\gamma_2} 4\pi r^2 \dd{r} \nonumber\\
&= \mSMBH{}\qty[3+2\frac{3-\gamma_1}{\gamma_2-3}\qty(1-\qty(\frac{r}{r_0})^{3-\gamma_2})].\end{aligned}$$ As a result, the dependence of dynamical friction (DF) timescale on radius is the following: $$\begin{aligned}
\tdf &= \tdfo \, x^{\gamma_2-3/2} \qty[1+\frac{2}{3}\frac{3-\gamma_1}{\gamma_2-3}(1-x^{3-\gamma_2})]^{3/2},\\
x &\equiv \frac{r}{r_0},\\
\tdfo &\equiv \frac{(3r_0)^{3/2}M_\mathrm{SMBH}^{1/2}}{1.08(3-\gamma_1)\ln\Lambda \, G^{1/2}m}\nonumber\\
&= \SI{3.8}{Gyr}\, \frac{1.5}{3-\gamma_1} \qty(\frac{m}{10\,\msun})^{-1} \qty(\frac{r_0}{\SI{3}{pc}})^{3/2} \nonumber\\
&\times \qty(\frac{\mSMBH}{\num{4e6}\msun})^{1/2}. \label{eq:tdf0}\end{aligned}$$
The equation of motion (\[eq:drdt\]) then becomes \[eq:dxdtau\] $$\begin{aligned}
\dv{x}{\tau} &= -x \qty(\frac{\tdf}{\tdfo})^{-1} \nonumber\\
&= -x^{5/2-\gamma_2} \qty[1+\frac{2}{3}(3-\gamma_1)\frac{1-x^{3-\gamma_2}}{\gamma_2-3}]^{-3/2},\\
\tau &\equiv \frac{t}{\tdfo}.\end{aligned}$$
[The radius $\rdf(m)$ from which objects of mass $m$ sink to within the radius of influence $r_0$ within time $T$ satisfies]{} \[eq:boundary\_conditions\] $$\begin{aligned}
x\qty(\tau=\frac{T}{\tdfo}) &= 1,\\
x(\tau=0) &= \frac{\rdf}{r_0}.\end{aligned}$$
The numerical solution of Eq. with boundary conditions for $\gamma_1=1.5$, $\gamma_2=3.2$ can be approximated (with $3\%$ accuracy for $5\msun<m<40\msun$) as $$\begin{aligned}
\label{eq:rdf-outside}
\rdf &= r_0 \qty(1+k_1\qty(\frac{T}{\tdfo})^{k_2})^{k_3},$$ where $k_1=1.016$, $k_2=0.740$, $k_3=0.654$. [Note that $\rdf$ depends on $m$ through $\tdfo$, as shown on Fig. \[fig:rdf\]. E.g in a MW-like galaxy, for $m=40\,\msun$ $\rdf\approx\SI{11}{pc}$ which is about the distance where the NSC starts dominating over the galactic background in the MW [if we take the MW stellar density from e.g. @Gnedin2014]. ]{}
After we substitute Eq. (\[eq:rdf-outside\]) into Eq. (\[eq:ntot\]), we find that the BH mass function within distance $r_0$ after a Hubble time is $$\begin{aligned}
\dv{\ntot}{m} &=
4\pi k\rho_0 r_0^3\fbhi \nonumber\\
&\times \left\{\frac{1}{3-\gamma_1} +
\frac{1}{\gamma_2-3}\qty[\qty(1-\frac{\rdf(m)}{r_0})^{3-\gamma_2}]\right\}.\end{aligned}$$ As $\fbhi$ and $k$ (Eq. \[eq:k\]) are highly uncertain, it is useful to calculate the relative increase in the number of BHs due to DF: [$$\begin{aligned}
\label{eq:rdf}
\XI &\equiv \frac{\dv*{\ntot}{m}}{\dv*{\ninit}{m}} =
1 + \frac{3-\gamma_1}{\gamma_2-3}\qty(1-\qty(\frac{\rdf(m)}{r_0})^{3-\gamma_2}) \nonumber\\
&= 1 + \frac{3-\gamma_1}{\gamma_2-3}
\qty[1-\qty{1+k_1\qty(\frac{T}{\tdfo})^{k_2}}^{(3-\gamma_2)k_3}]\end{aligned}$$]{}
So far we have assumed that all of the BHs in the Galactic center were born $T=\SI{12}{Gyr}$ ago. Additionally, we also consider the possibility in which the BHs are produced continuously in time with a constant rate. The true BH formation history may be expected to lie between these two extreme cases, and the corresponding estimates for the number of BHs or their merger rate may represent upper and lower limits, respectively. The corresponding expressions for $\XI$ are derived in Appendix \[appendix:xi-continuous\].
Figure \[fig:xi\] shows that the value of $\XI$ ranges between 1.5 and 3 in MW-like galaxies depending on the BH mass and the assumed BH formation history. However, as shown in the next section, $\XI$ is higher or lower in more or less massive galaxies, respectively.
Merger rate per galaxy {#section:mergerrate}
======================
The rate of GW captures in a single NSC, $\Gamma$, may be calculated by adding up the contribution of different radial shells around the SMBH [within its radius of influence]{}, accounting for the local flux of objects and the cross section to form binaries by GW emission for given BH masses. To do that, we adopt the formula from @Gondan2017 [Eq. 125]: $$\begin{aligned}
\label{eq:gamma0mamb}
\frac{\partial^2\Gamma_0}{\partial m_A \partial m_B} \approx& \frac{G^{17/14}}{c^{10/7}}
\frac{N_{\rm BH}^2 c_\eta^{2/7}}{\mSMBH^{11/14}\,r_0 ^{31/14}}
\nonumber \\ & \times
\frac{ 9 m_\mathrm{max}^2 - 6 p_0 m_\mathrm{max} m_\mathrm{tot}
+ 4 p_0^2 \mu m_\mathrm{tot} }{16\, m_\mathrm{max}^2 }
\nonumber \\ & \times
\frac{ m_\mathrm{tot}^{2-\beta} \mu^{-\beta} (1-\beta)^2 }{ \qty(m_\mathrm{max}
^{1-\beta} - m_\mathrm{min}^{1-\beta} )^2}
\nonumber \\ & \times
\frac{ 1 - (r_\mathrm{min}/r_0)^{ 11/14 - p_0 m_\mathrm{tot} / m_\mathrm{max}
} }{\frac{11}{14} - p_0 \frac{ m_\mathrm{tot} }{ m_\mathrm{max} } }.\end{aligned}$$
Here $m_{A,B}$ are the BH masses, $\rmin$ and $r_0$ are the minimum and maximum radii of the BH density distribution, $N_{\rm BH}$ is the total number of BHs within $\rmin<r<r_0$, $\beta$ is the BH mass function power-law (as in Eq. \[eq:fBH\]) and $m_{\rm min,\,max}$ are the minimum and maximum BH masses (in this paper we assume $\mmin=5\msun$, $\mmax=40\msun$). Also, $$\mtot \equiv m_A+m_B,\quad
\mu \equiv \frac{m_Am_B}{m_A+m_B},\quad
\eta \equiv \frac{\mu}{\mtot}$$ are the total mass, reduced mass, and symmetric mass ratio, respectively, and $c_\eta \equiv (340\pi/3)\eta$.
[ For illustrative purposes, we also derive the formulae for the rates of mergers between the smallest and the heaviest BHs as [$$\begin{aligned}
\Gamma_{\min} &= \left.\frac{\partial^2 \Gamma_0}{\partial\ln m_A\, \partial \ln m_B}\right|_{m_A=m_B=m_{\min}} \nonumber\\
&= \frac{\partial^2\Gamma_0}{\partial m_A \partial m_B} m_{\min}^2 \nonumber\\
&\approx C \cdot \frac{63}{22} \qty[\frac{\beta-1}{1-\qty(\frac{\mmin}{\mmax})^{\beta-1}}]^2 m_{\min}^2,\\
\Gamma_{\max} &= \left.\frac{\partial^2 \Gamma_0}{\partial\ln m_A\, \partial \ln m_B}\right|_{m_A=m_B=m_{\max}} \nonumber\\
&\approx C \qty[\frac{\beta-1}{1-\qty(\frac{\mmin}{\mmax})^{1-\beta}}]^2 \frac{14}{3}\qty(\frac{\rmin}{r_0})^{-3/14} m_{\max}^2,\\
C &\equiv \frac{G^{17/14}N_{\rm BH}^2 c_\eta^{2/7}}{c^{10/7}\mSMBH^{11/14}\,r_0 ^{31/14}}.\end{aligned}$$]{} Here we assume $p_0=0.5$ and $\mmin\ll\mmax$. As we can see from those expressions, the majority of mergers happen mergers between the smallest BHs and the heaviest BHs when $\beta\gtrsim2$ and $\beta\lesssim2$, respectively (which is later shown more rigorously on Fig. \[fig:mergerrate\]): [$$\begin{aligned}
\frac{\Gamma_{\min}}{\Gamma_{\max}} \approx \frac{27}{44} \qty(\frac{\mmin}{\mmax})^{4-2\beta} \qty(\frac{\rmin}{r_0})^{3/14} .\end{aligned}$$]{} ]{}
Eq. (\[eq:gamma0mamb\]) assumes a steady state mass-segregated radial 3D number density profile derived by @OLeary2009 using the Fokker-Planck equation: \[eq:n(m,r)\] n(m,r)r\^[--p\_0]{}, where $p_0\approx 0.5$ [@OLeary2009], i.e. the radial power law index varies from $-1.5$ for the lightest BHs to $-2$ for the heaviest ones. [However, we also examine different assumptions in Section \[section:total-merger-rate\].]{}
[l c c c]{} Reference & [@OLeary2009] & [@Kocsis2012] & [@Gondan2017]\
\[0.5ex\] $\rmin$ &
--------------------------------------
$\tgw(\rmin) = t_H$
\[1ex\] $\rmin \propto \mSMBH^{1/2}$
--------------------------------------
&
---------------------------------------
$\nbh(\rmin) = 1$
\[1ex\] $\rmin \propto \mSMBH^{-1/2}$
---------------------------------------
&
----------------------------------------
$\tgw(\rmin) = t_\mathrm{rel}(\rmin)$
\[1ex\] $\rmin \propto \mSMBH^{13/16}$
----------------------------------------
\
\[3ex\] $\Gamma$ & $\propto \mSMBH^{3/28}$ & $\propto \mSMBH^{9/28}$ &
-------------------------------------------------------------------------------------
$\propto \mSMBH^{3/28},\quad m_\mathrm{tot} < \frac{11}{7} m_\mathrm{max}$
\[1ex\] $\propto \mSMBH^{9/224},\quad m_\mathrm{tot} > \frac{11}{7} m_\mathrm{max}$
-------------------------------------------------------------------------------------
\
\[3ex\]\
\[table:comparison\]
As shown in [@Gondan2017], for any $\mSMBH\lesssim10^7\msun$ the relaxation time inside $r_0$ is shorter than the Hubble time. Given that most of the merger events come from the low-mass galaxies [@OLeary2009], this justifies [our assumption that the SBH sphere of influence is fully relaxed.]{} And since the BH density outside $r_0$ falls down quickly ($\propto r^{-3.2}$), we assume we can ignore the contribution of those BHs to the total merger rate.
Following [@Gondan2017], we define $\rmin$ as the radius where the GW inspiral time becomes shorter than the relaxation time: t\_ &=& 0.34\
&=& t\_ = ,\
&=& \^[1/3.62]{}\
&=& ()\^[0.69]{} ()\^[0.31]{}\
&& ()\^[0.28]{} ()\^[-0.28]{} ()\^[0.36]{} \[eq:rgw\] (see Appendix \[appendix:rmin\] for the derivation). This value is in a good agreement with @Gondan2017 [Figure 11]. However, previous papers [@OLeary2009; @Kocsis2012] have assumed different definitions of $\rmin$, as summarized in Table \[table:comparison\]. [^2] In [@Kocsis2012] $\rmin$ is the radius with only one BH inside it (as determined by $n(r)$). However, even the region containing $<1$ BH [*on average*]{} can still make a non-negligible contribution to the total merger rate due to its very high average BH density and orbital velocity. We extrapolate the number density equation into this region given that $t_{\rm rel}\leq t_{\rm GW}$, but warn the reader that the assumptions used to derive that equation (phase space distribution function is smooth and correlations are negligible) break there. And in @OLeary2009 $\rmin$ is the radius where $t_{\rm GW}=t_{\rm H}$, which is a more conservative assumption than ours given that the relaxation time is smaller than $t_{\rm H}$. In any case, the dependence of the merger rate on $\rmin$ is rather weak ($\Gamma\propto\rmin^{-3/14}$).
The effect of dynamical friction on the merger rate
---------------------------------------------------
As the merger rate defined by Eq. is proportional to the total numbers of BHs with masses $m_A$ and $m_B$, to account for the effects of DF we only have to multiply it by the corresponding BH number increase coefficients:
$$\begin{aligned}
\frac{\partial^2\Gamma}{\partial m_A \partial m_B} = \XI(m_A)\XI(m_B) \frac{\partial^2\Gamma_0}{\partial m_A \partial m_B} \label{eq:gammagamma0}\end{aligned}$$
We present the two-dimensional (2D) mass distributions of the GW capture rate as a function of total BH mass and mass ratio following @Gondan2017 and also calculate the marginalized 1D total mass distribution as discussed in Appendix \[appendix:ddGamma\].
To make a prediction for the total observed merger rate, we add up the local merger rates $\Gamma$ for every galaxy within the observable volume. For that purpose, it is useful to express $r_0$ in terms of the central supermassive BH mass using the $M-\sigma$ relation [@KormendyHo]: $$\begin{aligned}
\label{eq:r_0}
&\mSMBH = M_0 \qty(\frac{\sigma}{\sigma_0})^{\alpha_0},\\
&M_0 = \num{3.097e8} \msun \qc \sigma=\SI{200}{km/s} \qc \alpha_0 = 4.384,\\
&r_0 = \frac{G\mSMBH}{\sigma^2} = \SI{3.14}{pc} \,\qty(\frac{\mSMBH}{\num{4e6}\msun})^{0.543}.\end{aligned}$$ The value of $r_0$ in this formula for a MW mass galaxy matches the measured value ($\approx\SI{3}{pc}$). [As for $\gamma_{1,2}$, ]{}we assume they have MW values $\gamma_1=1.5$, $\gamma_2=3.2$ for all galaxies. Under these assumptions, Eq. (\[eq:tdf0\]) takes the form [$$\begin{aligned}
\label{eq:tdf0-mSMBH}
t_\mathrm{DF,0} = \SI{4.1}{Gyr}\, \frac{1.5}{3-\gamma_1} \qty(\frac{m}{10\,\msun})^{-1} \qty(\frac{\mSMBH}{\num{4e6}\msun})^{1.31}\end{aligned}$$]{} which is to be used in Eqs. – instead of Eq. . This shows that the DF time in more massive galaxies is longer, implying that $\XI$, the increase in BH number due to DF, is less, as shown in Figure \[fig:gg0\] (top left).
In analogy with $\XI$, we calculate the relative increase in merger rate due to [the DF]{} effects, which is shown in Figure \[fig:gg0\] for two different values of $\mSMBH$. In accord with Eq. (\[eq:tdf0-mSMBH\]), we can see that the enhancement is stronger for heavier BHs and lower-mass galaxies. As also shown in @OLeary2009 [@Kocsis2012; @Gondan2017], the dependence of merger rate per galaxy on the SMBH mass (without explicitly taking into account DF effects) is very weak – which implies that the total merger rate is dominated by numerous low-mass galaxies. And our results show that in these galaxies the merger rates per galaxy are up to 20 times higher due to dynamical friction. This increases the overall contribution of low-mass galaxies even further.
Total merger rate {#section:total-merger-rate}
=================
The total merger rate per unit volume may be calculated from the average merger rate per galaxy with a given SMBH mass and the distribution of the number density of galaxies with respect to the SMBH mass: &=& \_\^\
&& . Previously we assumed that all galaxies follow the $M-\sigma$ relation exactly. In practice there may be significant variations in the model parameters between galaxies so that
1. \[i:mSMBH\] = C\_[M]{} M\_0 (/\_0)\^[\_0]{}, C\_[M]{}= 1,
2. the $r_0$ containing the stellar mass $M_\ast=2\mSMBH$ satisfies \[i:r\_0\] r\_0 = ,
3. \[i:fBH\] the parameters of BH distribution ($m_\mathrm{min,max}$, $\beta$ and $\mcr$) as well as $\gamma_{1,2}$ may also vary over different galaxies, which also affect the total number of black holes in the NSC, $N_{\rm BH}$.
Since $\left\{C_{M\sigma},\cinf,m_\mathrm{min},m_\mathrm{max}, \beta,\mcr,\gamma_1,\gamma_2\right\}$ may vary from galaxy to galaxy, this variance can significantly change (usually increase) the average merger rate compared to its value calculated using the average parameter values: \[eq:zeta\] &&(C\_[M]{},,…)\
&=& \_[M]{}\_ \_ (C\_[M]{},,…), where $\xi_{M\sigma}$, $\xi_\mathrm{inf}$ and $\xi_\mathrm{other}$ are the enhancement coefficients due to the variance in $C_{M\sigma}$, $C_\mathrm{inf}$ and all the other factors, respectively: \_x . Here we have assumed there are no correlations between different galaxy parameters and that the dependence of $\Gamma$ on them is separable: = f\_1(C\_[M]{}) f\_2() …
To obtain the parameter dependencies we eliminate $\sigma$ from the definition of $r_0$ in using , and substitute the result in Eqs. and to obtain the scaling of $r_{\min}$ and $\tdf$ with $C_{M\sigma}$ and $C_{\rm inf}$: \[eq:cmsigma\] $$\begin{aligned}
r_0 &= C_{\rm inf} G\mSMBH \qty(\frac{\mSMBH}{C_{M\sigma}M_0})^{-2/\alpha_0}\\
\rmin &\propto r_0^{0.31} \propto C_\mathrm{inf}^{0.31} C_{M\sigma}^{0.14},
\\
\tdfo &\propto r_0^{3/2} \propto C_{\rm inf}^{3/2} C_{M\sigma}^{0.68}.\end{aligned}$$ According to [@KormendyHo], the intrinsic scatter of $M-\sigma$ relation is 0.29 dex. This implies $1.3\lesssim \xi_{M\sigma} \lesssim 1.5 $ depending on $m$ and $\mSMBH$ (Appexdix \[appendix:msigma\]). As for $\cinf$, given the stellar density profile, it only depends on the velocity anisotropy [for relaxed NSCs]{} [@BinneyTremaine Section 4.8.1]. We assume all galactic nuclei to be isotropic, which gives $\cinf=1$ and $\xi_{\rm inf}=1$.
@Tsang2013 made an estimate $\cinf=6.1$ based on the observed relation between $\rho_0$ and $\sigma$ and its scatter [@Merritt2007]. However, that is likely an upper limit to $\cinf$ as they have ignored the possible observational errors in both $\rho_0$ and $\sigma$ and also overestimated the spread in $\rho_0$ at fixed $\sigma$. Using the same plot from @Merritt2007, @OLeary2009 arrived at the rough estimate of $\xi=30$; however, they only accounted for the variance in $\rho_0$ and ignored the variance in $r_0$ which is in fact related to $\rho_0$ at a given $\mSMBH$ (Eq. \[eq:rho0\]).
The mass distribution of SMBHs is taken from [@Shankar2004][^3]: \[eq:dndM\] &=&\
&=& ()\^[+1]{}\
&& ,\
\_&=&\
M\_&=&\
&=&-1.11\
&=&0.49. Instead of merger rate per unit $m_A$ and $m_B$, we calculate a more illustrative merger rate per unit log total mass and mass ratio: $$\begin{aligned}
&\frac{\partial^2\mathcal{R}}{\partial\log{\mtot} \partial q} =
\frac{m_{\rm tot}^2}{(1+q)^2} \frac{\partial^2\mathcal{R}}{\partial m_A \partial m_B}\\
&= \SI{8.4e-16}{yr^{-1}}\, \frac{\xi}{1.4} \qty(\frac{\mtot/\msun}{1+q})^2
\nonumber\\&\times
\qty(9-6p_0\frac{\mtot}{\mmax}+4p_0^2\frac{\mu\mtot}{m_\mathrm{max}^2})\\
&\times \frac{ m_\mathrm{tot}^{2-\beta} \mu^{-\beta} (1-\beta)^2 }{ \qty(m_\mathrm{max}
^{1-\beta} - m_\mathrm{min}^{1-\beta} )^2}
\frac{c_\eta^{2/7}}{\frac{11}{14} - p_0 \frac{ m_\mathrm{tot} }{ m_\mathrm{max} }}
\qty(\frac{\mcr}{20\msun})^{-2.6} \\
&\times \int_\mSMBHmin^\mSMBHmax
\XI\qty(\frac{\mtot}{1+q},\mSMBH) \XI\qty(\frac{q\mtot}{1+q},\mSMBH)
\nonumber\\&\times
\qty(\frac{\mSMBH}{\num{4e6}\msun})^{3/28}\\
&\times \qty[1-\qty(\frac{\rmin}{r_0})^{\frac{11}{14} - p_0 \frac{ m_\mathrm{tot} }{ m_\mathrm{max} }}] \dv{n_\mathrm{gal}}{\mSMBH} \dd{\mSMBH},\\
\frac{\rmin}{r_0} &= \num{2.6e-5} \qty(\frac{\mSMBH}{\num{4e6}\msun})^{0.35}
\qty(\frac{\mtot/(1+q)}{20\msun})^{0.28},\\
\xi &\equiv \xi_{M\sigma}\xi_\mathrm{inf} \xi_\mathrm{other}.\end{aligned}$$
The results of the calculation are shown in Figure \[fig:mergerrate\]. Compared to Figure 9 of [@Gondan2017], the distribution is shifted towards higher BH masses due to DF. The SMBH mass function is almost log-uniform at low SMBH masses, and low-mass galaxies actually contribute more events due to enhanced DF (see Figure \[fig:gg0\], top left). Therefore, the total merger rate depends on how far into the low SMBH masses does the distribution extend, which is illustrated in Figure \[fig:totalmergerrate\] (left). This figure shows the total merger rate integrated over all BH masses and mass ratios: [$$\begin{aligned}
\mathcal{R} = \int_{2\mmin}^{2\mmax}\dd{\mtot} \int_{\mmin/\mmax}^1\dd{q}
\frac{\partial^2\mathcal{R}}{\partial\log{\mtot} \partial q}\end{aligned}$$]{} Figure \[fig:totalmergerrate\] (left panel) also shows that the merger rate is a decreasing function of $\beta$ (i.e. an increasing function of the average BH mass). Given $\xi=3$, $p_0=0.5$, $\mcr=20\msun$, and ranges $M_\mathrm{SMBH,min}=10^4-10^5\msun$ and $\beta=1-3$, the overall merger rate is $0.002-\SI{0.04}{Gpc^{-3}yr^{-1}}$.
Based on fits to the isotropic Fokker Planck results of @OLeary2009, which imply $p_0\approx0.5-0.6$, we have assumed $p_0=0.5$, i.e. a power-law BH radial density distribution given by Eq. which has a slope 1.5 for the lightest BHs and 2 for the heaviest ones. However, using different mass functions in isotropic Fokker-Planck models, @Keshet2009 [Figure 3] have shown that for $\beta\gtrsim4$ and $\mmax/\mmin\gtrsim10$ the heaviest BH density profile can be steeper, up to $r^{-3}$, which would correspond to $p_0=1.5$ in Eqs. and . There is indeed some observational evidence that the surface density distribution of massive O-stars (i.e. BH progenitors) in the Galactic center is $\propto R^{-1.4}$ [@Bartko2009], which implies 3D density $\propto r^{-2.4}$, i.e. $p_0\sim0.9$ [see, however, @Stostad2015 who claim that the young star density distribution is better described by broken power-law with the inner slope $\propto R^{-0.9}$]. Another possible reason for the density cusp to be steeper than $r^{-2}$ is binary disruption by the SMBH’s tidal field [@FragioneSari2018]. Figure \[fig:totalmergerrate\] (right) shows that such an increase in $p_0$ could increase the merger rate by orders of magnitude and therefore deserves further study.
Up to this point, we have assumed the BH density distribution to be spherical. However, the MW NSC is observed to be flattened with mean axis ratio 0.7-0.8 [@Schodel2014; @Fritz2016]. @Feldmeier+17 found it to be decreasing towards the center within $r<\SI{1}{pc}$, reaching 0.4 when $r\rightarrow0$. According to their orbit-based modelling, that corresponds to a triaxial density profile with axial ratios $c/a=0.28$, $b/a=0.64$ in the center. The triaxiality leads to increase in the average BH density $n_{\rm BH}$ within $r_0$ and, consequently, to an increase in the event rate $\mathcal{R}\propto n_{\rm BH}^2$. A crude upper limit estimate for this increase in $\mathcal{R}$ may be obtained as $\qty(a^2/bc)^2\approx30$. In addition, from a theoretical point of view, vector resonant relaxation in a multimass system causes the heaviest objects (BHs) to segregate from an initially spherical stellar distribution into a disk [@Szolgyen+18]. The final distribution of heavy BH angular momenta in Fig. 2 of @Szolgyen+18 implies the $\mathcal{R}$ increase by a factor of $\approx2.8$. How effective that phenomenon is in non-spherical systems is currently unknown and deserves further study.
In addition to the merger rate, we also calculate the universal dimensionless parameter \[eq:alpha\] = -\^2 which is independent of the BH mass function and is sensitive to the astrophysical process leading to the BH merger [@Kocsis2018]. Its value varies from 1.4 for the smallest BHs to $-6.3$ for the heaviest ones which is in good agreement with @Gondan2017; its dependence on $\mtot$ is shown in Figure \[fig:alpha\] (top). It turns out to be practically independent of $q$. This is different from, e.g., $\alpha=1$ for primordial BH binaries formed in the early universe [@Kocsis2018] or $\alpha=1.43$ for BHs in dark matter halos [@Bird2016].
Conclusions {#section:conclusions}
===========
We have calculated the number density of stellar-mass BHs in a galactic nucleus with a SMBH in the center using simplified isotropic models, where we assumed the BHs and stars reach a mass-segregated steady-state distribution inside the influence radius and took into account the dynamical friction bringing BHs inside that region. We have shown that dynamical friction can increase the BH number up to $\sim5$ times and also that its effect is much more pronounced in small galaxies.
We used this information to calculate the rate of GW captures in galactic nuclei taking into account the observed SMBH mass distribution and the scaling relations between the influence radius, SMBH mass and velocity dispersion. The total event rate turns out to be dominated by small galaxies, due to both the event rate per galaxy being weakly dependent on mass and the dynamical friction effect being more pronounced in smaller galactic nuclei.
The event rate is determined, on one hand, by the BH mass distribution parameters (their total mass fraction $\kappa$ and mass distribution slope $\beta$) and also by the SMBH mass function below $\sim10^7\msun$: the SMBH number density per unit $\log\mSMBH$ per unit comoving volume $N_\mathrm{SMBH}$ and the SMBH mass lower limit $M_\mathrm{SMBH, min}$. The approximate dependence of the total event rate on all these parameters is \[eq:final\] && ()\^2\
&& ()\^[-0.32]{} e\^[1.06(1-)]{}, which is similar to a previously published estimate of @Tsang2013 and below the rates cited in @OLeary2009, $0.6\,(\xi/30)\dots 45\,(\xi/30)\,\si{yr^{-1}Gpc^{-3}}$ for different galaxy models. However, Eq. assumes a certain BH number density distribution (Eq. \[eq:n(m,r)\] with $p_0=1/2$); a steeper ($p_0>1/2$) and/or non-spherical distribution could increase $\mathcal{R}$ by orders of magnitude, as discussed in Section \[section:total-merger-rate\].
The GW capture rates we calculated are much lower than the current estimates by LIGO [$24-112\,\si{yr^{-1}Gpc^{-3}}$, @ligo2018], therefore they’re unlikely to be the dominant source of BH mergers. The number of predicted aLIGO detections per year in our model is determined by the event rate per unit comoving volume $\mathcal{R}$ and the accessible volume: &=& \_0\^[z\_]{} ,\
&=& 4,\
d(z) &=& \_0\^z. Here $z_\mathrm{max}$ is the maximum accessible redshift (we ignore its dependence on the BH masses and other parameters of the binary), $\dv*{V_c}{z}$ is the comoving volume per unit redshift and $d(z)$ is the comoving distance. Figure \[fig:alpha\] (bottom) illustrates that dependence. In this equation we have not accounted for the possibility of $p_0$ in Eq. (\[eq:n(m,r)\]) being higher than 0.5 that could potentially increase the event rate up to a few orders of magnitude (Figure \[fig:totalmergerrate\], right).
A possible way to distinguish GW captures from the other channels is their high eccentricity in the LIGO frequency range ($e>0.1$ at $f>\SI{10}{Hz}$). However, eccentric mergers can also be produced in triple systems where a BH binary achieves extreme eccentricity through interaction with with a third body; those systems could be
- A binary BH orbiting around a SMBH in a galactic center can experience variations of inclination and eccentricity due to Kozai-Lidov effect, sometimes reaching $e>0.9999$ which quickly leads to coalescence through GW energy loss [@Antonini2012; @FragioneKL; @Hamers2018; @Hoang2018].
- Lidov-Kozai effect can also cause the inner BH binary coalescence in hierarchical BH triples which form through binary-binary interactions in globular clusters [@Antonini2016] or through the evolution of massive star triples in galactic field [@Antonini2017; @Silsbee2017].
- Triple BHs can also form via binary BH – single BH encounters in globular clusters [@Samsing+2018; @Zevin2019]. A merger subsequently happens as a result of GW emission during a close encounter between two of the three BHs while they temporarily form a bound three-body state [@Samsing2018].
Table \[table:mergerRates\] shows the merger rates for all those different mechanisms in different environments. The rates turn out to be comparable ($\sim0.01-0.1\,\si{Gpc^{-3}yr^{-1}}$) but it still might be possible to distinguish them by their eccentricity distribution.
GW capture (single-single interactions) Hierarchical truples (Kozai-Lidov effect) Binary-single interactions
----------------------- ----------------------------------------- ------------------------------------------- ----------------------------
Nuclear star clusters ? ?
Globular clusters ? 0.04 [@Antonini2016] 0.5 [@Rodriguez2018]
Galactic field 0? 0?
\[table:mergerRates\]
Apart from NSCs with a SMBH in the center we considered in this paper, other kinds of star clusters could also contribute to the GW capture merger rate: NSCs without SMBHs, globular clusters and open clusters. However, in those other systems the velocity dispersion is not as high which means the GW capture mergers in them are much less eccentric [Figure 6 in @OLeary2009]. It’s also worth noticing that the number density of NSCs with SMBHs in them can be up to $40\%$ higher due to ultracompact dwarf galaxies some of which are believed to be stripped NSCs of low-mass galaxies [@Voggel2018]. Finally, we assumed all BHs to be formed in-situ an haven’t accounted for the BHs brough into the Galctic Center via globular cluster inspiral [@ArcaSedda1; @ArcaSedda2].
Acknowledgements {#acknowledgements .unnumbered}
================
This work received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 638435 (GalNUC) and was supported by the Hungarian National Research, Development, and Innovation Office grant NKFIH KH-125675.
Increase in BH number in the assumption of continuous BH formation {#appendix:xi-continuous}
==================================================================
Let $t(m,r)$ be the time it takes for a BH of mass $m$ to reach $r_0$ starting from $r>r_0$. From Eq. (\[eq:rdf-outside\]) we know that [$$\begin{aligned}
t = \tdfo \qty(\frac{(r/r_0)^{1/k_3}-1}{k_1})^{1/k_2},\end{aligned}$$]{} where $k_{1,2,3}$ are the same as in Eq. (\[eq:rdf-outside\]). Assuming BHs are formed with a constant rate, only a fraction $1-t/T$ of all the BHs formed at radius $r$ will be born early enough to reach $r_0$ by present time. Therefore, Eq. (\[eq:ntot\]) for the total number of BHs at present now reads as follows: &=& \_0\^[r\_0]{} n(r)4r\^2 + \_[r\_0]{}\^[(m)]{} n(r)4r\^2 (1-), and the BH number increase coefficient is [$$\begin{aligned}
\XI &= 1 + (3-\gamma_1) \int_1^{\rdf/r_0} x^{2-\gamma_2} \qty(1-\frac{\tdfo}{T}\qty(\frac{x^{1/k_3}-1}{k_1})^{1/k_2}) \dd{x}.\end{aligned}$$]{}
Minimum radius of BH distribution {#appendix:rmin}
=================================
As mentioned in Section \[section:mergerrate\], the BHs are depleted due to their inspiral into the SMBH below a minimum radius $\rgw$: \[eq:rlx=gw\] 0.34 = where [$\sigma(r)=\sqrt{G\mSMBH/r}$ and]{} $n$ is the total number density of objects (both BHs and stars) and $\langle m^2\rangle$ is their average squared mass: \[eq:nm2\] n(r)m\^2= \_0\^ m\_\^2 + \_\^ \^2 = n\_(r)m\_\^2+ n\_(r) \^2At $r\ll r_0$, which is a reasonable assumption given Eq. (\[eq:rlx=gw\]), the BH term dominates the right-hand side of Eq. (\[eq:nm2\]). Indeed, \[eq:dn/dm\] = ()\^[-3/2-p\_0m/]{} C(m), where $C(m)$ is a function of $m$ that can be found from \_0\^[r\_0]{} 4r\^2 = = (m) , $\ninit(r_0)$ and $\nbh(r_0)$ being the initial and final numbers of BHs within the sphere of influence (Eq. \[eq:ninit\]), so that \[eq:dninit/dm\] = (r\_0) .
Eqs. (\[eq:dn/dm\])–(\[eq:dninit/dm\]) yield = ( - p\_0) ()\^[-3/2-p\_0m/]{} which implies that n\_(r) \^2 \_\^( - p\_0) ()\^[-3/2-p\_0m/]{} m\^[2-]{} . At the default values of $p_0=0.5$, $\mmax=40\msun$, $\mmin=5\msun$ and $\beta=2.3$ the dependence of the integral on $r/r_0$ can be approximated by a power law [(with $3\%$ accuracy for $r/r_0<10^{-3}$ which turns out to be a safe assumption)]{}: \[eq:nbh\] n\_(r) \^2 116\^2 ()\^[-1.88]{}.
For stellar number density we have n\_(r)m\_\*\^2&=& = ()\^[-3/2]{} From Eq. (\[eq:KroupaIMF\]) $\langle m_\ast\rangle = 0.30\msun$ and $\langle m_\ast^2\rangle = 0.79\msun^2$ , so that = ()\^[-0.38]{} 0.33 ()\^[-0.38]{} ()\^[-1.3]{}. We can conclude that BHs dominate the relaxation process everywhere inside $0.05r_0$ [similarly to what was reported in @OLeary2009 Fig. 2], and the stellar term can indeed be ignored in Eq. (\[eq:rlx=gw\]). Combined with Eqs. (\[eq:nbh\]) and (\[eq:lnLambda\]), Eq. (\[eq:rlx=gw\]) yields Eq. (\[eq:rgw\]).
Contribution of BHs inside $\rmin$ to the event rate {#appendix:insidermin}
====================================================
Inside the sphere of radius $\rmin$ where relaxation is ineffective compared to the GW-induced orbital radius shrinking (the orbits are assumed circular) the BH number density is defined by = 4r\^2 n(r) r\^5, so that $n(r) \propto r^3$. From [@Kocsis2012] we know that the merger rate per unit radius r\^[39/14]{} n\^2(r) Here we consider mergers between heaviest BHs that have $n(r)\propto r^{-2}$, $r>\rmin$. Then = C
()\^[-17/14]{}r>\
()\^[123/14]{}r<
C = . From this we can conclude that the contribution of BHs inside $\rmin$ to the total event rate is negligible: = = 0.02
The impact of the intrinsic scatter of $M-\sigma$ relation {#appendix:msigma}
==========================================================
The increase in the merger rate $\Gamma$ due to scatter in $C_{M\sigma}$ (distributed log-normally with mean 1 and standard deviation $\delta$) is \[eq:zetamsigma\] \_[M]{}= / As we can see from Eqs. and , $\Gamma$ depends on $C_{M\sigma}$ through $r_0$, $\rmin$ and $\XI$: && (m\_A)(m\_B)r\_0\^[-31/14]{} |1 - ()\^[11/14 - p\_0 m\_ / m\_]{}|\
&& (m\_A)(m\_B)
C\_[M]{}\^[-1.26+0.16/]{}m\_ > m\_\
C\_[M]{}\^[-1.01]{}m\_ < m\_
,\
&=& 1 + . Here we used Eqs. (\[eq:cmsigma\]), (\[eq:rdf\]) and (\[eq:tdf0-mSMBH\]) to determine the dependence of $r_0$, $\rmin$ and $\XI$ on $C_{M\sigma}$. Calculating the integral in Eq. (\[eq:zetamsigma\]) numerically, we see that $\xi_{M\sigma}$ is a decreasing function of $m_{A,B}$ and at $\delta=0.29\ln10$ it spans [a range of values $\xi_{M\sigma}=1.3-1.5$ for $m_{A,B}\in[5\msun,50\msun]$, $\mSMBH\in[10^5\msun,10^7\msun]$.]{}
Manipulation of merger rate distributions {#appendix:ddGamma}
=========================================
As derived in [@Gondan2017], the merger rate distributions per unit $\mtot$, $q\equiv m_B/m_A$, $\mathcal{M}$ and $\eta$ can be calculated as
$$\begin{aligned}
\frac{\partial^2\Gamma}{\partial \mtot \partial q} &= \frac{\mtot}{(1+q)^2} \frac{\partial^2\Gamma}{\partial m_A \partial m_B} , \\
\frac{\partial^2\Gamma}{\partial \mathcal{M} \partial \eta} &= \mathcal{M}\eta^{-6/5}(1-4\eta)^{-1/2} \frac{\partial^2\Gamma}{\partial m_A \partial m_B} .\end{aligned}$$
The merger rate distribution as a function of only one variable can be given by marginalizing one of these equations over the other variable, e.g. $$\begin{aligned}
\frac{\partial\Gamma}{\partial \mtot} = \int^1_\frac{\mmin}{\mmax} \frac{\partial^2\Gamma}{\partial \mtot q} \dd{q}
= \int^1_\frac{\mmin}{\mmax} \frac{\partial^2\Gamma_0}{\partial m_A \partial m_B} \frac{\mtot}{(1+q)^2}
\XI\qty(\frac{\mtot}{1+q}) \XI\qty(\frac{q\mtot}{1+q}) \dd{q},\end{aligned}$$ where $\Gamma_0$ is the merger rate calculated without taking into account the effects of DF (Eqs. \[eq:gamma0mamb\]).
[^1]: However, diffuse light measurements of @Schodel2018 give a lower value $\gamma_1=1.13\pm0.08$. The difference in the BH number between $\gamma_1=1.5$ and $\gamma_1=1.1$ is only $\sim15\%$.
[^2]: The reason for $\mSMBH$ dependence being different for different $\mtot$ in @Gondan2017 (as well as in this work) is that for low mass BHs whose density declines less steeply than $r^{-2}$, most of the mergers are contributed by the largest $r$ rather than the smallest ones.
[^3]: It is consistent within uncertainties with the other SMBH mass estimates in the literature [e.g. @Hopkins2007] as well as the SMBH masses inferred from the galaxy bulge mass distribution [@Thanjavur2016] with $\mSMBH/M_{\rm bulge}=0.003$.
|
Josephson qubits are among the most promising devices to implement solid state quantum computation [@b1; @b2]. Quantum manipulations of individual [@b3; @b4; @b5; @b6; @b7; @b8; @b9] and coupled [@b10] qubits has been demonstrated experimentally. At present, probably the main obstacle to the development of a larger-scale solid state quantum logic circuits is presented by decoherence. It is therefore important to develop strategies to minimize the effects of decoherence on the dynamics of the qubit systems.
Known approaches to reduction of decoherence include both error-correction and error-avoiding schemes that either employ symmetries of the qubit-environment interaction to create areas of the Hilbert space not affected by decoherence [@b11; @b12] or use rapid random dynamic perturbations of the system to average out the effects of external noise [@b13; @b14]. The error-avoiding approaches appear to be less suitable for the solid-state qubits. Indeed, noise in solid-state systems typically does not have any particular symmetry and its correlation time is short, so that the application of the control pulses within this time-scale, as required by the dynamic averaging schemes, is problematic. This leaves error-correction as the main strategy for suppression of decoherence in solid-state qubits. In this work, we suggest an implementation of one of the basic error-correction algorithms for the suppression of dephasing errors (which can be expected to be the dominant type of errors in solid-state circuits - see, e.g., Ref. [@b9]), and develop its quantitative description. Our scheme employs the Josephson-junction qubits that combine charge and flux dynamics [@b15; @b17; @b6], and requires only a small number of qubit transformations to operate.
From the perspective of the general theory of error-correction, an interesting feature of the scheme considered in this work is the possibility of developing its detailed quantitative description within the realistic model of the qubit-environment interaction and analyzing, for instance, the effect of the correlations in the noise acting on different qubits. While discussions of the error-correction rely typically in an essential way on independent noise models, environments of the solid-state qubits can be to a large degree correlated because of the finite distance between the qubits in a circuit. A clear illustration of this is provided by the background charge fluctuations that are the main source of dephasing in charge qubits [@b3; @b6; @b10; @b16]. Long-range nature of the Coulomb interaction creates noise correlations by coupling the qubits to the same charge fluctuators.
We consider specifically the problem of “quantum memory”, when the task is to preserve the stationary state of the qubit in the presence of dephasing noise. The qubit Hamiltonian contains then only the coupling to the environment. Under the assumption that the environment has many degrees of freedom each of which is only weakly coupled to qubits, it can be modeled as an ensemble of harmonic oscillators [@b18; @b19; @b20; @b21] (see however [@b16]), so that the Hamiltonian of the qubit register is: $$H=\sum_{j} \sigma_z^{(j)}\xi_j \, ,
\label{1}$$ where $\xi_j= \sum_{m,k} [\lambda_{m,j}(\omega) a_{m,\omega} +
\mbox{h.c.}]$. Here we assumed several independent ensembles of environmental oscillators (numbered by $m$), as needed to model different profiles of spatial correlations of random forces $\xi_j$. The index $j=1,2... $ in (\[1\]) numbers the qubits, and coefficients $\lambda_{m,j}(\omega)$ are coupling constants of the qubit $j$ to the oscillators of reservoir $m$ in the mode $\omega$ and creation/annihilation operators $a_{m,\omega},
a^{\dagger}_{m,\omega}$. Time evolution of the “qubits+environment” system is described conveniently in the interaction representation with respect to the interaction Hamiltonian of Eq. (\[1\]). The evolution operator $U(t)$ can then be calculated explicitly by separating the two non-commuting parts, $a_{m,\omega}$, and $a^{\dagger}_{m,\omega}$, of the qubit-oscillator coupling, and using the fact that their commutator is a $c$-number: $$U(t) = \exp \{ -i \sum_{j} \varphi_j(t) \sigma_z^{(j)} \} U_r(t)
\, ,
\label{2}$$
$$U_r(t) = \exp \{ i \sum_{m,\omega} \frac{\omega t-\sin
\omega t}{\omega^2} | \sum_{j} \lambda_{m,j}\sigma_z^{(j)} |^2
\} \, .$$ The first term in $U(t)$ represents fluctuating phases $\varphi_j(t)$ of the qubit basis states induced by the environmental forces $\xi_j(t)$: $\varphi_j(t) =\int_0^{t}
\xi_j(t') dt'$. The second term, $U_r(t)$, results from the renormalization of the qubit parameters by the qubit-environment interaction. To see this more explicitly, we note that the sum over frequencies $\omega$ in this exponent has a natural cut-off at large frequencies $\omega \simeq \tau_c^{-1}$, where $\tau_c$ is the time scale at which environment forces acting on different qubits are correlated. For weak decoherence we are interested in the time scales much larger that $\tau_c$. In this regime, the phase represented by $U_r(t)$ is dominated by the term that grows linearly with $t$, and can be viewed as arising from the renormalization of the qubit energy. Equation (\[2\]) shows that such a renormalization includes then the shift of the total energy of the register and creation of the qubit-qubit interaction. The total energy shift is irrelevant as long as we consider an individual register. Neglecting it, we see that $U_r(t)$ results from the Hamiltonian evolution with the Hamiltonian $$H_r = -\sum_{j,j'}V_{jj'}\sigma_z^{(j)}\sigma_z^{(j')} \, ,
\label{3}$$ and $V_{jj'} = 2\mbox{Re} \sum_{m,\omega} (\lambda_{m,j}(\omega)
\lambda^*_{m,j'}(\omega)/ \omega)$, if the sum over frequencies $\omega$ in this expression is converging at low frequencies. The qubit-qubit interaction strength $V_{jj'}$ is non-vanishing only if the same reservoir $m$ couples to more than one qubit, so that the reservoir forces $\xi_j$ at different qubits are correlated.
The time evolution with the Hamiltonian $H_r$, and more generally, the evolution operator $U_r$ in Eq. (\[2\]) represent deterministic part of the qubit evolution induced by the qubit-reservoir interaction. As a result, it can in principle be compensated for by adjusting the regular (non-dissipative) part of the Hamiltonian of the qubit register. This procedure, however, is impractical even in the case of constant $H_r$, since the interaction constants $V_{jj'}$ are apriori unknown and incommesurate quantities. This complexity means that a more appropriate approach is to treat the time evolution represented by $U_r$ as dephasing despite its deterministic character.
The time evolution of the density matrix $\rho (t)$ of the qubit register is obtained from Eq. (\[2\]) through the relation $\rho (t)=\mbox{Tr}_{env} \{ U^{\dagger} (t)
\sigma (0) U(t) \}$, where $\sigma$ is the total density matrix of the “qubits+environment” system. The environment will dephase the qubits if they are prepared initially in the state $\rho(0)$ that is uncorrelated with the state of the environment, $\sigma (0) = \rho_{env} \rho(0)$. Assuming that the environment is in thermal equilibrium at temperature $\Theta$, and no error correction procedure is applied, we get using the standard property of the Gaussian noise:
$$\rho(t) = \exp \{ -\frac{1}{2} \sum_{j,j'} \langle
\varphi_j(t) \varphi_{j'}(t)\rangle (\sigma_z^{(j)}- \bar{
\sigma}_z^{(j)}) (\sigma_z^{(j')}- \bar{\sigma}_z^{(j')})\}$$
$$\cdot U_r^{\dagger} (t) \rho(0) U_r(t) \, .
\label{4}$$
Here we introduced the convention that the bar over $\sigma_z$ operators means that they act on $\rho$ from the right. Qualitatively, Eq. (\[4\]) shows that the matrix elements of $\rho$ that are further away from the diagonal in the $\sigma_z$ basis decay faster. The diagonal elements (on which $\sigma_z-
\bar{\sigma}_z=0$) remain constant. In the case of one physical qubit, Eq. (\[4\]) gives $\rho (t) = e^{-\langle \varphi^2(t)
\rangle (1-\sigma_z \bar{\sigma}_z)} \rho(0)$, i.e., the off-diagonal elements of $\rho$ are suppressed with time as $e^{-2 \langle \varphi^2(t)\rangle } \equiv e^{-P(t)}$. If the environment density of states is Ohmic, i.e., $\sum_{m,
\omega} |\lambda_m (\omega)|^2 ... = g \int_0^\infty d\omega
\omega e^{-\omega \tau_0} ... \, $, direct evaluation for $\Theta \ll 1/\tau_0$ gives: $P(t) = 2g \ln [\sinh (\pi t
\Theta)/(\pi \tau_0 \Theta) ]$. At large $t$, when the random force $\xi$ appears $\delta$-correlated, $P(t)$ reduces to $P(t)= \Gamma t$, where $\Gamma= 2\pi g \Theta$ is the dephasing rate.
One can reduce the effective dephasing rate by the encoding that corrects the phase errors [@b22; @b23]. Generalized to $k$ errors, this encoding is: $$\alpha |0\rangle +\beta |1\rangle \rightarrow \alpha |
\underbrace{++...+}_{2k+1} \rangle + \beta |\underbrace{--...-
}_{2k+1} \rangle \, .
\label{6}$$ In Equation (\[6\]), a bit of quantum information is encoded in the state of the $2k+1$ physical qubits, and the $|\pm\rangle$ states of each of these qubits are obtained through the Hadamard transform $\hat{H}$ (the $\pi/2$-rotation around $y$ axis) from the $|0,1\rangle$ states. All of the $\sigma_z$ operators in the dephasing-induced time evolution (\[2\]) are changed by $\hat{H}$: $\hat{H}\sigma_z \hat{H} =\sigma_x$, so that for the states on the right-hand-side of Eq. (\[6\]) the dephasing looks like transitions between the $|\pm\rangle$ states of each qubit, and can be directly detected by measurements in this basis and corrected by applying simple pulses returning the qubit into the initial state. The error-detecting measurements, however, should not destroy the quantum information encoded in the state (\[6\]), i.e., they should not distinguish the $\alpha$ and $\beta$ parts of this state. This condition is not satisfied by measurements on individual qubits but can be satisfied by the measurements on pairs of the nearest-neighbor qubits comparing their states. Despite the apparent complexity of this scheme, it has quite natural implementation in the Josephson-junction qubits - see Fig. 1.
To describe this process quantitatively we assume that its measurement/correction part can be done on the time scale that is much shorter than the one set by the characteristic dephasing rate $\Gamma$. Different terms in the environment-induced evolution of the encoded state, Eq. (\[6\]), during the time interval $T$ between the successive application of the “measurement+correction” operations can be conveniently classified by the number of qubits flipped during this time interval. In the relevant regime of sufficiently short $T$: $P(T) \ll 1$, the probability amplitude of these terms decreases rapidly when this number increases. If we keep only the terms that flip up to $k$ qubits, we see directly from Eq. (\[2\]) that the time evolution at this level of accuracy (denoted by $U_k(T)$) preserves the superposition of the $\alpha$ and $\beta$ parts of the encoded state: $$U_k(T)[ \alpha | \oplus \rangle +\beta |\ominus \rangle
]= \sum_{q} [\alpha |\psi_q \rangle +\beta \hat{R} |\psi_q
\rangle ] u_q \, .
\label{7}$$ Here index $q$ runs over $2^{2k}$ different register states obtained from the state $|\oplus\rangle \equiv |+...+\rangle $ by flipping up to $k$ qubits, $u_q$ are the probability amplitudes of these states, $\hat{R}|\psi_q \rangle$ denotes the state $|\psi_q \rangle$ with all $2k+1$ qubits inverted, and $|\ominus \rangle \equiv |-...-\rangle $.
The measurements that compare the qubit states in all pairs of the nearest-neighbor qubits do not distinguish states $|\psi_q \rangle$ and $\hat{R}|\psi_q \rangle$, and therefore also preserve the superposition of the $\alpha$ and $\beta$ terms in Eq. (\[7\]). The $2^{2k}$ different outcomes (“equal” or “different”) of the $2k$ such measurements distinguish all terms with different $q$ in Eq. (\[7\]) and enable one to decide what qubits were flipped during the time interval $T$. Application of the correcting pulses should then bring the state of the qubit register back to its initial form (\[6\]) so that the encoded quantum state does not change in this approximation. The residual evolution of the encoded state is associated with the possibility that environment flips more that $k$ different qubits; for $P(T)\ll 1$ – precisely $k+1$ qubits. Following the same steps as above, we see that when $k+1$ qubits are flipped, the measurement/correction cycle interchanges the $\alpha$ and $\beta$ weights in the encoded state (\[6\]). Since the probability $p$ of this mistake is small, $p\ll 1$, the encoded state changes substantially only on the time scale larger than the period $T$ of one error-correction cycle, and its evolution on this scale can be conveniently described by the continuous equation for the density matrix $\rho^{(c)}$ in the basis of $|\oplus \rangle$ and $|\ominus \rangle$ states. The interchange of the $\alpha$ and $\beta$ terms in (\[6\]) leads to the following evolution of $\rho^{(c)}$: $$\dot{\rho}^{(c)}_{++}=\frac{\gamma_k}{2} (\rho^{(c)}_{--}
-\rho^{(c)}_{++}) \, ,\;\;\;\;
\dot{\rho}^{(c)}_{+-} =\frac{\gamma_k}{2}
(\rho^{(c)}_{-+} -\rho^{(c)}_{+-}) \, .
\label{8}$$ Here $\gamma_k \equiv 2p/T$ is the effective dephasing rate of the encoded quantum information, and the superscript $(c)$ indicates that $\rho^{(c)}$ is the reduced density matrix in the presence of error correction. Thus, our error-correcting procedure replaces the dephasing in the individual physical qubit with the dephasing of encoded quantum information at a smaller rate. Indeed, if one writes Eqs. (\[8\]) in the rotated basis $|\oplus \rangle \pm
|\ominus \rangle$, they explicitly acquire the form characteristic for pure dephasing: constant diagonal elements of the density matrix and decay of the off-diagonal elements with the rate $\gamma_k$. The dephasing rate $\gamma_k$ can be calculated from the evolution operator (\[2\]). Now we discuss several important limits.
For $k=1$, when the relevant errors flip 2 out of 3 qubits, we get: $$\gamma_1 = \frac{2}{T} \sum_{j>j'} (T^2V_{jj'}^2+ 2 \langle
\varphi_j(T) \varphi_{j'}(T)\rangle^2 + \langle \varphi_j^2(T)
\rangle \langle \varphi_{j'}^2(T)\rangle ) \, .$$ The first two terms in this expression represent contribution to dephasing from noise correlations at different qubits, while the last term exists also for uncorrelated noise. If the noise is $\delta$-correlated in time, $\gamma_1$ reduces to $\gamma_1 = T\sum_{j>j'} (2V_{jj'}^2 +\Gamma_{jj'}^2 +\Gamma_j
\Gamma_{j'}/2)$, where $\Gamma_j$ is the dephasing rate in the $j$th qubit, and $\Gamma_{jj'}$ is introduced through $2\langle \varphi_j(t) \varphi_{j'}(t)\rangle = \Gamma_{jj'}t$.
For $k=2$, the dephasing rate of the encoded state is: $$\gamma_2 = \frac{2}{T} \sum_{j>j'>j''} \langle \varphi_j^2
\varphi_{j'}^2 \varphi_{j''}^2\rangle \, .
\label{13}$$ Since the effective coupling induced by the environment – see Eq. (\[3\]), flips the qubits only in pairs, it does not contribute to $\gamma_2$. If the dephasing noise is $\delta$-correlated in time, its space correlations are non-vanishing only for the nearest-neighbor qubits, and the corresponding dephasing rates are the same for all qubits, Eq. (\[13\]) gives: $\gamma_2 = 5 \Gamma T^2 (\bar{\Gamma}^2 + \Gamma^2/2)$, where $\bar{\Gamma} \equiv \Gamma_{jj+1}$.
If the dephasing forces at different qubits are uncorrelated, the encoded dephasing rate can be easily calculated for arbitrary $k$: $$\gamma_k = \frac{1}{2^kT} \sum_{j_1>j_2> ...…>j_{k+1}}
P_{j_1}(T) P_{j_2}(T) ... P_{j_{k+1}}(T) \, ,
\label{9}$$ and one sees that $\gamma_k$ decreases exponentially with the “degree of encoding” $k$. When the probabilities $P(T)$ of dephasing errors in individual qubits can be expressed through the dephasing rate $\Gamma$, Eq. (\[9\]) reduces to $\gamma_k = \Gamma (\Gamma T)^k (2k+1)! /(2^k k!(k+1)!)$, if $\Gamma$ is the same for all qubits.
Exponential suppression of $\gamma_k$ with $k$ is limited in the scheme considered above by possible imperfections of the measurement/correction operations. The most important is direct dephasing of the encoded state by measurements, which, in contrast to correction steps, need to be performed each period $T$. For example, one of the specific non-idealities of measurement detectors that leads to direct dephasing of the encoded state is residual linear response coefficient of the quadratic detectors needed to perform pair-wise comparison of the qubit states – see Eq. (\[15\]) below. Linear terms couple the detector directly to the $|\pm \rangle$ states of individual qubits and introduce finite phase shifts between them. Since the number of required measurements is proportional to $k$, the rate of introduced dephasing should also be proportional to $k$, and can be denoted as $\bar{\gamma} k$. The effect of this dephasing on the evolution of the encoded quantum information is described then by adding the usual dephasing term to the equation for the off-diagonal element of the density matrix $\rho^{(c)}$ (\[8\]): $$\dot{\rho}^{(c)}_{+-} = \frac{\gamma_k}{2} (\rho^{(c)}_{-+}-
\rho^{(c)}_{+-}) -\bar{\gamma} k \rho^{(c)}_{+-}\, .
\label{14}$$ Qualitatively, the two types of dephasing processes in Eq. (\[14\]) have similar effect of suppressing the fidelity of the encoded state, but depend differently on $k$. The optimum degree of encoding is estimated crudely by minimizing the total dephasing rate: $k_{opt} \sim \ln (\bar{\gamma}/\Gamma)/\ln
(T\Gamma)$. One obvious result of this optimization is that for the considered scheme of the dephasing suppression to make sense, the dephasing introduced by imperfections of the correcting procedure should be much weaker than the original qubit dephasing $\Gamma$.
(3.2,2.15) (.25,.1)[=2.7in]{}
This condition can be satisfied in Josephson-junction qubits, where the dynamics of magnetic flux characterized by longer coherence times (at least tens of nanoseconds – see, e.g., [@b9]) can be used to suppress dephasing in charge-based qubits. The charge qubits have quite short decoherence times, $\sim 1$ ns [@b3; @b10], limited by the background charge fluctuations, but offer some advantages, e.g., demonstrated simplicity of the qubit-qubit coupling [@b10]. Therefore it would be of interest to use the approach discussed in this work to suppress dephasing of charge degrees of freedom with the help of controlled flux dynamics. A sketch of the possible set-up achieving this is shown in Fig. 1. Its main elements are the charge qubits, formed by two small tunnel junctions in series, enclosed in small superconducting loops threaded by magnetic flux $\Phi$ equal to half of the magnetic flux quantum $\Phi_0$. It can be shown [@b17] that the current in each loop represents the $\sigma_x$ component of the qubit dynamics, and its monitoring measures therefore the qubit in the $\sigma_x$ basis as needed for detection of the dephasing errors. Comparison of the states of the nearest-neighbor qubits can be achieved by measuring not directly the currents in the loops, but the square of the difference (or of the sum) between the currents. Such a quadratic detection measures the product operator $\sigma_x^{(j)}\sigma_x^{(j+1)}$: $$(\sigma_x^{(j)} \pm \sigma_x^{(j+1)})^2= 2(1 \pm
\sigma_x^{(j)}\sigma_x^{(j+1)}) \, ,
\label{15}$$ and provides information on whether the states of the two qubits are the same or not without measuring them. Quadratic detection can be realized by the usual magnetometers but operated at a point where the linear response coefficient vanishes. These measurements, subsequent classical calculations, and application of correction pulses, can be done with sufficient frequency by existing “SFQ” superconductor electronics [@b24] compatible with the qubits.
In summary, we suggested a simple scheme of performing basic error-correction in Josephson-junction qubits. The scheme suppresses dephasing errors and can be analyzed quantitatively within the realistic model of environment, including the possibility of noise correlations at different qubits. If the errors introduced by the correction procedure are negligible, the residual dephasing rate for the encoded quantum information decreases exponentially with the degree of encoding.
This work was supported in part by the NSF under grant \# 0121428, and by the NSA and ARDA under the ARO contract (D.V.A.), and by EC-grant IST-FET-SQUBIT (R.F.)
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|
---
author:
- Eric Primozic
title: Motivic Steenrod operations in characteristic $p$
---
Introduction {#introduction .unnumbered}
============
Voevodsky constructed motivic reduced power operations $P^{n}_{F}$ for $n \geq0$ where the base field $F$ is a perfect field with $\mathrm{char}(F)$ not equal to the characteristic $p>0$ of the coefficient field [@Voe]. These operations were used in the proof of the Bloch-Kato conejcture. Brosnan gave an elementary construction of Steenrod operations on mod $p$ Chow groups over a base field of characteristic $\neq p$ [@Bro]. Steenrod operations on Chow groups have been used succesfully in the study of quadratic forms over a base field of characteristic $\neq 2$ and to prove degree formulas in algebraic geometry as in [@EKM] and [@Mer].
For a prime $p$, Voevodsky’s construction of Steenrod operations for the coefficient field $\mathbb{F}_{p}$ uses the calculation of the motivic cohomology of $BS_{p}$. However, when defined over a base field $k$ of characteristic $p$, $B\mathbb{Z}/p$ is contractible [@MorVoe Proposition 3.3]. Hence, over the base field $k$, $H^{*,*}(BS_{p}, \mathbb{F}_{p})\cong H^{*,*}(k, \mathbb{F}_{p})$ and so one cannot carry out Voevodsky’s construction. It has also been an open problem to just define Steenrod operations on the mod $p$ Chow groups of smooth schemes over a field of characteristic $p$. Haution made progress on this problem by constructing the first $p-1$ homological Steenrod operations on Chow groups mod $p$ and $p$-primary torsion over any base field [@Hau4], defining the first Steenrod square on mod $2$ Chow groups over any base field [@Hau2], and constructing weak forms of the second and third Steenrod squares over a field of characteristic $2$ [@Hau5]. Note that in papers where Steenrod squares (or weak forms of Steenrod squares) on mod $2$ Chow groups are used, the $n$th Steenrod square on mod $2$ Chow groups corresponds to the $2n$th Steenrod square on mod $2$ motivic cohomology since the Bockstein homomorphism is $0$ on mod $2$ Chow groups.
For $p$ a prime, we use the results of Frankland and Spitzweck in [@FrankSpi] to define Steenrod operations $P_{k}^{n}: H^{i,j}(-, \mathbb{F}_p) \to H^{i+2n(p-1),j+n(p-1)}(-, \mathbb{F}_p) $ for $n \geq 0$ on the mod $p$ motivic cohomology of smooth schemes over a field $k$ of characteristic $p$. Note that some authors use the notation $H^{i}(-,\mathbb{Z}(j))$ in place of $H^{i,j}(-, \mathbb{Z})$ to denote motivic cohomology. For $ n \geq 1$, we show that $P^{n}_{k}$ is the $p$th power on $H^{2n,n}(-,\mathbb{F}_{p})=CH^{n}(-)/p$, and we also prove an instability result for the Steenrod operations. Restricted to mod $p$ Chow groups, we prove that the $P_{k}^i$ satisfy expected properties such as Adem relations and the Cartan formula. We also show that the operations $P^{n}_{k}$ agree with the operations $P^{n}_{K}$, constructed by Voevodsky for $\mathrm{char}(K)=0$, on the mod $p$ Chow rings of flag varieties in characteristic $0$.
In Section \[Rost\], we extend Rost’s degree formula [@Mer Theorem 6.4] to a base field of arbitrary characteristic. The degree formula we obtain at odd primes seems to be new. In Section \[S:appquad\], we use the new operations to study quadratic forms defined over a field of characteristic $2$. Previous results or proofs avoided the case of quadratic forms in characteristic $2$ since Steenrod squares were not available. We prove Hoffmann’s conjecture (a generalization including characteristic $2$ quadratic forms) on the possible values of the first Witt index of anisotropic quadratic forms for the case of nonsingular anisotropic quadratic forms over a field of characteristic $2$. In characteristic $\neq 2$, Hoffmann’s conjecture was proved by Karpenko in [@Kar]. Previously, Haution used a weak form of the first homological Steenrod square to prove a result on the parity of the first Witt index for nonsingular anisotropic quadratic forms over a field of characteristic $2$ [@Hau Theorem 6.2]. Haution’s result is a corollary of the case of Hoffmann’s conjecture proved in this paper. Using the Steenrod squares defined in this paper, it should be possible to extend other results on quadratic forms to the case where the base field has characteristic $2$.
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank Burt Totaro for suggesting this project to me and for providing advice. I am very grateful to Marc Hoyois for answering my questions and telling me the strategy used in the proof of Proposition \[T:pullbaclEB\] along with most of the details of the proof. I thank Markus Spitzweck for answering my questions. I thank Chuck Weibel for his comments. I thank Nikita Karpenko for telling me about some of the applications of the new Steenrod operations to quadratic forms in characteristic $2$ given in Section \[S:appquad\].
Prior results on the dual Steenrod algebra and setup
====================================================
Let $k$ be a field of characteristic $p>0$. For a base scheme $S$, we let $\textup{Sm}_{S}$ denote the category of quasi-projective separated smooth schemes of finite type over $S$, let $H(S)$ denote the unstable motivic homotopy category of spaces over $S$ defined by Morel-Voevodsky [@MorVoe], let $H_{\bullet}(S)$ denote the pointed unstable motivic homotopy category of spaces over $S$, and we let $SH(S)$ denote the stable motivic homotopy category of spectra over $S$ [@Voe3]. Let $$\Sigma^{\infty}_{+}:\textup{Sm}_{S} \to SH(S),$$ $$\Sigma^{\infty}_{+}:H(S) \to H_{\bullet}(S) \to SH(S)$$ denote the infinite $\mathbb{P}^{1}$-suspension functors.
We recall some results from [@Spi] and [@FrankSpi] that hold in the categories $H(k)$ and $SH(k)$. Let $B\mu_{p} \in H(k)$ denote the geometric motivic classifying space of the group scheme $\mu_{p}$ over $k$ of the $p$th roots of unity. Let $H\mathbb{F}^{k}_{p} \in SH(k)$ denote the motivic Eilenberg-MacLane spectrum representing mod $p$ motivic cohomology. Let $v \in H^{2,1}(B\mu_{p}, \mathbb{F}_p)$ denote the pullback of the first Chern class $c_{1}\in H^{2,1}(B\mathbb{G}_{m}, \mathbb{F}_p)$. From the computation of the motivic cohomology of $B\mu_{p}$ in [@Voe Theorem 6.10], there exists a unique $u \in H^{1,1}(B\mu_{p}, \mathbb{F}_p)$ such that $\beta(u)=v$ where $\beta$ denotes the Bockstein homomorphism on mod $p$ motivic cohomology. The class $\rho=-1$ in $H^{1,1}(k, \mathbb{F}_{p})=k^{*}/k^{* \, p}$ is $0$ and the class of $\tau \in H^{0,1}(k, \mathbb{F}_{p})=\mu_{p}(k)=0$ described in [@Voe Theorem 6.10] is also $0$. We need the following computation which can be deduced from [@Voe Theorem 6.10] by setting $\rho=0$ and $\tau=0$.
\[T:comp of coh bmu\] There is an isomorphism $$H^{*,*}(B\mu_{p}, \mathbb{F}_{p}) \cong H^{*,*}(k, \mathbb{F}_{p})\llbracket v,u\rrbracket /(u^{2}).$$
Note that $H^{*,*}(B\mu_{p}, \mathbb{F}_{p})$ is defined in [@Voe] as a limit of motivic cohomology rings of smooth schemes over the base field. This explains why power series appear in the above theorem.
Let $\mathcal{A}^{k}_{*,*} \coloneqq \pi_{*,*}(H\mathbb{F}^{k}_{p} \wedge H\mathbb{F}^{k}_{p})$. As described in [@Spi Chapter 10.2], there is a coaction map $$\label{eq:coaction} H^{*,*}(B\mu_{p}, \mathbb{F}_p) \to \mathcal{A}^{k}_{-*,-*} \widehat{\otimes}_{\pi_{-*,-*}H\mathbb{F}_{p}^{k}}H^{*,*}(B\mu_{p}, \mathbb{F}_p).$$ We use the left $H\mathbb{F}_{p}^{k}$-module structure on $H\mathbb{F}_{p}^{k}\wedge H\mathbb{F}_{p}^{k}$ for this coaction map. For $i\geq 0$ and $j \geq 1$, classes $\tau_{i} \in \mathcal{A}^{k}_{2p^{i}-1,p^{i}-1} $ and $\xi_{j} \in \mathcal{A}^{k}_{2p^{j}-2,p^{j}-1}$ are defined by the coaction map: $$u \mapsto u+ \Sigma _{i\geq 0}\tau_{i} \otimes v^{p^{i}},$$
$$v \mapsto v+ \Sigma _{j\geq 1}\xi_{j} \otimes v^{p^{j}}.$$
\[P:tau2=0\] $\tau_{i}^{2}=0$ for all $i \geq 0$.
We use the argument of [@Voe Theorem 12.6]. First, we assume that $\mathrm{char}(k)=2$. Under the coaction map \[eq:coaction\], $$u^{2}=0 \mapsto u^{2}+\Sigma _{i\geq 0}\tau^{2}_{i} \otimes v^{2^{i+1}}=0.$$ For $i \geq0$, the coefficient of $v^{2^{i+1}}$ equals $0=\tau^{2}_{i}$.
Now we assume that $p=\mathrm{char}(k)$ is odd. Let $i\geq0$. As $\mathcal{A}^{k}_{*,*}$ is graded-commutative under the first grading, we have $\tau^{2}_{i}=(-1)^{(2p^{i}-1)(2p^{i}-1)}\tau^{2}_{i}=-\tau^{2}_{i}$ which implies that $\tau^{2}_{i}=0$.
In this paper, we shall consider finite sequences $\alpha=(\epsilon_{0},r_{1},\epsilon_{1},r_{2},\ldots )$ of integers such that $\epsilon_{i}\in \{0,1\}$ and $r_{j} \geq 0$ for all $i \geq0$ and $j\geq 1$. From now on, it will be assumed that any sequence $\alpha$ in this paper satisfies these conditions. To a sequence $\alpha$, we associate a monomial $\omega(\alpha)=\tau_{0}^{\epsilon_{0}}\xi_{1}^{r_1}\tau_{1}^{\epsilon_{1}} \cdots \in \mathcal{A}^{k}_{*,*}$ of bidegree $(p_{\alpha},q_{\alpha})$. The sequences $\alpha$ induce a morphism $$\Psi_{k}: \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p} \to H\mathbb{F}^{k}_{p} \wedge H\mathbb{F}^{k}_{p}$$ of left $H\mathbb{F}^{k}_{p}$-modules. Frankland and Spitzweck proved the following theorem [@FrankSpi Theorem 1.1] which allows us to define Steenrod operations on mod $p$ motivic cohomology over the base $k$.
\[T:Psidef\] The morphism $$\Psi_{k}: \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p} \to H\mathbb{F}^{k}_{p} \wedge H\mathbb{F}^{k}_{p}$$ is a split monomorphism of left $H\mathbb{F}^{k}_{p}$-modules.
It is conjectured that $\Psi_{k}$ is an isomorphism. Frankland and Spitzweck proved this theorem by comparing $\Psi_{k}$ to the corresponding isomorphism $$\label{eq:char0dual}\Psi_{K}: \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{K}_{p} \to H\mathbb{F}^{K}_{p} \wedge H\mathbb{F}^{K}_{p}$$ of left $H\mathbb{F}^{K}_{p}$-modules for $\mathrm{char}(K)=0$. From now on, we will identify $\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{K}_{p}$ with $H\mathbb{F}^{K}_{p} \wedge H\mathbb{F}^{K}_{p}$ as left $H\mathbb{F}_{p}^{K}$-modules through $\Psi_{K}$ whenever $K$ is a field of characteristic $0$. Let $D$ be a complete unramified discrete valuation ring with closed point $i:\mathrm{Spec}(k) \to \mathrm{Spec}(D)$ and generic point $j:\mathrm{Spec}(K) \to \mathrm{Spec}(D)$ where $K=\mathrm{Frac}(D)$. For example, when $k=\mathbb{F}_{p}$, we take $D=\mathbb{Z}_{p}$ and $K=\mathbb{Q}_{p}$.
For a morphism $f: S_{1} \to S_{2}$ of base schemes, we let $f_{*}\coloneqq Rf_{*}: SH(S_{1}) \to SH(S_{2})$ and $f^{*} \coloneqq Lf^{*}:SH(S_{2}) \to SH(S_{1})$ denote the right derived pushforward and left derived pullback functors respectively. Pullback $f^{*}$ is strongly monoidal while $f_{*}$ is lax monoidal. Furthermore, $f_{*}$ commutes with all suspensions $\Sigma^{i,j}$ [@FrankSpi Lemma 7.5]. We also note that $f_{*}$ preserves coproducts [@FrankSpi Lemma 7.4].
For $S$ a separated Noetherian scheme of finite dimension, we let $\widehat{H}\mathbb{Z}^{S}\in SH(S)$ denote the motivic $E_{\infty}$ ring spectrum constructed by Spitzweck in [@Spi] and let $\widehat{H}\mathbb{F}^{S}_{p}\coloneqq \widehat{H}\mathbb{Z}^{S}/p$ . Let $D(\widehat{H}\mathbb{Z}^{S})$ denote the homotopy category of left $\widehat{H}\mathbb{Z}^{S}$-modules. See [@CisDeg Section 7.2] and [@FrankSpi Sections 2 and 3] for a discussion on the homotopy category of left $R$-modules $D(R)$ for a highly structured ring spectrum $R$. There is a forgetful functor $U_{S}:D(\widehat{H}\mathbb{Z}^{S}) \to SH(S)$.
The spectrum $\widehat{H}\mathbb{Z}^{S}$ enjoys a number of desirable properties. The spectrum $\widehat{H}\mathbb{Z}^{S}$ is Cartesian. This means that for a morphism $f:S_{1} \to S_{2}$ of base schemes, the induced morphism $f^{*}\widehat{H}\mathbb{Z}^{S_{2}} \to \widehat{H}\mathbb{Z}^{S_{1}}$ is an isomorphism in $SH(S_{1})$ of $E_{\infty}$ ring spectra [@Spi Chapter 9]. Throughout this paper, we will frequently identify $f^{*}\widehat{H}\mathbb{Z}^{S_{2}}$ with $\widehat{H}\mathbb{Z}^{S_{1}}$ whenever we are given a morphism $f:S_{1} \to S_{2}$ of base schemes. See also [@FrankSpi Section 2]. Hence, the square $$\begin{tikzcd}
D(\widehat{H}\mathbb{Z}^{S_{2}}) \arrow[r, "f^{*}"] \arrow[d,"U_{S_{2}}"]
& D(\widehat{H}\mathbb{Z}^{S_{1}}) \arrow[d, "U_{S_{1}}"]\\
SH(S_{2}) \arrow[r, "f^{*}"] & SH(S_{1})
\end{tikzcd}$$ commutes.
For $S=\mathrm{Spec}(F)$ with $F$ a field, $\widehat{H}\mathbb{Z}^{S}$ is isomorphic as an $E_{\infty}$ ring spectrum to the usual Eilenberg-MacLane spectrum $H\mathbb{Z}^{S}$ constructed by Voevodsky [@Spi Theorem 6.7]. For the discrete valuation ring $D$, $\widehat{H}\mathbb{Z}^{D}$ represents Bloch-Levine motivic cohomology as defined in [@Lev].
We briefly describe the definition of Bloch-Levine motivic cohomology in [@Lev] for a discrete valuation ring $D$. Let $X \to \mathrm{Spec}(D)$ be a morphism of finite type with $X$ irreducible. If the image of the generic point $\eta_{X}$ of $X$ is $\mathrm{Spec}(k)$, then we define dim$(X) \coloneqq$dim$(X_{\mathrm{Spec}(k)})$. Otherwise, we define dim$(X) \coloneqq$dim$(X_{\mathrm{Spec}(K)})+1$. For $n\geq 0$, let $\Delta^{n} \coloneqq \mathrm{Spec}(D[t_{0}, \ldots, t_{n}]/ \Sigma_{i}t_{i}-1)$ denote the algebraic $n$-simplex over $D$. Let $z_{q}(X, r)$ denote the free abelian group generated by all irreducible closed subschemes $C \subset \Delta^{r} \times _{\mathrm{Spec}(D)} X$ of dimension $r+q$ such that $C$ meets each face of $\Delta^{r} \times _{\mathrm{Spec}(D)} X$ properly. We then set $z^{q}(X, r)=z_{\mathrm{dim}(X)-q}(X, r)$ so that we get a pullback homomorphism $z^{q}(X, r) \to z^{q}(X, r-1)$ for each face of $\Delta^{r}$. Then the Zariski hypercohomology of the complex $z^{q}(X, *)$ with alternating face maps is Bloch-Levine motivic cohomology (with the appropriate shift).
\[T:defofpi0andpi\]
The morphism $H\mathbb{F}^{k}_{p}\cong i^{*}(\widehat{H}\mathbb{F}^{D}_{p}) \to i^{*}j_{*}H\mathbb{F}^{K}_{p} \cong i^{*}j_{*}j^{*}\widehat{H}\mathbb{F}^{D}_{p} $ in $D(H\mathbb{F}^{k}_{p})$ induced by adjunction induces a splitting $i^{*}j_{*}H\mathbb{F}^{K}_{p}\cong H\mathbb{F}^{k}_{p} \oplus \Sigma^{-1,-1}H\mathbb{F}^{k}_{p}$ in $D(H\mathbb{F}^{k}_{p})$. We let $\pi:i^{*}j_{*}H\mathbb{F}^{K}_{p} \to H\mathbb{F}^{k}_{p}$ and $\pi_{0}:i^{*}j_{*}H\mathbb{F}^{K}_{p} \to \Sigma^{-1,-1}H\mathbb{F}^{k}_{p}$ denote the projections induced by this splitting. There is also a splitting $i^{*}j_{*}H\mathbb{Z}^{K}\cong H\mathbb{Z}^{k} \oplus \Sigma^{-1,-,1}H\mathbb{Z}^{k}$ in $D(H\mathbb{Z}^{k})$ [@FrankSpi Lemma 4.10].
Let $\eta: id. \to j_{*}j^{*}$ denote the unit map. From now on, we shall denote all adjunction morphisms $i^{*}E \to i^{*}j_{*}j^{*}E$ for $E \in SH(D)$ by $i^{*}\eta$. We will also denote all $\Sigma^{s,t}\pi, \Sigma^{s,t}\pi_{0}$ by $\pi$ and $\pi_{0}$ respectively to make the text easier to read. The morphisms $\Psi_{k}$ and $\Psi_{K}$ lift to a morphism $$\Psi_{D}:\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}\widehat{H}\mathbb{F}^{D}_{p} \to \widehat{H}\mathbb{F}^{D}_{p} \wedge \widehat{H}\mathbb{F}^{D}_{p}$$ in $D(\widehat{H}\mathbb{F}^{D}_{p})$ [@FrankSpi Lemma 3.10]. Applying $i^{*}\eta$ to $\Psi_{D}$, we get a commuting square $$\label{PsiDcommute}
\begin{tikzcd}
\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p} \arrow[r, "\Psi_{k}"] \arrow[d, "i^{*}\eta"]
& H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p} \arrow[d, "i^{*}\eta"] \\
\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}i^{*}j_{*}H\mathbb{F}^{K}_{p} \arrow[r, "i^{*}j_{*}\Psi_{K}"]
& i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})
\end{tikzcd}$$ in $D(H\mathbb{F}^{k}_{p})$. Let $r:H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p} \to \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}$ be the retraction of $\Psi_{k}$ defined by the following composite [@FrankSpi Theorem 5.1]. $$\begin{tikzcd}
H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p} \arrow[r, "i^{*}\eta"]
& i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[r, "i^{*}j_{*}\Psi^{-1}_{K}"]
& \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}i^{*}j_{*}H\mathbb{F}^{K}_{p} \arrow[d, "\oplus \pi"] \\
& & \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}
\end{tikzcd}$$
For $S=k, K,$ or $D$ (use $\widehat{H}\mathbb{F}^{D}_{p}$), we let $\mu^{S}_{1}: H\mathbb{F}^{S}_{p} \wedge H\mathbb{F}^{S}_{p} \to H\mathbb{F}^{S}_{p}$ denote the multiplication morphism. There is also a multiplication morphism $$\mu^{S}_{2}:(H\mathbb{F}^{S}_{p} \wedge H\mathbb{F}^{S}_{p})\wedge (H\mathbb{F}^{S}_{p} \wedge H\mathbb{F}^{S}_{p}) \to H\mathbb{F}^{S}_{p} \wedge H\mathbb{F}^{S}_{p}$$ defined in the standard way by interchanging the two middle $H\mathbb{F}^{S}_{p}$ terms and then applying $\mu^{S}_{1} \wedge \mu^{S}_{1}$.
For a sequence $\alpha_{0}$, we define $i^{*}\eta_{\alpha_{0}}: H\mathbb{F}^{k}_{p} \wedge H\mathbb{F}^{k}_{p} \to \Sigma ^{p_{\alpha_{0}},q_{\alpha_{0}}}H\mathbb{F}^{k}_{p}$ in $D(H\mathbb{F}^{k}_{p})$ to be the composite $$\label{eq:retractexplicit r}
\begin{tikzcd}
H\mathbb{F}^{k}_{p} \wedge H\mathbb{F}^{k}_{p} \arrow[r, "i^{*}\eta"]
& i^{*}j_{*}(H\mathbb{F}^{K}_{p} \wedge H\mathbb{F}^{K}_{p}) \arrow[r, "i^{*}j_{*}\Psi^{-1}_{K}"]
& \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}i^{*}j_{*}H\mathbb{F}^{k}_{p} \arrow[r, "proj."]
& \Sigma^{p_{\alpha_{0}},q_{\alpha_{0}}}i^{*}j_{*}H\mathbb{F}^{K}_{p} \arrow[d, "\pi"]\\
& & & \Sigma^{p_{\alpha_{0}},q_{\alpha_{0}}}H\mathbb{F}^{k}_{p}.
\end{tikzcd}$$ The morphism $i^{*}\eta_{\alpha_{0}}$ is a retract of the morphism $H\mathbb{F}^{k}_{p} \wedge \omega(\alpha_{0}):\Sigma^{p_{\alpha_{0}},q_{\alpha_{0}}}H\mathbb{F}^{k}_{p} \to H\mathbb{F}^{k}_{p} \wedge H\mathbb{F}^{k}_{p}$.
From the work of Voevodsky [@Voe2] and Friedlander-Suslin [@FriSus Corollary 12.2], Bloch’s higher Chow groups are isomorphic to motivic cohomology as defined by Voevodsky. The isomorphism between motivic cohomology and Bloch’s higher Chow groups is compatible with pullback maps and product structures [@Spi Theorem 6.7]. See also [@KonYas].
\[voesusfri\] Let $F$ be a field and let $X \in \textup{Sm}_{F}$. Then $$H^{n,i}(X, \mathbb{Z}) \cong CH^{i}(X,2i-n)$$ for all $n$ and $i \geq 0$.
Let $n,i \geq 0$ such that $ n>2i$. From the above theorem, we get that $H^{n,i}(X,A)=0$ for any coefficient ring $A$ and $X \in \textup{Sm}_{F}$.
Definition of operations {#sec:def}
========================
In this section, we use the results of Frankland and Spitzweck in [@FrankSpi] to define new Steenrod operations $P^{n}_{k}$ for $n\geq 0$. Let $$i_{L}, i_{R}: H\mathbb{F}^{S}_{p} \to H\mathbb{F}^{S}_{p} \wedge H\mathbb{F}^{S}_{p}$$ denote the left and right $H\mathbb{F}_{p}^{S}$-module maps respectively for $S=D$ (use $\widehat{H}\mathbb{F}^{D}_{p}$), $k$, or $K$. Motivated by the corresponding duality in characteristic $0$, we want to define operations $P^{n}_{k}\in H\mathbb{F}^{k \, *,*}_{p} H\mathbb{F}^{k}_{p}$ for $n \geq 0$ by taking operations dual to the $\xi^{n}_{1}$.
Let $\alpha$ be a sequence. Define $P^{\alpha}_{k}\in H\mathbb{F}^{k \, *,*}_{p} H\mathbb{F}^{k}_{p}$ by $P^{\alpha}_{k}\coloneqq i^{*}\eta_{\alpha} \circ i_{R}$. For $n\geq 0$, we let $P^{n}_{k}=P^{(0,n,0,\ldots)}_{k}$. Let $\beta_{k}=P^{(1,0,\ldots)}_{k}$.
There are corresponding operations $P^{\alpha}_{K}$ in characteristic $0$ defined from \[eq:char0dual\] by $$\begin{tikzcd}
H\mathbb{F}^{K}_{p} \arrow[r, "i_{R}"] & H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p} \arrow[r, "proj."] & \Sigma^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{K}_{p}.
\end{tikzcd}$$
To define a homomorphism $\Phi:H\mathbb{F}^{K\, *,*}_{p} H\mathbb{F}^{K}_{p} \to H\mathbb{F}^{k\, *,*}_{p} H\mathbb{F}^{k}_{p}$ of graded additive groups, let $f: H\mathbb{F}^{K}_{p} \to \Sigma ^{k,l}H\mathbb{F}^{K}_{p}$ be given. Define $\Phi(f): H\mathbb{F}^{k}_{p} \to \Sigma ^{k,l}H\mathbb{F}^{k}_{p}$ by $\Phi(f)=\pi \circ i^{*}j_{*}(f) \circ i^{*}\eta$. $$\label{eq:phi def}
\begin{tikzcd}
H\mathbb{F}^{k}_{p} \arrow[r, "i^{*}\eta"]
& i^{*}j_{*}H\mathbb{F}^{K}_{p} \arrow[r, "i^{*}j_{*}(f)"]
& \Sigma^{k,l}i^{*}j_{*}H\mathbb{F}^{K}_{p} \arrow[r, "\pi"]
& \Sigma^{k,l}H\mathbb{F}^{k}_{p}
\end{tikzcd}$$
From the definition of $\Phi$, it is clear that $\Phi(id.)=id.$. The following lemma will be important for proving that the operations $P^{n}_{k}$ restricted to mod $p$ Chow groups satisfy the Adem relations and Cartan formula.
\[lemmafornexthm\]
Let $X \in \textup{Sm}_{k}$ and let $f: \Sigma^{\infty}_{+}X \to \Sigma^{2m,m}H\mathbb{F}_{p}^{k}$ be given.
1. Let $\alpha_{0}$ be a sequence. Consider the morphism $$g_{\alpha_{0}}:H\mathbb{F}^{k}_{p} \to \Sigma^{p_{\alpha_{0}}-1,q_{\alpha_{0}}-1}H\mathbb{F}^{k}_{p}$$ given by the following composite. $$\begin{tikzcd}
H\mathbb{F}^{k}_{p} \arrow[r, "i^{*}\eta"]
& i^{*}j_{*}H\mathbb{F}^{K}_{p} \arrow[r, "i^{*}j_{*}(P^{\alpha_{0}}_{K})"]
& i^{*}j_{*}\Sigma^{p_{\alpha_{0}},q_{\alpha_{0}}}H\mathbb{F}^{K}_{p} \arrow[r, "\pi_{0}"]
& \Sigma^{p_{\alpha_{0}}-1,q_{\alpha_{0}}-1}H\mathbb{F}^{k}_{p}.
\end{tikzcd}$$ Then $\Sigma^{2m,m}g_{\alpha_{0}} \circ f=0$.
2. The composite
$$\begin{tikzcd}
\Sigma^{\infty}_{+}X \arrow[r, "f"] & \Sigma^{2m,m}H\mathbb{F}_{p}^{k} \arrow[r, "i_{R}"] & \Sigma^{2m,m}H\mathbb{F}_{p}^{k} \wedge H\mathbb{F}_{p}^{k} \arrow[r, "i^{*}\eta"] & i^{*}j_{*}(\Sigma^{2m,m}H\mathbb{F}_{p}^{K} \wedge H\mathbb{F}_{p}^{K}) \arrow[d, "i^{*}j_{*}\Psi_{K}^{-1}"] \\
& & & \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha}+2m,q_{\alpha}+m}i^{*}j_{*}H\mathbb{F}^{K}_{p} \arrow[d,"\oplus \pi_{0}"] \\
& & & \bigoplus\limits _{\alpha} \Sigma ^{2m+p_{\alpha}-1,m+q_{\alpha}-1}H\mathbb{F}^{k}_{p}
\end{tikzcd}$$ is equal to $0$.
Note that for any sequence $\alpha$ of bidegree $(p_{\alpha},q_{\alpha})$, $p_{\alpha} \geq 2q_{\alpha}$ which implies that $p_{\alpha}-1> 2(q_{\alpha}-1)$. For $(1)$ and $(2)$, Theorem \[voesusfri\] implies that $$\textup{Hom}_{SH(k)}(\Sigma^{\infty}_{+}X, \Sigma^{2m+p_{\alpha}-1, m+q_{\alpha}-1}H\mathbb{F}_{p}^{k})=H^{2m+p_{\alpha}-1, m+q_{\alpha}-1}(X, \mathbb{F}_{p})=0$$ for any sequence $\alpha$.
\[T:phiproperties\]
1. We have $\Phi(H^{*,*}(K, \mathbb{F}_{p}))\subset H^{*,*}(k, \mathbb{F}_{p}).$
2. Let $\alpha$ be a sequence. Then $\Phi(P^{\alpha}_{K})=P^{\alpha}_{k}$. In particular, for the Bockstein $\beta_{K}$ and reduced power operations $P^{n}_{K}$ constructed by Voevodsky in characteristic $0$, $\Phi(P^{n}_{K})=P^{n}_{k}$ for $n \geq 0$ and $\Phi(\beta_{K})=\beta_{k}$. Also, $P^{0}_{k}$ is the identity since $P^{0}_{K}$ is the identity.
3. Let $X \in \textup{Sm}_{k}$ and let $f: \Sigma^{\infty}_{+}X \to \Sigma^{2m,m}H\mathbb{F}_{p}^{k}$ be given. Let $\alpha$ be a sequence and let $h:H\mathbb{F}_{p}^{K} \to \Sigma^{i,j}H\mathbb{F}_{p}^{K}$ be given. Then $$\Phi(h\circ P^{\alpha}_{K})(f)=\Phi(h)(P^{\alpha}_{k}(f)).$$
We first prove $(1)$. Let $a\in H^{*,*}(K, \mathbb{F}_{p})$. The element $a$ corresponds to a morphism $f_{a}: H\mathbb{F}^{K}_{p} \to\Sigma^{m,n}H\mathbb{F}^{K}_{p}$ in $D(H\mathbb{F}^{K}_{p})$. The functors $i^{*}$, $j_{*}$ restrict to functors $i^{*}:D(\widehat{H}\mathbb{F}^{D}_{p})\to D(H\mathbb{F}^{k}_{p})$ and $j_{*}:D(H\mathbb{F}^{K}_{p})\to D(\widehat{H}\mathbb{F}^{D}_{p})$. Hence, $i^{*}j_{*}(f_{a})$ is a morphism in $D(H\mathbb{F}^{k}_{p})$. From the definition of $\Phi$, it follows that $\Phi(f_{a})$ is a morphism in $D(H\mathbb{F}^{k}_{p}).$ Thus, $\Phi(a) \coloneqq \Phi(f_{a})\in H^{*,*}(k, \mathbb{F}_{p}).$
We now prove $(2)$. Let $\alpha$ be a sequence. Applying the natural transformation $i^{*} \to i^{*}j_{*}j^{*}$ to $i_{R}: \widehat{H}\mathbb{F}^{D}_{p} \to \widehat{H}\mathbb{F}^{D}_{p} \wedge \widehat{H}\mathbb{F}^{D}_{p}$, we obtain the following commuting square in $SH(k)$. $$\begin{tikzcd}
H\mathbb{F}^{k}_{p} \arrow[r, "i_{R}"] \arrow[d, "i^{*}\eta"]
& H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p} \arrow[d, "i^{*}\eta"] \\
i^{*}j_{*}H\mathbb{F}^{K}_{p} \arrow[r, "i^{*}j_{*}(i_{R})"]
& i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})
\end{tikzcd}$$ From the definition of $i^{*}\eta_{\alpha}$ \[eq:retractexplicit r\], the following diagram commutes. $$\begin{tikzcd}
H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p} \arrow[r, "i^{*}\eta_{\alpha}"] \arrow[d, "i^{*}\eta"]
& \Sigma^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p} \arrow[r, "id."]
& \Sigma^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p} \arrow[d, "id."] \\
i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[r, "proj."]
& i^{*}j_{*}\Sigma^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{K}_{p} \arrow[r, "\pi"]
& \Sigma^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}
\end{tikzcd}$$
Putting these 2 diagrams together, we get the following commuting diagram.
$$\label{eq:use for next ref}
\begin{tikzcd}
H\mathbb{F}^{k}_{p} \arrow[r, "i_{R}"] \arrow[d, "i^{*}\eta"]
& H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p} \arrow[r, "i^{*}\eta_{\alpha}"] \arrow[d, "i^{*}\eta"]
& \Sigma^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p} \arrow[r, "id."]
& \Sigma^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p} \arrow[d, "id."] \\
i^{*}j_{*}H\mathbb{F}^{K}_{p} \arrow[r, "i^{*}j_{*}(i_{R})"]
& i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[r, "proj."]
& i^{*}j_{*}\Sigma^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{K}_{p} \arrow[r, "\pi"]
& \Sigma^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}
\end{tikzcd}$$
The top row of this diagram gives $P^{\alpha}_{k}$ while the composite starting at $H\mathbb{F}^{k}_{p}$ in the top left and continuing along the bottom row gives $\Phi(P^{\alpha}_{K})$. Hence, $\Phi(P^{\alpha}_{K})=P^{\alpha}_{k}$.
Now, we prove $(3)$. Consider the following diagram. $$\label{eq:phi commut diag}
\begin{tikzcd}
\Sigma^{\infty}_{+}X \arrow[d, "f"] \\
\Sigma^{2m,m}H\mathbb{F}^{k}_{p} \arrow[r, "P^{\alpha}_{k}"] \arrow[d, "i^{*}\eta"]
&\Sigma^{2m+p_{\alpha},m+q_{\alpha}}H\mathbb{F}^{k}_{p} \arrow[r, "\Phi(h)"] \arrow[d, "i^{*}\eta"]
& \Sigma^{i+2m+p_{\alpha},j+m+q_{\alpha}}H\mathbb{F}^{k}_{p} \arrow[d, "i^{*}\eta"] \\
i^{*}j_{*}\Sigma^{2m,m}H\mathbb{F}^{K}_{p} \arrow[r, "i^{*}j_{*}P^{\alpha}_{K}"]
& i^{*}j_{*}\Sigma^{2m+p_{\alpha},m+q_{\alpha}}H\mathbb{F}^{K}_{p} \arrow[r, "i^{*}j_{*}h"]
& i^{*}j_{*}\Sigma^{i+2m+p_{\alpha},j+m+q_{\alpha}}H\mathbb{F}^{K}_{p} \arrow[d, "\pi"]\\
& & \Sigma^{i+2m+p_{\alpha},j+m+q_{\alpha}}H\mathbb{F}^{k}_{p}.
\end{tikzcd}$$
As $\Phi(P^{\alpha}_{K})=P^{\alpha}_{k},$ Lemma \[lemmafornexthm\] implies that the composite $$i^{*}\eta \circ P^{\alpha}_{k} \circ f:\Sigma^{\infty}_{+}X \to i^{*}j_{*}\Sigma^{2m+p_{\alpha},m+q_{\alpha}}H\mathbb{F}^{K}_{p}$$ in diagram \[eq:phi commut diag\] is equal to $$i^{*}j_{*}P^{\alpha}_{K} \circ i^{*}\eta \circ f.$$ Equivalently, $$\pi_{0} \circ i^{*}j_{*}P^{\alpha}_{K} \circ i^{*}\eta \circ f=0:\Sigma^{\infty}_{+}X \to \Sigma^{2m+p_{\alpha}-1,m+q_{\alpha}-1}H\mathbb{F}^{k}_{p} .$$
Thus, from diagram \[eq:phi commut diag\], $$\Phi(h)(P^{\alpha}_{k}(f))=\pi \circ i^{*}\eta \circ \Phi(h) \circ P^{\alpha}_{k} \circ f=\pi \circ i^{*}j_{*}(h)\circ i^{*}j_{*}(P^{\alpha}_{K}) \circ i^{*}\eta \circ f= \Phi(h\circ P^{\alpha}_{K})(f)$$ as desired.
We next prove that the operations $P^{n}_{k}$ commute with base change of the field $k$ on mod $p$ Chow groups. For a morphism of fields $f:\textup{Spec}(F_{1}) \to \textup{Spec}(F_{2})$, the pullback functor $f^{*}:SH(F_{2}) \to SH(F_{1})$ induces a homomorphism $H\mathbb{F}^{F_{2} \, *,*}_{p} H\mathbb{F}^{F_{2}}_{p} \to H\mathbb{F}^{F_{1} \, *,*}_{p} H\mathbb{F}^{F_{1}}_{p}$. For $\textup{char}(F_{2}) \neq p$, $f^{*}(P^{n}_{F_{2}})=P^{n}_{F_{1}}$ since the dual Steenrod algebra has the expected form in this case [@HKO Theorem 1.1]. However, for our situation where the base field is of characteristic $p$, we do not yet know the full structure of the dual Steenrod algebra.
Let $f_{1}:\textup{Spec}(k) \to \textup{Spec}(\mathbb{F}_{p})$ be the structure map. In the following commuting diagram, $f_{2}$, $f_{3}$, $i_{0}$, and $j_{0}$ are maps compatible with $f_{1}$. $$\begin{tikzcd}
\textup{Spec}(k) \arrow[r, "f_{1}"] \arrow[d, "i"] & \textup{Spec}(\mathbb{F}_{p}) \arrow[d, "i_{0}"] \\
\textup{Spec}(D) \arrow[r, "f_{2}"] & \textup{Spec}(\mathbb{Z}_{p}) \\
\textup{Spec}(K) \arrow[r, "f_{3}"] \arrow[u, "j"] & \textup{Spec}(\mathbb{Q}_{p}) \arrow[u, "j_{0}"]
\end{tikzcd}$$
\[P:pullback\] Let $X \in \textup{Sm}_{k}$ and let $g: \Sigma^{\infty}_{+}X \to \Sigma^{2m,m}H\mathbb{F}_{p}^{k}$ be given. Then $P^{n}_{k}(g)=f_{1}^{*}(P^{n}_{\mathbb{F}_{p}})(g)$ for all $n \geq 0$.
Let $\eta_{0}:1 \to j_{0 \,*}j^{*}_{0}$ denote the unit map. Let $f_{2}^{*}\widehat{H}\mathbb{F}_{p}^{\mathbb{Z}_{p}} \to f_{2}^{*}j_{0 \,*}H\mathbb{F}_{p}^{\mathbb{Q}_{p}}$ be the map $f_{2}^{*}\eta_{0}$ induced by the isomorphism $j_{0}^{*}\widehat{H}\mathbb{F}_{p}^{\mathbb{Z}_{p}} \to H\mathbb{F}_{p}^{\mathbb{Q}_{p}}$. The exchange transformation $f_{2}^{*}j_{0 \,*}\to j_{*}f_{3}^{*}$ induces a morphism $f_{2}^{*}j_{0 \,*}H\mathbb{F}_{p}^{\mathbb{Q}_{p}} \to j_{*}f^{*}_{3}H\mathbb{F}_{p}^{\mathbb{Q}_{p}}$. Let $f_{2}^{*}\widehat{H}\mathbb{F}_{p}^{\mathbb{Z}_{p}} \to j_{*}f^{*}_{3}H\mathbb{F}_{p}^{\mathbb{Q}_{p}}$ be the map $\eta f^{*}_{2}$ induced by the isomorphism $$j^{*}f^{*}_{2} \widehat{H}\mathbb{F}_{p}^{\mathbb{Z}_{p}}\cong f^{*}_{3}j^{*}_{0}\widehat{H}\mathbb{F}_{p}^{\mathbb{Z}_{p}} \to f_{3}^{*}H\mathbb{F}_{p}^{\mathbb{Q}_{p}}.$$ Putting these maps together, we get the following square which commutes by adjunction.
$$\label{diagpullback2}
\begin{tikzcd}
f_{2}^{*}\widehat{H}\mathbb{F}_{p}^{\mathbb{Z}_{p}} \arrow[r, "f_{2}^{*}\eta_{0}"] \arrow[d, "id."] & f_{2}^{*}j_{0 \,*}H\mathbb{F}_{p}^{\mathbb{Q}_{p}} \arrow[d, ""] \\
f_{2}^{*}\widehat{H}\mathbb{F}_{p}^{\mathbb{Z}_{p}} \arrow[r, "\eta f^{*}_{2}"] & j_{*}f^{*}_{3}H\mathbb{F}_{p}^{\mathbb{Q}_{p}}
\end{tikzcd}$$
Applying the exchange transformation $f_{2}^{*}j_{0 \,*}\to j_{*}f_{3}^{*}$ to $P^{n}_{\mathbb{Q}_{p}}$, we get the following commuting square.
$$\begin{tikzcd}
f_{2}^{*}j_{0 \,*}H\mathbb{F}_{p}^{\mathbb{Q}_{p}} \arrow[r,"f_{2}^{*}j_{0 \,*}P^{n}_{\mathbb{Q}_{p}}"] \arrow[d, ""] & f_{2}^{*}j_{0 \,*}\Sigma^{2n(p-1),n(p-1)}H\mathbb{F}_{p}^{\mathbb{Q}_{p}} \arrow[d,""] \\
j_{*}H\mathbb{F}_{p}^{K} \arrow[r, "j_{*}P^{n}_{K}"] & j_{*}\Sigma^{2n(p-1),n(p-1)}H\mathbb{F}_{p}^{K}
\end{tikzcd}$$ Applying $i^{*}$ (and the connection isomorphism $i^{*}f_{2}^{*} \cong f^{*}_{1}i^{*}_{0}$) to these two squares and combining with $g: \Sigma^{\infty}_{+}X \to \Sigma^{2m,m}H\mathbb{F}_{p}^{k}$ , we obtain the following commuting diagram.
$$\label{diagpullback3}
\begin{tikzcd}
\Sigma^{\infty}_{+}X \arrow[r, "g"] \arrow[d, "id."] & \Sigma^{2m,m}H\mathbb{F}_{p}^{k} \arrow[r,"f_{1}^{*}i_{0}^{*}\eta_{0}"] \arrow[d, "id."] & f_{1}^{*}i_{0}^{*} j_{0 \,*}\Sigma^{2m,m}H\mathbb{F}_{p}^{\mathbb{Q}_{p}} \arrow[r, "f_{1}^{*}i_{0}^{*} j_{0 \,*}P^{n}_{\mathbb{Q}_{p}}"]\arrow[d, ""]
& f_{1}^{*}i_{0}^{*} j_{0 \,*}\Sigma^{2(m+n(p-1)),m+n(p-1)}H\mathbb{F}_{p}^{\mathbb{Q}_{p}} \arrow[d, ""]\\
\Sigma^{\infty}_{+}X \arrow[r, "g"] & \Sigma^{2m,m}H\mathbb{F}_{p}^{k} \arrow[r, "i^{*}\eta"]& i^{*}j_{*}\Sigma^{2m,m}H\mathbb{F}_{p}^{K} \arrow[r, "i^{*}j_{*}P^{n}_{K}"] & i^{*}j_{*}\Sigma^{2(m+n(p-1)),m+n(p-1)}H\mathbb{F}_{p}^{K}
\end{tikzcd}$$
Let $\pi':i_{0}^{*} j_{0 \,*}H\mathbb{F}_{p}^{\mathbb{Q}_{p}} \to H\mathbb{F}_{p}^{\mathbb{F}_{p}}$ and $\pi_{0}':i_{0}^{*} j_{0 \,*}H\mathbb{F}_{p}^{\mathbb{Q}_{p}} \to \Sigma^{-1,-1}H\mathbb{F}_{p}^{\mathbb{F}_{p}}$ be projection morphisms induced by the isomorphism $i_{0}^{*} j_{0 \,*}H\mathbb{F}_{p}^{\mathbb{Q}_{p}} \cong H\mathbb{F}_{p}^{\mathbb{F}_{p}} \oplus \Sigma^{-1,-1}H\mathbb{F}_{p}^{\mathbb{F}_{p}}$ of Theorem \[T:defofpi0andpi\]. From Theorem \[voesusfri\], the two composites $\Sigma^{\infty}_{+}X \to \Sigma^{2(m+n(p-1))-1,m+n(p-1)-1}H\mathbb{F}_{p}^{k}$ given by the following diagram are equal to $0$.
$$\label{diagpullback4}
\begin{tikzcd}
& & & \Sigma^{2(m+n(p-1))-1,m+n(p-1)-1}H\mathbb{F}_{p}^{k}\\
\Sigma^{\infty}_{+}X \arrow[r, "g"] \arrow[d, "id."] & \Sigma^{2m,m}H\mathbb{F}_{p}^{k} \arrow[r,"f_{1}^{*}i_{0}^{*}\eta_{0}"] \arrow[d, "id."] & f_{1}^{*}i_{0}^{*} j_{0 \,*}\Sigma^{2m,m}H\mathbb{F}_{p}^{\mathbb{Q}_{p}} \arrow[r, "f_{1}^{*}i_{0}^{*} j_{0 \,*}P^{n}_{\mathbb{Q}_{p}}"]\arrow[d, ""]
& f_{1}^{*}i_{0}^{*} j_{0 \,*}\Sigma^{2(m+n(p-1)),m+n(p-1)}H\mathbb{F}_{p}^{\mathbb{Q}_{p}} \arrow[d, ""] \arrow[u, "f^{*}_{1}\pi_{0}'"]\\
\Sigma^{\infty}_{+}X \arrow[r, "g"] & \Sigma^{2m,m}H\mathbb{F}_{p}^{k} \arrow[r, "i^{*}\eta"]& i^{*}j_{*}\Sigma^{2m,m}H\mathbb{F}_{p}^{K} \arrow[r, "i^{*}j_{*}P^{n}_{K}"] & i^{*}j_{*}\Sigma^{2(m+n(p-1)),m+n(p-1)}H\mathbb{F}_{p}^{K} \arrow[d, "\pi_{0}"] \\
& & & \Sigma^{2(m+n(p-1))-1,m+n(p-1)-1}H\mathbb{F}_{p}^{k}
\end{tikzcd}$$
Consider the following diagram.
$$\label{diagpullback}
\begin{tikzcd}
& & & \Sigma^{2(m+n(p-1)),m+n(p-1)}H\mathbb{F}_{p}^{k}\\
\Sigma^{\infty}_{+}X \arrow[r, "g"] \arrow[d, "id."] & \Sigma^{2m,m}H\mathbb{F}_{p}^{k} \arrow[r,"f_{1}^{*}i_{0}^{*}\eta_{0}"] \arrow[d, "id."] & f_{1}^{*}i_{0}^{*} j_{0 \,*}\Sigma^{2m,m}H\mathbb{F}_{p}^{\mathbb{Q}_{p}} \arrow[r, "f_{1}^{*}i_{0}^{*} j_{0 \,*}P^{n}_{\mathbb{Q}_{p}}"]\arrow[d, ""]
& f_{1}^{*}i_{0}^{*} j_{0 \,*}\Sigma^{2(m+n(p-1)),m+n(p-1)}H\mathbb{F}_{p}^{\mathbb{Q}_{p}} \arrow[d, ""] \arrow[u, "f^{*}_{1}\pi'"]\\
\Sigma^{\infty}_{+}X \arrow[r, "g"] & \Sigma^{2m,m}H\mathbb{F}_{p}^{k} \arrow[r, "i^{*}\eta"]& i^{*}j_{*}\Sigma^{2m,m}H\mathbb{F}_{p}^{K} \arrow[r, "i^{*}j_{*}P^{n}_{K}"] & i^{*}j_{*}\Sigma^{2(m+n(p-1)),m+n(p-1)}H\mathbb{F}_{p}^{K} \arrow[d, "\pi"] \\
& & & \Sigma^{2(m+n(p-1)),m+n(p-1)}H\mathbb{F}_{p}^{k}
\end{tikzcd}$$
From Theorem \[T:phiproperties\], the composite $\Sigma^{\infty}_{+}X \to \Sigma^{2(m+n(p-1)),m+n(p-1)}H\mathbb{F}_{p}^{k}$ given by the upper half of diagram \[diagpullback\] is equal to $f_{1}^{*}(P^{n}_{\mathbb{F}_{p}})(g)$ and the composite $\Sigma^{\infty}_{+}X \to \Sigma^{2(m+n(p-1)),m+n(p-1)}H\mathbb{F}_{p}^{k}$ given by the lower half of diagram \[diagpullback\] is equal to $P^{n}_{k}(g)$. As diagram \[diagpullback3\] commutes and the $2$ composite morphisms from diagram \[diagpullback4\] are $0$, we then obtain that $f_{1}^{*}(P^{n}_{\mathbb{F}_{p}})(g)=P^{n}_{k}(g)$.
We can now prove that the Steenrod operations $P^{n}_{k}$ commute with base change on mod $p$ Chow groups. Let $f:\textup{Spec}(k_{1}) \to \textup{Spec}(k_{2})$ be given where $k_{1}, k_{2}$ are fields of characteristic $p$. Let $h:\textup{Spec}(k_{2}) \to \textup{Spec}(\mathbb{F}_{p})$ be the structure map.
\[corbasechange\] Let $X \in \textup{Sm}_{k_{2}}$. Let $n \geq 0$. The following square commutes.
$$\begin{tikzcd}
CH^{*}(X)/p \arrow[r, "P^{n}_{k_{2}}"] \arrow[d, "f^{*}"] &
CH^{*}(X)/p \arrow[d, "f^{*}"] \\
CH^{*}(X_{k_{1}})/p \arrow[r, "P^{n}_{k_{1}}"] & CH^{*}(X_{k_{1}})/p
\end{tikzcd}$$
From Proposition \[P:pullback\], $h^{*}P^{n}_{\mathbb{F}_{p}}$ agrees with $P^{n}_{k_{2}}$ on $CH^{*}(X)/p$ and $f^{*}h^{*}P^{n}_{\mathbb{F}_{p}}$ agrees with $P^{n}_{k_{1}}$ on $CH^{*}(X_{k_{1}})/p $. Let $g:\Sigma^{\infty}_{+}X \to \Sigma^{2m,m}H\mathbb{F}_{p}^{k_{2}}$ be given. Then $$f^{*}(P^{n}_{k_{2}}(g))=f^{*}(h^{*}P^{n}_{\mathbb{F}_{p}}(g))=f^{*}h^{*}(P^{n}_{\mathbb{F}_{p}})(f^{*}g)=P^{n}_{k_{1}}(f^{*}g)$$ as required.
The morphism $\beta_{k}=P^{(1,0,\ldots)}_{k}$ defined above is equal to the Bockstein homomorphism $\beta$ on mod $p$ motivic cohomology.
We let $\beta$ denote the Bockstein homomorphism on mod $p$ motivic cohomology over any base scheme. The Bockstein homomorphism $\beta$ in characteristic $0$ is known to be dual to $\tau_{0}$. Hence, $\beta=P^{(1,0,\ldots)}_{K}=\beta_{K}$. Applying the natural transformation $i^{*} \to i^{*}j_{*}j^{*}$ to the diagram $$\begin{tikzcd}
\widehat{H}\mathbb{Z}^{D} \arrow[r, "\cdot p"]
& \widehat{H}\mathbb{Z}^{D} \arrow[r, ""]
&(\widehat{H}\mathbb{Z}^{D})/p \arrow[r, ""] \arrow[rr, bend left, "\beta"]
& \Sigma^{1,0}\widehat{H}\mathbb{Z}^{D} \arrow[r, "proj."]
& \Sigma^{1,0}\widehat{H}\mathbb{Z}^{D}/p
\end{tikzcd}$$ in $SH(D)$, we get the following commuting diagram in $SH(k)$. $$\label{eq:bocktri}
\begin{tikzcd}
H\mathbb{Z}^{k} \arrow[r, "\cdot p"] \arrow[d, "i^{*}\eta"]
& H\mathbb{Z}^{k} \arrow[r, "proj."] \arrow[d, "i^{*}\eta"]
& H\mathbb{F}^{k}_{p} \arrow[r, ""] \arrow[rr, bend left, "\beta"] \arrow[d, "i^{*}\eta"]
& \Sigma^{1,0}H\mathbb{Z}^{k} \arrow[r, "proj."] \arrow[d, "i^{*}\eta"]
& \Sigma^{1,0}H\mathbb{F}^{k}_{p} \arrow[d, "i^{*}\eta"] \\
i^{*}j_{*}H\mathbb{Z}^{K} \arrow[r, "\cdot p"]
& i^{*}j_{*}H\mathbb{Z}^{K} \arrow[r, "proj."]
& i^{*}j_{*}H\mathbb{F}^{K}_{p} \arrow[r, ""] \arrow[rr, bend right, "i^{*}j_{*}\beta_{K}"]
& \Sigma^{1,0}i^{*}j_{*}H\mathbb{Z}^{K} \arrow[r, "proj."]
& \Sigma^{1,0}i^{*}j_{*}H\mathbb{F}^{K}_{p} \arrow[d, "\pi"] \\
& & & & \Sigma^{1,0}H\mathbb{F}^{k}_{p}
\end{tikzcd}$$ From Theorem \[T:phiproperties\], $\Phi(\beta_{K})=\beta_{k}$. The composite in diagram \[eq:bocktri\] that starts at $H\mathbb{F}^{k}_{p}$ in the top row and goes immediately down to $\Sigma^{1,0}H\mathbb{F}^{k}_{p}$ is equal to $\Phi(\beta_{K})$. As the diagram commutes and $\pi \circ i^{*}\eta=id.$, it follows that $\Phi(\beta_{K})=\beta=\beta_{k}$.
Adem relations
==============
In this section, we use the map $\Phi:H\mathbb{F}^{K\, *,*}_{p} H\mathbb{F}^{K}_{p} \to H\mathbb{F}^{k\, *,*}_{p} H\mathbb{F}^{k}_{p}$ \[eq:phi def\] and Theorem \[T:phiproperties\] to show that the operations $P^{n}_{k}$ for $n \geq 0$ satisfy the expected Adem relations when restricted to mod $p$ Chow groups. The proof uses the corresponding Adem relations in characteristic $0$ which can be found in [@Rio Théorème 4.5.1] for $p=2$ and [@Rio Théorème 4.5.2 ] for odd $p$. First, we state the Adem relations for $p=2$ over the base $K$ of characteristic $0$. Let $\tau \in H^{0,1}(K, \mathbb{F}_{2})$ denote the class of $-1 \in \mu_{2}(K)$ and let $\rho \in H^{1,1}(K, \mathbb{F}_{2})$ denote the class of $-1 \in K^{*}/K^{* \, 2}$. Set $\textrm{Sq}^{2n}_{k}\coloneqq P^{n}_{k}$ and $\textrm{Sq}^{2n+1}_{k}=\beta_{k}\textrm{Sq}^{2n}_{k}$ for $n \geq0$.
\[T:char0adem\] Let $a,b \in \mathbb{N}$ with $a<2b$.
1. $$\mathrm{Sq}^{a}_{K}\mathrm{Sq}^{b}_{K}=
\sum_{j=0}^{\lfloor \frac{a}{2} \rfloor}\binom{b-1-j}{a-2j}\mathrm{Sq}^{a+b-j}_{K}\mathrm{Sq}^{j}_{K} +\sum_{\substack{j=1 \\ j\, odd}}^{\lfloor \frac{a}{2} \rfloor}\rho \binom{b-1-j}{a-2j} \mathrm{Sq}^{a+b-j-1}_{K}\mathrm{Sq}^{j}_{K}$$ if $a$ is even and $b$ is odd.\
2. $$\mathrm{Sq}^{a}_{K}\mathrm{Sq}^{b}_{K}=
\sum_{\substack{j=0 \\ j\, odd}}^{\lfloor \frac{a}{2} \rfloor}\binom{b-1-j}{a-2j}\mathrm{Sq}^{a+b-j}_{K}\mathrm{Sq}^{j}_{K}$$ if $a$ and $b$ are odd.\
3. $$\mathrm{Sq}^{a}_{K}\mathrm{Sq}^{b}_{K}=
\sum_{\substack{j=0 \\ }}^{\lfloor \frac{a}{2} \rfloor}\tau^{j \, \textup{mod}\, 2}\binom{b-1-j}{a-2j}\mathrm{Sq}^{a+b-j}_{K}\mathrm{Sq}^{j}_{K}$$ if $a$ and $b$ are even.\
4. $$\mathrm{Sq}^{a}_{K}\mathrm{Sq}^{b}_{K}=
\sum_{\substack{j=0 \\ j\, even}}^{\lfloor \frac{a}{2} \rfloor}\binom{b-1-j}{a-2j}\mathrm{Sq}^{a+b-j}_{K}\mathrm{Sq}^{j}_{K} +\sum_{\substack{j=1 \\ j\, odd}}^{\lfloor \frac{a}{2} \rfloor}\rho \binom{b-1-j}{a-1-2j}\mathrm{Sq}^{a+b-j-1}_{K}\mathrm{Sq}^{j}_{K}$$ if $a$ is odd and $b$ is even.
Next, we state the characteristic $0$ Adem relations for $p$ odd.
1. Let $a,b \in \mathbb{N}$ with $a<pb$. Then $$P^{a}_{K}P^{b}_{K}=\sum^{\lfloor \frac{a}{p} \rfloor}_{j=0}(-1)^{a+j}\binom{(p-1)(b-j)-1}{a-pj}P^{a+b-j}_{K}P^{j}_{K}.$$
2. Let $a,b \in \mathbb{N}$ with $a\leq pb$. Then $$P^{a}_{K}\beta_{K}P^{b}_{K}=\sum^{\lfloor \frac{a}{p} \rfloor}_{j=0}(-1)^{a+j}\binom{(p-1)(b-j)-1}{a-pj}\beta_{K}P^{a+b-j}_{K}P^{j}_{K}+$$
$$\sum^{\lfloor \frac{a-1}{p} \rfloor}_{j=0}(-1)^{a+j+1}\binom{(p-1)(b-j)-1}{a-pj-1}P^{a+b-j}_{K}\beta_{K}P^{j}_{K}.$$
We can now prove the Adem relations for the operations $P^{n}_{k}$ restricted to mod $p$ Chow groups.
Let $X \in \textup{Sm}_{k}$ and let $x \in H^{2m,m}(X,\mathbb{F}_{p})=CH^{m}(X)/p$ for some $m \geq 0$. Let $a,b \in \mathbb{N}$ such that $a<pb$. Then $$P^{a}_{k}(P^{b}_{k}(x))=\sum^{\lfloor \frac{a}{p} \rfloor}_{j=0}(-1)^{a+j}\binom{(p-1)(b-j)-1}{a-pj}P^{a+b-j}_{k}(P^{j}_{k}(x)).$$
From Theorem \[T:phiproperties\], $P^{a}_{k}(P^{b}_{k}(x))=\Phi(P^{a}_{K}P^{b}_{K})(x)$. We then use the Adem relations in characteristic $0$ to rewrite $P^{a}_{K}P^{b}_{K} \in H\mathbb{F}^{K \, *,*}_{p} H\mathbb{F}^{K}_{p}$. Note that the Bockstein $\beta_{k}$ is the $0$ homomorphism on mod $p$ Chow groups. If $p=2$, $\Phi(\textup{Sq}^{n}_{K})(x)= \textup{Sq}^{n}_{k}(x)=0$ whenever $n$ is odd. Thus, applying Theorem \[T:phiproperties\], we get $$P^{a}_{k}(P^{b}_{k}(x))=\Phi(P^{a}_{K}P^{b}_{K})(x)=\Phi(\sum^{\lfloor \frac{a}{p} \rfloor}_{j=0}(-1)^{a+j}\binom{(p-1)(b-j)-1}{a-pj}P^{a+b-j}_{K}P^{j}_{K})(x)$$
$$=\sum^{\lfloor \frac{a}{p} \rfloor}_{j=0}(-1)^{a+j}\binom{(p-1)(b-j)-1}{a-pj}P^{a+b-j}_{k}(P^{j}_{k}(x)).$$
Coaction map for smooth $X$
============================
In this section, for $X\in \textup{Sm}_{k}$, we describe a coaction map $$\lambda_{X}:H^{*,*}(X, \mathbb{F}_{p})\to \pi_{-*,-*}(\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}) \otimes_{\pi_{-*,-*}H\mathbb{F}^{k}_{p}} H^{*,*}(X, \mathbb{F}_{p})$$ such that the actions of the cohomology operations $P^{n}_{k}$ defined in Section \[sec:def\] on $H^{*,*}(X, \mathbb{F}_{p})$ are determined by $\lambda_{X}$. We show that $\lambda_{X}$ is a ring homomorphism when restricted to mod $p$ Chow groups. This will allow us to prove the Cartan formula in the next section.
There is a multiplication morphism $$\label{eq:defofm} m:(\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}) \wedge (\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}) \to \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}$$ defined as $m=r \circ \mu_{2}^{k} \circ (\Psi_{k} \wedge \Psi_{k})$. The morphism $m$ defines multiplication on $$(\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p})^{*,*}(\Sigma^{\infty}_{+}X)$$ and $$\pi_{*,*}(\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}).$$ For sequences $\alpha_{1}, \alpha_{2}$, Proposition \[P:tau2=0\] allows us to calculate the product $$r_{*}(\omega(\alpha_{1}))r_{*}(\omega(\alpha_{2}))\in \pi_{*,*}(\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p})$$ in terms of another sequence $\alpha_{1}+\alpha_{2}$ by using the relations $\tau_{i}^{2}=0$ for $i\geq0$.
\[P:coactidentitensor\] The natural ring homomorphism $$\pi_{-*,-*}(\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}) \otimes_{\pi_{-*,-*}H\mathbb{F}^{k}_{p}}H^{*,*}(X, \mathbb{F}_{p}) \to (\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p})^{*,*}(\Sigma^{\infty}_{+}X)$$ is an isomorphism.
The suspension spectrum $\Sigma^{\infty}_{+}X \in SH(k)$ is compact. Hence, $$\mathrm{Hom}_{SH(k)}(\Sigma^{s,t}\Sigma^{\infty}_{+}X, \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}) \cong \bigoplus\limits _{\alpha}\mathrm{Hom}_{SH(k)}(\Sigma^{s,t}\Sigma^{\infty}_{+}X, \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p})$$ for all $s,t \in \mathbb{Z}$.
Using the isomorphism $$(\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p})^{*,*}(\Sigma^{\infty}_{+}X) \cong \pi_{-*,-*}(\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}) \otimes_{\pi_{-*,-*}H\mathbb{F}^{k}_{p}}H^{*,*}(X, \mathbb{F}_{p})$$ from Proposition \[P:coactidentitensor\] , define an additive homomorphism of graded abelian groups $$\lambda_{X}:H^{*,*}(X, \mathbb{F}_{p})\to \pi_{-*,-*}(\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}) \otimes_{\pi_{-*,-*}H\mathbb{F}^{k}_{p}} H^{*,*}(X, \mathbb{F}_{p})$$ by the composite $$\label{coacitonmapdef}
\begin{tikzcd}
H\mathbb{F}^{k \, *,*}_{p}(\Sigma^{\infty}_{+}X) \arrow[r, "i_{R \, *}"]
& (H\mathbb{F}^{k}_{p} \wedge H\mathbb{F}^{k}_{p})^{*,*}(\Sigma^{\infty}_{+}X) \arrow[d, "r_{*}"] \\
& \pi_{-*,-*}(\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}) \otimes_{\pi_{-*,-*}H\mathbb{F}^{k}_{p}}H^{*,*}(X, \mathbb{F}_{p}).
\end{tikzcd}$$
Restricted to mod $p$ Chow groups, $\lambda_{X}$ preserves multiplication. \[P:homringspec\]
Let $f: \Sigma^{\infty}_{+}X \to \Sigma^{2m,m}H\mathbb{F}_{p}^{k}$ and $g: \Sigma^{\infty}_{+}X \to \Sigma^{2n,n}H\mathbb{F}_{p}^{k}$ be given. We need to show that $\lambda_{X}(fg)=\lambda_{X}(f)\lambda_{X}(g).$ The right $H\mathbb{F}_{p}^{k}$ map $i_{R}$ is a morphism of commutative ring spectra. Hence, $i_{R *}$ is a homomorphism of rings. Hence, we need to prove that $r_{*}(i_{R *}(f)i_{R *}(g))=r_{*}(i_{R *}(f))r_{*}(i_{R *}(g)).$
Applying the natural transformation $i^{*} \to i^{*}j_{*}j^{*}$ to $\mu^{D}_{2}$, we get a commuting diagram. $$\label{eq:ringhomspectra1diag}
\begin{tikzcd}
(H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p})\wedge(H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p}) \arrow[r, "\mu^{k}_{2}"] \arrow[d, "i^{*}\eta"]
& H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p} \arrow[d, "i^{*}\eta"] \\
i^{*}j_{*}((H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})\wedge(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})) \arrow[r, "i^{*}j_{*}\mu^{K}_{2}"]
& i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[d, "\oplus \pi"] \\
& \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}
\end{tikzcd}$$ We will factor the left vertical morphism in this diagram. Consider the following triangle
$$\label{eq:triangle}
\begin{tikzcd}
(\widehat{H}\mathbb{F}^{D}_{p}\wedge \widehat{H}\mathbb{F}^{D}_{p})\wedge(\widehat{H}\mathbb{F}^{D}_{p}\wedge \widehat{H}\mathbb{F}^{D}_{p}) \arrow[r, "\eta \wedge \eta"] \arrow[d, "\eta"]
& j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})\wedge j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[dl, ""] \\
j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p} \wedge H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})
\end{tikzcd}$$
where the morphism on the hypotenuse is defined by the lax monoidal property of $j_{*}$. Note that the counit morphism $\epsilon:j^{*}j_{*} \to id.$ is an isomorphism since $j$ is open. By adjunction, the morphism on the hypotenuse of diagram \[eq:triangle\] is induced by the isomorphism $$\epsilon \wedge \epsilon:j^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})\wedge j^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \to (H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})\wedge (H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}).$$ The morphism $\eta$ on the left leg of the triangle \[eq:triangle\] is induced by the isomorphism $$j^{*}\eta:j^{*}((\widehat{H}\mathbb{F}^{D}_{p}\wedge \widehat{H}\mathbb{F}^{D}_{p})\wedge(\widehat{H}\mathbb{F}^{D}_{p}\wedge \widehat{H}\mathbb{F}^{D}_{p})) \to (H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})\wedge(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}).$$ Using that pullback is strongly monoidal, we then have the following commuting triangle.
$$\begin{tikzcd}
j^{*}(\widehat{H}\mathbb{F}^{D}_{p}\wedge \widehat{H}\mathbb{F}^{D}_{p})\wedge j^{*}(\widehat{H}\mathbb{F}^{D}_{p}\wedge \widehat{H}\mathbb{F}^{D}_{p}) \arrow[r, "j^{*}\eta \wedge j^{*}\eta"] \arrow[d, "j^{*}\eta"]
& j^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})\wedge j^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[dl, "\epsilon \wedge \epsilon"] \\
(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})\wedge(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})
\end{tikzcd}$$ Thus, by adjunction, the triangle \[eq:triangle\] commutes.
Applying $i^{*}$ to triangle \[eq:triangle\], we then see that the commuting diagram \[eq:ringhomspectra1diag\] is a sub-diagram of the commuting diagram $$\label{eq:bigcd}
\begin{tikzcd}
(H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p})\wedge(H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p}) \arrow[r, "\mu^{k}_{2}"] \arrow[d, "i^{*}\eta \wedge i^{*}\eta"]
& H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p} \arrow[dd, "i^{*}\eta"] \\
i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})\wedge i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[d, ""] \\
i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p} \wedge H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[r, "i^{*}j_{*}\mu^{K}_{2}"]
& i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[d, "\oplus \pi"] \\
& \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}.
\end{tikzcd}$$
From diagram \[PsiDcommute\], $$(i^{*}\eta \wedge i^{*}\eta)\circ (\Psi_{k}\wedge \Psi_{k}):(\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}) \wedge (\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}) \to i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \wedge i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})$$ is equal to the composite $(i^{*}j_{*}\Psi_{K} \wedge i^{*}j_{*}\Psi_{K}) \circ (i^{*}\eta \wedge i^{*}\eta).$ Hence, diagram \[eq:bigcd\] implies that the multiplication morphism $m=r \circ \mu_{2}^{k} \circ (\Psi_{k} \wedge \Psi_{k})$ on $$(\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}) \wedge (\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p})$$ is equal to the following composite.
$$\label{compositeform}
\begin{tikzcd}
(\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}) \wedge (\bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}) \arrow[d, "((i^{*}j_{*}\Psi_{K})\circ i^{*}\eta) \wedge ((i^{*}j_{*}\Psi_{K})\circ i^{*}\eta)"] \\
i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})\wedge i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[d, ""] \\
i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p} \wedge H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[r, "i^{*}j_{*}\mu^{K}_{2}"]
& i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[d, "\oplus \pi"] \\
& \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha},q_{\alpha}}H\mathbb{F}^{k}_{p}
\end{tikzcd}$$
From Lemma \[lemmafornexthm\], the composites
$$\begin{tikzcd}
\Sigma^{\infty}_{+}X \arrow[r, "f"]
& \Sigma^{2m,m}H\mathbb{F}_{p}^{k} \arrow[r, "i_{R}"]
& \Sigma^{2m,m}H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p} \arrow[r, "i^{*}\eta"]
& \Sigma^{2m,m}i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[d, "i^{*}j_{*}\Psi^{-1}_{K}"]\\
& & & \bigoplus\limits _{\alpha} \Sigma^{p_{\alpha}+2m,q_{\alpha}+m}i^{*}j_{*}H\mathbb{F}^{K}_{p} \arrow[d, "\oplus \pi_{0}"] \\
& & & \bigoplus\limits _{\alpha} \Sigma^{p_{\alpha}+2m-1,q_{\alpha}+m-1}H\mathbb{F}^{k}_{p}
\end{tikzcd}$$
and
$$\begin{tikzcd}
\Sigma^{\infty}_{+}X \arrow[r, "g"]
& \Sigma^{2n,n}H\mathbb{F}_{p}^{k} \arrow[r, "i_{R}"]
& \Sigma^{2n,n}H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p} \arrow[r, "i^{*}\eta"]
& \Sigma^{2n,n}i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[d, "i^{*}j_{*}\Psi^{-1}_{K}"]\\
& & & \bigoplus\limits _{\alpha} \Sigma^{p_{\alpha}+2n,q_{\alpha}+n}i^{*}j_{*}H\mathbb{F}^{K}_{p} \arrow[d, "\oplus \pi_{0}"] \\
& & & \bigoplus\limits _{\alpha} \Sigma^{p_{\alpha}+2n-1,q_{\alpha}+n-1}H\mathbb{F}^{k}_{p}
\end{tikzcd}$$
are equal to $0$. It follows that $i^{*}\eta \circ r \circ i_{R} \circ f=i^{*}\eta \circ i_{R}\circ f$ and $i^{*}\eta \circ r \circ i_{R} \circ g=i^{*}\eta \circ i_{R}\circ g$ in the following two diagrams. $$\label{diagX1}
\begin{tikzcd}
& \Sigma^{\infty}_{+}X \arrow[d, "f"] \\
& \Sigma^{2m,m}H\mathbb{F}_{p}^{k} \arrow[d, "i_{R}"] \\
& \Sigma^{2m,m}H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p} \arrow[dl, "r"] \arrow[d, "i^{*}\eta"] \\
\bigoplus\limits _{\alpha} \Sigma^{p_{\alpha}+2m,q_{\alpha}+m}H\mathbb{F}^{k}_{p} \arrow[r, "i^{*}\eta"]
& \Sigma^{2m,m}i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})
\end{tikzcd}$$
$$\label{diagX2}
\begin{tikzcd}
& \Sigma^{\infty}_{+}X \arrow[d, "g"] \\
& \Sigma^{2n,n}H\mathbb{F}_{p}^{k} \arrow[d, "i_{R}"] \\
& \Sigma^{2n,n}H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p} \arrow[dl, "r"] \arrow[d, "i^{*}\eta"] \\
\bigoplus\limits _{\alpha} \Sigma^{p_{\alpha}+2n,q_{\alpha}+n}H\mathbb{F}^{k}_{p} \arrow[r, "i^{*}\eta"]
& \Sigma^{2n,n}i^{*}j_{*}(H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})
\end{tikzcd}$$
To show that $r_{*}(i_{R *}(f)i_{R *}(g))=r_{*}(i_{R *}(f))r_{*}(i_{R *}(g))$, we consider the following commuting diagram where $\Delta$ is the diagonal morphism.
$$\label{diagramX}
\begin{tikzcd}
\Sigma^{\infty}_{+}X \arrow[d, "\Delta"] \\
\Sigma^{\infty}_{+}X \wedge \Sigma^{\infty}_{+}X \arrow[d, "f \wedge g"] \\
\Sigma^{2m,m}H\mathbb{F}_{p}^{k} \wedge \Sigma^{2n,n}H\mathbb{F}_{p}^{k} \arrow[d, "i_{R}\wedge i_{R}"]\\
(\Sigma^{2m,m}H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p})\wedge(\Sigma^{2n,n}H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p}) \arrow[d, "i^{*}\eta \wedge i^{*}\eta"] \arrow[r, "\mu_{2}^{k}"] & \Sigma^{2m,m}H\mathbb{F}^{k}_{p}\wedge \Sigma^{2n,n}H\mathbb{F}^{k}_{p} \arrow[dd, "i^{*}\eta"] \\
i^{*}j_{*}(\Sigma^{2m,m}H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})\wedge i^{*}j_{*}(\Sigma^{2n,n}H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[d, ""]\\
i^{*}j_{*}(\Sigma^{2(m+n),m+n}H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p} \wedge H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[r, "i^{*}j_{*}\mu_{2}^{K}"]
& i^{*}j_{*}(\Sigma^{2(m+n),m+n}H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[d, "\oplus \pi"] \\
& \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha}+2m+2n,q_{\alpha}+m+n}H\mathbb{F}^{k}_{p}
\end{tikzcd}$$
The composite $\oplus \pi \circ i^{*}\eta \circ \mu_{2}^{k} \circ (i_{R}\wedge i_{R}) \circ (f \wedge g)\circ \Delta$ in this diagram is equal to $r_{*}(i_{R *}(f)i_{R *}(g))$. From diagrams \[diagX1\] and \[diagX2\], the composite given by
$$\label{lastidagcoaction}
\begin{tikzcd}
\Sigma^{\infty}_{+}X \arrow[d, "\Delta"] \\
\Sigma^{\infty}_{+}X \wedge \Sigma^{\infty}_{+}X \arrow[d, "f \wedge g"] \\
\Sigma^{2m,m}H\mathbb{F}_{p}^{k} \wedge \Sigma^{2n,n}H\mathbb{F}_{p}^{k} \arrow[d, "i_{R}\wedge i_{R}"]\\
(\Sigma^{2m,m}H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p})\wedge(\Sigma^{2n,n}H\mathbb{F}^{k}_{p}\wedge H\mathbb{F}^{k}_{p}) \arrow[d, "r \wedge r"] \\
(\bigoplus\limits _{\alpha} \Sigma^{p_{\alpha}+2m,q_{\alpha}+m}H\mathbb{F}^{k}_{p}) \wedge (\bigoplus\limits _{\alpha} \Sigma^{p_{\alpha}+2n,q_{\alpha}+n}H\mathbb{F}^{k}_{p}) \arrow[d, "i^{*}\eta \wedge i^{*}\eta"]\\
i^{*}j_{*}(\Sigma^{2m,m}H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p})\wedge i^{*}j_{*}(\Sigma^{2n,n}H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[d, ""]\\
i^{*}j_{*}(\Sigma^{2(m+n),m+n}H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p} \wedge H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[r, "i^{*}j_{*}\mu_{2}^{K}"]
& i^{*}j_{*}(\Sigma^{2(m+n),m+n}H\mathbb{F}^{K}_{p}\wedge H\mathbb{F}^{K}_{p}) \arrow[d, "\oplus \pi"] \\
& \bigoplus\limits _{\alpha} \Sigma ^{p_{\alpha}+2m+2n,q_{\alpha}+m+n}H\mathbb{F}^{k}_{p}
\end{tikzcd}$$
is equal to the composite given by diagram \[diagramX\]. From diagram \[compositeform\], the composite given by diagram \[lastidagcoaction\] is equal to $\Sigma^{2(m+n),m+n}m\circ (r\wedge r)\circ (i_{R}\wedge i_{R})\circ (f \wedge g) \circ \Delta=r_{*}(i_{R *}(f))r_{*}(i_{R *}(g))$. Thus, $r_{*}(i_{R *}(f)i_{R *}(g))=r_{*}(i_{R *}(f))r_{*}(i_{R *}(g))$ as desired.
Cartan formula {#sectioncartan}
==============
In this section, we use the coaction map constructed in the previous section to prove a Cartan formula for the operations $P^{n}_{k}$ restricted to mod $p$ Chow groups. Let $X \in \textup{Sm}_{k}$. Let $\langle \cdot,\cdot \rangle$ denote the pairing between $\mathcal{A}^{k}_{*,*}$ and $H\mathbb{F}^{k \, *,*}_{p}H\mathbb{F}^{k}_{p}$. Let $n \geq 0$. For $x \in H^{*,*}(X,\mathbb{F}_{p})$ with $\lambda_{X}(x)= \Sigma y_{i} \otimes x_{i}$, we have $P^{n}_{k}(x)=\Sigma \langle y_{i},P^{n}_{k}\rangle x_{i}$.
Let $x,y\in CH^{*}(X)/p$ and $i \geq0$. Then $$P^{i}_{k}(xy)=\sum_{j=0}^{i}P^{j}_{k}(x)P^{i-j}_{k}(y).$$
From the definition of $P^{i}_{k}$, $\langle \xi_{1}^{i}, P^{i}_{k}\rangle=1$ and $\langle\omega(\alpha), P^{i}_{k}\rangle=0$ for all sequences $\alpha \neq (0,i,0,0,\ldots)$. Using the coaction map \[coacitonmapdef\], we write $$\lambda_{X}(x)=\sum_{q} \omega(\alpha^{1}_{q}) \otimes x_{q}$$ and $$\lambda_{X}(y)=\sum_{r} \omega(\alpha^{2}_{r}) \otimes y_{r}$$ for some sequences $\alpha^{1}_{q}, \alpha^{2}_{r}$. Then $$\lambda_{X}(xy)=\sum_{q, r} ((\omega(\alpha^{1}_{q}) \omega(\alpha^{2}_{r}) \otimes x_{q}y_{r}).$$ For any $2$ sequences $\alpha^{1}_{q}, \alpha^{2}_{r}$ appearing in these sums, we have $\omega(\alpha^{1}_{q}) \omega(\alpha^{2}_{r})=0$ if the relation $\tau_{m}^{2}=0$ from Proposition \[P:tau2=0\] applies for some $m \geq0$, or else $\omega(\alpha^{1}_{q}) \omega(\alpha^{2}_{r})=\pm \omega(\alpha^{1}_{q}+\alpha^{2}_{r})$.
From the definition of $\lambda_{X}$, $$P^{i}_{k}(xy)=\sum_{q, r} \langle (\omega(\alpha^{1}_{q}) \omega(\alpha^{2}_{r}), P^{i}_{k}\rangle x_{q}y_{r}.$$ Proposition \[P:tau2=0\] implies that if $\omega(\alpha_{1}) \omega(\alpha_{2})=a\xi_{1}^{i}$ for two sequences $\alpha_{1}, \alpha_{2}$ and $a\neq 0 \in H^{*,*}(k, \mathbb{F}_{p})$, then $a=1$ and $\omega(\alpha_{1})=\xi_{1}^{j}, \, \omega(\alpha_{2})=\xi_{1}^{i-j}$ for some $0 \leq j \leq i$. As $P^{i}_{k}$ is dual to $\xi_{1}^{i}$, the only terms for which $\langle\omega(\alpha^{1}_{q}+\alpha^{2}_{r}), P^{i}_{k}\rangle\neq 0$ are of the form $\omega(\alpha^{1}_{q_{j}})=\xi_{1}^{j}, \omega(\alpha^{2}_{r_{j}})=\xi_{1}^{i-j}$ for $0 \leq j \leq i$. Hence, $$\label{eq:cartanform1} P^{i}_{k}(xy)=\sum_{j=0}^{i} \langle\omega(\alpha^{1}_{q_{j}}+\alpha^{2}_{r_{j}}), P^{i}_{k}\rangle x_{q_{j}}y_{r_{j}}=\sum_{j=0}^{i}P^{j}_{k}(x)P^{i-j}_{k}(y)$$ as required.
$p$th power and instability
===========================
In this section, for $n \in \mathbb{N}$, we prove that $P^{n}_{k}$ is the $p$th power on $CH^{n}(-)/p$. Letting $f:\textup{Spec}(k) \to \textup{Spec}(\mathbb{F}_{p})$ denote the structure map, it suffices to prove that $f^{*}(P^{n}_{\mathbb{F}_{p}})(\iota_{n})=\iota^{p}_{n}$ for the canonical element $\iota_{n} \in H^{2n,n}(K_{n,k},\mathbb{F}_{p})$ where $K_{n,k} \in H(k)$ is the motivic Eilenberg-MacLane space representing $H^{2n,n}(-, \mathbb{F}_{p})$. Our proof makes use of Morel’s $S^1$-recognition principle.
We refer to [@EHK+ Section 3] as a reference for the $S^1$-recognition principle. For a base scheme $S$, let $\textrm{PSh}_{\textrm{nis}}(\textrm{Sm}_{S})$ denote the category of Nisnevich local presheaves of spaces on $\textrm{Sm}_{S}$. The unstable motivic homotopy category $H(S)$ can be described as the full subcategory of $\textrm{PSh}_{\textrm{nis}}(\textrm{Sm}_{S})$ of presheaves that are $\mathbb{A}^{1}$-invariant. Let $L_{\textrm{mot}}:\textrm{PSh}_{\textrm{nis}}(\textrm{Sm}_{S}) \to H(S)$ denote the $\mathbb{A}^{1}$-localization functor. Let $SH^{S^{1}}(S)$ denote the stable motivic homotopy category of $S^{1}$-spectra. For a morphism $f: S_{1} \to S_{2}$ of base schemes, we have the adjoint functors of pullback $f^{*} \coloneqq Lf^{*}$ and pushforward $f_{*} \coloneqq Rf_{*}$: $$f^{*}:H(S_{2}) \rightleftarrows H(S_{2}): f_{*}.$$ For $f: S_{1} \to S_{2}$ smooth, $f^{*}$ admits a left adjoint $f_{\#}$ such that $f_{\#}(X)=X \in H(S_{2})$ for any $X \in \textrm{Sm}_{S_{1}}.$
For $C=\textrm{PSh}_{\textrm{nis}}(\textrm{Sm}_{S})$ or $H(S)$, we consider the $n$-fold bar constructions $\textrm{B}^{n}_{C}$ that are adjoint to the $n$th $S^1$-deloopings $\Omega^{n}$:$$\textrm{B}^{n}_{C}: \textrm{Mon}_{\mathcal{E}_{n}}(C) \rightleftarrows C: \Omega^{n}.$$ For $C=\textrm{PSh}_{\textrm{nis}}(\textrm{Sm}_{S})$ or $H(S)$, we let $\textrm{Stab}(C) \coloneqq C \otimes \textrm{Spt}$ denote the $S^1$-stabilization of $C$. We also consider the infinite bar construction $$\textrm{B}_{C}^{\infty}: \textrm{CMon}(C)=\textrm{Mon}_{\mathcal{E}_{\infty}}(C)\rightleftarrows \textrm{Stab}(C): \Omega^{\infty}.$$ For $C=\textrm{PSh}_{\textrm{nis}}(\textrm{Sm}_{S})$, we denote $\textrm{B}^{n}_{C}$ by $\textrm{B}^{n}_{\textrm{nis}}$ and we denote $\textrm{B}^{\infty}_{C}$ by $\textrm{B}^{\infty}_{\textrm{nis}}$. Similarly, for $C=H(S)$, we denote $\textrm{B}^{n}_{C}$ by $\textrm{B}^{n}_{\textrm{mot}}$ and we denote $\textrm{B}^{\infty}_{C}$ by $\textrm{B}^{\infty}_{\textrm{mot}}$. For later use, we note that $\textrm{B}^{n}_{\textrm{nis}}$ and $\textrm{B}^{\infty}_{\textrm{nis}}$ commute with pullbacks.
Define $X \in \textup{Mon}(H(S))$ to be strongly $\mathbb{A}^{1}$-invariant if $\textup{B}_{\textup{nis}}X \simeq \textup{B}_{\textup{mot}}X$. Define $X \in \textup{CMon}(H(S))$ to be strictly $\mathbb{A}^{1}$-invariant if $\textup{B}^{n}_{\textup{nis}}X \simeq \textup{B}^{n}_{\textup{mot}}X$ for all $n\geq0$.
Most of the proof of the following proposition was suggested to us by Marc Hoyois.
\[T:pullbaclEB\] Let $k$ be a perfect field of characteristic $p$ and let $i:\textrm{Spec}(k) \to \textrm{Spec}(D)$ be a closed embedding where $D$ is a complete unramified DVR with generic point $j: \textrm{Spec}(K) \to \textrm{Spec}(D)$. Fix $n >0$. We let $K_{n,D} \coloneqq \Omega^{\infty}_{\mathbb{P}^{1}}\Sigma^{2n,n}\widehat{H}\mathbb{F}^{D}_{p}$. Then the morphism $i^{*}K_{n,D} \to K_{n,k}$ induced by $i^{*} \Sigma^{2n,n}\widehat{H}\mathbb{F}_{p}^{D} \cong \Sigma^{2n,n}H\mathbb{F}_{p}^{k}$ is an isomorphism in $H(k)$.
We first prove that $K_{n, D}$ is connected. Let $R$ be a Henselian local ring that is essentially smooth over $D$. From [@Gei Corollary 4.2], the Bloch-Levine Chow groups $CH^{m}(R)$ of $R$ vanish for $m \geq 1.$ Thus, $\pi_{0}^{\textup{nis}}(K_{n,D}(\textrm{Spec}(R))) \simeq * $ since $K_{n,D} \in H(D)$ represents the codimension $n$ mod $p$ Bloch-Levine Chow group.
Now we prove that $i^{*}K_{n, D}$ is connected. As $j:\textrm{Spec}(K) \to \textrm{Spec}(D)$ is smooth, $j^{*}K_{n,D} \simeq K_{n,K}$. Consider the homotopy pushout $P$ in $\textrm{PSh}_{\textrm{nis}}(\textrm{Sm}_{D})$ of the following diagram. $$\begin{tikzcd}
j_{\#}K_{n,K} \arrow[r, ""] \arrow[d, ""]
& K_{n,D} \\
\textrm{Spec}(K)
\end{tikzcd}$$ The morphism $j_{\#}K_{n,K} \to \textrm{Spec}(K)$ induces a bijection on $\pi^{\textrm{nis}}_{0}$. Hence, $\pi^{\textrm{nis}}_{0}(K_{n,D}) \simeq \pi^{\textrm{nis}}_{0}(P)$. From the gluing square [@MorVoe Theorem 2.21], $L_{\textrm{mot}}(P) \simeq i_{*}i^{*}(K_{n,D})$. From [@MorVoe Corollary 3.22], it follows that $i_{*}i^{*}(K_{n,D})$ is connected since $K_{n,D}$ is connected. Let $k \to S_{k}$ be an essentially smooth homomorphism of rings where $S_{k}$ is Henselian local. The ring $S_{k}$ admits a lift $S_{D}$ where $D \to S_{D}$ is essentially smooth and $S_{D}$ is Henselian local. Hence, $i^{*}(K_{n,D})(S_{k})\simeq i_{*}i^{*}(K_{n,D})(S_{D})$ is connected. Thus, $i^{*}K_{n,D} \in H(k)$ is connected. In particular, $\pi^{\textrm{nis}}_{0}(i^{*}(K_{n,D}))$ is strongly $\mathbb{A}^{1}$-invariant. The $S^{1}$-recognition principle [@EHK+ Theorem 3.1.12] then implies that $i^{*}K_{n,D}$ is strictly $\mathbb{A}^{1}$-invariant. Note that $K_{n,k}$ is also strictly $\mathbb{A}^{1}$-invariant since $\pi^{\textrm{nis}}_{0}(K_{n,k})$ is strongly $\mathbb{A}^{1}$-invariant.
From [@Spi Theorem 8.18], we have $$\textrm{B}^{\infty}_{\textup{mot}}i^{*}(K_{n,D})\cong i^{*}(\textrm{B}^{\infty}_{\textup{mot}}K_{n,D}) \cong i^{*}(\Omega^{\infty}_{\mathbb{G}_{m}} \Sigma^{2n,n}\widehat{H}\mathbb{F}^{D}_{p}) \cong \Omega^{\infty}_{\mathbb{G}_{m}} \Sigma^{2n,n}H\mathbb{F}^{k}_{p} \cong \textrm{B}^{\infty}_{\textup{mot}}K_{n,k}$$ in $SH^{S^{1}}(k)$. Then [@EHK+ Corollary 3.1.15] implies that $i^{*}K_{n,D} \cong K_{n,k}$ in $H(k)$.
\[P:pthpower\] Let $k$ be a field of characteristic $p$ with structure map $f:\textup{Spec}(k)\to \textup{Spec}(\mathbb{F}_{p})$ and let $\iota_{n} \in H^{2n,n}(K_{n,k}, \mathbb{F}_{p})$ be the canonical element. Then $f^{*}P^{n}_{\mathbb{F}_{p}}(\iota_{n})=\iota^{p}_{n}$.
First, we assume that $k$ is perfect. Let $D$ be a DVR having $k$ as a residue field with inclusion morphism $i:\textrm{Spec}(k) \to \textrm{Spec}(D)$ and generic point $j:\textrm{Spec}(K)\to \textrm{Spec}(D)$. From Proposition \[T:pullbaclEB\], $i^{*}K_{n,D} \cong K_{n,k}$. Over all base schemes $S$, we let $\iota_{n}$ denote the canonical element in $H^{2n,n}(K_{n,S}, \mathbb{F}_{p})$. Apply $i^{*} \to i^{*}j_{*}j^{*}$ to the natural morphism $\iota_{n}:\Sigma^{\infty}_{+}K_{n,D} \to \Sigma^{2n,n}\widehat{H}\mathbb{F}_{p}^{D}$ to get the following commuting square.
$$\begin{tikzcd}
\Sigma^{\infty}_{+}K_{n,k} \arrow[r, "\iota_{n}"] \arrow[d, "i^{*}\eta"]
& \Sigma^{2n,n}H\mathbb{F}_{p}^{k} \arrow[d, "i^{*}\eta"]\\
i^{*}j_{*}\Sigma^{\infty}_{+}K_{n,K} \arrow[r, "i^{*}j_{*}\iota_{n}"] & \Sigma^{2n,n}i^{*}j_{*}H\mathbb{F}_{p}^{K}
\end{tikzcd}$$ Apply $i^{*}\eta:i^{*} \to i^{*}j_{*}j^{*}$ to the morphism $\Sigma^{\infty}_{+}K_{n,D} \to \Sigma^{2pn,pn} \widehat{H}\mathbb{F}_{p}^{D}$ in $SH(D)$ corresponding to $\iota^{p}_{n}$ to get the commutative diagram
$$\label{diagrampthpower}
\begin{tikzcd}
\Sigma^{\infty}_{+}K_{n,k} \arrow[r, "\iota^{p}_{n}"] \arrow[d, "i^{*}\eta"]
& \Sigma^{2pn,pn} H\mathbb{F}_{p}^{k} \arrow[d, "i^{*}\eta"]\\
i^{*}j_{*}\Sigma^{\infty}_{+}K_{n,K} \arrow[r, "i^{*}j_{*}\iota^{p}_{n}"]
& i^{*}j_{*}\Sigma^{2pn,pn} H\mathbb{F}_{p}^{K} \arrow[r, "\pi"]& \Sigma^{2pn,pn} H\mathbb{F}_{p}^{k} .
\end{tikzcd}$$
From [@Voe Lemma 9.8], $i^{*}j_{*}\iota^{p}_{n}=i^{*}j_{*}P^{n}_{K}(\iota_{n})$. Hence, we can rewrite the bottom row of \[diagrampthpower\] as $$\begin{tikzcd}
i^{*}j_{*}\Sigma^{\infty}_{+}K_{n,K} \arrow[r,"i^{*}j_{*}\iota_{n}"]
& i^{*}j_{*}\Sigma^{2n,n}H\mathbb{F}_{p}^{K} \arrow[r, "i^{*}j_{*}P^{n}_{K}"] & i^{*}j_{*}\Sigma^{2pn,pn}H\mathbb{F}_{p}^{K} \arrow[r, "\pi"]& \Sigma^{2pn,pn} H\mathbb{F}_{p}^{k}.
\end{tikzcd}$$
From Theorem \[T:phiproperties\] and the above commuting diagrams, $P^{n}_{k}(\iota_{n})=\pi \circ (i^{*}j_{*}P^{n}_{K})\circ (i^{*}j_{*}\iota_{n})\circ i^{*}\eta$. Hence, from diagram \[diagrampthpower\], we get $P^{n}_{k}(\iota_{n})=\pi \circ (i^{*}j_{*}\iota_{n}^{p})\circ i^{*}\eta=\iota^{p}_{n}$.
For $k$ not perfect, we have an essentially smooth morphism $f: \textrm{Spec}(k) \to \textrm{Spec}(\mathbb{F}_{p})$ and $f^{*}(K_{n,\mathbb{F}_{p}})\cong K_{n,k}$ [@HKO Theorem 2.11]. As $\mathbb{F}_{p}$ is perfect, we then have $f^{*}(P^{n}_{\mathbb{F}_{p}}(\iota_{n}))=f^{*}(P^{n}_{\mathbb{F}_{p}})(\iota_{n})=f^{*}(\iota_{n}^{p})=\iota^{p}_{n}.$
From Proposition \[P:pullback\], we have the following corollary.
Let $X \in \textup{Sm}_{k}$. Then $P^{n}_{k}$ is the $p$th power on $CH^{n}(X)/p$.
Now that we know $f^{*}(P^{n}_{\mathbb{F}_{p}})$ is the $p$th power on $H^{2n,n}(-, \mathbb{F}_{p})$ for all $n \geq 1$, we can prove an instability result. Let $f:\textup{Spec}(k)\to \textup{Spec}(\mathbb{F}_{p})$ be the structure morphism.
Let $p, q, n \geq 0$ be integers such that $n>p-q$ and $n \geq q$. Let $X \in H(k)$ and let $x \in H^{p,q}(X, \mathbb{F}_{p})$. Then $f^{*}(P^{n}_{\mathbb{F}_{p}})(x)=0$.
Voevodsky’s proof in [@Voe Lemma 9.9] works here since $f^{*}(P^{n}_{\mathbb{F}_{p}})$ is the $p$th power on $H^{2n,n}(-, \mathbb{F}_{p})$ by Proposition \[P:pthpower\].
Let $X \in \textup{Sm}_{k}$. Then $P^{n}_{k}$ is the $0$ map on $CH^{m}(X)/p$ for $m<n$.
Proper pushforward
==================
In this section, we restrict our attention to mod $p$ Chow groups on $\text{Sm}_{k}$. The ring of mod $p$ Chow groups is an oriented cohomology pretheory in the sense of [@Pan Section 1] with perfect integration given by proper pushforward on Chow groups. Consider the total cohomological Steenrod operation $P_{k} \coloneqq P^{0}_{k}+P^{1}_{k}+P^{2}_{k}+ \cdots:CH^{*}(-)/p \to CH^{*}(-)/p.$ From the Cartan formula \[sectioncartan\], $P_{k}$ is a ring morphism of oriented cohomology pretheories in the sense of [@Pan Definition 1.1.7].
Let $\mathbb{Z}[[c_{1},c_{2}, \ldots]]$ denote the power series ring on Chern classes $c_{i}$ for $i \geq 1$ and let $w \in \mathbb{Z}[[c_{1},c_{2}, \ldots]]$ denote the total characteristic class corresponding to the polynomial $f(x)=1+x^{p-1}.$ For $p=2$, $w$ is the total Chern class. Let $X \in \textup{Sm}_{k}$. For a line bundle $L$ on $X$, $w(L)=1+c_{1}^{p-1}(L) \in CH^{*}(X)$. For a vector bundle $V$ on $X$ that has a filtration by subbundles with quotients given by line bundles $L_{1}, \ldots, L_{m}$, $w(V)=w(L_{1})\cdots w(L_{m}).$ Let $w_{i}$ denote the $i$th homogeneous component of $w$ for $i \geq 0$. We have $w_{i}=0$ if $p-1$ does not divide $i$. Define the total homological Steenrod operation $P^{X} \coloneqq w(-T_{X})\circ P_{k}:CH^{*}(X)/p \to CH^{*}(X)/p$ where $T_{X}$ is the tangent bundle on $X$. For $i \geq0$, let $P^{X}_{i}$ denote the $(p-1)i$th homogeneous component of $P^{X}$. The following proposition is a consequence of the general Riemann-Roch formulas proved by Panin in [@Pan]
\[pushforward\] Let $f:X \to Y$ be a morphism of smooth projective varieties over $k$. Then $$\begin{tikzcd}
CH^{*}(X)/p \arrow[d, "f_{*}"] \arrow[r, "P^{X}"]
& CH^{*}(X)/p \arrow[d, "f_{*}"]\\
CH^{*}(Y)/p \arrow[r, "P^{Y}"] & CH^{*}(Y)/p
\end{tikzcd}$$ commutes.
This is [@Pan Theorem 2.5.4]. See [@Pan Section 2.6] for a discussion relevant to our situation. The main ingredients are that the operations $P^{n}_{k}$ satisfy the Cartan formula and that $P^{n}_{k}$ is the $p$th power on $CH^{n}(-)/p$.
Restricting to the case $p=\textup{char}(k)=2$, we obtain a Wu formula from the work of Panin [@Pan Theorem 2.5.3]. Here, $w=c$ is the total Chern class and we let $\textup{Sq}$ denote the total Steenrod square $P_{k}$ on $CH^{*}(-)/2$.
Let $X, Y$ be smooth projective varieties over $k$, and let $i: X \xhookrightarrow{} Y$ be a closed embedding with normal bundle $N$. Then $$i_{*}(c(N))=\textup{Sq}([X])$$ in $CH^{*}(Y)/2$ where $[X] \in CH^{*}(Y)/2$ denotes the mod $2$ cycle class of $X$.
Rost’s degree formula {#Rost}
=====================
Now that we have Steenrod operations on mod $p$ Chow groups of $\textup{Sm}_{k}$, we can prove Rost’s degree formula [@Mer Theorem 6.4] without any restrictions on the characteristic of the base field. We closely follow the presentation of Merkurjev [@Mer] where Steenrod operations (assuming restrictions on the characteristic of the base field) are used to prove degree formulas. In [@Hau3], Haution extended the Rost degree formulas to base fields of characteristic $2$.
For a variety $X$ over $k$, let $n_{X}$ denote the greatest common divisor of $\textup{deg}(x)$ over all closed points $x \xhookrightarrow{} X$. Let $X \in \textup{Sm}_{k}$ be projective of dimension $d>0$. Applying Proposition \[pushforward\] to the structure morphism $X \to \textup{Spec}(k)$ and $[X] \in CH_{d}(X)/p$, we see that $p \mid \textup{deg}(w_{d}(-T_{X}))$.
Let $f:X \to Y$ be a morphism of projective varieties $X, Y \in \textup{Sm}_{k}$ of dimension $d>0$. Then $n_{Y} \mid n_{X}$ and $$\frac{\textup{deg}(w_{d}(-T_{X}))}{p} \equiv \textup{deg}(f)\cdot \frac{\textup{deg}(w_{d}(-T_{Y}))}{p} \mod n_{Y}.$$
The proof in [@Mer Theorem 6.4] works here. From Proposition \[pushforward\], $f_{*}(w_{d}(-T_{X}))\equiv \textup{deg}(f)w_{d}(-T_{Y}) \in CH_{0}(Y)/p.$ We then take the degree homomorphism to finish the proof.
Specialization map
==================
Fix a complete unramified DVR $D$ with residue field $i:\textup{Spec}(k) \to \textup{Spec}(D)$ and fraction field $j:\textup{Spec}(K) \to \textup{Spec}(D)$ as before. Let $X \in \textup{Sm}_{D}$ be projective with special fiber $X_{k}$ and generic fiber $X_{K}$. As described in [@Ful Chapter 20.3], there are specialization maps $\sigma_{n}:CH^{n}(X_{K}) \to CH^{n}(X_{k})$ defined for all $n \geq 0$. The specialization maps can be defined at the level of cycles. Namely, for an irreducible closed subvariety $Z_{K} \subset X_{K}$ of codimension $n$, we let $Z_{k}$ denote the special fiber of the reduced closed subscheme $\overline{Z_{K}} \subset X$ associated to $Z_{K} \subset X$. Then $\sigma_{n}(\left<Z_{K} \right>)=\left<Z_{k} \right> \in CH^{n}(X_{k})$. We also let $\sigma_{n}$ denote the specialization map induced on mod $p$ Chow groups.
We now show that the Steenrod operations $P^{n}_{k}$ defined on $CH^{*}(X_{k})$ are compatible with the operations $P^{n}_{K}$ defined on $CH^{*}(X_{K})$.
\[P:specialization\] Let $m \geq 0$ and let $Z_{K} \subset X_{K}$ be a closed subvariety of codimension $n$. Let $\left<Z_{K} \right> \in CH^{n}(X_{K})/p$ denote the mod $p$ cycle class of $Z_{K}$. Then $$P^{m}_{k}(\sigma_{n}(\left<Z_{K} \right>))=\sigma_{n+m(p-1)}(P^{m}_{K}(\left<Z_{K} \right>)) \in CH^{n+m(p-1)}(X_{k})/p.$$
The mod $p$ cycle class of $\overline{Z_{K}} \subset X$ induces a map $$f_{D}:\Sigma^{\infty}_{+}X \to \Sigma^{2n,n}\widehat{H}\mathbb{F}_{p}^{D}$$ in $SH(D)$. The map $i^{*}f_{D}$ gives the mod $p$ cycle class of $Z_{k}$ (the special fiber of $\overline{Z_{K}} \subset X$) and $j^{*}f_{D}$ gives the mod $p$ cycle class of $Z_{K}$. Applying the natural transformation $i^{*}\eta:i^{*} \to i^{*}j_{*}j^{*}$ to $f_{D}$, we get a commuting square.
$$\label{diag1special}
\begin{tikzcd}
\Sigma^{\infty}_{+}X_{k} \arrow[r, "i^{*}f_{D}"] \arrow[d, "i^{*}\eta"] & \Sigma^{2n,n}H\mathbb{F}_{p}^{k} \arrow[d, "i^{*}\eta"]\\
i^{*}j_{*}\Sigma^{\infty}_{+}X_{K} \arrow[r, "i^{*}j_{*}j^{*}f_{D}"] & i^{*}j_{*}\Sigma^{2n,n}H\mathbb{F}_{p}^{K}
\end{tikzcd}$$
From Theorem \[T:phiproperties\], $P^{m}_{k}=\Phi(P^{m}_{K})=\pi \circ i^{*}j_{*}P^{m}_{K}\circ i^{*}\eta$. Hence, from diagram \[diag1special\], we get that $$\pi \circ i^{*}\eta\circ P^{m}_{k} \circ i^{*}f_{D}=\pi \circ i^{*}j_{*}P^{m}_{K}\circ i^{*}j_{*}j^{*}f_{D}\circ i^{*}\eta$$ in the following commuting diagram.
$$\label{diag2special}
\begin{tikzcd}
\Sigma^{\infty}_{+}X_{k} \arrow[r, "i^{*}f_{D}"] \arrow[d, "i^{*}\eta"] & \Sigma^{2n,n}H\mathbb{F}_{p}^{k} \arrow[d, "i^{*}\eta"] \arrow[r, "P^{m}_{k}"] & \Sigma^{2(n+m(p-1)),n+m(p-1)}H\mathbb{F}_{p}^{k} \arrow[d, "i^{*}\eta"] \\
i^{*}j_{*}\Sigma^{\infty}_{+}X_{K} \arrow[r, "i^{*}j_{*}j^{*}f_{D}"] & i^{*}j_{*}\Sigma^{2n,n}H\mathbb{F}_{p}^{K} \arrow[r, "i^{*}j_{*}P^{m}_{K}"] & i^{*}j_{*}\Sigma^{2(n+m(p-1)),n+m(p-1)}H\mathbb{F}_{p}^{k} \arrow[d, "\pi"] \\
& & \Sigma^{2(n+m(p-1)),n+m(p-1)}H\mathbb{F}_{p}^{k}
\end{tikzcd}$$
Write $P^{m}_{K}(\left<Z_{K}\right>)=\sum_{l=1}^{q}a_{l}\left<Z^{l}_{K}\right>$ for some $q, a_{l} \in \mathbb{Z}$ and closed subvarieties $Z^{l}_{K} \subset X_{K}$ of codimension $n+m(p-1)$. Taking the associated reduced closed subschemes in $X$, we get an element $\sum_{l=1}^{q}a_{l}\left<\overline{Z}^{l}_{K}\right> \in H^{2(n+m(p-1)),n+m(p-1)}(X, \mathbb{F}_{p})$ which corresponds to a morphism $g:\Sigma^{\infty}_{+}X \to \ \Sigma^{2(n+m(p-1)),n+m(p-1)}\widehat{H}\mathbb{F}_{p}^{D}$. For $1 \leq l \leq q$, let $Z^{l}_{k}$ denote the special fiber of $\overline{Z}^{l}_{K}$. Taking pullbacks, $i^{*}g$ gives $\sum_{l=1}^{q}a_{l}\left<Z^{l}_{k}\right> \in H^{2(n+m(p-1)),n+m(p-1)}(X_{k}, \mathbb{F}_{p})$ and $j^{*}g=\sum_{l=1}^{q}a_{l}\left<Z^{l}_{K}\right>=P^{m}_{K}(\left<Z_{K}\right>)$. Applying $i^{*}\eta$ to $g$, we get a commuting diagram.
$$\label{diagspecial3}
\begin{tikzcd}
\Sigma^{\infty}_{+}X_{k} \arrow[r, "i^{*}g"] \arrow[d, "i^{*}\eta"] & \Sigma^{2(n+m(p-1)),n+m(p-1)}H\mathbb{F}_{p}^{k} \arrow[d, "i^{*}\eta"] \\
i^{*}j_{*}\Sigma^{\infty}_{+}X_{K} \arrow[r, "i^{*}j_{*}j^{*}g"] & i^{*}j_{*}\Sigma^{2(n+m(p-1)),n+m(p-1)}H\mathbb{F}_{p}^{K} \arrow[d, "\pi"] \\ & \Sigma^{2(n+m(p-1)),n+m(p-1)}H\mathbb{F}_{p}^{k}
\end{tikzcd}$$
From diagrams \[diag2special\] and \[diagspecial3\], we get $$i^{*}g=\sum_{l=1}^{q}a_{l}\left<Z^{l}_{k}\right>=\pi \circ i^{*}j_{*}j^{*}g \circ i^{*}\eta=\pi \circ i^{*}j_{*}(P^{m}_{K}(\left<Z_{K}\right>)) \circ i^{*}\eta\\$$ $$=\pi \circ i^{*}j_{*}P^{m}_{K} \circ i^{*}j_{*}j^{*}f_{D} \circ i^{*}\eta=P^{m}_{k}(\left<Z_{k}\right>)$$
as required.
We recall some facts about flag varieties, using [@Koc] as a reference. Let $G_{k}$ be a split reductive group over $k$ with Borel subgroup $B_{k}$ and Weyl group $W$. From the Bruhat decomposition, we have $$G_{k}/B_{k}=\coprod _{w \in W} B_{k}wB_{k}/B_{k}.$$ For $w \in W$, the closure $X^{w}_{k}$ of $B_{k}wB_{k}/B_{k}$ in $G_{k}/B_{k}$ is called a Schubert variety and $$B_{k}wB_{k}/B_{k} \cong \mathbb{A}^{l(w)}_{k}$$ where $l(w)$ is the length of $w$ in $W$. Let $P_{k} \supseteq B_{k}$ be a parabolic subgroup of $G_{k}$. We have $P_{k}=BW_{P}B$ for some subgroup $W_{P} \leq W$. There is a related $W^{P} \subset W$, such that for each $w \in W^{P}$, $B_{k}wB_{k}/B_{k}$ is isomorphic to $B_{k}wB_{k}/P_{k}$ under the quotient morphism $G_{k}/B_{k} \to G_{k}/P_{k}$ [@Koc Lemma 1.2]. We also have a cell decomposition $$G_{k}/P_{k}=\coprod _{w \in W^{P}} B_{k}wB_{k}/P_{k}.$$ This cell decomposition is independent of the field $k$. It follows that the total chow group $CH^{*}(G_{k}/P_{k})$ is freely generated as an additive group by the cycle classes $\left<Y^{w}_{k}\right>$ of the images $Y^{w}_{k}$ of the Schubert varieties $X^{w}_{k}$ for $w \in W^{P}$.
Chevalley [@Chev] and Demazure [@Dem] showed that the chow rings $$CH^{*}(G_{F_{1}}/P_{F_{1}})\; \textup{and} \; CH^{*}(G_{F_{2}}/P_{F_{2}})$$ are isomorphic for any two fields $F_{1}, F_{2}$. The isomorphism is given by mapping the class of a Schubert subscheme $Y^{w}_{F_{1}}$ to $Y^{w}_{F_{2}}$ for $w \in W^{P}$. We now prove that the Steenrod operations $P^{n}_{k}$ and $P^{n}_{K}$ give the same action on $H^{2*,*}(G_{k}/P_{k}, \mathbb{F}_{p}) \cong CH^{*}(G_{k}/P_{k})/p \cong CH^{*}(G_{K}/P_{K})/p \cong H^{2*,*}(G_{K}/P_{K}, \mathbb{F}_{p}).$
\[P:flagvar\] Let $n \geq 0$ and let $w_{0} \in W^{P}$. We have $$P^{n}_{K}(\left<Y^{K}_{w_{0}}\right>)=\sum_{w \in W^{P}} a_{w}\left<Y^{K}_{w}\right>$$ in $CH^{*}(G_{K}/P_{K})/p$ for some $a_{w} \in \mathbb{Z}$. Then $$P^{n}_{k}(\left<Y^{k}_{w_{0}}\right>)=\sum_{w \in W^{P}} a_{w}\left<Y^{k}_{w}\right>.$$
We refer to [@Con] for facts about integral models of split reductive groups. Let $w\in W$ and let $X^{w}_{D}$ be the reduced closed subscheme of $G_{D}/B_{D}$ associated to $B_{D}wB_{D}/B_{D}$. Note that $X^{w}_{D}$ is flat over $\mathrm{Spec}(D).$ For any field $F$ and morphism $\mathrm{Spec}(F) \to \mathrm{Spec}(D)$, the fiber $X^{w}_{D} \times _{\mathrm{Spec}(D)} \mathrm{Spec}(F)$ in $G_{F}/B_{F}$ is isomorphic to $X^{w}_{F}$ [@Ses Theorem 2]. The main point to check is that the fibers of $X^{w}_{D}$ over $\mathrm{Spec}(D)$ are reduced.
Now assume that $w\in W^{P}$. Let $Y^{w}_{D}$ denote the image of $X^{w}_{D}$ in $G_{D}/P_{D}$. Then $Y^{w}_{D} \times _{\mathrm{Spec}(D)} \mathrm{Spec}(F) \cong Y^{w}_{F}$ for any field $F$ and morphism $\mathrm{Spec}(F) \to \mathrm{Spec}(D)$. Proposition \[P:specialization\] then applies to finish the proof.
Applications to quadratic forms {#S:appquad}
===============================
In this section, we use the Steenrod squares $\textrm{Sq}^{2n}_{k}$ to prove new results about nonsingular quadratic forms over a field $k$ of characteristic $2$. The results we prove have analogues in characteristic $\neq 2$ conveniently found in [@EKM Sections 79-82] where the only missing ingredient for extending to characteristic $2$ was the existence of Steenrod squares satisfying expected properties.
Recall that a quadratic form $(q,V)$ over $k$ is nonsingular if the associated radical $V^{\perp}$ is of dimension at most $1$ and $q$ is nonzero on $V^{\perp}\setminus{0}$. Equivalently, $(q,V)$ is nonsingular if the associated projective quadric is smooth. Note that nonsingular quadratic forms are called nondegenerate in [@EKM]. In characteristic $2$, anisotropic quadratic forms are not necessarily nonsingular. Let $(q, V)$ be a nonsingular anisotropic quadratic form of dimension $D$ defined over $k$ and let $X$ be the associated quadric. Over some field extension $F$ of $k$, the quadric $X_{F}$ becomes split. A computation of $CH^{*}(X_{F})$ can be found in [@EKM Chapter XIII]. Let $h \in CH^{1}(X_{F})$ denote the pullback of the hyperplane class in $\mathbb{P}(V)$ and let $l_{d} \in CH_{d}(X_{F})$ denote the class of a $d$-dimensional subspace in $X_{F}$ where $d= \lfloor (D-1)/2 \rfloor$. Let $l_{i}=h^{i} \cdot l_{d}$ for $ 0 \leq i \leq d$.
As an additive group, $CH^{*}(X_{F})$ is freely generated by $h^{i}, l_{i}$ for $ 0 \leq i \leq d$. For the ring structure, $h^{d+1}=2l_{D-d-1}$, $l^{2}_{d}=0$ if $4$ does not divide $D$, and $l^{2}_{d}=l_{0}$ if $4$ divides $D$.
From Corollary \[P:flagvar\], the action of the Steenrod squares $\textrm{Sq}^{2n}_{F}$ on $CH^{*}(X_{F})/2$ agrees with the action of Steenrod squares on the mod $2$ Chow ring of a split quadric in characteristic $0$. We refer to [@EKM Corollary 78.5] for the calculation of the action of Steenrod squares on the mod $2$ Chow ring of a split quadric in characteristic $0$.
\[P:action on quad\] For any $0 \leq i \leq d$ and $j \geq0$, $$\mathrm{Sq}^{2j}_{F}(h^{i})=\binom{i}{j}h^{i+j} \, \,and \, \, \,
\mathrm{Sq}^{2j}_{F}(l_{i})=\binom{D+1-i}{j}l_{i-j}.$$
To state our results, we recall the definition of relative higher Witt indices. Let $\varphi$ be a nonsingular quadratic form over a field $F$ and let $F(\varphi)$ denote the function field of the associated quadric. Let $\varphi_{an}$ denote the anisotropic part of $\varphi$ and let $\frak{i}_{0}(\varphi)$, the Witt index of $\varphi$, denote the dimension of a maximal isotropic subspace for $\varphi$. Start with $\varphi_{0}\coloneqq \varphi_{an}$ and $F_{0} \coloneqq F$. Inductively define $F_{i} \coloneqq F_{i-1}(\varphi_{i-1})$ and $\varphi_{i} \coloneqq (\varphi_{F_{i}})_{an}$ for $i>0$. There exists an integer $\frak{h}(\varphi)$ which is called the height of $\varphi$ such that $\mathrm{dim}\varphi_{\frak{h}(\varphi)} \leq 1$. For $1\leq j \leq \frak{h}(\varphi)$, we then define the $j$th relative higher Witt index $\frak{i}_{j}(\varphi)$ to be $\frak{i}_{0}(\varphi_{F_{j}})-\frak{i}_{0}(\varphi_{F_{j-1}})$.
We recall Hoffmann’s conjecture on the possible values of the first Witt index of an anisotropic quadratic form. Hoffmann’s conjecture was originally restricted to quadratic forms over a field of characteristic $\neq 2$ but it makes sense to consider the conjecture in characteristic $2$ as well. For an integer $n$, let $v_{2}(n)$ denote the $2$-adic exponent of $n$.
Let $\varphi$ be an anisotropic quadratic form over a field $F$ such that $\mathrm{dim}\varphi \geq 2$. Then $\frak{i}_{1}(\varphi) \leq 2^{v_{2}(\mathrm{dim}\varphi-\frak{i}_{1}(\varphi))}.$
Hoffmann’s original conjecture for characteristic $\neq 2$ was proved by Karpenko in [@Kar]. Karpenko’s proof makes use of Steenrod squares on mod $2$ Chow groups. With our construction of Steenrod squares on the mod $2$ Chow groups over a base field of characteristic $2$, we can now prove Hoffmann’s conjecture for nonsingular anisotropic quadratic forms over a field of characteristic $2$.
\[P:newquad\] Let $\varphi$ be a nonsingular anisotropic quadratic form over $k$ such that $\mathrm{dim}\varphi \geq 2$. Then $\frak{i}_{1}(\varphi) \leq 2^{v_{2}(\mathrm{dim}\varphi-\frak{i}_{1}(\varphi))}.$
The proof of [@EKM Proposition 79.4] works in this case and uses the computation of the Steenrod squares on the mod $2$ Chow ring of a split quadratic given by Proposition \[P:action on quad\] along with Corollary \[corbasechange\] on base change of the Steenrod squares. From the Cartan formula \[sectioncartan\] and results on shell triangles in [@EKM Sections 72,73] that were proved in arbitrary characteristic, we see that the conclusion of [@EKM Lemma 79.3] holds for nonsingular anisotropic quadratic forms in characteristic $2$.
Proposition \[P:newquad\] provides further evidence for the validity of Hoffmann’s conjecture in characteristic $2$. Scully has proved that Hoffmann’s conjecture is valid for totally singular quadratic forms over a field of characteristic $2$ [@Scu].
To finish, we extend $3$ more results of Karpenko on quadratic forms in characteristic $ \neq 2$ to the case of nonsingular anisotropic quadratic forms in characteristic $2$. Let $\varphi$ be a nonsingular anisotropic quadratic form defined over a field $k$ of characteristic $2$ with relative higher Witt indices $\frak{i}_{j} \coloneqq \frak{i}_{j}(\varphi)$ as defined above for $j=1, \ldots, \frak{h}\coloneqq \frak{h}(\varphi).$
Assume that $\frak{h}>1$. Then $$v_{2}(\frak{i}_{1}) \geq \textup{min}(v_{2}(\frak{i}_{2}), \ldots, v_{2}(\frak{i}_{\frak{h}}))-1.$$
The analogue of this proposition in characteristic $\neq2$ can be found in [@EKM Corollary 81.19]. The proof of [@EKM Corollary 81.19] works over a base field of characteristic $2$ using the properties we have established for the Steenrod squares $\textup{Sq}^{2n}_{k}.$ The conclusions of [@EKM Lemma 80.1] and [@EKM Theorem 80.2] hold in our situation since $\textup{Sq}^{2n}_{k}$ acts by squaring on $CH^{n}(-)/2$ by Proposition \[P:pthpower\] and the total homological Steenrod square commutes with proper pushforward by Proposition \[pushforward\].
We next discuss the characteristic $2$ analogue of the “holes in $I^n$" result [@EKM Corollary 82.2]. For a field $F$, the quadratic Witt group $I_{q}(F)$ is defined as the quotient of the Grothendieck group of the monoid of isometry classes of even-dimensional nonsingular quadratic forms by the subgroup generated by the hyperbolic plane [@EKM Section 8]. There is an action of the Witt ring $W(F)$ of nondegenerate symmetric bilinear forms on $I_{q}(F)$. Let $I(F) \subset W(F)$ denote the fundamental ideal of $W(F)$ and set $I_{q}^{n}(F) \coloneqq I^{n-1}(F) \cdot I_{q}(F)$ for $n \geq 1$. Let $k$ be a field of characteristic $2$. Mimicking the proof [@EKM Corollary 82.2] with $I_{q}^{n}(k)$ used in place of $I^{n}(k)$, we get the following result. Let $n \geq 1$.
Let $\varphi \in I_{q}^{n}(k)$ be a nonsingular anisotropic quadratic form such that $\textup{dim} \varphi < 2^{n+1}.$ Then there exists $ 0 \leq i \leq n$ such that $\textup{dim} \varphi=2^{n+1}-2^{i+1}.$
Our last result concerns $u$-invariants of fields. Following [@EKM Section 36], the $u$-invariant $u(F)$ of a field $F$ is defined to be the smallest non-negative integer (or $\infty$ if there is no such integer) $u(F)$ such that every nonsingular locally hyperbolic quadratic form $\varphi$ over $F$ with $\textup{dim}\varphi>u(F)$ is isotropic. Over a field of finite characteristic, every quadratic form is locally hyperbolic.
In [@Vis], Vishik constructed characteristic $0$ fields of $u$-invariant $2^{r}+1$ for all $r \geq 3$. Karpenko used Steenrod squares on mod $2$ Chow groups to show that for any $r \geq 3$ and any field $F$ of characteristic $\neq 2$, $F$ is contained in a field of $u$-invariant $2^{r}+1$ [@Kar2]. Karpenko’s constructions in [@Kar2] now extend to fields of characteristic $2$ through the use of the Steenrod squares $\textup{Sq}^{2n}_{k}$ defined in this paper for $k$ of characteristic $2$.
Let $k$ be a field of characteristic $2$ and let $r \geq 3$. Then $k$ is a subfield of a field of $u$-invariant $2^{r}+1.$
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abstract: '[The tetragonal ferrimagnetic Mn$_3$Ga exhibits a wide range of intriguing magnetic properties. Here, we report the emergence of topologically nontrivial nodal lines in the absence of spin orbit coupling (SOC) which are protected by both mirror and $C_{4z}$ rotational symmetries. In the presence of SOC we demonstrate that the doubly degenerate nontrivial crossing points evolve into $C_{4z}$-protected Weyl nodes with chiral charge of $\pm$2. Furthermore, we have considered the experimentally reported noncollinear ferrimagnetic structure, where the magnetic moment of the Mn$_I$ atom (on the Mn-Ga plane) is tilted by an angle $\theta$ with respect to the crystallographic $c$ axis. The evolution of the Weyl nodes with $\theta$ reveals that the double Weyl nodes split into a pair of charge-1 Weyl nodes whose separation can be tuned by the magnetic orientation in the noncollinear ferrimagnetic structure.]{}'
author:
- 'Cheng-Yi Huang'
- Hugo Aramberri
- Hsin Lin
- Nicholas Kioussis
title: 'Noncollinear Magnetic Modulation of Weyl Nodes in Ferrimagnetic Mn$_3$Ga'
---
=1
INTRODUCTION
============
The discovery of topological states of matter represents a cornerstone of condensed-matter physics that may accelerate the development of quantum information and spintronics and pave the way to realize massless particles such as Dirac and Weyl fermions. A Weyl semimetal (WSM) is a topological semimetallic material hosting doubly-degenerate gapless nodes near the Fermi level in the three-dimensional (3D) momentum space[@PTang17; @BJYang14; @NPArmitage18; @MZHasan15]. The nodes correspond to effective magnetic monopoles or antimonopoles which carry nonvanishing positive and negative chiral charge $\pm q$. Typically, $q$ takes values of $\pm$1 corresponding to Weyl nodes, but is also possible to have integers, $q = \pm2, \pm3, \dots$ for double Weyl nodes, etc. [@Hasan16] The Weyl nodes gives rise to surface states which form open Fermi arcs rather than closed loops.
Compared to their Dirac semimetal counterparts, WSMs require the breakdown either of inversion symmetry or time reversal symmetry (TRS) to split each four-fold degenerate Dirac node into a pair of Weyl nodes. A number of WSMs that break inversion symmetry have been identified in the past few years[@PTang17; @BJYang14; @NPArmitage18; @MZHasan15]. Moreover the presence of crystalline symmetries can further protect multiple Weyl nodes with large chiral charge[@CFang12; @ZGao16; @BBradlyn17]. On the other hand, the discovery of their broken TRS counterparts, which link the two worlds of topology and spintronics, remains challenging and elusive[@CFang12]. Many potential TRS-breaking WSM have been proposed. Recently, three groups have provided unambiguous and direct experimental confirmation that Co$_3$Sn$_2$S$_2$[@Morali; @Belopolskiarxiv], which becomes a ferromagnet below 175 K, and Co$_2$MnGa, a room-temperature ferromagnet[@Belopolski], are TRS-breaking WSMs. The discovery of magnetic WSMs give rise to exotic quantum states ranging from quantum anomalous Hall effect to axion insulators[@NPArmitage18]. Another remarkable and highly promising class of magnetic materials is the Heusler family[@CFelser15; @JWinterlik12] which includes half metals,[@RGroot83] ferromagnets, ferrimagnets, antiferromagnets, and even topological insulators[@SChadov10; @HLin10] and Weyl semimetals. In particular the ferrimagnetic and antiferromagnetic compounds with antiparallel exchange coupling, have recently garnered intense interest because of the faster spin dynamics (in the terahertz range) compared to the gigahertz-range magnetization dynamics of their ferromagnetic counterparts.[@KCai2019]
The Mn$_3$X (X=Ga, Ge, Sn) Heusler compounds are considered prototypes with promising applications in the area of spintronics[@JWinterlik12; @DZhang13]. These compounds can be experimentally stabilized in either the hexagonal DO$_{19}$ structure ($\epsilon$ phase) or the tetragonal DO$_{22}$ structure ($\tau$ phase)[@SKhmelevskyi16]. The high-temperature hexagonal crystal structure is antiferromagnetic with a high Néel temperature ($\sim$ 470 K) and a noncollinear triangular magnetic structure. Recently, several experimental and theoretical studies have demonstrated[@Kiyohara16; @AKNayak16; @PKRout19; @HYang17; @KKuroda17; @ZHLiu17; @ELiu18] the emergence of large anomalous Hall effect (AHE) in the noncollinear AFM hexagonal Mn$_3$X family, whose origin lies on the nonvanishing Berry curvature in momentum space. In addition, [*ab initio*]{} calculations have revealed that these chiral AFM materials are topological Weyl semimetals[@HYang17]. On the other hand, the low-temperature tetragonal phase, which can be obtained by annealing the hexagonal phase, is ferrimagnetic at room temperature and shows a unique combination of magnetic and electronic properties, including low magnetization,[@Wu2009] high uniaxial anisotropy,[@Bai2012] high spin polarization ($\approx$ 88%),[@Mizukami2011; @Kurt2011; @Winterlik2008] low Gilbert damping constant,[@Mizukami2011] and high Curie temperature.[@Kren1970] Interestingly, neutron scattering experiments have reported[@KRode13] a noncollinear ferrimagnetic magnetic structure in Mn$_3$Ga, where the magnetic moment orientation of the Mn atoms on the Mn-Ga (001) plane is tilted by about 21with respect to the crystallographic $c$ axis.
The objective of this work is to carry out first-principles electronic structure calculations to investigate the emergence of topological nodal lines in the absence or presence of SOC in tetragonal ferrimagnetic Mn$_3$Ga. Furthermore, we present results of the effect of non-collinear magnetism on the evolution of the Weyl nodes.
METHODOLOGY
===========
The electronic structure calculations were carried out by means of first-principles spin-polarized collinear calculations within the density functional theory (DFT) framework as implemented in the VASP package [@Vasp]. The Perdew-Burke-Ernzerhof [@PBEgga] (PBE) implementation of the generalized gradient approximation (GGA) for the exchange-correlation functional was employed. The plane-wave cutoff energy was set to 400 eV, which was enough to yield well-converged results. The Brillouin zone (BZ) was sampled using a $\Gamma$-centered mesh of $10$x$10$x$10$ k-points. The structure was allowed to relax until residual atomic forces became lower than 0.01 eV/Å and residual stresses became smaller than 0.01 GPa. The spin-orbit coupling (SOC) of the valence electrons is in turn included self-consistently using the second-variation method employing the scalar-relativistic eigenfunctions of the valence states[@koelling], as implemented in VASP. Then, DFT derived wave functions both without and with SOC were in turn projected to Wannier functions using the wannier90 package [@Wannier90].
In the $D$O$_{22}$ structure (I4/mmm space group) the two (001) antiferromagnetically-coupled Mn sublattices, shown in Fig. \[struc\](a), consist of Mn$_I$ atoms at the Wyckoff positions 2b (0,0,1/2) \[Mn$_I$-Ga (001) plane\] and Mn$_{II}$ atoms at the 4d (0,1/2,1/4) positions \[Mn$_{II}$-Mn$_{II}$ (001) plane\]. For the noncollinear calculation, where the magnetic moment of the Mn$_I$ is rotated by an angle $\pi-\theta$ with respect to the \[001\] direction, the angular dependence of the Wannier Hamiltonian is determined from, $$H({\bf k},\theta)=H_0({\bf k})+U(\theta)H_{ex}({\bf k})U^{\dag}(\theta).$$ Here, $H_0({\bf k})$ is the TRS preserving Hamiltonian without or with SOC, \[$TH_0({\bf k})T^{-1}=H_0(-{\bf k})$\], $T$ is the TRS operator, $H_{ex}({\bf k})$ is the TRS-breaking exchange Hamiltonian, \[$TH_{ex}({\bf k})T^{-1}=-H_{ex}(-{\bf k})$\], $U(\theta)=e^{-i\frac{\pi-\theta}{2}\sigma_{y,Mn_{I}}}$ is the spin rotation operator, and $\sigma_{y,{\text{Mn$_I$}}}$ is the $y$ component of Pauli matrix acting on the spin degrees of freedom of Mn$_I$.
results and discussion
======================
**Nodal lines in the absence of SOC**
-------------------------------------
The calculated lattice parameters $a=b$=3.78 Å and $c$=7.08 Å, are in good agreement with previous calculations [@Balke2007; @KRode13; @Aqtash2015], which are, however, lower than the experimental values of $a=b$=3.92 Å and $c$=7.08 Å. Our calculated values of the magnetic moments of -2.83 $\mu_B$ and 2.30 $\mu_B$ for the Mn$_I$ and Mn$_{II}$ atoms, respectively, are in good agreement with previous DFT calculations[@Balke2007; @KRode13; @Aqtash2015].
![(Color online) (a) The tetragonal cell of the DO$_{22}$ ferrimagnetic structure with \[001\] spin polarization. Arrows denote the magnetic moments of Mn$_I$ (purple) and Mn$_{II}$ (red) sublattices which are coupled antiferromagnetically. (b) First Brillouin zone of the primitive cell shown in panel (a). (c) Spin-polarized band structure without SOC along the high symmetry directions of the primitive cell, where the spin-up (spin-down) bands are denoted by blue (red). \[struc\]](structureband.pdf){width="8.cm"}
Fig. \[struc\](c) shows the spin-polarized band structure of the majority- (blue) and minority-spin (red) bands of Mn$_3$Ga without SOC and with collinear spins along the symmetry lines of the Brillouin zone (BZ) of the primitive cell, shown in Fig. \[struc\](b). For each spin channel, the energy bands can be labeled by the eigenvalues of the crystalline symmetry operator of a particular high symmetry direction. The band structure along the M-$\Gamma$-M direction, shown in Fig. \[NL\](a), features several band crossings close to the Fermi level. Thus, throughout the remainder of the manuscript, we only focus on the crossing points, marked by black circles in Fig. \[NL\] (a), between the majority-spin bands along the $k_z$ ($\Gamma-M$) direction. These points are protected by both a mirror reflection symmetry normal to the \[110\] direction, $M_{[110]}$, and a four-fold rotational symmetry, $C_{4z}$, and hence can be labeled by the pair of eigenvalues, ($\pm 1$,$\pm 1$), of $M_{[110]}$ and $C_{4z}$, respectively. We have tracked the nodal lines on the $M_{[110]}$-invariant plane. The other nodal lines on the $M_{[1\bar{1}0]}$-invariant plane were determined using the $C_{4z}$ rotational symmetry. Fig. \[NL\](b) shows the 3D landscape of nodal lines in momentum space. We find that the nodal lines are topologically nontrivial characterized by the $\pi$ Berry phase[@RYu11; @CKChiu14; @CFang15]. The two black points denote the nontrivial crossing points as well as the intersecting points of nodal lines along the $k_z$ direction in Fig. \[NL\](b). Notably the crossing points remain gapless and robust against a distortion breaking either $M_{[110]}$ or $C_{4z}$.
![(Color online) (a) Spin polarized band structure along $k_z$-axis ($\Gamma-M$ direction) without SOC, where the blue (red) bands denote the spin-up (spin-down) states. The two nontrivial crossing points, denoted with the black dots, are labeled with the pair of eigenvalues, ($\pm1$,$\pm$1), of the mirror, $M_{[110]}$, and four-fold rotational, $C_{4z}$, symmetries, which protect them, (b) 3D landscape of the nodal lines where the two black dots denote the two nontrivial crossing points in (a). The color bar represents the energy of the nodal points relative to the Fermi energy. \[NL\]](NL.pdf){width="8.cm"}
Weyl Nodes in the Presence of SOC
---------------------------------
In the presence of SOC, the symmetry conservation depends on the magnetic orientation and the crystalline symmetries. More specifically the \[001\] collinear magnetic configuration is invariant under (1) inversion symmetry ($P$), (2) fourfold rotational symmetry about the $z$-axis ($C_{4z}$) and (3) mirror reflection symmetry normal to the $z$ direction ($M_z$). We next discuss the effect of magnetization orientation (collinear versus noncollinear) on the topological features of the band structure.
*Weyl Nodes in Collinear Ferrimagnetism—* In the presence of SOC, the mirror symmetry $M_{[110]}$ is no longer preserved when the magentization of the collinear ferrimagentic Mn$_3$Ga is along the \[001\] direction. Consequently, in general the nodal points in Fig. \[NL\](b) are gapped out except for those crossing points along $k_z$ which are protected by the $C_{4z}$ rotational symmetry. Thus, for the band structure along the $C_{4z}$-invariant $k_z$-axis, shown in Fig. \[WP\](a), we can identify the states by the eigenvalues of $C_{4z}$ and locate the nontrivial crossing points associated with different eigenvalues. The nontrivial crossing points, marked by red circles in Fig. \[WP\](a), are Weyl nodes protected by $C_{4z}$ symmetry, whose position along $k_z$, energy relative to E$_F$, ratio of conduction to valence band $C_{4z}$ eigenvalues, $u_c/u_v$, chiral charge, $C$, and dispersion are summarized in Table \[T1\]. Interestingly, the $C_{4z}$-protected Weyl fermion with $u_c/u_v$= -1 carries chiral charge +2 and has quadratic dispersion on the $k_x$-$k_y$ plane,[@CFang12] in sharp contrast to the double Weyl fermion with fourfold degeneracy and linear dispersion[@PTang17]. Its other parity partner has opposite chiral charge of -2. Fig. \[WP\](b) displays the two Fermi arcs on the (100) surface emerging from the two charge-2 Weyl nodes.
{width="90.00000%"}
[c>m[2cm]{}ccm[2.5cm]{}]{} $k_z$(Å$^{-1}$) & E-E$_F$ (meV) & $u_c/u_v$ & $C$ & Dispersion
on $k_x$-$k_y$ plane\
\
0.2811 & 229 & -1 & 2 & $k^2$\
-0.2811 & 229 & -1 & -2 & $k^2$\
*Evolution of Weyl Fermions in NonCollinear Ferrimagnetism—* Neutron scattering experiments have reported[@KRode13] a noncollinear ferrimagnetic magnetic structure in the DO$_{22}$ ferrimagnetic Mn$_3$Ga structure, where there is a significant in-plane magnetic moment, $\mu_x^{2b}$ = 1.19$\mu_B$ carried by the Mn$_I$ atoms \[on the Mn-Ga (001) plane\] leading to a 21$^{\circ}$ tilt of the Mn$_I$ moment from the crystallographic $c$ axis \[see Fig. \[wpevo\](a)\]. This noncollinear magnetic ordering spontaneously breaks both the $C_{4z}$ and $M_z$ symmetry operations while only preserving $P$. Consequently, the $C_{4z}$-protected double Weyl fermion on the $k_z$ axis for the case of collinear ferrimagntism splits into two charge-1 Weyl fermions which shift away from the $k_z$ axis.
In order to investigate this scenario, we have studied the evolution of the Weyl points upon rotation of all magnetic moments of the Mn$_I$ atoms at the Wyckoff positions 2b with respect to the crystallographic $z$ axis by the angle $\theta$, ${\bm {\mu}}^{2b} = \mu^{2b} (-\sin\theta\hat x+\cos\theta\hat z)$, while fixing the direction of the Mn$_{II}$ magnetic moments, as shown in Fig. \[wpevo\](a). Here, $\theta=$ 180 indicates the collinear (001) ferrimagnetism. Using the Wannier functions we find that at $\theta$ = 160 the magnitude of the calculated $x$-component of the magnetic moment of the Mn$_I$ atoms is 0.94$\mu_B$/Mn in good agreement with the corresponding experimental values of 1.12$\mu_B$. Fig. \[wpevo\](b) shows the evolution of the Weyl nodes as $\theta$ changes from 180 to 170 and finally to 160. Initially, at $\theta=$ 180, the two charge-2 Weyl nodes lie on the $k_z$-axis. As $\theta$ decreases each charge-2 Weyl node splits into two charge-1 Weyl nodes which move away from the $k_z$-axis, leading to the emergence of four charge-1 Weyl fermions in the case of noncollinear ferrimagnetism. Our electronic structure calculations of the Fermi arcs on the (100) surface for $\theta$ = 160 show that the noncollinear effect is small on the Fermi arcs in Fig. \[WP\](b), at least for small angle.
![(Color online) (a) Noncollinear ferrimagnetic DO$_{22}$ structure of Mn$_3$Ga,[@KRode13] where the Mn$_I$ atoms \[on the Mn-Ga (001) plane\] carry a substantial in-plane magnetic moment leading to a tilt of their moments from the crystallographic $c$ axis. (b) Evolution of Weyl nodes in the 3D BZ as a function of tilt angle $\theta$, where the red, green and blue circles denote the Weyl nodes at $\theta=$ 180, 170 and 160,respectively. Dashed arrows show the motion of Weyl points with decreasing $\theta$. At $\theta=$180, (collinear case) the two charge-2 Weyl nodes lie on the $C_{4z}$-protected $k_z$-axis. For $\theta \neq$180 each charge-2 Weyl node splits into two charge-1 Weyl nodes which in turn move away from the $k_z$-axis. The integer above each Weyl node denotes the chiral charge.\[wpevo\]](WPevo.pdf){width="8.5cm"}
CONCLUSION
==========
In summary, our [*ab initio*]{} electronic structure calculations have shown that in the absence of SOC, nontrivial nodal lines emerge in collinear ferrimagnetic tetragonal Mn$_3$Ga. The nodal lines are protected by both mirror reflection symmetry normal to the \[110\] direction, $M_{[110]}$, and a four-fold rotational symmetry, $C_{4z}$. The presence of SOC gaps out the nodal lines except for the nodal line intersecting points which become $C_{4z}$-protected charge-2 Weyl nodes with quadratic dispersion in the $k_x$-$k_y$ plane. The noncollinear magnetism associated with the Mn$_I$ atoms splits the double Weyl nodes Fermions into two charge-1 Weyl nodes moving away from the $k_z$ axis, whose separation can be selectively tuned by the noncollinarity angle.
The work at CSUN is supported by NSF-Partnership in Research and Education in Materials (PREM) Grant No. DMR-1828019. H.L. acknowledges the support by the Ministry of Science and Technology (MOST) in Taiwan under grant number MOST 109-2112-M-001-014-MY3.
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abstract: 'We provide a “first principles” description of scattering from open quantum systems subject to a Lindblad-type dynamics. In particular we consider the case that the duration of the scattering process is of similar order as the decoherence time of the scatterer. Under rather general conditions, the derivations lead to the the following new result: The irreversible time-evolution may cause a reduction of the system’s transition rate being effectuated by scattering. This is tantamount to a shortfall of scattering intensity. The possible connection with striking experimental results of neutron and electron Compton scattering from protons in condensed matter is mentioned.'
---
C. Aris Chatzidimitriou-Dreismann$^1$ and Stig Stenholm$^2$
$^1$ Institute of Chemistry, Sekr C2, Technical University Berlin, D-10623 Berlin,\
Germany. *Email:* dreismann@chem.tu-berlin.de
$^2$Department of Physics, Royal Institute of Technology, SE-10691 Stockholm, Sweden. *Email:* stenholm@atom.kth.se
Keywords: irreversible dynamics, entanglement, decoherence, neutron Compton scattering, electron-proton Compton scattering
Introduction
============
The counter-intuitive phenomenon of entanglement [@QE] between two or more quantum systems has emerged as the most emblematic feature of quantum mechanics. Experiments investigating entnaglement, however, are mainly focused on collections of few simple (two- or three-level) quantum systems thoroughly isolated from their environment (e.g., atoms in high-$Q$ cavities and optical lattices). These experimental conditions are necessary due to the decoherence of entangled states. In short, decoherence refers to the suppression of quantum superpositions caused by the environment. By contrast, entanglement in condensed and/or molecular matter at ambient conditions is usually assumed to be experimentally inaccessible. However, two new scattering techniques operating in the sub-femtosecond time scale provided results indicating that short-lived entangled states may be measurable in condensed matter even at room temperature [@PRL97; @highlights].
In this paper we provide a first-principles treatment of scattering from “small” open quantum systems in condense-matter environments, in the “time window” of decoherence of the scattering system. That is, the focus is in “fast” scattering processes with a duration (usually denoted scattering time, $\tau_{sc}$) of the order to the scatterer’s decoherence time, $\tau_{dec}$. This may be considered to represent an “extension” of standard scattering theory — as applied e.g. to neutron physics [@vHove; @Squires] or electron scattering [@Weigold] — in which the concepts of entanglement and decoherence play essentially no role. The first part of the derivations are analogous to the standard (often denoted) “van Hove formalism” [@vHove]; see also the textbook [@Squires]. Then a reduced open quantum system, i.e. a micro- or mesoscopic system characterized by a set of preferred coordinates, is introduced. This corresponds to the “small” physical system that scatters a neutron (electron, etc.) with a sufficiently large momentum transfer. Its dynamics is described by a simple Lindblad-type master equation [@Lindblad; @Stig1] (which, for the sake of simplicity, contains only one Lindblad operator, $X$), thus including explicitly the effect of decoherence into the formalism.
The striking result of the derivation may be summarized as follows: The irreversible time-evolution (owing to the Lindblad operator $X$) may cause a reduction of the transition rate of the system (from its initial to its final state). In “experimental” terms, this is tantamount to an effective reduction of the system’s cross-section density and thus a shortfall of scattering intensity.
Scattering in brief
===================
We assume an N-body Hamiltonian $H_{total}=H_0+V$ with an interaction of the form $$V(\mathbf{r})=\lambda \,n(\mathbf{r}) \label{a1}$$ where $\,n(\mathbf{r})$ is the particle density operator and $\lambda $ is the rest of the interaction (contact potential). For example, in the case of neutron scattering from a system consisting of $N$ particles with the same scattering length $b$ one may put $$n(\mathbf{r}) = \frac{1}{V}\sum_j \delta(\mathbf{r}-\mathbf{R}_j)
\label{a1-1}$$ where $V$ is the volume, $\mathbf{R}_j$ is the spatial position of the $j$-th particle, and $$\lambda = \frac{2 \pi \hbar^2}{m}\, b \ ,$$ $m$ being the neutron mass. (For further details about scattering from “bound” and “free” particles, see the textbook [@Squires].)
In the *interaction* picture, the Schrödinger equation is now (putting for simplicity $\hbar =1$) $$i\partial _{t}\Psi =\lambda\, n(\mathbf{r,}t)\Psi$$ with the perturbative solution $$\Psi (t)=\Psi (0)-i\lambda \int_{0}^{t}n(\mathbf{r},t^{\prime
})dt^{\prime }\Psi (0). \label{a2}$$
We write the transition probability $W(t)$ between initial states $\psi_{i}$ (with probability $P_{i}$ that the scattering system is in the state $\psi_{i}$) and final states $\psi _{f}$ of the scattering system to be given by $$W(t) = \sum_{i,f}\mid \langle \psi _{f}\mid \lambda
\int_{0}^{t}n(\mathbf{r}, t^{\prime })dt^{\prime }\mid \psi
_{i}\rangle \mid ^{2}P_{i}.
\label{a3}$$ It should be noted that $\psi_{i}$ and $\psi _{f}$ are eigenstates of the unperturbed N-body Hamiltonian $H_0$ [@Squires; @vHove]. This allows us to write the transition probability in the form $$W(t) = \lambda ^2\int_0^tdt^{\prime }\int_0^tdt^{\prime \prime}
\sum_f \langle \psi _f\mid n(\mathbf{r,}t^{\prime })\,
\rho\, n(\mathbf{r,}
t^{\prime \prime })\mid \psi _f\rangle , \label{a3+1}$$ where $$%\[ \]
\rho =\sum_i\mid \psi _i\rangle P_i\langle \psi _i\mid ,
\label{rho}$$ by noting that $n^\dag(\mathbf{r},t) = n(\mathbf{r},t)$, since $\mathbf{R}_j$ and $\mathbf{r}$ are Hermitian operators.
In an actual scattering experiment from condensed matter, we do not measure the cross-section for a process in which the scattering system goes from a specific initial state $\psi _{i}$ to another state $\psi _{f}$, both being unobserved states of the many-body system. Therefore, one takes an appropriate average over all these states [@Squires; @vHove], as done in Eq. (\[a3\]).
Furthermore, the initial ($\mathbf{k}_0$) and final ($\mathbf{k}_1$) momenta of an impinging probe particle (neutron) may be assumed to be well defined [@Squires; @vHove]. Introducing the momentum transfer $\mathbf{q}= \mathbf{k}_0 -
\mathbf{k}_1$ from the probe particle to the scattering system, the Fourier transform of the particle density reads $$n(\mathbf{r},t) = \frac{1}{(2\pi)^3}
\int\!d\mathbf{q}\, n(\mathbf{q},t)
\exp\{i\,\mathbf{q}\cdot\mathbf{r}\}$$ where, in the case of neutron scattering, cf. Eq.(\[a1-1\]), $$n(\mathbf{q},t)= \sum_j \exp\{-i \mathbf{q}\cdot \mathbf{R}_j(t)
\} \ .$$ Since $n(\mathbf{r},t) $ is Hermitian it holds $
n^\dag(\mathbf{q},t) = n(-\mathbf{q},t)
$ and one obtains from Eq. (\[a3\]) $$W(t) = \lambda ^2\int_0^tdt^{\prime }\int_0^tdt^{\prime \prime}
\sum_f \langle \psi _f|
n(\mathbf{q},t^{\prime })\,\rho \,n(-\mathbf{q},t^{\prime \prime
})
|\psi _f\rangle , \label{a4}$$
At this stage one traditionally assumes that the sum over $\psi
_f$ runs over all possible eigenstates of $H_0$ which constitute a complete set, i.e. $\Sigma_f |\psi _f\rangle\langle \psi _f|
=\mathbf{1}$; see [@Squires; @vHove]. Hence $$\sum_f \langle \psi _f|
n(\mathbf{q},t^{\prime })\,\rho \,n(-\mathbf{q},t^{\prime \prime
})
|\psi _f\rangle =
Tr\left[ n( \mathbf{q},t^{\prime })\,\rho
\,n(-\mathbf{q},t^{\prime \prime })\right]
\label{a-ignore}$$ where $Tr[...]$ denotes the trace operation. As done in standard theory [@Squires; @vHove], in Eq. (\[a-ignore\]) one first sums over all final states, keeping the initial state $\psi _{i}$ fixed, and then averages over all $\psi_{i}$ (see e.g. [@Squires], p. 19). The right-hand-side of Eq. (\[a-ignore\]) contains the density operator $\rho$ of the system before collision, Eq. (\[rho\]), which is a well known result.
By introducing a measurement time (the so-called scattering time) $\tau_{sc}$, that is the duration of the scattering process, we find $$\begin{aligned}
W(\tau_{sc}) &=&\lambda ^2\int_0^{\tau_{sc}}dt^{\prime
}\int_0^{\tau_{sc}} dt^{\prime \prime }Tr\left[ n(
\mathbf{q},t^{\prime })\,\rho \,n(-\mathbf{q},t^{\prime \prime
})\right]
\nonumber\\
&=&\lambda ^2 \tau_{sc} \int_0^{\tau_{sc}}d\tau \,Tr\left[
n(\mathbf{q},t^{\prime })\,\rho \,n(-\mathbf{q},t^{\prime } + \tau
)\right] ,
\label{a5}\end{aligned}$$ where the stationary property of the correlation function has been used [@Squires]. Now one can introduce the transition rate, say $\dot{W}$, which is defined as $$\begin{aligned}
\dot{W} \equiv \frac{W(\tau_{sc})}{\tau_{sc}}
&=&\lambda
^2\int_0^{\tau_{sc}}d\tau \,Tr\left( n(\mathbf{k,} t^{\prime
})\,\rho \,n(-\mathbf{k,}t^{\prime }+\tau )\right)
\nonumber\\
&\equiv& \lambda
^2\int_0^{\tau_{sc}}d\tau \,C(\mathbf{q},\tau ).
\label{a6}\end{aligned}$$ Here the correlation function $$C(\mathbf{q},t)= Tr[n(\mathbf{q}, 0)\,\rho \,n(-\mathbf{q},t)]
\label{a7}$$ is introduced, which is analogous to the so-called intermediate function of neutron scattering theory [@Squires].
Irreversible dynamics
=====================
We now introduce a set of preferred coordinates $\{\, | \xi
\rangle\}$, cf. [@Zeh; @Zurek]. These are the relevant degrees of freedom coupled to the neutron probe. The density matrix needed in (\[a5\]) is then the *reduced* one in the space spanned by these states, and it is obtained by tracing out the (huge number of the) remaining degrees of freedom belonging to the “environment” of the microscopic scattering system (e.g. a proton and its adjacent particles). To simplify notations, we denote this reduced density matrix by $\rho$ too.
In the *subspace* spanned by the preferred coordinates (also denoted ’pointer basis’), we assume the relevant density matrix to obey a Lindblad-type equation of the form [@Lindblad; @Stig1] $$\partial _t\rho =-i\left[ H,\rho \right] +\mathcal{R}\rho \equiv \mathcal{L}%
\rho \label{b1}$$ with the formal solution $$\rho (t)=e^{\mathcal{L}t}\rho (0).$$
Let us look at a time-dependent expectation value $$\langle A(t)\rangle \equiv Tr\left( \rho (t)A\right) =Tr\left( e^{\mathcal{L}%
t}\rho (0)A\right) =Tr\left( \rho (0)e^{\mathcal{L}^{\dagger
}t}A\right) \ ,$$ where we define $\mathcal{L}^{\dagger }$ by setting $$Tr\left( \left( \mathcal{L}X\right) Y\right) =Tr\left( X\left( \mathcal{L}%
^{\dagger }Y\right) \right) \ .$$ Thus we obtain a Lindblad time evolution for the operators too by writing $$\partial _tA(t)=\mathcal{L}^{\dagger }A(t).$$ This form was actually the original Lindblad result. Note that this works as long as $\mathcal{L}$ does not depend on time. For time-dependent generators of the evolution, a somewhat more elaborate scheme is needed.
Now we find that we may use this formalism to calculate correlation functions like the one in (\[a6\]). We write $$\langle A(t)B\rangle =Tr\left[ \rho (0)\left(
e^{\mathcal{L}^{\dagger
}t}A\right) B\right] =Tr\left[ Ae^{\mathcal{L}t}\left( B\rho (0)\right) %
\right] \equiv Tr\left( A\rho _B(t)\right) ,
\label{b2}$$ where $\rho _B(t)$, as defined in Eqs. (\[b2\]), obeys the equation $$\partial _t\rho _B(t)=\mathcal{L}\rho _B(t)$$ and the initial condition $$\rho _B(0)=B\rho (0).$$ Thus, except for the initial condition, we have to solve the same equation of motion as for the density matrix (\[b1\]).
Application to scattering
=========================
We here assume a simple Lindblad-type ansatz for the master equation in the relevant subspace. We set $$\partial _t\rho =-i\left[ H,\rho \right] -K\left[ X,\left[ X,\rho \right] %
\right] =\mathcal{L\rho },
\label{c1}$$ where $K>0$ and $H$ is the reduced (or relevant Hamiltonian) of a microscopic or mesoscopic scattering system and the double commutator term describes decoherence. For simplicity of the further calculations, we here assume that $$\begin{array}{lll}
H\mid \xi \rangle & = & \mathcal{E}_\xi \mid \xi \rangle \\
& & \\
X\mid \xi \rangle & = & \xi \mid \xi \rangle .%
\end{array}
\label{b3}$$ With Eq. (\[a7\]) we have $$\begin{aligned}
C(\mathbf{q},\tau )
&= &Tr [ n(\mathbf{q},0)\,\rho
\,n(-\mathbf{q},\tau )]
= Tr\left( n(\mathbf{q},0)\,\rho
e^{\mathcal{L}^{\dagger }t}n(-\mathbf{q}, 0)\,\right) \nonumber \\
& =& Tr\left( n(-\mathbf{q},0)\,e^{\mathcal{L}t}\left( n(\mathbf{q},
0)\,\rho \right) \right) \ .\end{aligned}$$ This is equivalent with the expression $$C(\mathbf{q},\tau
)=\sum_{\xi ,\xi ^{\prime }}\langle \xi \mid n(-\mathbf{q},0)\mid
\xi ^{\prime }\rangle \langle \xi ^{\prime }\mid \rho _n(t)\mid
\xi \rangle .$$
With the equation (\[c1\]), one easily finds the well known solution $$\begin{aligned}
\langle \xi ^{\prime }\mid \rho _n(t)\mid \xi \rangle
&=&
\exp\left[ -i\left( \mathcal{E}_{\xi ^{\prime }}-\mathcal{E}_\xi
\right)
t \right]
\exp \left[ -K\left( \xi ^{\prime }-\xi \right) ^2t\right]
\nonumber\\
& &\times \ \langle \xi ^{\prime }\mid n( \mathbf{q},0)\,\rho(0)
\mid \xi \rangle .\end{aligned}$$ Inserting this into the expression (\[a6\]) for the transition rate we find $$\begin{aligned}
\dot{W} &= &\lambda ^2\int_0^{\tau_{sc}} d\tau \,\sum_{\xi ,\xi
^{\prime }}\exp\left[ -i\left( \mathcal{E}_{\xi ^{\prime
}}-\mathcal{E}_\xi \right) t\right] \ \exp \left[ -K\left( \xi
^{\prime }-\xi \right) ^2t\right]
\nonumber \\
&&\hspace{2cm}
\times\langle \xi \mid n(
-\mathbf{q},0)\mid \xi ^{\prime }\rangle \langle \xi ^{\prime
}\mid n(\mathbf{ q},0)\,\rho(0) \mid \xi \rangle \ .
\label{b-end}\end{aligned}$$
Decoherence and decrease of transition rate
===========================================
Obviously, the decoherence-free limit of this result, i.e. with $K=0$, corresponds to the conventional result of scattering theory.
The oscillating factors $\exp\left[ -i\left( \mathcal{E}_{\xi
^{\prime }}-\mathcal{E}_\xi \right) t\right]$ are characteristic for the ’unitary-type’ dynamics caused by the commutator part $-i\left[ H,\rho \right]$ of the master equation (\[c1\]) for the reduced (or: relevant) density matrix $\rho$. These factors have the absolute value 1 and do not affect the numerical value of the transition rate.
On the other hand, the restrictive factors $\exp ( -K\left( \xi
^{\prime }-\xi \right) ^2t) \leq 1$, which are due to the decoherence, can be seen to cause a decrease of the transition rate and thus of the associated cross-section. This can be illustrated in physical terms as follows:
Let us first assume that the reduced density operator $\rho(0)$ can be chosen to be *diagonal* in the preferred $\xi-$representation (which corresponds to the usual random phase approximation at $t=0$). Then each term of Eq. (\[b-end\]) is of the form $$\begin{aligned}
\lefteqn{
\langle\xi| n(-\mathbf{q},0)| \xi ^{\prime }\rangle
\langle \xi ^{\prime }| n(\mathbf{ q},0)\,\rho(0) | \xi \rangle
= } \nonumber\\
& & \hspace{0.5cm}
\langle \xi |n( -\mathbf{q},0)\mid \xi ^{\prime }\rangle
\langle \xi ^{\prime }| n(\mathbf{
q},0)|\xi\rangle\langle\xi|\rho(0) | \xi \rangle = \nonumber \\
& & \hspace{2cm}
|\langle \xi \mid n( -\mathbf{q},0)\mid \xi ^{\prime }\rangle|^2
\langle\xi|\rho(0)| \xi \rangle \geq 0 \ .\end{aligned}$$ The last inequality is valid because it holds $\langle\xi|\rho(0)|
\xi \rangle \geq 0$.
If the assumed diagonal form of $\rho(0)$ would be considered as being ’too strong’, one may note the following: The exponentials $\exp ( -K\left( \xi ^{\prime }-\xi \right) ^2t)$ due to decoherence imply that only terms with $\xi \approx \xi'$ contribute significantly to the transition rate. Thus we may conclude that, by continuity, all associated terms with $\xi
\approx \xi'$ in Eq. (\[b-end\]) should be positive, too. The further terms with $\xi$ being much different from $\xi'$ can be positive or negative. But they may be approximately neglected, since they decay very fast and thus contribute less significantly to $\dot{W}$.
The main conclusion from the preceding considerations is that the time average in Eq. (\[b-end\]) always decreases the numerical value of $\dot{W} \equiv W(\tau_{sc})/\tau_{sc}$, due to the presence of the exponential factors $\exp(-K\left( \xi ^{\prime
}-\xi \right) ^2t) \leq 1$. In other words, the effect of decoherence during the experimental time window $\tau_{sc}$ plays a crucial role in the scattering process and leads to an ’anomalous’ decrease of the transition rate and the associated scattering intensity. This result is in line with that of Ref. [@Schema1], which investigated the standard expression of the double differential cross-section of neutron scattering theory by ad hoc assuming decoherence of final and initial states of the scatterer.
Very interesting is also the conclusion that, in the limit of very slow decoherence ($K\rightarrow 0$), this ’anomaly’ disappears, i.e. the scattering results are expected to agree with conventional theoretical expectations. This is contrary to the associated prediction of the theoretical model of Ref. [@Karlsson1]
Additional remarks
==================
A related effect (i.e., a shortfall of scattering intensity) was observed in recent neutron-proton Compton scattering (NCS) and electron-proton Compton scattering (ECS) experiments in condensed matter [@PRL97; @highlights], in which the experimental scattering time lies in the sub-femtosecond time scale. This coincides with the characteristic time of electronic re-arrangements accompanying the breaking (or formation) of a chemical bond. Note that in these experiments the energy transferred to a proton is large enough to break the bond (C–H and O–H).
Some remarks about the possible selection, definition and/or physical meaning of the preferred coordinates may be appropriate. In the case of conventional NCS theory, for example, one uses momentum eigenstates of the scattering particle (e.g. proton) — as well as for the neutron — as the appropriate basis [@Watson]. In the light of the preceding derivations, however, one may observe the following: Due to the strong (Coulomb) interactions of the scattering proton with its adjacent particles (electrons, and probably also other nuclei), $\{|\xi\rangle\}$ can not be one-body states but they should rather be considered to represent momentum states being strongly “dressed” (and entangled) with degrees of freedom of adjacent particles.
Further work will deal with the more general — and experimentally relevant — case, in which the preferred states $\{| \xi \rangle\}$ are not eigenstates of the “reduced” energy and Lindbald operators, Eqs. (\[b3\]). In that case, the result of Eq. (\[b-end\]) will become less simple.
Acknowledgments
===============
This work was partially supported by the EU RT-network QUACS (Quantum Complex Systems: Entanglement and Decoherence from Nano- to Macro-Scales).
[00]{} A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. **47**, 777 (1935). L. van Hove, Phys. Rev. **95**, 249 (1954). G. L. Squires, *Introduction to the Theory of Thermal Neutron Scattering* (Dover, Mineola, 1996).
E. Weigold and I. E. McCarthy, *Electron Momentum Spectroscopy* (Kluwer Academic/Plenum, New York, 1999).
C. A. Chatzidimitriou-Dreismann et al., Phys. Rev. Lett. [**79**]{}, 2839 (1997); and [**91**]{}, 057403 (2003). (a) Physics Today, Physics Update, p. 9, September 2003; (b) Scientific American, p. 20, October 2003; (c) The AIP Bulletin of Physics News, Physics New Update No. 648, 31 July 2003.
G. Lindblad, Comm. Math. Phys. **48**, 119 (1976). S. M. Barnett and S. Stenholm, Phys. Rev. A **64**, 033808 (2001).
\(a) H. D. Zeh, Found. Phys. **3**, 109 ( 1973); (b) O. Kübler and H. D. Zeh, Ann. Phys. (NY) **76**, 405 (1973). (a) W. H. Zurek, Phys. Rev. D **24**, 1516 (1981); and **26**, 1862 (1982); (b) W. H. Zurek, Rev. Mod. Phys. **75**, 715 (2003). C. A. Chatzidimitriou-Dreismann, preprint (2003); and J. Alloys Compd. **356-357**, 244 (2003) E. B. Karlsson and S. W. Lovesey, Phys. Rev. A **61**, 062714 (2000); Phys. Scr. **65**, 112 (2002). G. I. Watson, J. Phys. Condens. Matter [**8**]{}, 5955 (1996).
|
---
abstract: 'We study coherent pion production in neutrino-nucleus scattering in the energy region relevant to neutrino oscillation experiments of current interest. Our approach is based on a combined use of the Sato-Lee model of electroweak pion production on a nucleon and the $\Delta$-hole model of pion-nucleus reactions. Thus we develop a model which describes pion-nucleus scattering and electroweak coherent pion production in a unified manner. Numerical calculations are carried out for the case of the $^{12}$C target. All the free parameters in our model are fixed by fitting to both total and elastic differential cross sections for $\pi-^{12}$C scattering. Then we demonstrate the reliability of our approach by confronting our prediction for the coherent pion photo-productions with data. Finally, we calculate total and differential cross sections for neutrino-induced coherent pion production, and some of the results are (will be) compared with the recent (forthcoming) data from K2K, SciBooNE and MiniBooNE. We also study effect of the non-locality of the $\Delta$-propagation in the nucleus, and compare the elementary amplitudes used in different microscopic calculations.'
author:
- 'S. X. Nakamura'
- 'T. Sato'
- 'T.-S. H. Lee'
- 'B. Szczerbinska'
- 'K. Kubodera'
title: 'Dynamical Model of Coherent Pion Production in Neutrino-Nucleus Scattering'
---
[^1]
Introduction {#intro}
============
The detailed theoretical study of neutrino-nucleus reactions is of great current importance due to the ever increasing precision of neutrino oscillation experiments (recently carried out, on-going and forthcoming). Since most of these experiments measure the neutrino flux through neutrino-nucleus scattering, reliable theoretical estimates of the relevant cross sections are prerequisite for the accurate interpretation of the data. Some of these experiments (T2K, MiniBooNE, etc.) use neutrinos in an energy range within which the dominant processes are the quasi-elastic nucleon knockout and the quasi-free single-pion production through the excitation of the $\Delta$ (1232) resonance. Meanwhile, coherent single-pion production in this energy region (albeit not a dominant process) is also of considerable interest, since it allows us to study, with no ambiguity concerning the final nuclear state, the details of the $\Delta$-excitation mechanism and medium effects on the pion; the knowledge of these details is essential for predicting the dominant quasi-free pion production processes. In this paper we focus on the coherent single-pion production process.
There have indeed been quite active experimental efforts to investigate neutrino-induced coherent single-pion production in the $\Delta$-excitation region. K2K[@hasegawa] and SciBooNE[@hiraide] investigated charged-current (CC) coherent pion production, while MiniBooNE[@miniboone] studied neutral-current (NC) coherent pion production. Furthermore, results for the anti-neutrino-induced coherent pion-production processes are expected to become available soon from MiniBooNE for the NC process [@anti_nc], and from SciBooNE for the CC process [@anti_cc]. It is to be remarked, however, that the recent experimental results offer a rather puzzling situation. The experiments at K2K[@hasegawa] and SciBooNE[@hiraide] report that the CC process is not observed, whereas the MiniBooNE experiment [@miniboone] concludes that the NC process is observed. Now, from the isospin factors, we expect an approximate relation $\sigma_{CC}\sim 2\sigma_{NC}$. Although the muon mass can reduce the phase space for the CC process at low energies, we still expect that $\sigma_{CC}$ should be of a significant size compared with $\sigma_{NC}$, and hence the above experimental results seem quite puzzling. In this connection it is to be noted that the MiniBooNE’s use of the Rein-Sehgal (RS) model [@RS] in analyzing the NC data has recently be questioned [@amaro]: for a critical review of the RS model, see Refs. [@amaro; @hernandez]. The CC data analyses in Refs.[@hasegawa; @hiraide] did not rely on a particular theoretical model for coherent pion production itself but, in dealing with some other neutrino-nucleus reactions that entered into the analyses, certain models whose reliability was open to debate needed to be invoked.
The theoretical treatment of coherent pion production can be categorized into two types: a PCAC-based model and a microscopic model. In the former approach, the hadronic matrix element for neutrino-induced pion production is related to the pion-nucleus (or pion-nucleon) scattering amplitude through the PCAC relation. Meanwhile, in the microscopic approach, the hadronic matrix element is calculated by summing the elementary amplitude for weak pion-production off a single nucleon embedded in a nuclear environment.
A prominent example of the PCAC-based approach is the model due to Rein and Sehgal (RS model) [@RS]. Because of its success in the high energy neutrino process [@rs_success] ($E_\nu$2 GeV, where $E_\nu$ is the incident neutrino energy) and its simplicity, the RS model has been extensively used in analyzing data in neutrino-oscillation experiments. Several authors, however, have recently pointed out that the RS model does not give a reasonable description for relatively low-energy neutrino processes ($E_\nu$2 GeV)[@amaro; @hernandez], and that the use of the RS model may have led to the puzzling experimental situation currently facing us. There have been several proposals [@hernandez; @RS2; @paschos; @bs_pcac] to remedy some of the possible insufficiencies in the original RS model.
Meanwhile, in order to build a quantitatively reliable microscopic approach, it is obviously of primary importance to start with a model that can describe with sufficient accuracy electroweak pion production off a free single nucleon. Furthermore, for pion production off a nuclear target, we need to consider medium effects such as the final-state interactions (FSI) between the outgoing pion and nucleus, etc. Recently there have been several microscopic calculations[@amaro; @singh; @valencia1; @valencia2], the most elaborate one being that by Amaro et al. [@amaro]. These calculations differ in the way the elementary process ($\nu_\mu N\to \mu^+ N \pi$) is modeled and/or in the way the medium effects are taken into account. For example, only the resonant $\Delta$-excitation mechanism is considered in Refs. [@singh; @valencia1], while the non-resonant mechanism is additionally considered in Refs. [@amaro; @valencia2]. It was shown in Refs. [@amaro; @valencia2] that the inclusion of the non-resonant mechanism leads to a reduction of the cross section by a factor of $\sim 2$, even though both models are constructed in such a manner that the data for the elementary process are reproduced fairly well. [^2] This result indicates the importance of modelling the elementary process with a sound and systematic approach which has been extensively tested by available data.
The purpose of the present article is to develop an alternative microscopic model for coherent pion production. An important ingredient of our formalism is a reliable dynamical model for the elementary process, and for that we shall employ the Sato-Lee (SL) model [@SL; @SUL]. The SL model was first developed as a systematic framework for studying the resonance properties by analyzing data on pion production in photon (electron)-nucleon scattering in the $\Delta$-resonance region [@SL; @SL2]. The SL model treats the resonant and non-resonant mechanisms on the same footing, and is known to provide a reasonably accurate description of an extensive set of pion production data. The SL model was further extended to the weak sector in Ref. [@SUL], and was shown to be able to reproduce data for neutrino-induced pion production off a nucleon. As has been done in the previous microscopic calculations, we also need to incorporate the nuclear medium effects. In the energy region of our interest, the $\Delta$-hole approach has proved to be successful in describing various processes involving pion-nucleus dynamics. These situations motivate us to develop a model for coherent pion production by combining the SL model and the $\Delta$-hole model, and this is what we attempt in this article. We shall limit ourselves here to a case where the target nucleus (and hence the final nucleus also) has spin 0, and employ a simplified $\Delta$-hole model proposed in Ref. [@karaoglu]. As for concrete numerical calculations, we concentrate on the $^{12}$C target, which has been and will continue to be an important nuclear target in many of neutrino-oscillation experiments. To test the reliability of our approach, we first calculate observables for coherent photo-pion production on $^{12}$C using the same theoretical framework and show that the calculated results agree well with data. We then proceed to calculate observables for coherent neutrino-pion production on $^{12}$C and present numerical results that can be compared with the recent data from K2K and SciBooNE. We shall also present theoretical predictions for those quantities for which experimental data will soon become available.
The fact that the previous microscopic calculations exhibit rather large model-dependence makes it particularly interesting to use the SL model, which has been highly successful in the single nucleon sector. The SL model provides a consistent set of amplitudes for pion production and pion-nucleon scattering on a single nucleon; all these amplitudes are obtained in a systematic manner from the same Lagrangian. In our approach this consistency can be further taken over to the description of the FSI between the final pion and the nucleus. Thus, based on the SL amplitudes, we can construct a pion-nucleus optical potential that is consistent with the transition operators for electroweak pion production off a nucleus. To the best of our knowledge, our approach is the first to provide a consistent framework for treating the medium effect on the pion and electroweak pion production on the same footing. This point is worth emphasizing because it is this consistency that enables us to [*predict*]{} cross sections for electroweak coherent pion production [*with no adjustable parameters*]{}, once we fix certain parameters (see below) relevant to medium effects by fitting to the pion-nucleus scattering data.
Another point to be noted is that our model takes into account the non-local effect for in-medium $\Delta$-propagation. For neutrino-induced coherent pion production, neither the RS-based nor previous microscopic models have included this effect. As pointed out in Ref. [@non-local], the non-local effect could reduce the cross section by a factor of $\sim$ 2 ($\sim$ 1.7) for $E_\nu$ = 0.5 (1) GeV. We consider it important to take due account of the possibly large non-local effect.
Our calculation adopts the following procedure. We first construct a pion-nucleus optical potential, employing the SL $\pi N$ scattering (on- and off-shell) amplitudes as basic ingredient. The medium modification of the $\Delta$-propagation in a nucleus is considered with the use of the $\Delta$-hole model [@karaoglu]. All the free parameters in our model (spreading potential, phenomenological terms in the optical potential) are fixed by fitting to pion-nucleus scattering data. After these parameters are determined, we are in a position to make prediction on the coherent pion production process. Before calculating the neutrino-induced process, we test the reliability of our model by comparing our predictions for the photo-induced process with data. After finding satisfactory results for the photo-process, we proceed to calculate neutrino-induced coherent pion production.
The organization of this paper is as follows. Sec. \[sec\_model\] is dedicated to the explanation of our approach. We first introduce the elementary amplitudes of the SL model. We then give expressions for calculating the electroweak coherent pion production amplitudes in terms of the SL amplitudes and derive the cross section formulae. The expression for the constructed optical potential and its relation with the scattering amplitude are also given there. We present numerical results in Sec. \[sec\_result\], and give a conclusion in Sec. \[sec\_conclusion\]. Appendix \[app\_mutlipole\] provides the definition of the multipole amplitudes, while Appendix \[app\_2cm\] explains the Lorentz transformation used in our calculation. In Appendix \[app\_misc\] we give expressions for quantities that appear in the $\Delta$-hole model.
Formulation {#sec_model}
===========
The kinematics of the reactions under consideration is as follows. We consider coherent pion production in neutrino($\nu_\ell$)-nucleus($t$) scattering: $\nu_\ell (p_\nu)+ t (p_t) \to \ell^- (p_\ell^\prime) +
\pi^+ (k) + t (p_t^\prime)$ for the CC process, and $\nu_\ell (p_\nu)+ t (p_t) \to \nu_\ell (p_\ell^\prime) +
\pi^0 (k) + t (p_t^\prime)$ for the NC process; we also consider the antineutrino-counterparts. The four-momentum for each particle in the laboratory frame (LAB) is given in the parentheses. The four-momentum transfer from the leptons is denoted by $q^\mu\equiv p_\nu^\mu-p_\ell^{\prime\,\mu}$. We choose a right-handed coordinate system in which the $z$-axis lies along the incident neutrino momentum $\bm{p}_\nu$, and the $y$-axis is taken along $\bm{p}_\nu\times\bm{p}_\ell^\prime$. In evaluating a nuclear matrix element, it is convenient to work in the pion-nucleus center-of-mass frame (ACM). The kinematical variables in ACM are denoted by $\bm{q}_A$, $\bm{k}_A$, etc. We also work in the pion-[*nucleon*]{} CM frame (2CM), when calculating the elementary SL amplitudes. The kinematical variables in 2CM are denoted by $\bm{q}_2$, $\bm{k}_2$, etc. When working in ACM (2CM), we choose a coordinate system in which the $z$-axis lies along $\bm{q}_A$ ($\bm{q}_2$) and the $y$-axis is along $\bm{p}_{\nu,A}\times\bm{p}_{\ell,A}^\prime$ ($\bm{p}_{\nu,2}\times\bm{p}_{\ell,2}^\prime$).
The SL Model
------------
We express nuclear transition amplitudes for coherent pion production in terms of the elementary amplitudes derived from the SL model [@SUL]. In this section, therefore, we introduce the SL amplitudes. The differential cross section in the LAB frame for pion production in the neutrino-[*nucleon*]{} CC reaction, $\nu_\ell (p_\nu)+ N (p_N) \to \ell^- (p_\ell^\prime) +
\pi^+ (k) + N (p_N^\prime)$, is given by (cf. Eq. (10) of Ref. [@SUL]) $$\begin{aligned}
{\label{eq:xs1}}
\frac{d^5\sigma}{d E_\ell^\prime d\Omega_\ell^\prime d\Omega_\pi}
=\frac{G_F^2\cos^2\theta_c}{2}
\left({ {|\bm{k}| \over \omega_\pi} +
{|\bm{k}| - {\hat{\bm{k}}\cdot (\bm{p}_\nu -
\bm{p}_\ell^\prime) }\over E_N^\prime}}\right)^{-1}
\frac{|\bm{p}_\ell^\prime|}{|\bm{p}_\nu|}
{|\bm{k}|^2 m_N^2 \over \omega_\pi E_N E_N^\prime}
\frac{L^{\mu\nu}W_{\mu\nu}}{(2\pi)^5} \ ,
$$ where $G_F=1.16637 \times 10^{-5}$ GeV$^{-2}$ is the Fermi constant, and $\theta_c$ is the Cabbibo angle ($\cos\theta_c=0.974$). $E_\ell^\prime$ and $\omega_\pi$ are the energies of the final lepton and pion, respectively, $m_N$ is the nucleon mass, and $E_N$ ($E_N^\prime$) is the initial (final) nucleon energy. $W_{\mu\nu}$ and $L^{\mu\nu}$ represent the hadron and the lepton tensors, respectively, and their definitions are found in Ref. [@SUL] \[Eqs. (11), (12)\]. The above cross section can be written as $$\begin{aligned}
{\label{eq:xs1-2}}
\frac{d^5\sigma}{d E_\ell^\prime d\Omega_\ell^\prime d\Omega_\pi}
&=&\frac{G_F^2\cos^2\theta_c}{2}
\left({ {|\bm{k}| \over \omega_\pi} +
{|\bm{k}| - {\hat{\bm{k}}\cdot (\bm{p}_\nu -
\bm{p}_\ell^\prime) }\over E_N^\prime}}\right)^{-1}
\frac{|\bm{p}_\ell^\prime||\bm{k}|^2}{(2\pi)^5|\bm{p}_\nu|}
{E_{\ell,2}^\prime p_{\nu,2}}
\\\nonumber
&\times& {1\over 2}\sum_{s_N
s_N^\prime}\sum_{s_\ell^\prime} |\Gamma_{2L} (F^V-F^A)|^2 \ ,\end{aligned}$$ where $F^V$ and $F^A$ are the transition amplitudes in which the hadronic vector and the axial-vector currents are respectively contracted with the leptonic current. The symbol $s_N$ ($s_N^\prime$) is the z-component of the initial (final) nucleon spin, while $s_\ell^\prime$ denotes the final lepton spin. The energies of the final lepton and the initial neutrino in 2CM are denoted by $E_{\ell,2}^\prime$ and $p_{\nu,2}$, respectively. $F^V$ and $F^A$, including both hadronic and lepton currents, are calculated in 2CM, and then embedded in the cross section expression given in LAB. The factor $\Gamma_{2L}$ arises from the relevant Lorentz transformation (see Appendix B): $$\begin{aligned}
\Gamma_{2L} = \sqrt{\omega_{\pi,2}E_{N,2}E^\prime_{N,2}\over
\omega_{\pi}E_{N}E^\prime_{N}} \ ,\end{aligned}$$ where $\omega_{\pi,2}$, $E_{N,2}$ and $E^\prime_{N,2}$ are the energies of the pion, the incident nucleon and the final nucleon in 2CM. The spin structure of $F^V$ and $F^A$ can be parametrized as $$\begin{aligned}
F^V = - i\vec{\sigma}\cdot \vec{\epsilon}_\perp F_1^V
- \vec{\sigma}\cdot \hat{k}_2 \vec{\sigma}\cdot\hat{q}_2\times\vec{\epsilon}_\perp
F_2^V
- i\vec{\sigma}\cdot\hat{q}_2\hat{k}_2\cdot\vec{\epsilon}_\perp F_3^V
- i\vec{\sigma}\cdot\hat{k}_2\hat{k}_2\cdot\vec{\epsilon}_\perp F_4^V
\nonumber \\
- i\vec{\sigma}\cdot\hat{q}_2\hat{q}_2\cdot\vec{\epsilon} F_5^V
- i\vec{\sigma}\cdot\hat{k}_2\hat{q}_2\cdot\vec{\epsilon} F_6^V
+ i \vec{\sigma}\cdot\hat{k}_2 \epsilon_0 F_7^V
+ i \vec{\sigma}\cdot\hat{q}_2\epsilon_0 F_8^V\,,
{\label{eq:fvec}}\end{aligned}$$ where $\vec{\epsilon}_\perp = \hat{q}_2\times(\vec{\epsilon} \times \hat{q}_2)$ and $$\begin{aligned}
F^A = - i\vec{\sigma}\cdot\hat{k}_2\vec{\sigma}\cdot \vec{\epsilon}_\perp F_1^A
- \vec{\sigma}\cdot\hat{q}_2\times\vec{\epsilon}_\perp
F_2^A
- i\vec{\sigma}\cdot\hat{k}_2
\vec{\sigma}\cdot\hat{q}_2\hat{k}_2\cdot\vec{\epsilon}_\perp F_3^A
- i\hat{k}_2\cdot\vec{\epsilon}_\perp F_4^A
\nonumber \\
- i\vec{\sigma}\cdot\hat{k}_2
\vec{\sigma}\cdot\hat{q}_2\hat{q}_2\cdot\vec{\epsilon} F_5^A
- i\hat{q}_2\cdot\vec{\epsilon} F_6^A
+ i \epsilon_0 F_7^A
+ i \vec{\sigma}\cdot\hat{k}_2\vec{\sigma}\cdot\hat{q}_2 \epsilon_0 F_8^A\,.
{\label{eq:faxi}}\end{aligned}$$ The lepton-current matrix element $\epsilon^\mu$ is given by $\epsilon^\mu=\bra{\ell}\bar{\psi}_l\gamma^\mu(1-\gamma_5)\psi_\nu\ket{\nu_\ell}$. We have introduced parametrization for $F^A$ simply via $F^A=\vec{\sigma}\cdot\hat{k}_2F^V$. The amplitudes, $F_i^V$ and $F_i^A$, are expressed in terms of the multipole amplitudes $E_{l\pm}^{V,A},M_{l\pm}^{V,A},
S_{l\pm}^{V,A}$ and $L_{l\pm}^A$, which are functions of $q^2$ and $W$ (the $\pi N$ invariant mass) and computed in 2CM. Their explicit expressions are presented in Appendix \[app\_mutlipole\].
In a coherent process on a spin-zero target under consideration, only the spin non-flip terms of the transition amplitudes contribute. We therefore can work with $\bar{F}^{V(A)}$ defined by $$\begin{aligned}
{\label{eq:non-flip}}
\bar{F}^{V(A)} = {1\over 2} {\rm Tr} [F^{V(A)}] \ ,\end{aligned}$$ where the trace is taken for nucleon spin space. Their explicit forms are $$\begin{aligned}
{\label{eq:bar_fv}}
\bar{F}^V =
- \hat{k}_2 \cdot\hat{q}_2\times\vec{\epsilon}_\perp F_2^V \ ,\end{aligned}$$ and $$\begin{aligned}
{\label{eq:bar_fa}}
\bar{F}^A &=& - i\hat{k}_2\cdot \vec{\epsilon}_\perp F_1^A
- i\hat{k}_2\cdot\hat{q}_2
\hat{k}_2\cdot\vec{\epsilon}_\perp F_3^A
- i\hat{k}_2\cdot\vec{\epsilon}_\perp F_4^A \nonumber \\
&& - i\hat{k}_2\cdot\hat{q}_2
\hat{q}_2\cdot\vec{\epsilon} F_5^A
- i\hat{q}_2\cdot\vec{\epsilon} F_6^A
+ i \epsilon_0 F_7^A
+ i \hat{k}_2\cdot\hat{q}_2 \epsilon_0 F_8^A\ .\end{aligned}$$ In particular, the resonant parts of the elementary amplitudes are given by $$\begin{aligned}
\bar{F}^V_R - \bar{F}^A_R
&=& \left( -2 \hat{k}_2\cdot\hat{q}_2\times\vec{\epsilon}_\perp M_{R\ 1+}^{(3/2),V}
-2 i \hat{k}_2\cdot \vec{\epsilon}_\perp E_{R\ 1+}^{(3/2),A}\right.\\\nonumber
&&\left. -4 i \hat{k}_2\cdot\hat{q}_2 \epsilon_0 S_{R\ 1+}^{(3/2),A}
+4 i \hat{k}_2\cdot\hat{q}_2 \hat{q}_2\cdot\vec{\epsilon} L_{R\ 1+}^{(3/2),A}
\right)\Lambda^{3/2}_{ij} \ ,\end{aligned}$$ where the suffix “$R$” stands for the resonant parts of the corresponding multipole amplitudes associated with the excitation of the $\Delta$ resonance. From the resonant amplitude we can factor out the $\Delta$-propagator, $D(W)$, as $$\begin{aligned}
{\label{eq:amp-res}}
\bar{F}^V_R - \bar{F}^A_R = {N(k_2,q_2)\over D(W)} \ ,\end{aligned}$$ and $$\begin{aligned}
{\label{eq:delta-prop}}
D(W)=W-m_\Delta-\Sigma_\Delta(W) \ ,\end{aligned}$$ where $m_\Delta$ and $\Sigma_\Delta$ are the bare mass and self energy of the $\Delta$-resonance, respectively.
We next discuss the T-matrix element for $\pi N$ scattering, which serves as an input for constructing an optical potential for pion-nucleus scattering. A calculational procedure for the $\pi N$ T-matrix within the SL model can be found in Ref. [@SL]. A distorted wave obtained with this optical potential will be used to take account of the final-state interaction in coherent pion production. The T-matrix is decomposed into the resonant ($t_R$) and non-resonant ($t_{nr}$) parts as $$\begin{aligned}
{\label{eq:t-mat}}
t^{(c)}_{\pi N} = t^{(c)}_R + t^{(c)}_{nr} \ ,\end{aligned}$$ where the superfix $c$ specifies a channel; in our model the resonance amplitude exists only for the $P_{33}$ channel. The on-shell component of the T-matrix given in [Eq. [(\[eq:\]]{})]{}[t-mat]{} is related to the phase shift by $$\begin{aligned}
{\label{eq:phase}}
t^{(c)}_{\pi N} = - {W\over \pi \omega_{\pi,2} E_{N,2}}
{e^{2i\delta^{(c)}}-1\over 2i k_2^o} \ ,\end{aligned}$$ where $W$ is the invariant mass of the $\pi N$ system, and $\omega_{\pi,2}$ =$\sqrt{k_2^{o\,2}+m_\pi^2}$ and $E_{N,2}=\sqrt{k_2^{o\,2}+m_N^2}$ are the on-shell energies of the pion and the nucleon in 2CM, respectively. The resonant amplitude is expressed as $$\begin{aligned}
{\label{eq:t-res}}
t^{(P_{33})}_R (k_2^\prime,k_2; W) = - {F_{\pi N\Delta}(k_2^\prime)
F_{\pi N\Delta}(k_2)\over D(W)} \ ,\end{aligned}$$ where $F_{\pi N\Delta}(k_2)$ is the dressed $\pi N\Delta$ vertex, and $D(W)$ is the $\Delta$ propagator introduced in [Eq. [(\[eq:\]]{})]{}[delta-prop]{}. We note that the four-momenta, $k_2$ and $k_2^\prime$, are in general off-energy-shell.
Coherent pion production in neutrino-nucleus scattering
-------------------------------------------------------
Similarly to [Eq. [(\[eq:\]]{})]{}[t-mat]{} for the $\pi N$ scattering amplitude, the weak amplitudes $\bar{F}^{V(A)}$, defined in [Eqs. (\[eq:non-flip\])–(\[eq:bar\_fa\])]{}, also have a resonant $\bar{F}^{V(A)}_{R}$ and non-resonant $\bar{F}^{V(A)}_{nr}$ parts. Accordingly, the transition amplitudes of coherent pion production on nuclei have the resonant and non-resonant parts. We now describe how these two components are calculated in our approach.
### transition matrix element: resonant part {#sec_res}
The main task in calculating the resonant part of coherent pion production on nuclei is to account for the medium effects on $\Delta$ propagation in the elementary resonant amplitudes $F^{V(A)}_R$. Here we follow the procedure of the $\Delta$-hole model of pion-nucleus reactions by modifying the $\Delta$ propagator in [Eq. [(\[eq:\]]{})]{}[delta-prop]{}. Thus it is useful to first briefly explain how the $\Delta$-hole model is formulated by considering the elastic pion-nucleus scattering; for a full account of the formulation see Refs. [@annal99; @annal108; @annal120; @taniguchi].
The $\Delta$-hole model is formulated within the projection operator formalism[@annal99]. The nuclear Fock space is divided into four spaces; $P_0$, $P_1$, $D$ and $Q$. The $P_0$-space is spanned by the pion and the nuclear ground state, the $P_1$-space by the pion and one-particle one-hole states, the $D$-space by the one-$\Delta$ one-hole configurations, and $Q=1-P_0-P_1-D$ contains the reminder of the full space. A projected Hamiltonian is written as, e.g., $H_{P_0D}= P_0 H D$. Starting with the Schrödinger equation in the full space ($H\ket{\Psi} = E\ket{\Psi}$), we can apply the standard projection operator techniques[@annal99] to obtain an equation, defined only in the $P_0$-space, to describe the pion-nucleus elastic scattering T-matrix. In the $\Delta$-hole model, one further imposes the condition that the $D$-space is the doorway of the transitions between $P=P_0+P_1$ and $Q$ spaces; namely $H_{PQ}=H_{QP}=0$. The pion-nucleus scattering amplitude due to the $\Delta$ excitation can then be written as $$\begin{aligned}
{\label{eq:elastic_amp}}
T_{P_0P_0}(E) = H_{P_0D} G_{\Delta h}(E) H_{DP_0} \ ,\end{aligned}$$ where the total energy defined in ACM ($E + A m_N$) is given by $$\begin{aligned}
E + A m_N = q_A^0 + \sqrt{\bm{q}_A^2 + (Am_N)^2}
= \sqrt{\bm{k}_A^2 + m_\pi^2} + \sqrt{\bm{k}_A^2 + (Am_N)^2} \ ,\end{aligned}$$ where $A$ is the mass number. The $\Delta$-hole propagator $G_{\Delta h}$ in [Eq. [(\[eq:\]]{})]{}[elastic\_amp]{} is defined by $$\begin{aligned}
{\label{eq:nuclear-delta-prop}}
G_{\Delta h}^{-1} =
D(E-H_\Delta)-W_{el} - \Sigma_{\rm pauli} - \Sigma_{\rm spr} \ .\end{aligned}$$ Here $D(E-H_\Delta)$ can be calculated from [Eq. [(\[eq:\]]{})]{}[delta-prop]{} with $H_\Delta$ being the Hamiltonian for the $\Delta$-particle in the nuclear many-body system. The effects due to the $Q$-space are included in the so-called spreading potential, $\Sigma_{\rm spr}$. A microscopic calculation of the spreading potential is very complicated since it involves the calculation of pion absorption by two or more nucleons. It is therefore a common practice to determine $\Sigma_{\rm spr}$ phenomenologically by fitting to the pion-nucleus scattering data. Excitations to the $P_1$-space are included in the $\Delta$ self energy $\Sigma_\Delta(W)$ of $D(E-H_\Delta)$ \[see [Eq. [(\[eq:\]]{})]{}[delta-prop]{}\] with a correction due to the Pauli blocking ($\Sigma_{\rm pauli}$). De-excitation to the $P_0$-space is the rescattering in the elastic mode, and is denoted by $W_{el}$. In our actual calculation, we expand $G_{\Delta h}$ in term of $W_{el}$, and the expansion series is resummed by solving the Lippmann-Schwinger equation.
The calculations of the pion-nucleus scattering amplitude in [Eq. [(\[eq:\]]{})]{}[elastic\_amp]{} require a diagonalization of the $\Delta$-hole propagator $G_{\Delta h}$ of [Eq. [(\[eq:\]]{})]{}[nuclear-delta-prop]{}. For the diagonalization, it is practically convenient to work with the oscillator basis for the $\Delta$ state, defined by the Hamiltonian $H_\Delta$, and the nucleon hole state. This diagonalization is a difficult numerical task. Although an efficient method using the doorway state expansion has been developed [@annal120], the diagonalization of $G_{\Delta h}$ is still difficult, particularly for heavier nuclei. In Ref. [@karaoglu], Karaoglu and Moniz (KM) proposed a simplified calculation with the $\Delta$-hole model in which $G_{\Delta h}$ is calculated with a local density approximation rather than a diagonalization. In their simplified treatment, $\Sigma_{\rm pauli}$ is calculated by a nuclear matter calculation[@pauli], and their result is given in Appendix \[app\_misc\]. Their parametrization of the spreading potential $\Sigma_{\rm spr}$ in terms of a central and a spin-orbit terms are also given in Appendix \[app\_misc\]. Each term of the spreading potential has a complex strength, which are determined by fitting to the pion-nucleus scattering data. KM applied their approach to $\pi$-$^{16}$O scattering, and found a good agreement between their calculation with data, and also with the full $\Delta$-hole calculation [@annal108; @annal120] except for the most central partial waves. Encouraged by this success, we follow this simplified version of the $\Delta$-hole model to include the medium effects on the $\Delta$ propagation in defining the electroweak pion production matrix elements.
Schematically, the resonant part of the transition matrix element, ${\cal M}^A_R$, of weak coherent pion production on nuclei induced by the charged current can be obtained by replacing the initial $H_{DP_0}$ of [Eq. [(\[eq:\]]{})]{}[elastic\_amp]{} by $H_{DP_0^\prime}$ where $P_0^\prime$ is the space spanned by the (axial-)vector current and the nucleus in the ground state. In terms of the single particle wave functions $\psi_j(\bm{p}_N)$ of the nucleons in the initial and final nuclear states, we thus have[^3] $$\begin{aligned}
{\label{eq:mr2}}
{\cal M}^A_{R} &=& \sum_{j}
\int {d^3p_N \over (2\pi)^3}{d^3p_N^\prime \over (2\pi)^3}
\psi_j^*(\bm{p}_N^\prime)
{\Gamma_{2A}
N(k_2,q_2)
(2\pi)^3 \delta(\bm{p}_N+\bm{q}_A-\bm{p}_N^\prime-\bm{k}_A)\over
D(E+m_N-H_\Delta) - \Sigma_{\rm pauli} - \Sigma_{\rm spr} }
\psi_j(\bm{p}_N) \nonumber \\
&=& \sum_{j}
\int {d^3p_\Delta \over (2\pi)^3}
\psi_j^*(\bm{p}_N^\prime)
{\Gamma_{2A}
N(k_2,q_2)
\over D(E+m_N-H_\Delta) - \Sigma_{\rm pauli} - \Sigma_{\rm spr} }
\psi_j(\bm{p}_N) \ ,\end{aligned}$$ where $\bm{p}_\Delta=\bm{p}_N+\bm{q}_A=\bm{p}_N^\prime+\bm{k}_A$; the index $j$ denotes single particle quantum numbers including the isospin . The summation ($\sum_j$) is taken over the occupied states of the nucleus. The factor $\Gamma_{2A}$ is defined by $$\begin{aligned}
\Gamma_{2A} = \sqrt{\omega_{\pi,2}E_{N,2}E^\prime_{N,2}\over
\omega_{\pi,A}E_{N,A}E^\prime_{N,A}} \ ,\label{eq:G2A}\end{aligned}$$ where $\omega_{\pi}$, $E_{N}$ and $E^\prime_{N}$ are the energies of the pion, the incoming nucleon and the outgoing nucleon, respectively, and the quantities in the numerator (denominator) refer to 2CM (ACM). This factor arises from the fact that $\bar{F}^{V(A)}_R$ computed in 2CM are to be embedded in ${\cal M}^A_R$ evaluated in ACM. To evaluate the numerator in the integrand of [Eq. [(\[eq:\]]{})]{}[mr2]{}, we clearly need a prescription for relating variables in 2CM to those in ACM. Here we use the commonly used prescription [@gmitro; @chumbalov] to fix the nucleon momenta with the lepton momentum transfer $\bm{q}_A$ and outgoing pion momentum $\bm{k}_A$ as $$\begin{aligned}
{\label{eq:p_fix}}
\bm{p}_N = - {\bm{q}_A\over A} - {A-1\over 2A}(\bm{q}_A-\bm{k}_A)\ ,
\qquad
\bm{p}_N^\prime = - {\bm{k}_A\over A} + {A-1\over 2A}(\bm{q}_A-\bm{k}_A)
\ ,\end{aligned}$$ and write the $\pi N$ invariant mass as $$\begin{aligned}
{\label{eq:inv_mass}}
W = \sqrt{(E_{N\,A}+q_A^0)^2 - (\bm{p}_N+\bm{q}_A)^2
}\ ,\end{aligned}$$ with $E_{N\,A}=\sqrt{\bm{p}_N^2+m_N^2}$. Having specified all the relevant variables in ACM, we can derive the corresponding variables in 2CM via a Lorentz transformation to obtain $N(k_2,q_2)$ of [Eq. [(\[eq:\]]{})]{}[mr2]{}. For more details about this Lorentz transformation (including the discussion of a somewhat different treatment of an off-shell pion momentum), see Appendix \[app\_2cm\]. Note that, in treating the wave functions, $\psi(\bm{p}_N)$ and $\psi(\bm{p}'_N)$, and the $\Delta$ kinetic term in the denominator in the integrand of [Eq. [(\[eq:\]]{})]{}[mr2]{}, we do [*not* ]{} use the prescription given in [Eqs. (\[eq:p\_fix\]) and (\[eq:inv\_mass\])]{}; thus the important recoil effects on $\Delta$-propagation are not neglected in our calculations.
We incorporate the recoil effect on the $\Delta$ self-energy in the first order approximation. This is done by linearizing the $\Delta$-propagator with the following expansion[@taniguchi]: $$\begin{aligned}
{\label{eq:linearize}}
D(E+m_N-H_\Delta) &\sim& D(W)-\gamma(W) (H_\Delta - e_\Delta^0) , \\
{\label{eq:e_delta}}
E+m_N &=& W + e_\Delta^0 \ , \\
\gamma(W) &=& \partial D(W) / \partial W , \\
{\label{eq:delta_H}}
H_\Delta &=& {\bm{p}_\Delta^2\over 2 \mu_\Delta} + V_\Delta + V_\Delta^C +
e_N , \\
1/\mu_\Delta &=& 1/m_\Delta + 1/(A-1)m_N \ ,\end{aligned}$$ where $V_\Delta$ ($V_\Delta^C$) is the $\Delta$ (Coulomb) potential in the nucleus, and $e_N$ is the hole energy. The $\Delta$ potential is taken to be the same as that for the nucleon; its explicit expression is given in Appendix \[app\_misc\]. [Equation [(\[eq:\]]{})]{}[e\_delta]{} defines $e_\Delta^0$. To carry out the integration over the $\Delta$ momentum $\bm{p}_\Delta$ in [Eq. [(\[eq:\]]{})]{}[mr2]{}, we express the nucleon wave function $\psi_j(\bm{p})$ in terms of its coordinate-space form $\phi_j(\bm{r})$. We note that with the prescription in [Eqs. (\[eq:p\_fix\]) and (\[eq:inv\_mass\])]{}, the numerator $N(k_2,q_2)$ of [Eq. [(\[eq:\]]{})]{}[mr2]{} is independent of the variable $\bm{p}_\Delta$ and can be factorized out of the integration. With this factorization approximation and with the use of the linearized form in [Eq. [(\[eq:\]]{})]{}[linearize]{}, the integration over $\bm{p}_\Delta$ leads to the following $r$-space expression: $$\begin{aligned}
{\label{eq:mr3}}
{\cal M}^A_{R} &=& - \left(\mu_\Delta \Gamma_{2A} N(k_2,q_2)\over 2\pi\gamma\right)
\sum_{j}
\int d^3r d^3r^\prime \phi_j^*(\bm{r}^\prime)
e^{-i \bm{k}_A\cdot\bm{r}^\prime}
{e^{i K_\Delta |\bm{r}^\prime-\bm{r}|}
\over |\bm{r}^\prime-\bm{r}|
}
e^{i \bm{q}_A\cdot\bm{r}}\phi_j(\bm{r}) \ ,\end{aligned}$$ where $$\begin{aligned}
{\label{eq:delta_k2}}
K_\Delta^2 = {2\mu_\Delta\over \gamma}\left\{
W -m_\Delta-\Sigma_\Delta(W) + \gamma(E-W+m_N)
- \gamma \left[e_N + V_\Delta + V_\Delta^C \right]
- \Sigma_{\rm pauli} - \Sigma_{\rm spr}
\right\} \ .\end{aligned}$$ Following the procedure described in Ref. [@karaoglu] \[see Eqs. (25)–(39) therein\], and subsequently applying the Lorentz transformation from ACM to LAB, we obtain the following expression for the transition matrix element ${\cal M}^L_{R}$ in LAB $$\begin{aligned}
{\label{eq:mr4}}
&&{\cal M}^L_{R} = {16\sqrt{1+|\lambda|}\pi \over 3} {\mu_\Delta D(W)\over\gamma}
\Gamma_{2L}
\\\nonumber &\times&
\left( -2 \hat{k}_2\cdot\hat{q}_2\times\vec{\epsilon}_\perp M_{R\ 1+}^{(3/2),V}
-2 i \hat{k}_2\cdot \vec{\epsilon}_\perp E_{R\ 1+}^{(3/2),A}
-4 i \hat{k}_2\cdot\hat{q}_2 \epsilon_0 S_{R\ 1+}^{(3/2),A}
+4 i \hat{k}_2\cdot\hat{q}_2 \hat{q}_2\cdot\vec{\epsilon} L_{R\ 1+}^{(3/2),A}
\right)\\\nonumber
&\times& \sum_{N=p,n} (1+{\lambda\tau_N\over 2})
\int s^2 ds R^2 dR j_0(pR)j_0(Ps)
{e^{i \bar{K}_\Delta s}\over s}
\left\{1 + {i \mu_\Delta s\over \bar{K}_\Delta}
\left[\bar{e_N} + H_N\right]\right\}
\rho_N(R)\hat{j}_1(k_F s) \ ,\label{eq:MLR}\end{aligned}$$ where $p=|\bm{k}_A-\bm{q}_A|$, $P=|\bm{k}_A+\bm{q}_A|/2$, $s=|\bm{r}^\prime-\bm{r}|$, $R=|\bm{r}^\prime+\bm{r}|/2$, and $\bar{K}_\Delta$ is obtained from $K_\Delta$ by replacing $e_N$ with its average value, $\bar{e}_N$; we choose $\bar{e}_N = 16$ MeV. The 2CM variables $k_2$ and $q_2$ are obtained from $k_A$ and $q_A$ using the Lorentz transformation as mentioned above. The variable $\lambda$ denotes the charge state of the outgoing pion, while $\tau_N=1\, (-1)$ for $N$ = proton (neutron). The factor $\Gamma_{2L}$ is from the Lorentz transformation from 2CM to LAB and is defined by $$\begin{aligned}
\Gamma_{2L} = \sqrt{\omega_{\pi,2}E_{N,2}E^\prime_{N,2}\over
\omega_{\pi,L}E_{N,L}E^\prime_{N,L}} \ .\end{aligned}$$ In [Eq. [(\[eq:\]]{})]{}[mr4]{}, $j_\ell(x)$ is the spherical Bessel function of order $\ell$, and $\hat{j}_1(x) \equiv {3\over x} j_1(x)$; $k_F$ is the Fermi momentum $$\begin{aligned}
{\label{eq:fermi_mom}}
k_F^3(R) = {3\pi^2\over 2}\rho_N(R) \ .\end{aligned}$$ The proton (neutron) matter density is denoted by $\rho_p$ ($\rho_n$), and is normalized to the total number of protons (neutrons) inside the target. For the proton matter form factor we use the empirical nuclear charge form factor [@charge_formfac] divided by the proton charge form factor [@nucl_formfac]. The neutron matter density is assumed to be the same as the proton matter density. The single nucleon Hamiltonian appearing in [Eq. [(\[eq:\]]{})]{}[mr4]{} is given by $$\begin{aligned}
H_N = - {\nabla^2_s\over 2m_N} - {\nabla^2_R\over 8m_N}
+V\left[(R^2+s^2/4)^{1/2}\right] \ ,\end{aligned}$$ where $V$ is the single particle potential \[[Eq. [(\[eq:\]]{})]{}[nucl\_potential]{}\].
To take account of the final pion-nucleus interactions, we convolute the matrix element ${\cal M}^L_{R}$ of [Eq. [(\[eq:\]]{})]{}[mr4]{} with the pion distorted wave which is expanded in partial waves: $$\begin{aligned}
{\label{eq:pi_wave2}}
\chi_\lambda^* (\bm{k}_A^\prime)
= \sum_{l_\pi m_\pi}
\chi_{\lambda\,l_\pi}^* (k_A^\prime) Y_{l_\pi m_\pi}^* (\hat{\bm{k}}_A)
Y_{l_\pi m_\pi} (\hat{\bm{k}}_A^\prime) \ ,\end{aligned}$$ where $k_A^\prime$ is the off-shell momentum. We note that the pion distorted wave also depends on the pion charge ($\lambda$). More details on our calculations of the pion wave functions are given in Sec. \[sec\_opt\].
By performing the partial wave decomposition of ${\cal M}^L_{R}$ (now defined by the off-shell pion momentum by setting $\bm{k}_A \rightarrow \bm{k}_A^\prime$) and using [Eq. [(\[eq:\]]{})]{}[pi\_wave2]{}, the amplitude ${\cal M}^L_{R}$ with pion-nucleus FSI takes the following form: $$\begin{aligned}
{\label{eq:mr5}}
{\cal M}^L_{R} &=& \epsilon_A^\mu
\sum_{l_\pi} \left[ P_{l_\pi}^1 (x_A)\left(
\cos\phi^\pi_A I^{l_\pi\, 1}_{E\, \mu}
-i\sin\phi^\pi_A I^{l_\pi\, 1}_{M\, \mu} \right)
\right.\\\nonumber
&&\left.
+ P_{l_\pi}^1 (x_A)\left(
\sin\phi^\pi_A I^{l_\pi\, 2}_{E\, \mu}
+i\cos\phi^\pi_A I^{l_\pi\, 2}_{M\, \mu} \right)
-2 P_{l_\pi} (x_A) I^{l_\pi\, 3}_{L\, \mu}
+2 P_{l_\pi} (x_A) I^{l_\pi\, 0}_{S\, \mu}
\right] \ ,\end{aligned}$$ where $x_A=\hat{q}_A\cdot\hat{k}_A$, $\phi^\pi_A$ is the azimuthal angle of the pion, and $\epsilon_A^\mu$ is the lepton current matrix element in ACM. The associated Legendre function of degree $l_\pi$ and order 0 (1) is denoted by $P_{l_\pi}$ ($P_{l_\pi}^1$). We have introduced the quantities $I^{l_\pi\, \nu}_{X\, \mu}$ defined by $$\begin{aligned}
{\label{eq:i_mtx}}
&& I^{l_\pi\, \nu}_{X\, \mu} = - i {32\sqrt{1+|\lambda|}\pi \mu_\Delta\over 3}
\int dk_A^\prime k_A^{\prime 2}\
\chi_{\lambda\,l_\pi}^*(k_A^\prime)\
\int dx_A^\prime \Lambda^\nu_\mu\Gamma_{AL}^\chi\Gamma_{2AL}
\gamma^{-1} X_{R}\xi^X_{1 l_\pi}(x_A^\prime)
\\\nonumber
&&\times \sum_{N=p,n} (1+{\lambda\tau_N\over 2})\int s^2 ds R^2 dR
j_0(pR)j_0(Ps)
{e^{i \bar{K}_\Delta s}\over s}
\left\{1 + {i \mu_\Delta s\over \bar{K}_\Delta}
\left[\bar{e_N} + H_N\right]\right\}\rho_N(R) \hat{j}_1(k_F s) \ ,
$$ where $x_A^\prime= \hat{q}_A\!\cdot\!\hat{k}_A^\prime$, $x_2^\prime = \hat{q}_2\!\cdot\!\hat{k}_2^\prime$, and $$\begin{aligned}
\xi^X_{\ell l_\pi}(x_A^\prime) &=&
\left\{ \begin{array}{ll}
{\displaystyle 2 l_\pi+1 \over\displaystyle 2 l_\pi (l_\pi+1)}
P_{\ell}^1 (x_2^\prime)
P_{l_\pi}^1 (x_A^\prime) \ ,& \qquad (X=E, M) \\[5mm]
{\displaystyle 2 l_\pi+1 \over\displaystyle 2}
P_{\ell} (x_2^\prime)
P_{l_\pi} (x_A^\prime) \ ,&\qquad (X=L, S) \end{array}
\right.\end{aligned}$$ and $$\begin{aligned}
{X_{R}\over D(W)} = E_{R\ 1+}^{(3/2),A}\ ,\ M_{R\ 1+}^{(3/2),V}\ ,\ L_{R\
1+}^{(3/2),A}\ ,\ S_{R\ 1+}^{(3/2),A} \ ,\end{aligned}$$ for $X=E,M,L,S$.
The Lorentz transformation factors coming from the electroweak amplitudes ($\Gamma_{2AL}$) and the wave function ($\Gamma^\chi$) in [Eq. [(\[eq:\]]{})]{}[mr5]{} are respectively $$\begin{aligned}
{\label{eq:gam_3}}
\Gamma_{2AL} = \sqrt{\omega^\prime_{\pi,2}E^\prime_{N,2}E^{i}_{N,2}
\over \omega^\prime_{\pi,A}E^\prime_{N,L}E^{i}_{N,L}} \ , \qquad
\Gamma^\chi = \sqrt{\omega_{\pi,A}E^{\prime\prime}_{N,A}E^{f}_{N,A}
\over \omega_{\pi,L}E^{\prime\prime}_{N,L}E^{f}_{N,L}} \ ,\end{aligned}$$ where $\omega^\prime_{\pi}$ is the pion energy in the intermediate state, $E^i_{N}$ and $E^f_{N}$ are the nucleon energies in the initial and final states while $E^\prime_{N}$ and $E^{\prime\prime}_{N}$ are those in the intermediate states; in general, $E^\prime_{N}$ and $E^{\prime\prime}_{N}$ can be different. As before, the suffices {$2,A,L$} attached to the energies specify reference frames. It is noted that the multipole amplitudes ($X_R^{A}$) depend on $x_A^\prime$ because the $\pi N$ invariant mass in the intermediate state depends on it \[[Eqs. (\[eq:p\_fix\]) and (\[eq:inv\_mass\])]{}\]. We also have introduced the Lorentz matrix $\Lambda^\nu_\mu$ defined by $\epsilon_2^\nu=\Lambda^\nu_\mu \epsilon_A^\mu$; $\Lambda^\nu_\mu$ also depends on $x_A^\prime$; the same Lorentz matrix relates $q_A$ ($k^\prime_A$ ) to $q_2$ ($k^\prime_2$ ). A procedure for deriving the Lorentz matrix and the transformation factors in [Eq. [(\[eq:\]]{})]{}[gam\_3]{} are explained in Appendix \[app\_2cm\].
### transition matrix element: non-resonant part
We assume that there is no medium effect on the non-resonant part, $\bar{F}^{V}_{nr}-\bar{F}^{A}_{nr}$, of the weak pion production amplitude on a nucleon in nuclei. Including the final pion-nucleus interactions and using the same factorization approximation based on the choice [Eq. [(\[eq:\]]{})]{}[p\_fix]{} of the nucleon momenta to evaluate $\bar{F}^{V}_{nr}-\bar{F}^{A}_{nr}$, the non-resonant coherent pion production matrix element ${\cal M}^L_{nr}$ can be written as $$\begin{aligned}
{\label{eq:mnr1}}
{\cal M}^L_{nr} = \sum_{N=p,n}
\int d^3k_A^\prime \chi_\lambda^*(\bm{k}_A^\prime)
\Gamma_{AL}^\chi\Gamma_{2AL}
F_N(\bm{k}_A^\prime-\bm{q}_A) (\bar{F}^{V,\zeta}_{nr}-\bar{F}^{A,\zeta}_{nr}) \ ,\end{aligned}$$ where $\bar{F}^{V,\zeta}_{nr}$ ($\bar{F}^{A,\zeta}_{nr}$) is the non-resonant part of $\bar{F}^V$ ($\bar{F}^A$) given in [Eqs. (\[eq:bar\_fv\]) and (\[eq:bar\_fa\])]{}. $\bar{F}^{V (A)}_{nr}$ depends on $N$ and $\lambda$ \[[Eq. [(\[eq:\]]{})]{}[amp\_iso]{}\], and the set $(N, \lambda)$ is collectively denoted by $\zeta$. The nuclear form factor $F_N(\bm{p})$ is given by $$\begin{aligned}
F_N(\bm{p}) = \int d^3r \rho_N (\bm{r}) e^{i \bm{p}\cdot \bm{r}} \ .\end{aligned}$$ After the partial wave expansion of the pion distorted wave, we arrive at $$\begin{aligned}
{\label{eq:mnr2}}
{\cal M}^L_{nr} &=& \epsilon_A^\mu
\sum_{l_\pi} \left[ P_{l_\pi}^1 (x_A)\left(
\cos\phi^\pi_A J^{l_\pi\, 1}_{E\, \mu}
-i\sin\phi^\pi_A J^{l_\pi\, 1}_{M\, \mu} \right)
\right.\\\nonumber
&&\left.
+ P_{l_\pi}^1 (x_A)\left(
\sin\phi^\pi_A J^{l_\pi\, 2}_{E\, \mu}
+i\cos\phi^\pi_A J^{l_\pi\, 2}_{M\, \mu} \right)
- P_{l_\pi} (x_A) J^{l_\pi\, 3}_{L\, \mu}
+ P_{l_\pi} (x_A) J^{l_\pi\, 0}_{S\, \mu}
\right] \ ,\end{aligned}$$ where we have introduced $J^{l_\pi\, \nu}_{X\, \mu}$ defined by $$\begin{aligned}
{\label{eq:j_mtx}}
J^{l_\pi\,\nu}_{X\,\mu} &=&-4\pi i \int dk_A^\prime k_A^{\prime 2}\
\chi_{\lambda\,l_\pi}^*(k_A^\prime)\
\int dx_A^\prime \Lambda^\nu_\mu\Gamma_{AL}^\chi\Gamma_{2AL} \nonumber \\
&\times &
\sum_\ell \xi^X_{\ell l_\pi}(x_A^\prime)
\sum_{N=p,n}
X^{\ell,\zeta}_{nr}
\int r^2 dr \rho_N(r)j_0(pr) \ ,\end{aligned}$$ for $X=E,M,L,S$. The multipole amplitudes are included in $X^{\ell,\zeta}_{nr}$ as $$\begin{aligned}
{\label{eq:x_ell}}
X^{\ell,\zeta}_{nr} &=& (\ell+1)^2 X_{nr\;\ell +}^{A,\zeta} + \ell^2 X_{nr\;\ell -}^{A,\zeta} \
, \end{aligned}$$ for $X=L,S$, and $$\begin{aligned}
{\label{eq:e_ell}}
E^{\ell,\zeta}_{nr} &=& (\ell+1)E_{nr\;\ell +}^{A,\zeta} - \ell E_{nr\;\ell -}^{A,\zeta} \
, \\
{\label{eq:m_ell}}
M^{\ell,\zeta}_{nr} &=& (\ell+1)M_{nr\;\ell +}^{V,\zeta} + \ell M_{nr\;\ell -}^{V,\zeta} \ ,\end{aligned}$$ for $X=E,M$. The $\zeta$ dependence of the multipole amplitudes is indicated explicitly. For example, $E_{nr\;\ell +}^{A,\zeta}$ is the non-resonant part of $E_{\ell +}^{A}$ which has been introduced previously. The same rule applies to the other multipole amplitudes.
### Cross Section
Having written the transition amplitude for the coherent process in terms of the SL multipole amplitudes, we can proceed to calculate the cross section for the CC process. First, we write the transition amplitudes in [Eqs. (\[eq:mr5\]) and (\[eq:mnr2\])]{} as $$\begin{aligned}
{\cal M}^L_{R}&=&\bar{{\cal M}}^L_{R,\mu}\epsilon^\mu_A \nonumber \\
{\cal M}^L_{nr}&=&\bar{{\cal M}}^L_{nr,\mu}\epsilon^\mu_A \ . \nonumber\end{aligned}$$ In the Laboratory frame, the differential cross sections for $\nu_\ell (p_\nu)+ t (p_t) \to \ell^- (p_\ell^\prime) +
\pi^+ (k) + t (p_t^\prime)$ is then given by $$\begin{aligned}
{\label{eq:d5s}}
\frac{d^5\sigma}{d E_\ell^\prime d\Omega_\ell^\prime d\Omega_\pi}
&=&\frac{G_F^2\cos^2\theta_c}{2}
\left({ {|\bm{k}| \over \omega_\pi} +
{|\bm{k}| - {\hat{\bm{k}}\cdot (\bm{p}_\nu -
\bm{p}_\ell^\prime) }\over E_t^\prime}}\right)^{-1}\!\!
\frac{|\bm{p}_\ell^\prime||\bm{k}|^2}{(2\pi)^5|\bm{p}_\nu|}
{E_{\ell,A}^\prime p_{\nu,A}}\label{CrossSec}
\\\nonumber&\times&
\sum_{s_\ell^\prime} |(\bar{{\cal M}}^L_{R,\mu} +\bar{{\cal M}}^L_{nr,\mu})
\epsilon^\mu_A |^2 \ ,
$$ where $E_t^\prime \left(= \sqrt{\bm{p}^2_t + (A m_N)^2}\right)$ is the total energy of the nucleus in the final state in LAB, and $E_{\ell,A}^\prime$ and $p_{\nu,A} $ are the energies of the final lepton and the initial neutrino in ACM. Note that the calculation of $\sum_{s_\ell^\prime} |(\bar{{\cal M}}^L_{R,\mu} +\bar{{\cal M}}^L_{nr,\mu})
\epsilon^\mu_A |^2$ of [Eq. [(\[eq:\]]{})]{}[d5s]{} can make use of the following property: $$\begin{aligned}
{\label{eq:lepton_tensor}}
L^{\mu\nu}_A \equiv {E_{\ell,A}^\prime p_{\nu,A}\over 2}
\sum_{s_\ell^\prime} \epsilon^\mu_A\epsilon^{\nu *}_A
= p^\mu_{\nu,A}p^{\prime\,\nu}_{\ell,A} + p^\nu_{\nu,A}p^{\prime\,\mu}_{\ell,A}
-g^{\mu\nu}p_{\nu,A}\cdot p^\prime_{\ell,A}
\pm i\epsilon^{\mu\nu\rho\sigma}p_{\nu,A\, \rho}p^\prime_{\ell,A\, \sigma} \ ,\end{aligned}$$ where $g^{\mu\nu}$ is the geometric tensor and $\epsilon^{\mu\nu\rho\sigma}$ is the antisymmetric tensor with $\epsilon^{0123}=1$. The plus (minus) sign in the last term is for the (anti-)neutrino process.
To obtain the cross section formula for the neutrino NC process, $\nu + t \to \nu + \pi^0 + t$, we make the following changes in Eq.(\[CrossSec\]): Remove the Cabbibo angle. Set the lepton mass equal to zero. Set the pion charge index $\lambda$ (and $\zeta$) to zero in $I^{l_\pi\, \nu}_{X\, \mu}$ and $J^{l_\pi\, \nu}_{X\, \mu}$ ($X=E,M,L,S$) in [Eqs. (\[eq:i\_mtx\]) and (\[eq:j\_mtx\])]{}. (Note that the pion wave function ($\chi_{\lambda\,l_\pi}$) also contains $\lambda$-dependence.) Finally, multiply the multipole amplitudes $M_{\ell +}^{(3/2,1/2),V}$ with $(1-2\sin^2\theta_W)$, where $\theta_W$ is the Weinberg angle ($\sin^2\theta_W = 0.23$), and multiply $M_{\ell +}^{(0),V}$ with $(-2\sin^2\theta_W)$.
For the anti-neutrino CC process, the result for the neutrino CC process is modified as follows. Set the pion charge index $\lambda$ (and $\zeta$) to $-1$ in $I^{l_\pi\, \nu}_{X\, \mu}$ and $J^{l_\pi\, \nu}_{X\, \mu}$. Replace the lepton current by the one for the anti-neutrino process, which amounts to adopting the negative sign in the leptonic tensor, [Eq. [(\[eq:\]]{})]{}[lepton\_tensor]{}. What modifications are needed for getting the cross section for the anti-neutrino NC process is now obvious.
Coherent Pion Photo-Production
------------------------------
With the same derivation given above, we can also get an expression for the differential cross section of the coherent $\pi^0$ photo-production process $\gamma(q)+ t (p_t) \to \pi^0 (k) + t (p_t^\prime)$ in the LAB frame: $$\begin{aligned}
{\label{eq:xs4}}
\frac{d^2\sigma}{d\Omega_\pi}
=\frac{\alpha}{2\pi}
\left( {|\bm{k}|\over \omega_\pi} + {|\bm{k}| - \hat{k}\cdot\bm{q}\over E_t^\prime}
\right)^{-1}
{|\bm{k}|^2 \over |\bm{q}|\omega_\pi}
{1\over 2} \sum_\epsilon |{\cal M}^\gamma_R + {\cal M}^\gamma_{nr}|^2 \ ,\end{aligned}$$ where $\alpha$ is the fine structure constant, and ${1\over 2}\sum_\epsilon$ stands for averaging over the photon polarization. The transition amplitudes ${\cal M}^\gamma_R$ and ${\cal M}^\gamma_{nr}$ for the photo-process are obtained from [Eqs. (\[eq:mr5\]) and (\[eq:mnr2\])]{} by retaining only the vector current, setting $\phi^\pi_A = 0$, and regarding $\epsilon^\mu_A$ as the polarization vector of the incident photon. Finally, the pion charge index ($\lambda$) is set to zero in $I^{l_\pi\, \nu}_{X\, \mu}$ and $J^{l_\pi\, \nu}_{X\, \mu}$ \[[Eqs. (\[eq:i\_mtx\]) and (\[eq:j\_mtx\])]{}\].
Optical Potential for Pion-Nucleus Scattering {#sec_opt}
---------------------------------------------
We calculate the pion-nucleus scattering using the computer code, PIPIT [@pipit] by appropriately modifying the optical potential there to accommodate the dynamical features of the $\Delta$-hole model and the SL model. In the original PIPIT, the optical potential ($U$), which is derived within the multiple scattering formalism by Kerman, McManus and Thaler (KMT) [@kmt], is given by[^4] $$\begin{aligned}
U(\bm{k}_A^\prime,\bm{k}_A)={A-1\over A}
\left\{\rho_p(\bm{q}) t_{\pi p}(\bm{k}_A^\prime,\bm{k}_A;k^o_A)
+\rho_n(\bm{q})t_{\pi n}(\bm{k}_A^\prime,\bm{k}_A;k^o_A)
\right\} \ ,\end{aligned}$$ where $\bm{k}_A$ ($\bm{k}_A^\prime$) is the incoming (outgoing) pion momentum in ACM, and $k^o_A$ the magnitude of the on-shell momentum. The quantities $\rho_p(\bm{q})$ ($\rho_n(\bm{q})$) is the form factor of the proton (neutron) matter distribution for $\bm{q}=\bm{k}_A-\bm{k}_A^\prime$, and $t_{\pi p}$ ($t_{\pi n}$) is the pion-proton (pion-neutron) scattering T-matrix whose normalization has been defined in [Eq. [(\[eq:\]]{})]{}[phase]{}. It is to be noted that this original optical potential does not take account of $\Delta$-propagation in nuclei. In Ref. [@karaoglu], KM separated $t_{\pi p}$ ($t_{\pi n}$) into the resonant and non-resonant parts, took the non-resonant and the Coulomb parts of the optical potential from the PIPIT code, and combined it with the resonant part derived from a simplified $\Delta$-hole model. A phenomenological s-wave potential which is proportional to the square of the nuclear density ($\rho_t=\rho_p+\rho_n$) was also included to account for the pion absorption by two nucleons through non-$\Delta$ mechanisms. Thus the KM optical potential is given by $$\begin{aligned}
{\label{eq:separation}}
U(\bm{k}_A^\prime,\bm{k}_A)=
U_{nr}+U_{R}+U_{ph}(\rho_t^2) \ ,\end{aligned}$$ where $U_{nr}$, $U_{R}$ and $U_{ph}$ are the non-resonant, resonant and phenomenological parts, respectively.
In constructing our optical potential, we follow the same separation as in [Eq. [(\[eq:\]]{})]{}[separation]{}. The non-resonant part of the optical potential is obtained from the PIPIT code by replacing the non-resonant T-matrices in the original code with those derived from the SL model. It is worth emphasizing that the SL model provides both on-shell and off-shell T-matrix elements. Another difference from the original PIPIT code is that we use a different prescription for the Lorentz transformation from ACM to 2CM, as explained in Appendix \[app\_2cm\]. Regarding the resonant part, we use the resonant part of $\pi N$ T-matrix from the SL model, basically following the procedure used in Ref. [@karaoglu] (apart from a more elaborate treatment of kinematics (Lorentz transformation, etc.)). First, we expand the optical potential into partial waves as $$\begin{aligned}
{\label{eq:partial_potential}}
U(\bm{k}_A^\prime,\bm{k}_A)= {2\over \pi} \sum_{l_\pi m_\pi}
V^{l_\pi}(k_A^\prime,k_A) Y^*_{l_\pi m_\pi}(\hat{k}_A^\prime)
Y_{l_\pi m_\pi}(\hat{k}_A) \ .\end{aligned}$$ The resonant part of the potential is (cf. Eq. (39) of Ref. [@karaoglu]) $$\begin{aligned}
{\label{eq:v_res}}
&&V^{l_\pi}_R(k_A^\prime,k_A) = {A-1\over A}
{8\pi^2 \mu_\Delta\over 3}
\int dx_A \Gamma_{A2}\gamma^{-1}
x_2 P_{l_\pi}(x_A)
F_{\pi N\Delta}(k_2^\prime)F_{\pi N\Delta}(k_2) \\\nonumber
&\times&
\sum_{N=p,n} (1+{\lambda\tau_N\over 2})\int s^2 ds R^2 dR
j_0(pR)j_0(Ps)
{e^{i \bar{K}_\Delta s}\over s}
\left\{1 + {i \mu_\Delta s\over \bar{K}_\Delta}
\left[\bar{e_N} + H_N\right]\right\}\rho_N(R) \hat{j}_1(k_F s) \ ,\end{aligned}$$ where $\bm{k}_2$ ($\bm{k}_2^\prime$) is the incoming (outgoing) pion momentum in 2CM, and $x_A=\hat{k}_A\cdot\hat{k}_A^\prime$, $x_2=\hat{k}_2\cdot\hat{k}_2^\prime$, $P=|\bm{k}_A+\bm{k}_A^\prime|/2$, $p=|\bm{k}_A-\bm{k}_A^\prime|$. The dressed $\pi N\Delta$ coupling ($F_{\pi N\Delta}$) has been introduced in [Eq. [(\[eq:\]]{})]{}[t-res]{}. The Lorentz transformation of the T-matrix from 2CM to ACM gives rise to the factor $\Gamma_{A2}$ defined by $$\begin{aligned}
\Gamma_{A2} = \sqrt{\omega_{\pi,2}\,\omega_{\pi,2}^\prime\,
E_{N,2}E_{N,2}^\prime \over
\omega_{\pi,A}\,\omega_{\pi,A}^\prime\, E_{N,A}E_{N,A}^\prime} \ ,\end{aligned}$$ with $\omega_{\pi,2}^{(\prime)}=\sqrt{\bm{k}_2^{(\prime)\,2}+m_\pi^2}$, $\omega_{\pi,A}^{(\prime)}=\sqrt{\bm{k}_A^{(\prime)\,2}+m_\pi^2}$, $E_{N,2}^{(\prime)}=\sqrt{\bm{k}_2^{(\prime)\,2}+m_N^2}$ and $E_{N,A}^{(\prime)}=\sqrt{\bm{p}_{N,A}^{(\prime)\,2}+m_N^2}$. The values of $\bm{k}_2^{(\prime)}$ and $\bm{p}_{N,A}^{(\prime)}$ are fixed according to the prescription explained in Appendix \[app\_2cm\]. The other quantities have already been introduced in Sec. \[sec\_res\].
Finally, we discuss the phenomenological term, $U_{ph}$. We assume that in coordinate space $U_{ph}$ can be parametrized as $$\begin{aligned}
{\label{eq:rho2-term}}
U_{ph}(\bm{r}) = B \left({\rho_t(r)\over\rho_t(0)}\right)^2 \ ,\end{aligned}$$ where $B$ is the partial wave dependent strength of the potential. The corresponding partial wave potential in momentum space is given by $$\begin{aligned}
{\label{eq:wave}}
V^{l_\pi}_{ph}(k_A^\prime, k_A) = {A-1\over A} 4\pi^3 B_{l_\pi}
\int^1_{-1}dx_A P_{l_\pi}(x_A)\int dr r^2 j_0(pr)
\left({\rho_t(r)\over\rho_t(0)}\right)^2 \ .\end{aligned}$$ In the present calculation we include $V^{0}_{ph}$ and $V^{1}_{ph}$ and treat their strengths $B_0$ and $B_1$ as adjustable parameters. Thus our model contains as free parameters $B_0$ and $B_1$ (complex numbers) in addition to the couplings in the spreading potential.
Given the optical potential, we solve the Lippmann-Schwinger equation $$\begin{aligned}
{\label{eq:lippman}}
T^\prime_{l_\pi}(k_A^\prime,k_A;k_A^o)
= V_{l_\pi}(k_A^\prime,k_A;k_A^o)
+{2\over\pi}\int
{V_{l_\pi}(k_A^\prime,\bar{k}_A;k_A^o)T^\prime_{l_\pi}(\bar{k}_A,k_A;k_A^o)
\bar{k}_A^2 d\bar{k}_A
\over \omega_\pi(k_A^o)+E_t(k_A^o)-\omega_\pi(\bar{k}_A)-E_t(\bar{k}_A)
+ i\epsilon} \ .\end{aligned}$$ The solution to this equation will be used in two contexts. First, we use it to calculate pion-nucleus elastic and total scattering cross sections, and compare them with data to find the optimal values of the free parameters in our model. The solution to [Eq. [(\[eq:\]]{})]{}[lippman]{} is also used to compute the pion distorted wave function that features in the matrix elements in [Eqs. (\[eq:i\_mtx\]) and (\[eq:j\_mtx\])]{}. For the former purpose, we obtain the full T-matrix of pion-nucleus scattering from $T^\prime$ in [Eq. [(\[eq:\]]{})]{}[lippman]{} using the relation $$\begin{aligned}
T = {A\over A-1}T^\prime \ .\end{aligned}$$ For charged-pion scattering, corrections for the finite range Coulomb potential are incorporated with the use of the Vincent-Phatak method [@vincent]. The procedure for calculating scattering observables from $T$ is detailed in Ref. [@pipit]. For the latter purpose, we calculate the pion distorted wave $\chi_{l_\pi}^*(k_A)$ associated with $T^\prime$ using the relation $$\begin{aligned}
{\label{eq:pi_wave}}
\chi_{l_\pi}^*(k_A) = {\delta(k_A-k_A^o)\over k_A^2 }
+ {T^\prime_{l_\pi}(k_A^o,k_A;k_A^o)\over
\omega_\pi(k_A^o)+E_t(k_A^o)-\omega_\pi(k_A)-E_t(k_A)
+ i\epsilon} \ ,\end{aligned}$$ where, for notational simplicity, dependence on the pion charge ($\lambda$) is suppressed. Following the KMT formalism [@kmt], we use $ \chi_{l_\pi}^*(k_A) $ in evaluating the matrix elements in [Eqs. (\[eq:i\_mtx\]) and (\[eq:j\_mtx\])]{}. This wave function is related to the full wave function by $$\begin{aligned}
\chi_{l_\pi}^{\rm (full)*} =
-{1\over A-1} + {A\over A-1} \chi_{l_\pi}^* \ .\end{aligned}$$ For charged-pion scattering, $\chi_{l_\pi}^{\rm (full)*}$ does not have the correct normalization, because the Coulomb potential has been cut off at a finite distance; this entails the necessity of multiplying $\chi_{l_\pi}^{\rm (full)*}$ with a normalization factor (call it $\kappa$). We note that it is $\chi_{l_\pi}^*$ rather than $\chi_{l_\pi}^{\rm (full)*}$ that enters into our calculation, and we choose to use the same normalization factor $\kappa$ for $\chi_{l_\pi}^*$ as for $\chi_{l_\pi}^{\rm (full)*}$. Thus, in evaluating the matrix elements in [Eqs. (\[eq:i\_mtx\]) and (\[eq:j\_mtx\])]{}, we use $\kappa\chi_{l_\pi}^*$ instead of $\chi_{l_\pi}^*$. In the $\Delta$ resonance region of our interest, it turns out that $|\kappa -1| \ltap 0.01$. (For neutral pion scattering, $\kappa = 1$.)
Numerical Results {#sec_result}
=================
Pion-Nucleus Scattering
-----------------------
As explained in the previous sections, our model contains four complex free parameters. Two of them are the central ($V_C$) and LS ($V_{LS}$) parts of the spreading potential \[see [Eq. [(\[eq:\]]{})]{}[spr]{}\], and the other two are the strengths, $B_0$ and $B_1$, of the s-wave and p-wave phenomenological terms in the optical potential \[see [Eq. [(\[eq:\]]{})]{}[wave]{}\]. These free parameters are optimized to fit the pion-nucleus scattering data. Since our aim here is to calculate coherent pion production off $^{12}$C, we should use the $\pi\! -\!^{12}$C scattering data to fix these parameters. Adjusting them to reproduce the total cross sections and the elastic differential cross sections for $\pi\! -\!^{12}$C scattering, we obtain: $$\begin{aligned}
V_C = 48.0 - 34.5 i\ {\rm MeV} \ ,\qquad
V_{LS} = -3.0 - 2.0 i\ {\rm MeV} \nonumber\\
B_0 = 5.1 + 5.2 i\ {\rm MeV} \ ,\qquad
B_1 = 2.8 - 5.7 i\ {\rm MeV} \ .\end{aligned}$$ We note that our calculations include the pion-nucleus partial waves up to $l_\pi \le 9$ \[[Eq. [(\[eq:\]]{})]{}[lippman]{}\], and $s$- and $p$-waves (and all possible spin-isospin states) for the elementary $\pi N$ scattering.[^5] Figures. \[fig\_total\] and \[fig\_elastic\] illustrate the quality of fit to the $\pi-{}^{12}$C scattering data achieved in our model (with our optical potential).
![\[fig\_total\] Total cross sections for $\pi^- - {}^{12}$C scattering. The solid curve is obtained with our full calculation, while the dashed curve is obtained without the spreading potential. The data are from Ref. [@pi-nucleus1]. ](tot_piA){width="90mm"}
![\[fig\_elastic\] $\pi - {}^{12}$C elastic differential cross sections. The solid curve is obtained with our full calculation while the dashed curve is obtained without the phenomenological terms in [Eq. [(\[eq:\]]{})]{}[rho2-term]{}. The data are from Ref. [@pi-nucleus2] for (a), Ref. [@pi-nucleus3] for (b) and Ref. [@pi-nucleus1] for (c)-(f). ](dsig_1){width="80mm"}
![\[fig\_elastic\] $\pi - {}^{12}$C elastic differential cross sections. The solid curve is obtained with our full calculation while the dashed curve is obtained without the phenomenological terms in [Eq. [(\[eq:\]]{})]{}[rho2-term]{}. The data are from Ref. [@pi-nucleus2] for (a), Ref. [@pi-nucleus3] for (b) and Ref. [@pi-nucleus1] for (c)-(f). ](dsig_2){width="80mm"}
In Fig. \[fig\_total\], the total cross sections for $\pi^-$-$^{12}$C scattering are shown as a function of the pion kinetic energy $T_\pi$ in the laboratory frame. The results of our full calculation are given by the solid curve and, for comparison, the results obtained without the spreading potential are also shown in the dashed curve. We observe a large reduction in the total cross section as we go from the dashed to solid lines, which is mainly caused by the strong pion absorption simulated by the spreading potential. In connection with fitting to the pion-nucleus scattering data, it is worthwhile to make the following comment. In the calculation of coherent pion production, the final-state interaction is nothing but elastic scattering between the pion and nucleus. One might therefore think that a phenomenological adjustment of the pion-nucleus optical potential to fit the elastic pion-nucleus scattering data will be good enough. However, in our consistent model building, the spreading potential enters not only into the optical potential but also into the pion production operators, and hence it is important to control its strength using the total cross section data. The fact that the spreading potential has a very large effect on the total cross sections makes this point particularly important.
Our results for the differential cross sections are shown in Fig. \[fig\_elastic\]. In addition to our full calculation shown in the solid curve, we also show in the dashed curve the results obtained without the phenomenological term $U_{ph}$ \[see [Eq. [(\[eq:\]]{})]{}[rho2-term]{}\]. We see that this phenomenological $\rho^2$ term, which simulates absorption of $s$-wave and $p$-wave pions by two-nucleons within our model, is not large in the considered $T_\pi > $ 40 MeV region for $\pi-^{12}$C elastic scattering. However it is known that $U_{ph}$ can play an important role for many observables in low-energy pion-nucleus scattering. As an example to shed light on this point, we have calculated $\pi-{}^{16}O$ elastic scattering at $T_{\pi}$ = 50 MeV using the same model (only the nuclear density is different). We have found that, in reproducing the data satisfactorily in our approach, the $\rho^2$ term plays an important role, its size being almost as large as that found in Fig. 4(a) of Ref. [@karaoglu]. Overall, the results of our full calculation satisfactorily reproduce the data for both the total and elastic cross sections.
Coherent Pion Photo-Production
------------------------------
We are now in a position to perform a parameter-free calculation of the cross sections for coherent pion production. The photo-process, for which extensive data are available, provides a good testing ground for checking the reliability of our approach. We compare in Fig. \[fig\_krusche\] our numerical results for the differential cross sections for $\gamma + {}^{12}{\rm C}_{g.s.} \to \pi^0 + {}^{12}{\rm C}_{g.s.}$ with the existing data [@gothe; @krusche].
![\[fig\_krusche\] Differential cross sections for $\gamma + {}^{12}{\rm C}_{g.s.} \to \pi^0 + {}^{12}{\rm C}_{g.s.}$ for different incident photon energies (indicated in each panel). The solid lines represent the results of the full calculation. The dashed lines are obtained without the FSI and without the medium effects on the $\Delta$-propagation, while the dotted lines are obtained with the medium effects on the $\Delta$ included. The dash-dotted curves correspond to a case in which the pion production operator includes only the $\Delta$ mechanism. For more detailed explanations for the different cases, see the text. The data are from Ref. [@gothe] for (a) and from Ref. [@krusche] for (b)-(d). ](gothe){width="77mm"}
![\[fig\_krusche\] Differential cross sections for $\gamma + {}^{12}{\rm C}_{g.s.} \to \pi^0 + {}^{12}{\rm C}_{g.s.}$ for different incident photon energies (indicated in each panel). The solid lines represent the results of the full calculation. The dashed lines are obtained without the FSI and without the medium effects on the $\Delta$-propagation, while the dotted lines are obtained with the medium effects on the $\Delta$ included. The dash-dotted curves correspond to a case in which the pion production operator includes only the $\Delta$ mechanism. For more detailed explanations for the different cases, see the text. The data are from Ref. [@gothe] for (a) and from Ref. [@krusche] for (b)-(d). ](krusche-200){width="77mm"}
![\[fig\_krusche\] Differential cross sections for $\gamma + {}^{12}{\rm C}_{g.s.} \to \pi^0 + {}^{12}{\rm C}_{g.s.}$ for different incident photon energies (indicated in each panel). The solid lines represent the results of the full calculation. The dashed lines are obtained without the FSI and without the medium effects on the $\Delta$-propagation, while the dotted lines are obtained with the medium effects on the $\Delta$ included. The dash-dotted curves correspond to a case in which the pion production operator includes only the $\Delta$ mechanism. For more detailed explanations for the different cases, see the text. The data are from Ref. [@gothe] for (a) and from Ref. [@krusche] for (b)-(d). ](krusche-290){width="77mm"}
![\[fig\_krusche\] Differential cross sections for $\gamma + {}^{12}{\rm C}_{g.s.} \to \pi^0 + {}^{12}{\rm C}_{g.s.}$ for different incident photon energies (indicated in each panel). The solid lines represent the results of the full calculation. The dashed lines are obtained without the FSI and without the medium effects on the $\Delta$-propagation, while the dotted lines are obtained with the medium effects on the $\Delta$ included. The dash-dotted curves correspond to a case in which the pion production operator includes only the $\Delta$ mechanism. For more detailed explanations for the different cases, see the text. The data are from Ref. [@gothe] for (a) and from Ref. [@krusche] for (b)-(d). ](krusche-350){width="77mm"}
The long-dash lines are obtained without FSI and without the medium effects on $\Delta$-propagation.[^6] With the medium effects on the $\Delta$ included, the short-dash lines are obtained, and the results of our full calculation are given by the solid lines. Figure \[fig\_krusche\] indicates that the medium effects are quite sizable, and they play an important role in bringing the calculated differential cross sections in agreement with the data. Particularly noteworthy is the drastic reduction of the cross section in the $\Delta$ region \[Fig. \[fig\_krusche\] (c)\], a feature that reflects the fact that a significant part of the medium effects simulate pion absorption. The good general agreement seen in Fig. \[fig\_krusche\] indicates the basic soundness of the method we have used in determining the spreading potential.
It is true that, for higher incident energies, in the large angle region beyond the peak position, there are noticeable discrepancies between the results of our full calculation and the data. However, as noted in Ref. [@krusche], the data in this region are likely to be substantially contaminated by incoherent processes in which the final nucleus is in its low-lying excited states. The effects of this type of contamination are expected to grow for higher incident photon energies and for larger momentum transfers (the large angle region) because of increased nuclear excitations. We therefore take the viewpoint that the discrepancy found in Figs. \[fig\_krusche\] (b)-(d) does not necessarily signal a failure of our model, and that our model describes coherent pion photo-production reasonably well.
Figure \[fig\_krusche\] also shows (in the dash-dotted lines) the results corresponding to a case in which the pion production operator includes only the $\Delta$ mechanism (the non-resonant mechanism turned off); [^7] the distorted pion wave function incorporating FSI is the same as that used for the full calculation. These results serve to demonstrate the importance of the non-resonant mechanism. Fig. \[fig\_krusche\] (a) indicates that, near threshold, the contributions from the resonant and non-resonant mechanisms are comparable, a feature that is not surprising away from the resonance peak. A remarkable feature is that even near the resonance energy \[see Fig. \[fig\_krusche\] (c)\] the contribution from the non-resonant mechanism is quite significant. This is partly because the resonant contribution is considerably suppressed by pion absorption (the spreading potential) and the non-local effect of $\Delta$ propagation (the $\Delta$ kinetic term).[^8] To summarize this section, the results for the coherent photo-pion production process establish to a satisfactory degree the reliability of our present approach ([*i.e.,*]{} combined use of the SL model and the $\Delta$-hole model) and motivate us to apply the same approach to neutrino-induced coherent pion production.
Neutrino-Induced Coherent Pion Production
-----------------------------------------
![\[fig\_tot\] The $E_\nu$-dependence of the total cross section for $\nu_\mu + {}^{12}{\rm C}_{g.s.} \to \mu^- + \pi^+ + {}^{12}{\rm
C}_{g.s.}$ (solid line), $\nu + {}^{12}{\rm C}_{g.s.} \to \nu + \pi^0 + {}^{12}{\rm C}_{g.s.}$ (dashed line), $\bar{\nu}_\mu + {}^{12}{\rm C}_{g.s.} \to \mu^+ + \pi^- + {}^{12}{\rm
C}_{g.s.}$ (dotted line) and $\bar{\nu} + {}^{12}{\rm C}_{g.s.} \to \bar{{\nu}} + \pi^0 + {}^{12}{\rm
C}_{g.s.}$ (dash-dotted line). ](nu_tot){width="85mm"}
We now present the numerical results of our calculations for neutrino-induced coherent pion production on the $^{12}$C target. We consider the CC and NC processes induced by a neutrino or an anti-neutrino: $$\begin{aligned}
\nu_\mu + {}^{12}{\rm C}_{g.s.} &\to&
\mu^- + \pi^+ + {}^{12}{\rm C}_{g.s.}\nonumber\\
\nu + {}^{12}{\rm C}_{g.s.} &\to&
\nu + \pi^0 + {}^{12}{\rm C}_{g.s.}\\
\bar{\nu}_\mu + {}^{12}{\rm C}_{g.s.} &\to&
\mu^+ + \pi^- + {}^{12}{\rm C}_{g.s.} \nonumber\\
\bar{\nu} + {}^{12}{\rm C}_{g.s.} &\to&
\bar{{\nu}} + \pi^0 + {}^{12}{\rm C}_{g.s.} \nonumber\end{aligned}$$ Figure \[fig\_tot\] gives the total cross sections for these processes as functions of the incident neutrino (anti-neutrino) energy in the laboratory system, $E_\nu$. It is seen that, for higher incident energies, the ratio $\sigma_{CC}/ \sigma_{NC}$ approaches 2, a value expected from the isospin factor. For lower incident energies ($E_\nu$500 MeV), however, $\sigma_{NC}$ is larger than $\sigma_{CC}$, reflecting the fact that the phase space for the CC process is reduced significantly by the muon mass. It is well known that interference between the vector and axial-vector currents can lead to different cross sections for the neutrino and anti-neutrino processes. However, since the coherent process is dominated by the contribution of the axial current (see Fig. \[fig\_v-a\]), the role of the interference term is diminished drastically. This explains why in Fig. \[fig\_tot\] the cross sections for the neutrino and anti-neutrino processes are almost the same.
To compare our results with data, we need to evaluate the total cross sections averaged over the neutrino fluxes that pertain to the relevant experiments. We choose to use the fluxes up to $E_\nu \le$ 2 GeV and neglect the fluxes beyond that limit based on the following consideration. Since our model includes no resonances other than the $\Delta$, it is expected to be reliable only for $W$1.4 GeV. The fact that even at $E_\nu$ = 1 GeV coherent pion production can involve contributions coming from the $W \!>\!$ 1.4 GeV region is disquieting, but we can still expect that the $\Delta$-excitation contribution is predominant for the total cross section for the coherent process. \[This feature can be seen in, [*e.g.*]{}, Fig. \[fig\_tpi.1gev\] to be discussed later.\] For $E_\nu \sim$ 2 GeV, we do expect that $\Delta$ dominance gets significantly less pronounced but that $\Delta$ still gives the most important contribution. Meanwhile, the region $E_\nu$1.5 GeV belongs to the tail of the neutrino flux used in MiniBooNE. We therefore consider it reasonable to compare with data our theoretical cross section averaged over the neutrino flux up to $E_\nu$ = 2 GeV. For the CC process, we use the flux reported in Ref. [@K2K] and deduce $$\begin{aligned}
\sigma_{\rm ave}^{CC} = 6.3 \times 10^{-40} {\rm cm}^2 \ .
\label{SCCave}\end{aligned}$$ A K2K experiment [@hasegawa] reports the upper limit $$\begin{aligned}
\sigma_{\rm K2K} < 7.7 \times 10^{-40} {\rm cm}^2 \ .\end{aligned}$$ In fact, this upper limit corresponds to events satisfying the muon momentum cut, $p_\mu > 450$ MeV and the cut on the momentum transfer squared, $Q_{\rm rec}^2 < 0.1$ GeV$^2$; $Q_{\rm rec}^2$ is calculated as $$\begin{aligned}
{\label{eq:q_rec}}
Q_{\rm rec}^2 = 2 E_\nu^{\rm rec} (E_\mu - p_\mu\cos\theta_\mu) -
m_\mu^2 \ ,\end{aligned}$$ where the reconstructed neutrino energy ($E^{\rm rec}_\nu$) is calculated from the muon kinematics \[the energy ($E_\mu$) and the scattering angle ($\theta_\mu$)\] assuming the quasi-elastic kinematics: $$\begin{aligned}
{\label{eq:e_rec}}
E^{\rm rec}_\nu = {1\over 2}{(m_p^2 - m_\mu^2) - (m_n - V)^2 + 2E_\mu
(m_n - V)\over (m_n - V) - E_\mu + p_\mu\cos\theta_\mu} \ ,\end{aligned}$$ where $m_p$, $m_n$ and $m_\mu$ are the masses of the proton, neutron and muon, respectively and the nuclear potential ($V$) is set to 27 MeV. Our result in Eq.(\[SCCave\]) is also obtained with these cuts, and is consistent with the K2K data. We note that a recent report from SciBooNE [@hiraide] gives a similar empirical upper limit.
For the NC process, we use the flux reported by MiniBooNE in Ref. [@miniboone_flux] and arrive at $$\begin{aligned}
\sigma_{\rm ave}^{NC} = 2.8 \times 10^{-40} {\rm cm}^2 \ .\end{aligned}$$ This is to be compared with $$\begin{aligned}
\sigma_{\rm MiniBooNE} = 7.7 \pm 1.6\pm 3.6 \times 10^{-40} {\rm cm}^2 \ , \end{aligned}$$ given in Ref. [@raaf]. Our result is consistent with the empirical value within the large experimental errors, even though the theoretical value is rather visibly smaller than the empirical central value. It is to be noted however that Ref. [@raaf] is a preliminary report, and that, as discussed in great detail in Ref. [@amaro], $\sigma_{\rm MiniBooNE}$ may be overestimated due to the use of the RS model[@RS] in the analysis.
![\[fig\_tpi.0p5gev\] Same as in Fig. \[fig\_tpi.1gev\] but for $E_\nu$ = 0.5 GeV. ](pmom.med.cc.1gev.eps){width="77mm"}
![\[fig\_tpi.0p5gev\] Same as in Fig. \[fig\_tpi.1gev\] but for $E_\nu$ = 0.5 GeV. ](pmom.med.cc.0p5gev.eps){width="77mm"}
![\[fig\_q2.nc\] Same as in Fig. \[fig\_q2.cc\] but for the NC process at $E_\nu = 1$ GeV. ](q2.cc.eps){width="77mm"}
![\[fig\_q2.nc\] Same as in Fig. \[fig\_q2.cc\] but for the NC process at $E_\nu = 1$ GeV. ](q2.nc.eps){width="77mm"}
We now proceed to present our results for differential observables. In view of the fact that the event rates (cross section times flux) in the K2K, MiniBooNE and SciBooNE experiments [@K2K; @miniboone] have been reported to have a peak around $E_\nu \sim$ 1 GeV, we shall often use this energy as a representative in the following presentation. Meanwhile, since the neutrino flux in the planned T2K experiment is expected to have a peak around $E_\nu =$ 0.6 $\sim$ 0.7 GeV[@kato], we shall also present results for lower neutrino energies when that seems useful.
The pion momentum spectrum for CC neutrino-induced coherent pion production is shown in Fig. \[fig\_tpi.1gev\] (Fig. \[fig\_tpi.0p5gev\]) for $E_\nu$ = 1 GeV (0.5 GeV). The importance of the medium effects manifests itself here in the same manner as in the photo-process (Fig. \[fig\_krusche\]). In the $\Delta$ region, strong pion absorption is seen to reduce the cross sections significantly, and FSI shifts the peak position. The dash-dotted line corresponds to a case in which the pion production operator contains only the $\Delta$ mechanism (without non-resonant contributions), while the pion optical potential is kept unchanged. We note that, at $E_\nu$ = 1 GeV (0.5 GeV), the dash-dotted line corresponds to 82% (64%) of the solid line (the results of the full calculation). We have seen in the photo-process that the non-resonant mechanism is more important for a smaller energy transfer. To what extent the neutrino case should share this feature is not obvious because the axial-vector current contributions dominate here (see Fig. \[fig\_v-a\]). However, we can see in Figs. \[fig\_tpi.1gev\] and \[fig\_tpi.0p5gev\] that, in the neutrino case as well, the differential cross sections with smaller pion momenta are more enhanced by the non-resonant mechanism, and that this feature is more prominent for a smaller value of $E_\nu$. A similar tendency is seen for the NC process also. These results indicate that the non-resonant amplitudes in our model, which are dressed by the rescattering, play a significant role in coherent pion production; their role is particularly important for $E_\nu$0.5 GeV. This characteristic feature of our model should be contrasted with the fact that (tree-level) non-resonant mechanisms play essentially no role in any of the previous microscopic calculations for neutrino-induced coherent pion production. A more detailed comparison of the elementary amplitudes used in our present calculation and the previous microscopic-model calculations will be given later in Sec. \[sec\_comp\].
We show in Fig. \[fig\_q2.cc\] (Fig. \[fig\_q2.nc\]) the $Q^2$-distribution for the CC (NC) process. Note that $Q^2$ defined by $Q^2 \equiv -q^2 \equiv- (p_\nu-p_\ell^\prime)^2$ is different from $Q^2_{\rm rec}$ defined in [Eq. [(\[eq:\]]{})]{}[q\_rec]{}. Because of the nuclear form factor effect, the distribution rises sharply as $Q^2$ approaches 0; for the CC process, however, $Q^2$-distribution becomes zero at $Q^2 = 0$ due to the finite muon mass. Here again we show the results corresponding to a case in which the pion production operator contains only the $\Delta$ effect (with non-resonant contributions turned off). The non-resonant mechanism is seen to change the spectrum shape significantly and lead to a sharper peak.
It is informative to examine the individual contributions of the vector and axial-vector currents. We show in Fig. \[fig\_v-a\] these individual contributions to the neutrino CC process. We find strong dominance of the axial-vector current. The nuclear form factor causes the drastic suppression of non-forward pion production. This aspect combined with the fact that the transverse photon coupling of the vector current \[[Eq. [(\[eq:\]]{})]{}[bar\_fv]{}\] forbids forward pion production leads to strong suppression of the vector current contribution. By contrast, since the vertex structure of the axial-vector current favors forward pion production, the strong suppression mechanism at work for the vector current does not apply here. This is the reason why the axial-vector current dominates. This result may be used to argue that incoherent pion production processes in which a nucleus does not break up but transits to excited states, are much less important than coherent pion production in the neutrino-nucleus scattering. As seen in Fig. \[fig\_krusche\], the incoherent processes give considerable contributions to the total pion production in the photo-process,[^9] a feature that may lead to the expectation that the incoherent processes are considerable in the neutrino process as well. However, the mechanism responsible for the axial-vector dominance in the neutrino process works for the photo process in such a manner that coherent photo-pion production is strongly suppressed. Also, the inelastic transition form factor has a peak at a non-zero momentum transfer. As a result, for the photo reaction, the contributions from the incoherent processes become comparable to those from the coherent process. Thus the importance of the incoherent processes relative to the coherent process can be very different between the photo and neutrino processes. Takaki et al. [@takaki] used a similar argument to explain a significant (very small) contribution from the incoherent processes in the photo-pion production (pion-nucleus scattering), compared to the coherent process. This argument may serve as a justification for the assumption currently used in data analyses that the incoherent processes need not be taken into account explicitly .
![\[fig\_non\_local\] The effect of the non-locality of the $\Delta$-propagation for $\nu_\mu \!+\! {}^{12}{\rm C}_{g.s.}
\to \mu^- \!+\! \pi^+\! +\! {}^{12}{\rm C}_{g.s.}$. The ratio ${\cal R}$ of the total cross sections obtained with and without taking account of the non-local effect. ](v-a.cc.1gev.eps){width="77mm"}
![\[fig\_non\_local\] The effect of the non-locality of the $\Delta$-propagation for $\nu_\mu \!+\! {}^{12}{\rm C}_{g.s.}
\to \mu^- \!+\! \pi^+\! +\! {}^{12}{\rm C}_{g.s.}$. The ratio ${\cal R}$ of the total cross sections obtained with and without taking account of the non-local effect. ](non_local){width="77mm"}
Finally, we examine the effect of the non-locality of $\Delta$-propagation in nuclei; because we employed the local density approximation for evaluating the $\Delta$ Green function \[[Eq. [(\[eq:\]]{})]{}[nuclear-delta-prop]{}\], this effect arises only from the $\Delta$ kinetic term in the $\Delta$ Hamiltonian \[[Eq. [(\[eq:\]]{})]{}[delta\_H]{}\]. Although, as mentioned in the introduction, this subject has been studied in Ref. [@non-local], that study only included the $\Delta$-mechanism without considering FSI or the medium effects on the $\Delta$. It is thus interesting to revisit this problem in the framework of our significantly extended treatment. In the local approximation, we neglect the kinetic term in the $\Delta$-Hamiltonian \[[Eq. [(\[eq:\]]{})]{}[delta\_H]{}\], which means that the $\Delta$ is considered to be so heavy that it does not propagate in nuclear medium. To facilitate our discussion, we introduce the ratio ${\cal R}(E_\nu)$ defined by (E\_)(E\_) /\_[local]{}(E\_), where $\sigma(E_\nu)$ represents the total cross section for $\nu_\mu \!+\! {}^{12}{\rm C}_{g.s.}
\to \mu^- \!+\! \pi^+\! +\! {}^{12}{\rm C}_{g.s.}$ calculated with the $\Delta$-propagator including the $\Delta$ kinetic term, whereas $\sigma_{\rm local}(E_\nu)$ is that obtained in the local approximation. Figure \[fig\_non\_local\] shows ${\cal R}(E_\nu)$ calculated for the various cases. The long-dash curve corresponds to the $\Delta$-only case (without FSI or the medium effects on the $\Delta$; see footnote \[foot:medium\] ) and the solid line to the case that includes the non-resonant components, medium effects on the $\Delta$ and FSI. To make comparison with Ref. [@non-local], we first consider the long-dash line; ${\cal R}(E_\nu)$ in this case is found to be $0.55$, $1.03$ and $1.14$ at $E_\nu$ = 0.5, 1.0 and 1.5 GeV. Meanwhile, Ref. [@non-local] reports ${\cal R}(E_\nu)$ = $0.5$, $0.6$ and 1 at $E_\nu$ = 0.5, 1.0 and 1.5 GeV. Although both calculations indicate that the non-local effects are important, our results are qualitatively different from those of Ref. [@non-local]. This difference originates from different ways of treating the energy in the $\Delta$-propagator. In Ref. [@non-local], the in-medium $\Delta$-propagator is assumed to be the same as the free $\Delta$-propagator, whereas our $\Delta$-propagator \[$G_{\Delta h}$, [Eq. [(\[eq:\]]{})]{}[nuclear-delta-prop]{}\] is a nuclear many-body operator [@taniguchi] (with some of the medium effects switched off). To illustrate this point, we include in Fig. \[fig\_non\_local\] (dash-dotted line) the results obtained with the use of the free $\Delta$-propagator. In this case, we find ${\cal R}(E_\nu)$ = $0.4$, $0.76$ and $0.88$ at $E_\nu$ = 0.5, 1.0 and 1.5 GeV, which is fairly close to the results in Ref. [@non-local]. The result shown by the solid line indicates that, after the sophistication of the calculation, the non-locality due to the kinetic term is still important over the entire range of $E_\nu$ under consideration. In the previous microscopic calculations for neutrino-induced coherent pion production, the non-locality has not been explicitly taken into account. However, this does not necessarily mean that the earlier results are off by an amount suggested by comparison of the curves in Fig. \[fig\_non\_local\], for it is possible that the non-locality effects are partly included with the use of the spreading potential fitted to observables. In view of the importance of the non-local effect, however, we consider it preferable to take it into account explicitly, rather than include it operationally in the $\Delta$ mass shift.
An additional point of interest is that it was reported in Ref. [@non-local] that the non-locality changes the shapes of the differential cross sections. We remark that our results (not shown here) agree with that finding.
Comparison with SciBooNE and MiniBooNE data
-------------------------------------------
The SciBooNE collaboration has been pursuing a further analysis of the data on neutrino and anti-neutrino CC coherent pion production, and some preliminary results have appeared in Refs.[@hiraide_nuint09; @anti_cc]. These results contain detailed information on the differential observables for the pion and muon, and it seems informative to present our theoretical results in a manner that allows ready comparison with these data. To this end, we need to take into account the muon momentum cut ($p_\mu >$ 350 MeV) and the momentum transfer cut ($Q^2_{rec} <$ 0.1 GeV$^2$) adopted in the SciBooNE experiment; $Q^2_{rec}$ has been defined in [Eq. [(\[eq:\]]{})]{}[q\_rec]{}. The theoretical results we present in the following take account of these cuts unless otherwise stated. We will present the results at $E_\nu$ = 1 GeV around which the event rate has a peak. Although, for direct comparison, we need to convolute the observables with the (anti- )neutrino flux used in the SciBooNE experiment, the flux has not been released yet. We therefore present our results at a representative value of $E_\nu$ = 1 GeV. In Fig. \[fig\_theta\_p.sci\], we show the $\cos\theta_\pi$-distribution for the neutrino and anti-neutrino CC processes. In the recent data analysis by the SciBooNE collaboration, events are classified according to the pion emission angle ($\theta_\pi$). Their preliminary results exhibit a rather clear excess yield for $\theta_\pi<35^\circ$, which is thought to be ascribable to coherent pion production. In our model, 85% of the pions are emitted in $\theta_\pi<35^\circ$ for the neutrino CC process at $E_\nu =$ 1 GeV, a feature that is in fair agreement with the preliminary SciBooNE result .
Next we show in Fig. \[fig\_q2\_rec.sci\] (solid line) the $Q^2_{\rm rec}$ distribution for the neutrino reaction.[^10] Only the $p_\mu $ cut is applied here for an obvious reason. We can see that the contribution from above $Q^2_{\rm rec}$ = 0.1 GeV$^2$ (the value adopted for the $Q^2_{\rm rec}$ cut) constitutes only a small fraction of the entire contribution (3% for the solid curve). The decomposition of the total contribution (solid curve) into two parts according to whether $\theta_\pi$ is smaller or larger than $35^\circ$ is shown by the dashed curve ($\theta_\pi \!< \!35^\circ$) and the dotted curve ($\theta_\pi \!>\! 35^\circ$).
![\[fig\_q2\_rec.sci\] The $Q^2_{\rm rec}$ distribution for $\nu_\mu + {}^{12}{\rm C}_{g.s.} \to \mu^- + \pi^+ + {}^{12}{\rm C}_{g.s.}$ at $E_\nu = 1$ GeV. ](theta_p.nu.cc.sci.1gev.eps){width="77mm"}
![\[fig\_q2\_rec.sci\] The $Q^2_{\rm rec}$ distribution for $\nu_\mu + {}^{12}{\rm C}_{g.s.} \to \mu^- + \pi^+ + {}^{12}{\rm C}_{g.s.}$ at $E_\nu = 1$ GeV. ](q2_rec.nu.cc.sci.1gev.eps){width="77mm"}
The pion and muon momentum distributions are shown in Figs. \[fig\_pmom.sci\] and \[fig\_lmom.sci\]. The upper (lower) end of the pion (muon) momentum distribution is sharply cut off because of the muon momentum cut ($p_\mu >$ 350 MeV).
![\[fig\_lmom.sci\] The muon momentum distribution for $\nu_\mu + {}^{12}{\rm C}_{g.s.} \to \mu^- + \pi^+ + {}^{12}{\rm C}_{g.s.}$ at $E_\nu = 1$ GeV. ](pmom.nu.cc.sci.1gev.eps){width="77mm"}
![\[fig\_lmom.sci\] The muon momentum distribution for $\nu_\mu + {}^{12}{\rm C}_{g.s.} \to \mu^- + \pi^+ + {}^{12}{\rm C}_{g.s.}$ at $E_\nu = 1$ GeV. ](lmom.nu.cc.sci.1gev.eps){width="77mm"}
The muon scattering angle distribution is shown in Fig. \[fig\_theta\_l.sci\].
![\[fig\_theta\_l.sci\] The muon scattering angle distribution for $\nu_\mu + {}^{12}{\rm C}_{g.s.} \to \mu^- + \pi^+ + {}^{12}{\rm C}_{g.s.}$ at $E_\nu = 1$ GeV. ](theta_l.nu.cc.sci.1gev.eps){width="77mm"}
Figures \[fig\_theta\_p.sci\]–\[fig\_theta\_l.sci\] clearly show the characteristics of coherent pion production, i.e., sharply forward scattering (emission) of the muon (pion) with small momentum transfers. Finally, we show in Fig. \[fig\_coplanar.sci\] the spectrum with respect to the coplanar angle difference, $\Delta\phi$, which is defined by $\Delta\phi = \phi_\pi -\pi$, where $\phi_\pi$ is the pion azimuthal angle in the LAB frame. (See Fig. \[fig\_coplanar\] for a graphical representation of $\Delta\phi$.)
![\[fig\_coplanar\] Graphical definition for the coplanar angle difference ($\Delta\phi$). ](coplanar.nu.cc.sci.1gev.eps){width="77mm"}
![\[fig\_coplanar\] Graphical definition for the coplanar angle difference ($\Delta\phi$). ](coplanar.eps){width="77mm"}
Fig. \[fig\_coplanar.sci\] shows slight asymmetry in the $\Delta\phi$ distribution around $\Delta\phi=0$. It is interesting to note that this asymmetry is generated mostly by the contribution from the non-resonant amplitudes. To demonstrate this point, we present in the same figure the results obtained with the non-resonant amplitudes turned off, (dash-dotted curve). We also remark that the asymmetry arises mostly from the kinematical region satisfying $\theta_\pi > 35^\circ$ (see the dotted curve). A similar asymmetry also arises for the anti-neutrino process.
The SciBooNE collaboration have recently presented their preliminary results corresponding to Figs. \[fig\_theta\_p.sci\]–\[fig\_coplanar.sci\] for both of the neutrino and anti-neutrino CC coherent pion production reactions [@hiraide_nuint09; @anti_cc]. When the flux prediction for the SciBooNE experiment becomes available, we will be able to convolute the results of our calculation with the flux and make direct comparison with the data.
Meanwhile, the MiniBooNE collaboration has been investigating the NC process in\
(anti-)neutrino-nucleus scattering, and some results for the neutrino process have been published [@miniboone], and more results are expected to be released. Since the neutrino flux information for the MiniBooNE experiment is available [@miniboone_flux], we can give the theoretical values of relevant observables convoluted with the flux. At present, data are publicly available only for the $\eta$-distribution \[$\eta \equiv E_\pi (1-\cos\theta_\pi)$\], and we compare our calculation for this quantity with the data. In the analysis of the MiniBooNE NC data, the $\eta$-distribution was used to distinguish coherent pion production from other processes contributing to the $\pi^0$-production events. To be more specific, MiniBooNE used the “shape“ of the $\eta$-distribution obtained from the RS model [@RS] with the momentum reweighting function applied. It has been found, however, that a microscopic calculation in Ref. [@amaro] gives an $\eta$-distribution appreciably different from that obtained in the RS model, and the authors of Ref. [@amaro] have pointed out that the MiniBooNE might have substantially overestimated the NC events. Figure \[fig\_eta\_average\] shows the “average” $\eta$-distribution obtained by convoluting the $\eta$-distribution given by our present calculation with the MiniBooNE neutrino flux [@miniboone_flux]. For comparison, the figure also shows the MiniBooNE Monte Carlo results ([*cf.*]{} Fig. 3b of Ref. [@miniboone]), arbitrarily rescaled to match the theoretical curve at $\eta$ = 0.005 GeV. We remark that the $\eta$-distribution we have obtained is fairly close to that given in Ref. [@amaro], because the non-resonant amplitudes do not change the shape of the $\eta$-distribution significantly. Therefore, we arrive at the same conclusion as in Ref. [@amaro] that it is possible that MiniBooNE substantially overestimated the NC events.
To facilitate a comparison of our calculation with data that are expected to be become available soon from MiniBooNE, we present theoretical predictions for some more quantities that are likely to be relevant. Figure \[fig\_pi0\_miniboone\] shows the flux-convoluted $\pi^0$ momentum distribution predicted by our calculation. As far as observables for the anti-neutrino process are concerned, the flux-convoluted $\eta$-distribution resulting from our calculation is given in Fig. \[fig\_bnu\_eta\_average\], and the flux-convoluted $\pi^0$ momentum distribution obtained in our model is shown in Fig. \[fig\_bnu\_pi0\_miniboone\].
![\[fig\_pi0\_miniboone\] The flux-convoluted $\pi^0$ momentum distribution for $\nu + {}^{12}{\rm C}_{g.s.} \to \nu + \pi^0 + {}^{12}{\rm C}_{g.s.}$. The neutrino flux is taken from MiniBooNE [@miniboone_flux]. ](eta_nu.ave.eps){width="77mm"}
![\[fig\_pi0\_miniboone\] The flux-convoluted $\pi^0$ momentum distribution for $\nu + {}^{12}{\rm C}_{g.s.} \to \nu + \pi^0 + {}^{12}{\rm C}_{g.s.}$. The neutrino flux is taken from MiniBooNE [@miniboone_flux]. ](pmom_nu.ave.eps){width="77mm"}
![\[fig\_bnu\_pi0\_miniboone\] The flux-convoluted $\pi^0$ momentum distribution for $\bar{\nu} + {}^{12}{\rm C}_{g.s.} \to \bar{\nu} + \pi^0 + {}^{12}{\rm C}_{g.s.}$. The anti-neutrino flux is taken from MiniBooNE [@miniboone_flux]. ](eta_bnu.ave.eps){width="77mm"}
![\[fig\_bnu\_pi0\_miniboone\] The flux-convoluted $\pi^0$ momentum distribution for $\bar{\nu} + {}^{12}{\rm C}_{g.s.} \to \bar{\nu} + \pi^0 + {}^{12}{\rm C}_{g.s.}$. The anti-neutrino flux is taken from MiniBooNE [@miniboone_flux]. ](pmom_bnu.ave.eps){width="77mm"}
Comparison of Microscopic Models {#sec_comp}
--------------------------------
As mentioned, there are mainly two different theoretical approaches to coherent pion production in neutrino-nucleus scattering; a PCAC-based model and a microscopic model. The relation between the RS model (a PCAC-based model) and a microscopic model has been discussed in great detail in Ref. [@amaro], and comparison of those two models, including some improvement of the RS model, has been made in Refs. [@amaro; @hernandez]. The authors of Refs. [@amaro; @hernandez] have emphasized that it can be problematic to use the RS model for $E_\nu$2 GeV. To shed some more light on this issue, we consider it useful to make comparison of different microscopic models. In particular, we focus here on comparison between our model and the model of Amaro et al. [@amaro], which is the most sophisticated among the existing microscopic models for neutrino-induced coherent pion production. [^11] The other microscopic calculations in the literature lack one or more aspects that are obviously important, such as the distortion of the final pion and the non-resonant mechanism for the weak currents.
Here, we particularly focus on the elementary amplitudes for pion production off the nucleon. Our approach employs the SL model while Amaro et al. [@amaro] used a model developed in Ref. [@hnv] (to be referred to as HNV). Both SL and HNV include the resonant and non-resonant amplitudes. A point to be noted, however, is that, although both models reproduce reasonably well the data for the $\nu_\mu + N \to \mu^- + \pi^+ + N$ reactions after an appropriate adjustment of the axial-$N\Delta$ coupling, the two models involve rather different reaction mechanisms. In the SL model, we derive a set of tree diagrams from a given Lagrangian with the use of a unitary transformation, and then we embed these tree diagrams in the Lippmann-Schwinger equation, which is solved exactly to yield a non-perturbative pion production amplitude that satisfies $\pi$-$N$ two-body unitarity. In HNV, on the other hand, a set of tree diagrams are calculated from a chiral Lagrangian. Then the sum of the contributions of these tree diagrams is identified with the pion production amplitude. At the tree level, the SL and the HNV models have essentially the same non-resonant mechanisms; a contact vertex in HNV may be interpreted as the vector meson exchange mechanism in SL. However the role of the non-resonant amplitude appears differently in the two models. In the SL model, non-resonant amplitude contributes constructively (destructively) to the resonant amplitudes below (above) the resonance energy. For $\nu_\mu + p \to \mu^- + \pi^+ + p$, the interference of the non-resonant amplitude with the resonant amplitude changes in the SL model the total cross sections by a factor of 1.5, 1.02, 0.96 at $E_\nu =$ 0.5, 1, 1.5 GeV [^12], while the interference in the HNV always enhances the total cross sections; e.g., enhancement of a factor of 1.1 at $E_\nu =$ 1.5 GeV. The difference of the non-resonant mechanism appears also in the coherent pion production on ${}^{12}$C, where only the spin and isospin non-flip amplitude contributes. Whereas the non-resonant amplitude plays an important role in our model (as seen in Figs. \[fig\_tpi.1gev\] and \[fig\_tpi.0p5gev\]), it plays essentially no role in the HNV model. In the neutrino CC coherent pion production, the full (tree) non-resonant amplitude increases the total cross section by 36% (19%) at $E_\nu$ = 0.5 GeV and 18% (0.4%) at $E_\nu$ = 1 GeV in our model. Thus the non-resonant mechanism in the spin-isospin non-flip amplitude is enhanced by the rescattering process. In the SL model, the non-resonant and resonant $\pi N$ dynamics in the $\Delta$ resonance region has been tested using the extensive data of $(\gamma,\pi)$ and $(e,e^\prime\pi)$ reactions. Although the SL model, which provides a unified description of the electroweak pion production reactions, describes very well the available data of the $(\nu,\ell\pi)$ processes, the current data do not yet allow to test the details of the reaction mechanism.
Furthermore, utilizing the consistency of $(\nu,\ell\pi)$, $(e,e^\prime\pi)$ and $(\pi,\pi)$ reactions in the SL model, we have developed a model which treats photo- and neutrino-induced coherent pion production processes in a unified manner. Thus we were able to calibrate the reliability of our approach with data for the photo-processes, which is an aspect specific to our approach.
Conclusions {#sec_conclusion}
===========
We have developed a microscopic dynamical model for describing neutrino-induced coherent pion production on nuclei. Because experimental data for neutrino (both elementary and nuclear) processes are rather limited, it is not straightforward to assess the reliability of theoretical calculations. A reasonable strategy to take seems to develop a model which describes strong and electroweak processes in a unified way, and then to test the model extensively by comparing with a large collection of data for the strong-interaction and photo-induced processes and with limited available data for weak processes. We have carried out this program here for the case of the neutrino-induced coherent pion production process. By virtue of the mentioned strategy, our model is probably the most extensively tested one among the existing models for this process. To achieve the stated goal, we need a theoretical framework that provides a unified description for the elementary $(\pi,\pi')$, $(\gamma,\pi)$ and $(\nu,\ell\pi)$ processes on a single nucleon. We have adopted the SL model, which is known to give satisfactory descriptions of these elementary amplitudes. We then have combined the SL model with the $\Delta$-hole model to construct a theoretical framework that can describe in a unified way pion-nucleus scattering and electroweak coherent pion production. The unified nature of this approach allows us to fix free parameters in the model using the data for pion-nucleus scattering, which in turn enables us to make parameter-free predictions on electroweak coherent pion production off a nucleus. Another benefit of the present unified approach is that we can assess the reliability of our model by comparing the results for coherent pion photo-production with data. Our model is found to describe reasonably well both pion-nucleus scattering and coherent photo-processes, which establishes a basis for applying the same model to the neutrino-induced processes.
Comparing our numerical results with the recent data on neutrino-induced coherent pion production, we have found that the result for the CC process is consistent with the upper limit from K2K[@hasegawa], and that the result for the NC process is somewhat smaller than the preliminary experimental value from MiniBooNE[@raaf]. However, as discussed in the literature, MiniBooNE’s analysis may have overestimated the cross section due to the use of the RS model in their analysis. We have examined to what extent the various aspects of physics involved in our model individually affect the cross sections. We have shown that the medium effect on the $\Delta$ (the spreading potential effect in particular) and the FSI change the cross sections significantly. It is to be noted, however, that these rather drastic changes in the cross sections due to the medium effects are well under control because: (i) the spreading potential and the pion distorted wave function have been fitted to and tested by the empirical total and elastic cross sections for pion-nucleus scattering in and around the $\Delta$ region; (ii) the medium effects of a similar magnitude for the photo-process have been shown to bring our calculation into good agreement with the data. An interesting feature of our model is that the unitarized non-resonant amplitudes give a significant contribution to the cross sections. This is in sharp contrast with the results of the previous calculations; for instance, the calculations in Refs. [@amaro; @valencia2], which considered a tree-level non-resonant mechanism, found almost no contribution from it. It is worth emphasizing that this noticeable difference should not be taken as a measure of uncontrollable model dependence because (as we confirmed) the difference arises largely from unitarization of the non-resonant amplitude, which clearly needs to be implemented.
We have reexamined the non-local effect in $\Delta$-propagation in nuclei. It was emphasized in Ref. [@non-local] that this non-local effect, despite its large size, was not considered explicitly in any of the existing models for neutrino-induced coherent pion production (whether based on a microscopic model or the RS model). The authors of Ref. [@non-local] made this remark based on their calculation that only included the $\Delta$ mechanism. Our present calculation, which additionally incorporates the spreading potential and FSI, also indicates that the non-locality gives a large effect. Thus, regardless of the level of sophistication in the treatment of medium effects, one should always include the non-locality effect explicitly.
Because it is expected that the SciBooNE and the MiniBooNE collaborations will report more detailed data on (anti)neutrino-induced coherent CC and NC pion productions, we have presented numerical results relevant to these experiments.
Finally, we made a comparison of the elementary amplitude (HNV[@hnv]) used by Amaro et al.[@amaro] and ours (SL [@SL; @SUL]) to clarify similarities and differences between them. The noteworthy points are: (i) At tree-level, both SL and HNV have essentially the same non-resonant mechanism; (ii) In the SL model, a unitary pion-production amplitude is obtained by solving the Lippmann-Schwinger equation in which the tree-diagrams are embedded, whereas, in the HNV model, the sum of the tree-diagrams are identified with the pion-production amplitude; (iii) The non-resonant amplitudes of SL and HNV work differently both for the elementary processes (e.g., $\nu_\mu+p\to \mu^-+\pi^++p$), and for coherent pion production; (iv) In SL, the rescattering contribution contained in the non-resonant amplitude considerably enhances the cross section for coherent pion production.
S. X. N. acknowledges informative discussions with Hidekazu Tanaka and Hirohisa Tanaka about the SciBooNE and MiniBooNE experiments. S. X. N. also thanks Akira Konaka and Issei Kato for stimulating discussions. This work is supported by the Natural Sciences and Engineering Research Council of Canada and Universidade de São Paulo (SXN), by the U.S. Department of Energy, Office of Nuclear Physics, under contract DE-AC02-06CH11357 (TSHL), by the Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research(C) 20540270 (TS), and by the U.S. National Science Foundation under contract PHY-0758114 (KK).
multipole amplitudes {#app_mutlipole}
====================
The amplitudes $F_i^V, F_i^A$ in [Eqs. (\[eq:fvec\]) and (\[eq:faxi\])]{} are expressed in terms of multipole amplitudes $E_{l\pm}^{V,A},M_{l\pm}^{V,A},
S_{l\pm}^{V,A}$ and $L_{l\pm}^A$ as $$\begin{aligned}
F_1^V = \sum_l[
P_{l+1}' E_{l+}^V + P_{l-1}' E_{l-}^V
+ lP_{l+1}'M_{l+}^V + (l+1)P_{l-1}' M_{l-}^V]\,,\\
F_2^V = \sum_l[
(l+1)P_l'M_{l+}^V + lP_l' M_{l-}^V]\,,\\
F_3^V = \sum_l[
P_{l+1}'' E_{l+}^V + P_{l-1}'' E_{l-}^V
- P_{l+1}'' M_{l+}^V + P_{l-1}'' M_{l-}^V]\,,\\
F_4^V = \sum_l[
- P_{l}'' E_{l+}^V - P_{l}'' E_{l-}^V
+ P_{l}'' M_{l+}^V - P_{l}'' M_{l-}^V]\,,\\
F_5^V = \sum_l[
(l+1) P_{l+1}' L_{l+}^V - l P_{l-1}' L_{l-}^V]\,, \\
F_6^V = \sum_l[
-(l+1) P_{l}' L_{l+}^V + l P_{l}' L_{l-}^V]\,, \\
F_7^V = \sum_l[
-(l+1) P_{l}' S_{l+}^V + l P_{l}' S_{l-}^V]\,, \\
F_8^V = \sum_l[
(l+1) P_{l+1}' S_{l+}^V - l P_{l-1}' S_{l-}^V]\,,\end{aligned}$$ and $$\begin{aligned}
F_1^A = \sum_l[
P_{l}' E_{l+}^A + P_{l}' E_{l-}^A
+ (l+2)P_{l}'M_{l+}^A + (l-1)P_{l}' M_{l-}^A]\,,\\
F_2^A = \sum_l[
(l+1)P_{l+1}'M_{l+}^A + lP_{l-1}' M_{l-}^A]\,,\\
F_3^A = \sum_l[
P_{l}'' E_{l+}^A + P_{l}'' E_{l-}^A
+ P_{l}'' M_{l+}^A - P_{l}'' M_{l-}^A]\,,\\
F_4^A = \sum_l[
- P_{l+1}'' E_{l+}^A - P_{l-1}'' E_{l-}^A
- P_{l+1}'' M_{l+}^A + P_{l-1}'' M_{l-}^A]\,,\\
F_5^A = \sum_l[
-(l+1) P_{l}' L_{l+}^A + l P_{l}' L_{l-}^A] \,,\\
F_6^A = \sum_l[
(l+1) P_{l+1}' L_{l+}^A - l P_{l-1}' L_{l-}^A]\,, \\
F_7^A = \sum_l[
(l+1) P_{l+1}' S_{l+}^A - l P_{l-1}' S_{l-}^A] \,,\\
F_8^A = \sum_l[
-(l+1) P_{l}' S_{l+}^A + l P_{l}' S_{l-}^A] .\end{aligned}$$ $P_L(x)$ is the Legendre function and $x=\hat{k}\cdot\hat{q}$; $\bm{k}$ and $\bm{q}$ are the pion momentum and the momentum transfer to the nucleon, respectively.
The multipole amplitudes from isovector currents are further decomposed according to the total isospin ($T$) in the final $\pi N$ state as $$\begin{aligned}
{\label{eq:amp_iso}}
X_{l\pm}^{V,A} = \sum_{T=1/2,3/2} X_{l\pm}^{(T)V,A}\Lambda^{T}_{ij}\ ,\end{aligned}$$ with $X$ being $E$, $M$, $L$ or $S$. We have introduced the projection operator $\Lambda^{T}_{ij}$ defined by $$\begin{aligned}
\Lambda^{3/2}_{ij} = {2 \delta_{i,j}-i\epsilon_{ijk}\tau_k \over 3}\\
\Lambda^{1/2}_{ij} = {\delta_{i,j}+i\epsilon_{ijk}\tau_k \over 3} \ ,\end{aligned}$$ where the indexes $i$ and $j$ refer to the final pion isospin state and the component of the isovector current, respectively. For electromagnetic or NC processes, $M_{l\pm}^{(0)V}\tau_i$, which is due to an isoscalar current, is also added to [Eq. [(\[eq:\]]{})]{}[amp\_iso]{}.
In the main text we sometimes use the notation $X_{l\pm}^{V(A),\zeta}$, where $\zeta$ collectively denotes the pion charge and the nucleon isospin state; $X_{l\pm}^{V(A),\zeta}$ is a matrix element (in isospin space) of [Eq. [(\[eq:\]]{})]{}[amp\_iso]{}. Since we are only concerned with coherent pion production, the specification of the pion charge determines $i$ and $j$ in [Eq. [(\[eq:\]]{})]{}[amp\_iso]{}. We can find the matrix element (in isospin space) of [Eq. [(\[eq:\]]{})]{}[amp\_iso]{} by specifying the nucleon isospin state.
Lorentz transformation from ACM to 2CM {#app_2cm}
======================================
In coherent pion production in neutrino-nucleus scattering ($\nu_\ell + t \to \ell^- + \pi^+ + t$), the elementary process is $W^+ (q_A) + N (p_N) \to \pi^+(k_A) + N(p_N^\prime)$, where the four-momenta in the pion-nucleus center-of-mass frame (ACM) are given in the parentheses. We suppose here that the pion momentum is on-shell. In a prescription we employ, the nucleon momenta are fixed as $$\begin{aligned}
{\label{eq:p_fix2}}
\bm{p}_N = - {\bm{q}_A\over A} - {A-1\over 2A}(\bm{q}_A-\bm{k}_A)\ ,
\qquad
\bm{p}_N^\prime = - {\bm{k}_A\over A} + {A-1\over 2A}(\bm{q}_A-\bm{k}_A)
\ ,\end{aligned}$$ and the invariant mass ($W$) of the pion and nucleon is $$\begin{aligned}
{\label{eq:w_app}}
W = \sqrt{(p_N^0+q_A^0)^2 - (\bm{p}_N+\bm{q}_A)^2}\ ,\end{aligned}$$ where $p^0_N$ is the nucleon energy on the mass-shell. We note that $W$ depends on $x_A (\equiv \hat{k}_A\cdot\hat{q}_A)$ as well as $|\bm{q}_A|$ and $|\bm{k}_A|$. For convenience, we write $W(|\bm{q}_A|,|\bm{k}_A|,x_A)$.
We perform the standard Lorentz transformation from ACM to the $\pi N$ CM frame (2CM). An arbitrary four-momentum in 2CM ($p_2$) is written with the corresponding four-momentum in ACM ($p_A$) as $$\begin{aligned}
{\label{eq:lorentz}}
\bm{p}_2 &=& \bm{p}_A - {p_A^0\over W} \bm{P} + {P^0-W\over W}
(\bm{p}_A\cdot\hat{P})\hat{P} \ , \\\nonumber
p^0_2 &=& {P^0 p_A^0 - \bm{p}_A\cdot\bm{P}\over W} \ ,\end{aligned}$$ with $P = p_N + q_A$.
We now consider a case in which the pion momentum is off-shell ($k_A^\prime$). We encounter this situation when we consider the final-state interaction in the coherent process. As before, the nucleon momenta are fixed using [Eq. [(\[eq:\]]{})]{}[p\_fix2]{} with $k_A$ replaced by $k_A^\prime$. However, we do not use the nucleon energy on the mass-shell. Instead, we take $p_N^0$ so that $$\begin{aligned}
W(|\bm{q}_A|,|\bm{k}_A^\prime|,x^\prime_A)=W(|\bm{q}_A|,|\bm{k}_A|,x_A)
\qquad {\rm for}\ x^\prime_A=x_A\ , \end{aligned}$$ where $W$ is obtained with [Eq. [(\[eq:\]]{})]{}[w\_app]{}. With the nucleon four-momentum ($p_N$) obtained in this way, we can perform the Lorentz transformation as [Eq. [(\[eq:\]]{})]{}[lorentz]{}. This prescription greatly reduces the amount of labor involved in our numerical calculation, because the SL amplitudes need to be calculated at each value of $W$. With the variables obtained above, we can calculate $\Gamma_{2AL}$ used in [Eqs. (\[eq:i\_mtx\]) and (\[eq:j\_mtx\])]{}: $$\begin{aligned}
{\label{eq:gam_4}}
\Gamma_{2AL} = \sqrt{\omega^\prime_{\pi,2} p^{\prime\,0}_{N,2}p^{0}_{N,2}
\over \omega^\prime_{\pi,A}p^{\prime\,0}_{N,L}p^{0}_{N,L}} \ ,\end{aligned}$$ with $\omega^\prime_{\pi,A}=\sqrt{\bm{k}_A^\prime+m_\pi^2}$.
Finally, we discuss the factor $\Gamma^\chi$,used in [Eqs. (\[eq:i\_mtx\]) and (\[eq:j\_mtx\])]{}, which originates from the pion wave function due to the Lorentz transformation. Among the final-state interactions, the simplest process is the scattering of the pion off a single nucleon $\pi(k_A^\prime) + N(p_N^{\prime\prime}) \to \pi(k_A) + N(p_N^f)$, where the variables in ACM are shown in the parentheses; only $k_A$ is on-shell. Similarly to [Eq. [(\[eq:\]]{})]{}[p\_fix2]{}, we fix the nucleon momenta as $$\begin{aligned}
\bm{p}_N^{\prime\prime} = - {\bm{k}^\prime_A\over A} - {A-1\over 2A}(\bm{k}^\prime_A-\bm{k}_A)\ ,
\qquad
\bm{p}_N^f = - {\bm{k}_A\over A} + {A-1\over
2A}(\bm{k}^\prime_A-\bm{k}_A) \ .\end{aligned}$$ We assume here that the energies of all the nucleons are on the mass-shell. For the Lorentz transformation from ACM to LAB specified this way, we can calculate the Lorentz factor as $$\begin{aligned}
{\label{eq:gam_5}}
\Gamma^\chi = \sqrt{\omega_{\pi,A}E^{\prime\prime}_{N,A}E^{f}_{N,A}
\over \omega_{\pi,L}E^{\prime\prime}_{N,L}E^{f}_{N,L}}
\simeq \sqrt{\omega_{\pi,A}\over \omega_{\pi,L}} \ ,\end{aligned}$$ Although the actual final-state interaction includes multiple scattering processes, it is beyond our framework to calculate $\Gamma^\chi$ with multiple scattering taken into account. We therefore use $\Gamma^\chi$ calculated for the elementary process in [Eqs. (\[eq:i\_mtx\]) and (\[eq:j\_mtx\])]{}. Actually, the Lorentz factor for the plane wave term in [Eq. [(\[eq:\]]{})]{}[pi\_wave]{} is given by the the rightmost expression in [Eq. [(\[eq:\]]{})]{}[gam\_5]{}. Because the approximate equality in [Eq. [(\[eq:\]]{})]{}[gam\_5]{} is quite accurate for $\bm{k}^\prime_A=\bm{k}_A$, we use the middle expression in [Eq. [(\[eq:\]]{})]{}[gam\_5]{} to evaluate the matrix elements in [Eqs. (\[eq:i\_mtx\]) and (\[eq:j\_mtx\])]{}.
Expressions for some components in the $\Delta$ propagator {#app_misc}
==========================================================
Pauli correction to the $\Delta$ self energy
--------------------------------------------
We follow Ref. [@pauli] to calculate the Pauli correction to the $\Delta$ self energy ($\Sigma_{\rm Pauli}$). The $\pi N\Delta$ coupling is from the SL model. $$\begin{aligned}
\Sigma_{\rm Pauli} &=& {m_N\over W}
\left[2\theta(k_F-\beta)\int_0^{k_F-\beta} dq q^2
{\omega_\pi(q) F^{\rm bare}_{\pi N\Delta}(q) F_{\pi N\Delta}(q)\over
K^2 - q^2 + i\epsilon} \right.\\\nonumber
&+&\left.\int_{|k_F-\beta|}^{k_F+\beta} dq q^2
\left(1 - {q^2+\beta^2-k_F^2\over 2q\beta}\right)
{\omega_\pi(q) F^{\rm bare}_{\pi N\Delta}(q) F_{\pi N\Delta}(q)\over
K^2 - q^2 + i\epsilon}
\right] \ ,\end{aligned}$$ where $\theta(x)$ is the step function, $k_F$ is the Fermi momentum \[[Eq. [(\[eq:\]]{})]{}[fermi\_mom]{}\], $W$ is the $\pi N$ invariant mass \[[Eq. [(\[eq:\]]{})]{}[inv\_mass]{}\], $\omega_\pi(q)=\sqrt{q^2+m_\pi^2}$, and $$\begin{aligned}
K^2 &=& {m_N\over W} \left[(W-m_N)^2 - m_\pi^2\right] \ .\end{aligned}$$ Furthermore, for electroweak pion production amplitude \[[Eq. [(\[eq:\]]{})]{}[i\_mtx]{}\], $$\begin{aligned}
\bm{\beta} &=& {m_N\over W} (\bm{p}_N + \bm{q}_A) \ ,\end{aligned}$$ where $\bm{p}_N$ is fixed using [Eq. [(\[eq:\]]{})]{}[p\_fix]{}, and $\bm{q}_A$ is the momentum transfer to a nucleus in ACM; for the optical potential \[[Eq. [(\[eq:\]]{})]{}[v\_res]{}\], $\bm{q}_A$ is replaced with $\bm{k}_A$ (the incoming pion momentum). We use the on-shell pion momentum to fix $\bm{p}_N$. The dressed $\pi N\Delta$ vertex ($F_{\pi N\Delta}$) is taken from [Eq. [(\[eq:\]]{})]{}[t-res]{}, and the bare $\pi N\Delta$ vertex denoted by $F^{\rm bare}_{\pi N\Delta}$ is given as [@SL] $$\begin{aligned}
F^{\rm bare}_{\pi N\Delta}(q) = -i {f_{\pi N\Delta}\over m_\pi}
\sqrt{E_N(q)+m_N\over 24\pi^2E_N(q)\omega_\pi(q)}
\left(\Lambda^2_{\pi N\Delta}\over \Lambda^2_{\pi N\Delta} + q^2
\right)^2 \!q \ .\end{aligned}$$
$\Delta$ spreading potential
----------------------------
We consider the following spreading potential consisting of the central and the LS parts: $$\begin{aligned}
{\label{eq:spr}}
\Sigma_{\rm spr} &=& V_C {\rho_t(r)\over \rho_t(0)}
+ V_{LS} f_{LS}(r) 2 \bm{L}_\Delta\cdot\bm{\Sigma}_\Delta \ , \\
f_{LS}(r) &=& \mu r^2 e^{-\mu r^2} \ ,\end{aligned}$$ with $\mu=0.3$ fm$^{-2}$. We have two complex coupling constants $V_C$ and $V_{LS}$ which are fitted to pion-nucleus scattering data. The radial dependence of the LS spreading potential is taken from Ref. [@LS_spr]. We implement the spreading potential \[[Eq. [(\[eq:\]]{})]{}[spr]{}\] in the $\Delta$-propagator after evaluating the doorway state expectation value of the LS term. Thus, the LS term provides an L-dependent shift of the resonance mass and width as[@LS_spr] $$\begin{aligned}
\Sigma_{LS}^L = -5 V_{LS}{\bra{\phi_L}\rho_t f_{LS}k^2 -
\left(\rho_tf_{LS}\right)^\prime {d\over dr}
+ {L(L+1)\over 2r}\left(\rho_tf_{LS}\right)^\prime\ket{\phi_L}
\over
\bra{\phi_L}\rho_t k^2 -
(\rho_t)^\prime {d\over dr}
\ket{\phi_L}} \ ,\end{aligned}$$ with the plane wave pion function $\phi_L(r)=j_L(kr)$.
$\Delta$ (nucleon) potential
----------------------------
$$\begin{aligned}
{\label{eq:nucl_potential}}
V_{\Delta}(r)= V(r) = (-55\, {\rm MeV}) \left( {\rho_t(r)\over\rho_t(0)}\right) \ .\end{aligned}$$
$\Delta$ Coulomb potential
--------------------------
$$\begin{aligned}
&&\hspace{7mm} (r \ge R_e)
\hspace{30mm} (r < R_e) \\[5mm]\nonumber
V^C_{\Delta}(r)&=&
\left\{ \begin{array}{ccl}
{\displaystyle 2 (Z-1) \alpha \over\displaystyle r} \ , \qquad
& - {\displaystyle (Z-1)\alpha r^2 \over\displaystyle R_e^3}
+ {\displaystyle 3 (Z-1)\alpha \over\displaystyle R_e}
\ , \qquad
&(\pi^++p\to\Delta^{++})\\[5mm]
{\displaystyle Z \alpha \over\displaystyle r} \ , \qquad
&- {\displaystyle Z\alpha r^2 \over\displaystyle 2R_e^3}
+ {\displaystyle 3 Z\alpha \over\displaystyle 2R_e}
\ , \qquad
&(\pi^++n\to\Delta^{+})\\[5mm]
0 \ , \qquad
&
0 \ , \qquad
&(\pi^-+p\to\Delta^{0})\\[5mm]
-{\displaystyle Z \alpha \over\displaystyle r} \ , \qquad
& {\displaystyle Z\alpha r^2 \over\displaystyle 2R_e^3}
- {\displaystyle 3 Z\alpha \over\displaystyle 2R_e}
\ , \qquad
&(\pi^-+n\to\Delta^{-})
\end{array}
\right.\end{aligned}$$
In the above $Z$ is the atomic number. The equivalent square well radius, denoted by $R_e$, is related to the mean square radius ($\langle r^2 \rangle$) of a nucleus by $$\begin{aligned}
R_e = \sqrt{{5\over 3}\langle r^2 \rangle} \ .\end{aligned}$$
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[^1]: Current affiliation: Excited Baryon Analysis Center (EBAC), Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
[^2]: Unfortunately, conclusive data for the elementary neutrino process are still lacking, which leads to theoretical uncertainty.
[^3]: In Ref. [@koch_moniz], the authors carried out a calculation for photon-induced coherent pion production by diagonalizing $G_{\Delta h}$.
[^4]: PIPIT also includes a finite-range Coulomb interaction, and corrections from the truncated part of the Coulomb interaction are taken into account using the Vincent-Phatak method [@vincent].
[^5]: Hereafter, we include the same set of partial waves ($l_\pi$) in the amplitudes for both pion-nucleus scattering and pion production off a nucleus. For the non-resonant elementary pion production amplitudes, we include the partial waves up to $\ell\le 4$ in [Eq. [(\[eq:\]]{})]{}[j\_mtx]{}.
[^6]: \[foot:medium\] The “medium effects on the $\Delta$” here refer to the combined effects of the Pauli blocking of $\Delta$-decay ($\Sigma_{\rm pauli}$), the spreading potential ($\Sigma_{\rm spr}$), and the terms in the square bracket in [Eq. [(\[eq:\]]{})]{}[delta\_k2]{}.
[^7]: \[footnote:non-res\] In the SL model, the resonant amplitude itself contains the non-resonant mechanism. We refer to the purely non-resonant amplitudes as “non-resonant amplitudes”, and it is only these non-resonant amplitudes that we turn off here and later in Figs. \[fig\_tpi.1gev\]-\[fig\_q2.nc\] and \[fig\_non\_local\].
[^8]: We come back to the non-local effect due to the $\Delta$ kinetic term later when we discuss the neutrino-induced processes.
[^9]: The contributions from the incoherent processes are larger than they appear in Fig. \[fig\_krusche\] because $\sin\theta_\pi$ needs to be multiplied in integrating over $\theta_\pi$.
[^10]: As discussed earlier, the neutrino and anti-neutrino cross sections differ only slightly.
[^11]: A rather extensive comparison of numerical results from various calculations for the neutrino-induced coherent pion production, including those of Amaro et al. [@amaro], recent PCAC-based models [@paschos; @bs_pcac] and ours, has been presented at NuInt09 by Boyd et al. [@nuint09].
[^12]: See footnote \[footnote:non-res\].
|
---
abstract: |
**Background:** 50% of the heavy element abundances are produced via slow neutron capture reactions in different stellar scenarios. The underlying nucleosynthesis models need the input of neutron capture cross sections.\
**Purpose:** One of the fundamental signatures for active nucleosynthesis in our galaxy is the observation of long-lived radioactive isotopes, such as $^{60}$Fe with a half-life of $2.60\times10^6$yr. To reproduce this $\gamma$-activity in the universe, the nucleosynthesis of $^{60}$Fe has to be understood reliably.\
**Methods:** A $^{60}$Fe sample produced at the Paul-Scherrer-Institut was activated with thermal and epithermal neutrons at the research reactor at the Johannes Gutenberg-Universität Mainz.\
**Results:** The thermal neutron capture cross section has been measured for the first time to $\sigma_{\text{th}}\,=\,0.226 \ (^{+0.044}_{-0.049}) \ \text{b} $. An upper limit of $\sigma_{\text{RI}} < 0.50\ \text{b}$ could be determined for the resonance integral.\
**Conclusions:** An extrapolation towards the astrophysicaly interesting energy regime between and 100keV illustrates that the s-wave part of the direct capture component can be neglected.
author:
- 'T. Heftrich'
- 'M. Bichler'
- 'R. Dressler'
- 'K. Eberhardt'
- 'A. Endres'
- 'J. Glorius'
- 'K. Göbel'
- 'G. Hampel'
- 'M. Heftrich'
- 'F. Käppeler'
- 'C. Lederer'
- 'M. Mikorski'
- 'R. Plag'
- 'R. Reifarth'
- 'C. Stieghorst'
- 'S. Schmidt'
- 'D. Schumann'
- 'Z. Slavkovská'
- 'K. Sonnabend'
- 'A. Wallner'
- 'M. Weigand'
- 'N. Wiehl'
- 'S. Zauner'
title: 'The thermal neutron capture cross section of the radioactive isotope $^{60}$Fe'
---
Introduction
============
The decays of the unstable isotopes $^{60}$Fe and $^{26}$Al in the Milky Way, which have been observed with satellite-based $\gamma$-ray telescopes [@Smi03; @HKJ05], are considered as a clear signature of ongoing stellar nucleosynthesis [@TWH95].
The production of $^{60}$Fe in the slow neutron capture process ($s$-process) [@TWH95] is hampered by the rather short-lived precursor $^{59}$Fe ($t_{1/2}\,=\,44.495$d [@Bag02]), which acts as a branch point of the $s$-process path as illustrated in Fig. \[path\]. Accordingly, high neutron densities are required to avoid that the reaction flow is bypasses $^{60}$Fe via the decay of $^{59}$Fe. Once $^{60}$Fe is reached, it can also be destroyed by neutron capture or - on longer time scales - by $\beta^{-}$-decay. High neutron densities are generally accompanied by very high temperatures, but the synthesis of $^{60}$Fe requires an upper limit of about ($T_{9}$ = 2), because photodisintegration reactions such as and $^{59}$Fe($\gamma, \text{n}$) start to dominate otherwise.
![(Color online) The $s$-process reaction path between Fe and Ni. The isotope $^{60}$Fe is produced via a sequence of (n,$\gamma$) reactions starting at the stable iron isotopes. Because of the short half-life of $^{59}$Fe ($t_{1/2}~\approx~45$ d), the production of $^{60}$Fe depends critically on the stellar neutron density.[]{data-label="path"}](56feTO62ni.pdf){width="49.00000%"}
{width="99.00000%"}
There are two different astrophysical scenarios where $^{60}$Fe can be produced [@LCh06]: during the He-shell burning phase in low-mass thermally pulsing asymptotic giant branch (AGB) stars and during the convective burning in massive pre-supernova stars. In AGB stars, neutron densities of and temperatures around ($T_8=~2.5$) are reached, whereas in massive stars neutron densities of at temperatures of up to $T_8 = 10$ during C-shell burning are reached [@RLK14]. According to detailed stellar model calculations by Limongi and Chieffi [@LCh06], about 65% of the total yield of $^{60}$Fe are in fact synthesized in the pre-supernova stage of massive stars and 18% are contributed by the He burning shell of less massive stars. A third major component is eventually produced by explosive shell burning during the supernova itself. These contributions to the total $^{60}$Fe yield are strongly affected by the respective masses and metallicities of the stars involved and may vary correspondingly.
A crucial input for the production of $^{60}$Fe in AGB stars and massive pre-supernova stars are the neutron capture cross sections at the respective stellar temperatures. So far, an activation measurement of the $^{60}$Fe($\text{n}, \gamma$)$^{61}$Fe cross section at neutron energies corresponding to a thermal energy of $kT$ = 25keV (typical for AGB stars) was performed at Forschungszentrum Karlsruhe, Germany. The Maxwellian averaged cross section (MACS) at $kT$ = 30keV was determined to ($5.15\pm1.4$) mb [@URS09]. The direct capture (DC) component of the cross section at this temperature constitutes an important information for the extrapolation towards the astrophysically interesting temperatures in massive stars around $kT$ = 90keV. In this respect, the thermal cross section provides a constraint for the s-wave component of the DC cross section. Therefore, the previously unknown thermal cross section of $^{60}$Fe was measured using the irradiation facility at the TRIGA (Training, Research, Isotopes, General Atomic) type research reactor at Johannes Gutenberg-Universität Mainz, Germany [@EbK00; @HET06].
Experiment
==========
The $^{60}$Fe sample was produced at the Paul-Scherrer-Insitut (PSI) in Villigen, Switzerland [@SND10]. In order to compensate for the limited amount of $^{60}$Fe the only possible method for the determination of the neutron capture cross section was an integral activation measurement at high neutron fluxes. Compared to the more generally applicable time-of-flight technique, the activation method has the advantage of excellent sensitivity [@RLK14], which allows neutron capture measurements even on very small samples [@RAH03; @ReK02]. This technique has the additional advantage that it does not require isotopically enriched samples, because the capture reactions can be identified via the $\gamma$-decay characteristics
Isotope $t_{1/2}$ E$_\gamma$ /keV I$_\gamma$ /% Reference $\epsilon_{\gamma}$
----------- ---------------------- --------------------- -------------------- ----------- -----------------------
$^{60}$Co ($1925.28\pm0.14$) d $1173.228\pm 0.003$ $99.85\pm0.03$ [@Bro13] $0.024\pm 0.0002^{a}$
$1332.492\pm 0.004$ $99.9826\pm0.0006$ $0.022\pm 0.0002^{a}$
$^{97}$Zr ($16.749\pm0.008$) h $743.36\pm 0.03$ $93.09\pm 0.16$ [@Nic10] $0.069\pm 0.002^{b}$
$^{95}$Zr ($64.032\pm0.006$) d $724.195\pm 0.004$ $44.27\pm 0.22$ [@BMS10] $0.070\pm 0.001^{b}$
$756.728\pm 0.012$ $54.38\pm 0.22$ $0.068\pm 0.002^{b}$
$^{61}$Fe ($5.98\pm0.06$) min $297.90\pm 0.07$ $22.24\pm2.88$ [@Bha99] $0.087\pm 0.002^{b}$
$1027.42\pm 0.11$ $42.73\pm4.92$ $0.056\pm 0.002^{b}$
$1205.07\pm 0.12$ $43.60\pm4.50$ $0.051\pm 0.001^{b}$
\[CalStandard\] $ ^{a}$ measured with a HPGe detector at Goethe-Universität Frankfurt (used for 60Fe determination).\
$^{b}$ measured with a different HPGe detector at the research reactor at Johannes Gutenberg-Universität Mainz (used for $^{61}$Fe and Zirconium determination)
Measurements and calibration
----------------------------
The induced activities were measured using a HPGe detector (CANBERRA-GX7020) with a relative efficiency of 72.3%. The output signals from the preamplifier were converted with a flash-ADC (CAEN module V1724). The dead time corrections were determined using a $^{137}$Cs sample, which was placed at a fixed distant position during all activity measurements. The corresponding corrections were negligibly small. Because of the contamination of $^{55}$Fe in the $^{60}$Fe sample, the activity of $^{55}$Fe was suppressed by a lead foil 1mm in thickness.
The efficiency was determined by a calibrated solution containing the standard single- or double-line $\gamma$-ray emitters $^{60}$Co, $^{85}$Sr, $^{88}$Y, $^{113}$Sn, $^{137}$Cs, $^{139}$Ce, and $^{203}$Hg. The uncertainty of the $\gamma$-emission rate was given with 3% (2$\sigma$). This multi-nuclide solution was absorbed in a pure graphite disc 6mm in diameter and 1mm in thickness to match the properties of the $^{60}$Fe sample used in the measurement (see below). For all $\gamma$-activity measurements, the samples were placed 7.4mm in front of the Ge crystal. Because of the small distance between sample and detector, cascade corrections were necessary for the decays of $^{60}$Co and $^{89}$Y. Those corrections were based on the simulations performed using the [@GEA93; @DPA04]. The corresponding correction for the emission line of $^{60}$Co at the energy of was 30%, at 31% and for $^{88}$Y at the energy of and 27% and 29%, respectively. As shown in Fig. \[xleff\] the measured efficiencies could be reproduced within the experimental uncertainties of $\pm3.5$% over the energy range from 150keV to 1900keV by the expression $$\epsilon_{\gamma}=A\ {\rm exp}{\Large [}-B\ \ln \{ E_{\gamma}-C+D\ {\rm exp}({F\times E_{\gamma}) \} {\Large ]}}.$$
![(Color online) The activity of the $^{60}$Fe sample was determined by the $\gamma$-ray cascade with energies of 1173keV and 1332keV emitted after the $\beta$-decay of the daughter nucleus $^{60}$Co. Data from [@RFK09; @Bro13].[]{data-label="decay60fe"}](60fe_decay.jpg){width="45.00000%"}
Sample preparation {#SamplePreparation}
------------------
The $^{60}$Fe was extracted from slices of a cylindrical copper beam dump, which was previously irradiated with 590MeV protons at PSI [@SND10]. In addition to $^{60}$Fe activity, the initial copper sample of 3g also contained 150MBq of $^{60}$Co, 100MBq of $^{55}$Fe, and 2MBq of $^{44}$Ti. Details of the chemical separation of the $^{60}$Fe fraction are described in [@SND10]. The final purification was performed shortly before the experiment using liquid-liquid extraction into methyl-isobutyl ketone from 7 M HCl solution and following back-extraction with diluted HCl. This solution was evaporated on a graphite disc with 6 mm diameter and 1 mm thickness.
![(Color online) $\gamma$-ray spectrum for the determination of the $^{60}$Co activity taken 34 months after purification of the sample. The measurement time was .[]{data-label="sampleParticles"}](plot_Teilchenzahl.jpg){width="50.00000%"}
The number of $^{60}$Fe atoms in the sample was determined via the increasing $^{60}$Co activity according to Fig. \[decay60fe\]. The activity of $^{60}$Co nuclei increases as $$A(^{60}\rm{Co}) = \left( 1-\rm{exp} [-\lambda(^{60}\rm{Co}) t ]\right) A(^{60}\rm{Fe}),$$ where $\lambda$ is the decay constant. The related $\gamma$-activity at time $t$ can be derived using the integrated
$$\begin{aligned}
A_t(^{60}\text{Co}) = &\frac{C_{\gamma}}{0.9976\ \epsilon_{\gamma} \ I_{\gamma}} \nonumber \\
& \ \frac{\lambda}{\mbox{\rm{exp}}\{-\lambda t\}-\mbox{\rm{exp}}\{-\lambda(t+t_{\text{m}})\}},\end{aligned}$$
where $\lambda$ is the decay constant of $^{60}$Co and with the factor $0.9976\pm 0.0003$ [@Bro13] for the fraction of $^{60}$Co$^m$, that decays to the ground state of $^{60}$Co, the measurement time $t_\text{m}$, and the detection efficiencies for the and $\gamma$-transitions, respectively. For the analysis of the emission line at the energy of , the decay of $^{60}$Co$^\text{m}$ has to be corrected. The decay intensities $I_\gamma$ and efficiencies $\epsilon_{\gamma}$ are listed in which summarizes all decay characteristics adopted in the data analysis. With $A=\lambda N$, the number of $^{60}$Fe atoms becomes $$N(^{60}\rm{Fe}) = \frac{A_t(^{60}\rm{Co})}{1-\rm{exp} \{-\lambda(^{60}\rm{Co}) t \} } \frac{1}{\lambda(^{60}\rm{Fe})}.$$
The activity measurement of $^{60}$Co was carried out at the GoetheÐUniversität Frankfurt after the purification using an HPGe detector of 98% relative efficiency (see Fig. \[sampleParticles\]). Background due to the activity of the $^{55}$Fe contamination in the sample was suppressed by a lead foil 1mm in thickness. The number of $^{60}$Fe atoms in the sample $$N(^{60}\text{Fe})=(7.77\pm0.11_{\text{\tiny{stat}}}\pm0.42_{\text{\tiny{syst}}} )\times 10^{14}$$ has been determined as a weighted average comprising both $^{60}$Co lines. The systematic uncertainty is determined by the $\gamma$-ray detection efficiency, the decay intensities, and the half-life (see Table \[CalStandard\]). As the half-life of $^{60}$Fe a value was used.
$^{94}$Zr $^{96}$Zr
-------------------- ------------------- -------------------
$N$ $2.7331\pm0.044$ $0.440\pm 0.014$
$N_{\rm{Cd}}$ $2.7101\pm0.044$ $0.437\pm 0.014$
$\sigma_{\rm{th}}$ $0.0494\pm0.0017$ $0.0229\pm0.0010$
$\sigma_{\rm{RI}}$ $0.280\pm0.010$ $5.28\pm0.11$
: Number of atoms $N$ (in units of 10$^{19}$) and cross sections (in barn) of the neutron fluence monitors $^{94}$Zr and $^{96}$Zr . \[tablemonitor\]
Cross sections were obtained from Ref. [@Mug06].
Reactor activations
-------------------
In view of the short half-life of the produced $^{61}$Fe nuclei ($t_{1/2}~=~5.98$min [@Bha99]), the activations at the TRIGA research reactor were performed using a pneumatic transport system between the irradiation position and the counting room [@EbK00; @HET06].
The $^{60}$Fe sample was activated for with and without cadmium foils surrounding the sample in both cases. This so-called cadmium-difference-method allows the distinction between the thermal neutron capture cross section and the resonance integral, which takes into account the epithermal component of the reactor neutron spectrum. The reactor spectrum can be described as the sum of a thermal component, *i.e.*, a Maxwell-Boltzmann distribution corresponding to $kT=~25.3$meV, and an epithermal component following an $1/E$-dependence. Due to the very large thermal capture cross section of cadmium, a proper cadmium shielding of the sample results in a significantly different response to thermal and epithermal neutrons. In the ideal case, all thermal neutrons would be absorbed in the cadmium, while the epithermal spectrum remains undisturbed.
without Cd with Cd
-------------------------- ------------------------- -------------------------
$N(^{95}\text{Zr})$ $^b$ $1.517\pm0.005\pm0.018$ $0.510\pm0.006\pm0.008$
$N(^{97}\text{Zr})$ $^b$ $1.171\pm0.001\pm0.018$ $1.067\pm0.001\pm0.016$
$\Phi_{\rm{th}}$ $^c$ $8.60\pm0.03\pm0.38$ $1.21\pm0.01\pm0.16$
$\Phi_{\rm{epi}}$ $^c$ $0.467\pm0.002\pm0.014$ $0.458\pm0.005\pm0.014$
: The amount of Zr nuclei produced in the activations and the resulting neutron fluences without and with cadmium shielding$^a$.[]{data-label="fluences"}
$^a$ Uncertainties are statistical and systematic, respectively.\
$^b$ In units of $10^{9}$.\
$^c$ In units of $10^{14}$cm$^{-2}$.
![(Color online) The $\gamma$-ray spectra of the Zr monitor foils used for the neutron fluence analysis normalized for different measuring times. Due to the smaller capture cross section of $^{96}$Zr in the thermal energy regime, the cadmium shielding affects the $^{97}$Zr signature to a much lower degree than that of $^{95}$Zr.[]{data-label="Zrfluence"}](131211Zr_Nfluss.jpg){width="50.00000%"}
The number of product nuclei after the activation $N(^{A+1}X)$ can be expressed in terms of the thermal cross section $\sigma_{\text{th}}$, the resonance integral $\sigma_{\text{RI}}$ ( with the cutoff energy ), and the epithermal ($\Phi_{\text{epi}}$) and thermal neutron fluences ($\Phi_{\text{th}}$) in units of cm$^{-2}$, $$N(^{A+1}X) =\ N(^AX) \ (\Phi_{\text{th}} \sigma_{\text{th}} + \Phi_{\text{epi}} \sigma_{\text{RI}}),$$ where $N(^{A}X)$ is the number of target nuclei in the irradiated sample.
{width="100.00000%"}
Natural zirconium provides a well suited monitor for the epithermal and the thermal flux. The activation of the isotopes $^{94}$Zr and $^{96}$Zr exhibit significantly different ratios $\sigma_{\text{RI}}/\sigma_{\text{th}}$ (Table \[tablemonitor\]). The uncertainties in the number of Zr atoms are due to the sample weight (0.2%) and to the isotopic abundances (1.6 and 3.2% for $^{94}$Zr and $^{96}$Zr, respectively) [@BeW11]. Two sets of Zr foils were used in the activations with and without cadmium shielding.
Results
=======
Determination of the neutron fluence
------------------------------------
The zirconium foils used as neutron monitors are 0.127mm in thickness and 6mm in diameter. The foils are thin enough that neutron self-absorption losses during $\gamma$-spectroscopy could be neglected. The fluences in units of 1/cm$^2$ are $$\Phi_{\text{epi}}=\frac{ N^{97} - N^{96} \sigma_{\rm{th}}^{96} \Phi_{\rm{th}}}{N^{96} \sigma_{\rm{RI}}^{96}}
\label{eqn:NflussEPI}$$ and
$$\begin{aligned}
\Phi_{\rm{th}} = \frac{N^{96} N^{95} \sigma_{\rm{RI}}^{96} - N^{94} N^{97} \sigma_{\rm{RI}}^{94}}
{N^{94} N^{96}
[\sigma_{\rm{RI}}^{96}\sigma_{\rm{th}}^{94} - \sigma_{\rm{RI}}^{94} \sigma_{\rm{th}}^{96}]},
\label{eqn:NflussTH}\end{aligned}$$
where the indices are referring to the various Zr isotopes. Figure \[Zrfluence\] shows the $\gamma$-ray spectra of the monitor foils normalized to equal neutron fluence. Because of the small neutron capture cross section of $^{96}$Zr in the thermal energy regime, the $^{97}$Zr signal is only marginally affected by the cadmium shielding, whereas $^{95}$Zr exhibits a clear effect due to the larger thermal cross section of $^{94}$Zr. The number of produced Zr nuclei is
$$N(^{i}X) = \frac{C_{\gamma}}{\epsilon_{\gamma} I_{\gamma} f_\text{a} f_\text{w} f_\text{m}}
\label{ProducedAtoms}$$
where $$\begin{aligned}
f_\text{a} = & \frac{1-\exp{(-\lambda_i} t_{\rm a})}{\lambda_i t_{\rm a}}, \\
f_\text{w} = & \exp{(-\lambda_i t_{\rm w})},\\
f_\text{m} = & 1-\exp{(-\lambda_i t_{\rm m})}\end{aligned}$$ are the corrections for the decay during the activation $f_\text{a}$, during the waiting time between activation and measurement $f_\text{w}$, and during the measurement $f_\text{m}$, respectively. The correction for the deadtime was of the order of 0.5%. The systematic uncertainty is again determined the , the decay intensities, and the half-life (Table \[CalStandard\]). The resulting neutron fluences for the two activations are listed in Table \[fluences\].
--------------------- ---------------------- ---------------
$\gamma$-ray energy
/$\text{keV}$ without Cd with Cd
1027 $1.54\pm0.19\pm0.18$ $ < 0.179$
1205 $1.48\pm0.20\pm0.16$ $ < 0.206$
Weighted average $1.51\pm0.14\pm0.24$ $ < 0.179^b $
--------------------- ---------------------- ---------------
: The number of $^{61}$Fe nuclei (in units of 10$^5$) produced in the activations.[]{data-label="61FeProd"}
$^a$ Uncertainties are statistical and systematic, respectively.\
$^b$ Adopted upper limit for further discussion.
Thermal ($\text{n}, \gamma$) cross section of $^{60}$Fe
-------------------------------------------------------
The $\gamma$-spectrum measured after the activation of the $^{60}$Fe sample without cadmium shielding (Fig. \[61feWithoutCd\]) clearly exhibits the $\gamma$-transitions of $^{61}$Fe at 297.9keV, 1027keV, and 1205keV. However, only the last two ones were used in the analysis because of the poor signal-to-background ratio of the 298keV line. The systematic uncertainty is calculated by the error of the efficiency, the $I_{\gamma}$, the half-lifes, and the neutron fluences (see Table \[CalStandard\], Table \[fluences\], and Table \[61FeProd\]).\
In the corresponding spectrum measured after the activation with cadmium shielding, the $^{61}$Fe lines are completely missing as illustrated in Fig. \[61feWithCd\] for the 1027 keV line as an example. In this case, only an upper limit can be determined for the resonance integral. The numbers of produced $^{61}$Fe nuclei are listed in Table \[61FeProd\].
The number ratio of $^{61}$Fe and $^{60}$Fe after the activation without cadmium is $$N(^{61}\text{Fe})/N(^{60}\text{Fe}) = \Phi_{\text{th}} \sigma_{\text{th}} + \Phi_{\text{epi}} \sigma_{\text{RI}}.
\label{EgnWithoutCd}$$
The thermal cross section $$\sigma_{\text{th}}(^{60}\text{Fe}) =\frac{N(^{61}\text{Fe})}{N(^{60}\text{Fe})} \frac{1}{\Phi_{\text{th}}} - \sigma_{\text{RI}} \frac{\Phi_{\text{epi}}}{\Phi_{\text{th}}}
\label{XSth}$$ is determined by the number of sample atoms $N(^{60}\text{Fe})$ (Sec. \[SamplePreparation\]), the neutron fluences $\Phi_{\rm{th}}$ and $\Phi_{\rm{epi}}$ from the Zr monitor measurements (Table \[fluences\]), and the number of $^{61}$Fe nuclei produced during the activations $N(^{61}\text{Fe})$ (Table \[61FeProd\]). The resonance integral $$\sigma_{\text{RI}} =
\frac{
\frac{N^{\text{Cd}}(^{61}\text{Fe})} {N(^{60}\text{Fe})}
-\frac{\Phi_{\text{th}}^{\text{Cd}}} {\Phi_{\text{th}}}
\frac{N(^{61}\text{Fe})} {N(^{60}\text{Fe})}
}
{
\Phi_{\text{epi}}^{\text{Cd}}
-\frac{\Phi_{\text{th}}^{\text{Cd}} \Phi_{\text{epi}}}
{\Phi_{\text{th}}}
},
\label{Ires}$$
![(Color online) A detailed view into the region around 1027keV of the $\gamma$-ray spectrum after activation with the cadmium shielding, illustrating the absence of $^{61}$Fe lines. Therefore, only an upper limit can be deduced from the data.[]{data-label="61feWithCd"}](EPI_61Fe_1027.jpg){width="50.00000%"}
{width="100.00000%"}
is obtained accordingly. Since the epithermal fluences were almost equal in both activations, and because the number of $^{61}$Fe nuclei produced with the cadmium absorber is much smaller than without absorber, an upper limit for the resonance integral can be defined as $$\sigma_{\text{RI}} < \frac{N^{\text{Cd}}(^{61}\text{Fe})}{N(^{60}\text{Fe})}\frac{1}{\Phi_{\text{epi}}^{\text{Cd}}}.
\label{epiXS}$$ Assuming a 1$\sigma$ confidence level as a constraint for the resonance integral derived from the 1027keV line, one finds $$0 < \sigma_{\text{RI}} < 0.50\ \text{b}
\label{ri_simple_limit}$$ for calculating the thermal cross section using Eq. (\[XSth\]).
A variation of the resonance integral within these limits affects the thermal cross section by about 10%. We assume the resonance integral here explicitly as $$\sigma_{\text{RI}} = 0.00 ^{+0.50}_{-0.00}\ \text{b},
\label{ri_nice_limit}$$ consistent with Eq. (\[ri\_simple\_limit\]), and treat this range as a systematic uncertainty. Should the resonance integral be improved in the future, the thermal cross section can be re-evaluated accordingly. With Eqs. (\[XSth\]) and (\[ri\_nice\_limit\]) the thermal cross section of $^{60}$Fe becomes $$\sigma_{\text{th}}(^{60}\text{Fe})
= (0.226 \pm 0.021_{\text{{stat}}} (^{+0.039}_{-0.045})_{\text{syst}} ) \ \text{b}.$$
Summary and Discussion
======================
Within this work, we characterized the $^{60}$Fe sample to contain $N(^{60}\text{Fe})=(7.77\pm0.11_{\text{\tiny{stat}}}\pm0.42_{\text{\tiny{syst}}} )\times 10^{14}$ atoms. Using the cadmium-difference-method two activations of that sample have been performed at the TRIGA research reactor at Johannes Gutenberg-Universität Mainz, Germany. The neutron capture cross section of $^{60}$Fe at thermal energies and an experimental upper limit for the resonance integral could be determined for the first time: $$\sigma_{\text{th}}(^{60}\text{Fe})
= 0.226 \ (^{+0.044}_{-0.049}) \ \text{b}$$
and $$\sigma_{\text{RI}} < 0.50 \ {\rm b}.$$
Figure \[fig:comparison\] shows a comparison of our data with evaluated cross sections (TENDL-2014 [@KoR12]) and the so far only experimental value of $5.7\pm 1.4$ mb at $kT~=~25\,$keV [@URS09; @kadonis2009]. Under the assumption that the MACS in the meV-regime is dominated by an s-wave direct capture component, an extrapolation towards higher energies via $1/\sqrt{E}$ is possible. Together with the measurement of the total capture cross section at $kT$= 25keV, it is then possible to disentangle the direct and the resonant contribution in the astrophysically interesting energy regime. It turns out that the direct capture component is almost negligible, ranging from less than 10% to less than 1% between 10keV and 100keV. The comparison of the experimental data with the latest release of TENDL indicates that the resonant component is well described, but the direct capture component is overestimated.
We are very grateful for the excellent support by the entire team of the TRIGA reactor in Mainz. This work was supported by the Helmholtz Young Investigator project VH-NG-327, the BMBF project 05P12RFFN6, the Helmholtz International Center for FAIR and HGS-HIRe. K.S. acknowledges support by DFG . C.L. acknowledges support from the Austrian Science Fund (FWF): J3503.
|
---
abstract: 'A learning classifier must outperform a trivial solution, in case of imbalanced data, this condition usually does not hold true. To overcome this problem, we propose a novel data level resampling method - Clustering Based Oversampling for improved learning from class imbalanced datasets. The essential idea behind the proposed method is to use the distance between a minority class sample and its respective cluster centroid to infer the number of new sample points to be generated for that minority class sample. The proposed algorithm has very less dependence on the technique used for finding cluster centroids and does not effect the majority class learning in any way. It also improves learning from imbalanced data by incorporating the distribution structure of minority class samples in generation of new data samples. The newly generated minority class data is handled in a way as to prevent outlier production and overfitting. Implementation analysis on different datasets using deep neural networks as the learning classifier shows the effectiveness of this method as compared to other synthetic data resampling techniques across several evaluation metrics.'
author:
- |
Naman Deep Singh\
Abhinav Dhall\
bibliography:
- 'bibliography.bib'
title: Clustering and Learning from Imbalanced Data
---
Introduction {#intro}
============
The amount of data generated is increasing every day, this also increases the demand for learning systems which can predict, classify and analyse the data efficiently. Classification is the type of problem most commonly solved by predictive systems, it can be of two types: binary classification [@mccormick2012dynamic] and multi-class classification [@wu2004probability; @sun2006boosting]. When a general classifier encounters imbalanced data, it favors the majority class sample. Imbalanced data is a scenario when the number of instances of one class are scanty in comparison to other classes. This causes classical classifier systems to neglect minority class instances and emphasize on majority class, resulting in a skewed classification accuracy. This accuracy might be high but minority class is misclassified. Imbalance problem can either be multi class or binary class classification. Most of the multi class imbalance problems [@har2003constraint] are generally converted to binary class imbalance problem (using binarization etc.) and then solved. This work is carried out for binary class imbalance problem.\
Class imbalance problem has its applications in many real world problems. It is imperative in certain cases to correctly classify the minority class correctly. Finding fraudlent cases in health insurance, automobile insurance fraud detection [@1277822], credit card fraud detections [@chan1998toward] and other cases related to financial sector are most common. The need of better classification in imbalance problem also arises while classsifying medical datasets ([@bhattacharya2017icu]). In cancer detection positive class cases can be very rare (1:100), sometimes even more adverse [@weiss2004mining; @pearson2003imbalanced] and detection of these cases is very important for obvious reasons. Improper classification in such cases can have serious repercussions.\
Nowadays, researchers working on this problem have emphasized on creating learning classifiers that are better suited to handle class imbalance [@Zhang2015] and less attention is being given to data level imbalance rectification techniques. All the data level resampling techniques fall under either oversampling, which regenerates data of the minority class to make it in commensurate measures to the majority class through some algorithms; or undersampling, which resamples the majority class to level it with the instances of the minority class. As mentioned, imbalance class problem is also present in multi-class classification problems [@sun2006boosting; @abe2004iterative] but the proposed method has been showcased only for the bi-class imbalance classification. At algorithmic level, cost enhanced versions of learning algorithms like SVM, neural networks have been used but to some fruition. These cost sensitive methods are known to make learning classifiers more susceptible to variations in minority class samples.\
In our proposed method, we have incorporated the randomness factor of random sampling and nearness approach of synthetic sampling technique for clustering and generation of new samples. This combination ensures reduction of overfitting and better performance. State of the art methods suffer from two problems. The first is the curse of dimensionality, it has been seen that methods like SMOTE [@chawla2002smote] and the one’s built on top of it lag in performance when applied to high dimensional data. Secondly, these methods in one way or the other interfere with the data distributions of majority class space. The proposed method, CBOS (Clustering based oversampling) has been conceived to tackle these two fundamental problems. CBOS has been compared with the most commonly used oversampling techniques like SMOTE [@chawla2002smote], SMOTE-ENN [@batista2004study] and ADASYN [@he2008adasyn]. We have used several datasets and used deep neural networks as our classifier. Although existing methods show promising results, our method improves upon these methods across different performance metrics, as can be seen in experiment and discussions section.\
The rest of the paper is organized as follows. In Section \[sec2\], the past work done related to the class imbalance problem is reviewed. Section \[sec3\] gives a detailed description of the proposed method. Descriptive analysis of learning algorithms and the performance metrics used has been done in Section \[sec4\]. Experimental study, results obtained and related discussions have been made in Section \[sec5\]. Finally, the paper is concluded in Section \[sec6\].
Related Work {#sec2}
============
A lot of work on imbalance class problem is present in literature. More recent work focuses on making learning algorithms adept to learn from imbalanced distributions whereas distant literature has a lot of work on data level resampling. Data level resampling is one of the many ways to handle class imbalance problem. As we propose a clustering based oversampling method, Lin et al. proposed a clustering based undersampling technique with Multi layer perceptron in [@lin2017clustering]. Random oversampling and random undersampling are the two most basic techniques in this category. Random oversampling suffers from overfitting whereas random undersampling leads to underfitting due to loss of data. Synthetic resampling techniques[@chawla2002smote; @mani2003knn; @batista2004study] were proposed to overcome this problem. In 2002, a sampling based algorithm called SMOTE (Synthetic Minority Over-Sampling Technique) was introduced. SMOTE [@chawla2002smote] balances the class distribution of original data sets by incrementing some virtual samples. While generating virtual examples SMOTE does not take into account the neighboring examples from other classes, which results in an increase in class overlapping and may introduce noise. Some new techniques, which are built upon SMOTE have come up, SMOTE-ENN[@batista2004study], SMOTE-Tomek [@batista2004study]. Adaptive Synthetic (ADASYN) [@he2008adasyn] is based on the idea of adaptively generating minority data samples according to their distributions using K nearest neighbor. A detailed review of all synthetic techniques is given by [@1755-1315-58-1-012031].\
Cost Sensitive learning techniques for imbalance use a cost-matrix for different types of errors to aid learning from imbalanced data sets. It does not affect the data distribution but different cost matrices are used for misclassification of minority class samples than for majority class samples [@sun2007cost]. Cost-sensitive neural network models have been deeply studied in [@zhou2006training]. A threshold-moving technique has been used in this method to adjust the output threshold toward majority classes, such that high-cost (minority) instances are less likely to be misclassified. Liu et al. [@liu2006influence] study the effects of imbalanced data on cost sensitive learning and different cost sensitive learning techniques have been compared in [@thai2010cost] using several imbalanced datasets. According to [@wang2016training], these methods are only applicable when the specific cost values of misclassification are known.\
Kernel based methods have also been used to study class imbalance problem. In [@wu2005kba], Wu et al. propose a method to adjust the class boundary by adapting the kernel matrix in SVMs according to the distribution of the data. Ensemble models are also known to improve performance while facing the challenge of imbalance data. Bagging [@quinlan1996bagging] forms another class of techniques for handling imbalanced data sets. Random Forests, a form of bagging are not good to handle imbalance data as they are based on decision trees, which are adversely susceptible to class imbalance, but when used with suitable resampling technique they can be effective. An ensemble algorithm based on balanced Random Forest for buyer prediction problem is given by Yagci et al. [@7495927]. A class-wise weighted voting approach to Random Forest for class imbalance problem for medical data has been given in [@8246503]. Recently, unsupervised learning [@NIAN201658] techniques have been used to balance data based on unsupervised spectral rankings in auto insurance frauds.\
A new weight adjustment factor is used by Wonji et al. [@LEE201792] in SVMs, and the SVM has been used as a base classifier in AdaBoost algorithm. Supposedly, this method [@LEE201792] solves class overlapping and small disjuncts problems which are commonly seen in clssification problems. According to [@galar2012review], algorithmic level and cost-sensitive approaches are more problem dependent, whereas data level and ensemble learning methods are more pervasive as they can be used in independence to the learning classifier. A comprehensive comparison of ensemble methods for class imbalance can be seen in [@galar2012review]. Class switching, a mechanism to produce training perturbed sets, has proven to perform well in slightly imbalanced scenarios. In [@GONZALEZ201712] class switching’s potential to deal with highly imbalanced data has been analysed and a new ensemble approach based on switching using nearest enemy distance has also been proposed.\
The propitiousness of deep learning has led researchers to study the effect of class imbalance in deep neural nets as well. Authors of [@buda2017systematic] study the effects of imbalance on convolutional neural networks and conclude that oversampling is the best-suited resampling technique and does not lead to overfitting in CNNs. Further motivating us to explore balancing data using oversampling. Some other techniques to tackle class imbalance problem include the generation of new metrics [@boughorbel2017optimal] for optimization while working with class imbalanced data. One has been proposed in [@wang2016training], the metrics proposed are based on mean false positive and false negative error rates, and are used for optimization in neural networks and are better performing than mean squared error. We only discuss data level resampling (oversampling) techniques henceforth, which are befitting the scope of CBOS.\
Clustering Based Oversampling {#sec3}
==============================
Several synthetic data resampling methods have had fair amount of success. We propose a new data level oversampling method, Clustering Based Oversampling (CBOS) which has clustering as its basis. The success of techniques like SMOTE and ADASYN, which are predominantly based on kNN and also the sporadic success of randomly oversampling minority class motivated us to come up with a new data level oversampling technique. The proposed technique uses the euclidean distance in k-Means clustering [@kanungo2002efficient] to generate the cluster centroid and a distance normalization technique is used to generate the number of new data samples per existing minority class sample to be created. Although promising for low dimensional data space, the proposed algorithm is best suited for high dimensional data setting. This is due to the fact that with less number of attributes, the $Random$ parameter of our algorithm might sometimes generate similar samples leading to slight overfiting of the data.\
Our algorithm does not take the distribution of majority class into consideration while oversampling the minority class samples and hence does not effect learning as done from majority class space. The main differences between this method and other algorithms like SMOTE and ADASYN are: first, the way this algorithm decides how many new samples are to be generated for each existing minority class sample; second, clustering helps to add the spatial structure of minority class into the new generated data samples; thirdly, the non-dependence of our technique on clustering technique used makes it more stable and finally, prevention of any change in the learning performance of the majority class. The proposed algorithm is explained below.\
------------------------------------------------------------------------
**Input.** Training Data having $ K $ samples,\
$$K = K_{l} + K_{m} \quad (K_{m} >> K_{l})$$
where, $ K_{l} $ is the number of samples belonging to the minority class, and, $ K_{m} $ is the number of samples belonging to the majority class.\
$\textbf{1}.$ Assume the number of clusters in minority class data.\
$\textbf{2}.$ Find the respective cluster centroid ($ C $) for each minority class sample using k-Means clustering algorithm.\
$\textbf{3}.$ For each $ x_{i} \in K_{l} $ calculate the euclidean distance ($ dist$) between the respective cluster centroid ($ C $) of $x_{i} $ and $ x_{i} $ itself, defined by,\
$$dist_{i} = Euclidean(x_{i},C)$$ $\textbf{4}.$ Normalize the distances over the whole data :\
$ \quad \forall x_{i} \in K_{l} $ do\
$$dist_{i}=\frac{dist_{i}}{\sum\limits_{i=1}^{K_{l}} dist_{i}}$$ $\textbf{5}.$ $ \forall x_{i} \in K_{l} $ Calculate the number of samples ($n_{i}$) to be generated for each
original minority class sample. $$n_{i}=integer\big(dist_{i} \times (K_{m}-K_{l})\times \eta\big)$$\
here, $integer()$ is a function to round the resulting value, $K_{m}-K_{l}$ is the difference between the number of samples of majority and the minority class, $ \eta $ is the level of balance to be generated 1 meaning 100%, 0.5 meaning 50% and so on.\
$\textbf{6}.$ Generate $n_{i}$ new samples for each $ x_{i} $\
**for** $ i \in K_{l} $ **do**\
**for** $ j \in n_{i} $ **do**\
$$d = abs(sample[i] - C)$$ $$newsample[i][j] = sample[i] \pm d\times Random(0,1)$$ end **for**\
end **for**\
$C$ is the respective cluster centroid vector of $x_{i}$, $ newsample[i][j]$ is the $j^{th}$ new sample vector for $x_{i}$ sample which has $ sample[i]$ as its original sample vector from minority class $K_{l}$ .\
$abs()$ is the absolute element wise difference of the two vectors. $Random(0,1)$ is a random float between the $0,1$. For some cases the range can be reduced as well.\
After generating all the new samples, their attribute values are reduced or increased to the original maximum and minimum values associated with each feature in original sample vector respectively. This is done to prevent generation of outliers. This step is also necessary to tackle the effect of overfitting as often seen in oversampling techniques.\
CBOS can also be implemented by using a lower range for the $Random$ variable in step 5 of the algorithm, like ($0.2,0.4$) making range of $Random$ a very important parameter to be tuned for the proposed algorithm.\
The idea behind this algorithm is that the more distant a sample point is from its cluster centroid, the lower the number of new points of this sample will be generated; and the less distant a sample point is from cluster centroid, the more is the number of new samples associated with this sample to be generated. This can be thought of as a centroid distance based weight metric, where the weights are used to find the number of new sample points to be generated.\
The way our algorithm decides on the number of new samples for each original instance makes sure that the more important samples have higher representation than far lying samples in the new balanced minority class space. Also, as our algorithm does not take into consideration any effect of majority classs samples on minority samples, the classification accuracy of majority class is not affected at all.\
Learning Algorithm & Performance Metrics {#sec4}
========================================
Deep Neural Network {#sec4a}
-------------------
We have used Deep Neural Network (DNN) to learn the feature representation from the data. DNN are becoming state-of-the-art as more data is being generated, its power lies in the fact that they work better on large datasets. The DNN used refers to an artificial neural network with more than one hidden layer(s). We have implemented DNNs with hidden layers ranging from 2 to 4, with different number of nodes in each layer and have reported the best results achieved for each resampling technique.
Assessment Metrics {#sec4b}
------------------
Most of the classification performances are measured using accuracy or mean squared error. For imbalance class problem metrics taken from confusion matrix like F-score and the Area under ROC curve (AUC) [@fawcett2006introduction] have been quite often used.\
The left column in the confusion matrix represents positive instances of the data set and the right column represents the negative ones. Therefore, the proportion of the two columns is representative of the class distribution of the data set, and any metric that uses values from both of these columns shall be inadvertently sensitive to imbalances. Other accuracy measures namely, *[precision, recall, F-score]{} *have also been used for imbalance class problem. F-score is a weighted avergae of precision and recall, and has been used as one of the performance metrics in this work. $$F-score = (1+\beta^{2})\frac{Recall.Precision}{\beta^{2}.Recall+Precision}{}$$ $\beta$ is generally taken to be 1. To compare the performance of different classifier systems over a range of sample distributions, G-mean has also been calculated.\
$$G-Mean = \sqrt{\frac{TP}{TP+FN}\times\frac{TN}{TN+FP}}$$**
Experiments and Results {#sec5}
=======================
In this section, we have experimentally shown the performance of different resampling techniques as compared to the proposed method.
Experimental setting and data {#sec5a}
-----------------------------
All the six datasets used for this study are binary classification data. The very famous MNIST hand written digits dataset has been used. From the MNIST [@lecun1998gradient] dataset, two classes are selected randomly and imbalance is induced in these resulting datasets. Specifically, four datasets comprising of the digit pairs (7, 5), (9, 8), (6, 2) and (1, 4) have been used and an imbalance of 12%, 10%, 8% and 6% has been induced in them, respectively. These datasets are labeled as Data-1, Data-2, Data-3 and Data-4.\
The fifth dataset is a psychometric dataset from the SAPA [@978d379eca944e5581293c611919a87c] project called as SPI, extracted from R data repository and imbalance level is at 5%. The SPI data has 145 attributes and it has been labeled as Data-5. A bi-classification auto insurance fraud dataset from UCI ML repository (represented as Data - 6) having an imbalance rate of 6% and 32 atributes has also been used.\
[p[ 0.10]{}p[0.14]{}p[0.12]{}p[0.10]{}p[0.11]{}p[0.10]{}p[0.12]{}]{} **Dataset** & **Algorithm** & **Precision** & **Recall** & **Accuracy** & **F-score** & **G-Mean**
------------------------------------------------------------------------
------------------------------------------------------------------------
\
------------------------------------------------------------------------
& Imbalanced & 0.609 & 0.980 & 0.838 & 0.754 & 0.781
------------------------------------------------------------------------
------------------------------------------------------------------------
\
& SMOTE & 0.983 & 0.973 & 0.985 & **0.988** & 0.989\
------------------------------------------------------------------------
& SMT-ENN & 0.984 & 0.970 & 0.982 & 0.977 & 0.983\
------------------------------------------------------------------------
& ADASYN & 0.987 & 0.975 & 0.984 & 0.981 & 0.986\
& CBOS & **0.989** & **0.993** & **0.993** & 0.987 & **0.992**
------------------------------------------------------------------------
\
& Imbalanced & 0.475 & 0.975 & 0.776 & 0.639 & 0.686
------------------------------------------------------------------------
------------------------------------------------------------------------
\
& SMOTE & 0.967 & 0.977 & 0.973 & 0.971 & 0.974\
------------------------------------------------------------------------
& SMT-ENN & 0.952 & **0.982** & 0.973 & 0.964 & 0.970\
------------------------------------------------------------------------
& ADASYN & 0.952 & 0.979 & 0.970 & 0.962 & 0.961
------------------------------------------------------------------------
\
& CBOS & **0.972** & 0.977 & **0.981** & **0.977** & **0.979**
------------------------------------------------------------------------
\
& Imbalanced & 0.523 & 0.981 & 0.744 & 0.684 & 0.721
------------------------------------------------------------------------
------------------------------------------------------------------------
\
& SMOTE & 0.930 & 0.992 & 0.961 & 0.963 & 0.963\
------------------------------------------------------------------------
& SMT-ENN & 0.942 & 0.989 & 0.965 & 0.964 & 0.964\
------------------------------------------------------------------------
& ADASYN & 0.936 & **1.000** & **0.972** & 0.963 & 0.966
------------------------------------------------------------------------
\
& CBOS & **0.947** & 0.997 & 0.971 & **0.973** & **0.969**
------------------------------------------------------------------------
\
& Imbalanced & 0.484 & 0.941 & 0.704 & 0.624 & 0.683
------------------------------------------------------------------------
------------------------------------------------------------------------
\
& SMOTE & 0.910 & 0.971 & 0.932 & 0.933 & 0.928\
------------------------------------------------------------------------
& SMT-ENN & 0.902 & 0.979 & 0.927 & 0.924 & 0.919\
------------------------------------------------------------------------
& ADASYN & 0.916 & **0.971** & 0.938 & 0.937 & 0.931
------------------------------------------------------------------------
\
& CBOS & **0.922** & 0.970 & **0.941** & **0.943** & **0.939**
------------------------------------------------------------------------
\
& Imbalanced & 0.471 & 0.794 & 0.784 & 0.596 & 0.668
------------------------------------------------------------------------
------------------------------------------------------------------------
\
& SMOTE & 0.867 & 0.851 & 0.901 & 0.859 & 0.908\
------------------------------------------------------------------------
& SMT-ENN & 0.840 & 0.761 & 0.852 & 0.794 & 0.849\
------------------------------------------------------------------------
& ADASYN & **0.903** & 0.937 & **0.948** & 0.919 & **0.936**
------------------------------------------------------------------------
\
& CBOS & 0.892 & **0.960** & **0.948** & **0.920** & 0.929
------------------------------------------------------------------------
\
& Imbalanced & 0.538 & 0.841 & 0.824 & 0.628 & 0.782
------------------------------------------------------------------------
------------------------------------------------------------------------
\
& SMOTE & 0.931 & **0.901** & 0.901 & **0.918** & 0.915\
------------------------------------------------------------------------
& SMT-ENN & 0.914 & 0.891 & 0.908 & 0.903 & 0.912\
------------------------------------------------------------------------
& ADASYN & **0.943** & 0.889 & **0.919** & 0.913 & **0.941**
------------------------------------------------------------------------
\
& CBOS & 0.906 & 0.897 & 0.902 & 0.901 & 0.914
------------------------------------------------------------------------
\
\[tab:3\]
Based on the performance metrics discussed in Section \[sec4b\], Table \[tab:3\] presents the comparison between the proposed CBOS and other resampling techniques. The results shown are averaged over 10 runs. The resampling algorithms are then applied on the training data and tested on the test set. Best parameters for SMOTE, SMOTE-ENN and ADASYN have been selected. The DNN has been tuned to get the best peformance for all resampling techniques similarly.\
Analysing the Performance {#sec5b}
-------------------------
From Table \[tab:3\], it can be easily inferred that the proposed method CBOS is generally the best across all test benches. Performance results from the original imbalanced dataset have also been included and all the resampling methods improve upon the original imbalanced data. The original data show good performance in certain metrics recall. This is because only one class is imbalanced and directly using a learning classifier on original data results in good classification of majority class only.\
As the level of imbalance becomes severe from Data-1 to Data-4, we see fall in the value of mostly all the metrics. This implies that more severe the imbalance, the less effective the resmpling techniques are. Comparing the results of resampling techniques with respect to the majority class in Data-1, we see that the Recall score of all the techniques sans CBOS decreases as compared to the original imbalanced data. This shows that the way in which CBOS resamples minority class data does not in any way hinder in classification of the majority class whereas other techniques in some way have an effect on the majority class data in addition to the minority class data. The best performing technique for each metric has been highlighted, and the most highlighted across all datasets is the CBOS technique.\
On comparing the performance results of Data-5 and 6, we see that the attribute to sample ratio of Data-5 is very large than Data-6. This results in an increase in performance of resampling techniques as SMOTE and ADASYN, but, CBOS also produces results in a tantamount range. This leads us to say that SMOTE, ADASYN and other similar resampling techniques perform better in low dimesnional setting. The proposed method, CBOS performs comparable to others for low dimensional data but is more efficient in case of high dimensional data space.
Discussions {#sec5c}
-----------
The strength of CBOS lies in the fact that in addition to the oversampling minority classs accurately, CBOS produced new samples do not effect majority class space in any way. We use the randomness in an effective way by restraining the maximum and minimum values of the newly genrated samples. Also rather than using kNN for generation of new data points we use cluster centroids. This means our proposed algorithm also incorporates the distribution structure of the minority class data in the newly resampled data, not commonly seen in other synthetic resampling techniques. The proposed algorithm is more suited for medium and high dimensional data because of the $Random$ function used in the algorithm. If the need of $Random$ for CBOS is reduced in future, CBOS can be made adaptable to very low dimensional data as well.\
The results shown are only for deep neural networks but CBOS algorithm can be used with other learning algorithms as well. Ensemble models have proven to be great classifier systems and in future CBOS can be integrated with ensemble models to further enhance the classification prowess of CBOS. As most of the imabalance problems are dominated by two class classification, this work only studies CBOS for two class classification problem but CBOS can also be used for multi class imbalance problems. More imbalanced datasets can be used to test the efficacy of the proposed algorithm. We have shown CBOS when used with k-Means clustering but CBOS’s results when used with other clustering algorithms are expected to be in comparable ranges. In future, CBOS can be implemented with other clustering models for example Mean-shift clustering or the recently introduced, expectation maximization clustering using Gaussian mixture models [@reynolds2015gaussian]. An algorithm is considered novel and robust when it has lower number of parameters it requires to be tuned on, in this regard, we would like to reduce the efffect that $Random$ parameter has in generating new data samples, making CBOS more stable and robust to varied type of data.
Conclusion {#sec6}
==========
A lot of work has been done for solving class imbalance problem using data resampling, but most of these methods in some way effect the majority class space as well. CBOS method prevents generation of outliers and it does not affect majority class set in any way as CBOS’s oversampling technique does not take interdependence of the two classes into consideration. CBOS uses a completely different way to decide on the number of new samples to be generated for each existing minority class sample. CBOS uses clustering and cluster centroids for developing new data points meaning overall distribution of minority class is also taken into consideration. The distance normalization technique used for calculation of the number of new sample points to be generated is designed in a way that it lets the data points nearer to centroid have more say in the generation of new sample points. Motivated by the results in this paper, we believe that CBOS might prove a powerful method in multi class imbalanced learning as well.
|
---
abstract: |
Hindman and Leader first introduced the notion of Central sets near zero for dense subsemigroups of $((0,\infty),+)$ and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone-$\breve{C}$ech compactification, Bayatmanesh and Tootkabani generalized and extended this combinatorial theorem to the central theorem near zero. Algebraically one can define quasi-central set near zero for dense subsemigroup of $((0,\infty),+)$, and they also satisfy the conclusion of central sets theorem near zero. In a dense subsemigroup of $((0,\infty),+)$, C-sets near zero are the sets, which satisfies the conclusions of the central sets theorem near zero. Like discrete case, we shall produce dynamical characterizations of these combinatorically rich sets near zero.
AMS subjclass \[2010\] : 37B20; 37B05; 05B10.
address: 'Sourav Kanti Patra, Department of Mathematics, Ramakrishna Mission Vidyamandira, Belur Math, Howrah-711202, West Bengal, India'
author:
- Sourav Kanti Patra
title: DYNAMICAL CHARACTERIZATIONS OF COMBINATORIALLY RICH SETS NEAR ZERO
---
introduction
============
Furstenberg, defined the concept of a central subset of positive integers \[4, Definition 8.3\] and proved several important properties of such sets using notions from topological dynamics.\
**Definition 1.1** A dynamical system is a pair $(X,\langle T_s\rangle _{s \in S })$ such that,
\(i) $X$ is compact Hausdorff space,
\(ii) $S$ is a semigroup,
\(iii) for each $s\in S$, $T_s:X\rightarrow X$ and $T_s$ is continuous, and
\(iv) for all $s,t$ , $T_s\circ T_t=T_{st}$.\
Inspired by the fruitful interaction between Ramsey theory and ultrafilters on semigroups, Bergelson and Hindman, with the assistance of B. Weiss, later proved on algebraic characterization of central sets in $\mathbb{N}$ \[3, Section 6\]. Using this algebraic characterization a as a definition enabled them easily to extend the notion of a central set to any semigroup.\
Let us now give a brief description of the algebraic structure of $\beta S$ for a discrete semigroup $(S,\cdot)$. We take the points of $\beta S$ to be the ultrafilters on $S$, identifying the principal ultrafilters with the points of $S$ and thus pretending that $S\subseteq \beta S$. Given $A\subseteq S$ let us set, $\bar{A}=\{ p\in \beta S : A\in p\}$. Then the set $\{ \bar{A} : A\subseteq S\}$ is a basis for a topology on $\beta S$. The operation $\cdot$ on $S$ can be extended to the Stone-cech compactification $\beta S$ of $S$ so that $(\beta S,\cdot)$ is a compact right topological semigroup(meaning that for any $p\in \beta S$), the function $\rho_p:\beta S$ defined by, $\rho_p(q)=q\cdot p$ is continuous) with $S$ contained in its topological center (meaning that for any $x\in S$ the function $\lambda_x : \beta S \rightarrow \beta S$ defined by $\lambda_x (q)=x\cdot q$ is continuous). Given $p,q \in \beta S $ and $A \in S$ $A \in p\cdot q$ if and only if $\{ x \in S :x^{-1}A \in q\} \in p$, where $x^{-1}A=\{ y \in S: x\cdot y \in A \}$. A non-empty subset $I$ of a semigroup $(T,\cdot)$ is called a left ideal of $S$ if $T\cdot I \subseteq I$, a right ideal of $S$ if $I\cdot T \subseteq I$ and a two sided ideal (or simply an ideal) if it is both a left and a right ideal. A maximal left ideal is a left ideal that does not contain any proper left ideal. Similarly we can define minimal right ideal and smallest ideal. Any compact Hausdorff right topological semigroup $(T,\cdot)$ has a smallest two sided ideal.\
$$\begin{array}{ccc}
K(T) & = & \bigcup\{L:L \text{ is a minimal left ideal of } T\} \\
& = & \,\,\,\,\,\bigcup\{R:R \text{ is a minimal right ideal of } T\}\\
\end{array}$$\
We now present Bergelson’s Characterizations of Central sets.\
**Definition 1.2.** Let $S$ be a discrete semigroup and let $C$ be a subset of $S$. Then $C$ is centers if and only if there is an idempotent $p$ in $K(\beta S)$ such that $C \in p$.\
To give a dynamical characterization of central set in arbitrary semigroup $S$, we need the following definitions from now on, $P_f(X)$ is the set of finite non-empty subsets of $X$, for any set $X$.\
**Definition 1.3.**(\[Definition 3.1, 6\]) Let $S$ be a semigroup and let $A\subseteq S$.
\(a) The set $A$ is syndetic if and only if there is some $G \in P_f(S)$ such that $S=\cup_{t \in G}t^{-1}A $.
\(b) the set $A$ is piece-wise syndetic if and only if there is some $G \in P_f(S)$ such that for an $F \in P_f(S)$ there is some $x \in S$ with $Fx\subseteq \cup_{t \in G}t^{-1}A $\
Recall the definitions of proximality and uniform recurrence in a dynamical system from \[3, Definition 1.2(b)\] and \[3, Definition 1.2(c)\]\
**Definition 1.4.** Let $(X,\langle T_s\rangle _{s \in S})$ is a dynamical system.
\(a) A point $y \in S$ is uniformly recurrent if and only if , for every neighborhood $U$ of $y$, $\{ s\in S : T_s(y) \in U \}$ is syndetic.
\(b) The points $x$ and $y$ of $X$ are proximal if and only if for every neighborhood $U$ of the diagonal in $X\times X$, there is some $s \in S$ such that $(T_s(x), T_s(y)) \in U$.\
By \[Theorem 2.4, 10\], a subset $C$ of a semigroup $S$ is central if and only if there exist a dynamical system $(X,\langle T_s\rangle _{s \in S})$, points $x$ and $y$ of $X$ and a neighborhood $U$ of $y$ such that $y$ is uniformly recurrent, $x$ and $y$ are proximal and $C=\{s \in S: T_s(x) \in U \}$.\
We now state basic definitions, conventions and results for dynamical characterization of members of certain idempotent ultrafilters.\
**Definition 1.5.**(\[8, Definition 2.1\])Let $S$ be a nonempty discrete space and $\mathcal{K}$ is a filter on $S$.
\(a) $\bar{\mathcal{K}}= \{p \in \beta S : \mathcal{K} \subseteq p \}$
\(b) $L(\mathcal{K})=\{A\subseteq S: S\setminus A \not \in \mathcal{K}\}$\
As is well known, the function $\mathcal{K} \rightarrow \bar{\mathcal{K}}$ is a bijection from the collection of all filters on $S$ onto the collection of all compact subspace of $\beta S$ \[7, Theorem 3.20\]. We also have the following important theorem relating the above two concepts.\
**Theorem 1.6.** Let $S$ be a nonempty discrete space and $\mathcal{K}$ a filter on $S$.
\(a) $\bar{\mathcal{K}}=\{ p \in \beta S: A \in L(\mathcal{K})$ for all $A \in p$ }
\(b) Let $\beta \subseteq L(\mathcal{K})$ be closed under finite intersections then there exists a $p \in \beta S$ with $\beta \subseteq p \subseteq L(\mathcal{K})$.\
**Proof.** Both of these assertions follows from \[7,Theorem 3.11\]\
**Definition 1.7.**(8, Definition 3.1) Let $(X,\langle T_s\rangle _{s \in S})$ be a dynamical system, $x$ and $y$ points in $X$, and $\mathcal{K}$ a filter on $S$. The pair $(x,y)$ is called jointly $\mathcal{K}$-recurrent if and only if for every neighborhood $U$ of $y$ we have $\{ s \in S: T_s(x) \in U$ and $T_s(y) \in U \} \in L(\mathcal{K})$.\
Following theorem is \[8, Theorem 3.3\].\
**Theorem 1.8.** Let $(S,.)$ be a semigroup, let $\mathcal{K}$ be a filter on $S$ such that $\bar{\mathcal{K}}$ is a compact subsemigroup of $\beta S$, and let $A \subseteq S$. Then $A$ is a member of an idempotent in $\bar{\mathcal{K}}$ if and only if there exists a dynamics system $(X,\langle T_s\rangle _{s \in S})$ with points $x$ and $y$ in $X$ and there exists a neighborhood $U$ of $y$ such that the pair $(x,y)$ is jointly $\mathcal{K}$-recurrence and $A=\{ s \in S: T_s(x) \in U \}$.\
**Definition 1.9.**(\[6, Definition 1.2\]) Let S be a discrete semigroup and let $C$ be a subset of $S$. Then $C$ is quasi-central if and only if there is an idempotent $p$ in Cl $K(\beta S)$ such that $C \in p$.\
Now recall \[8, Definition 4.1 and Definition 4.4\]\
**Definition 1.10.** Let $S$ be a semigroup.
\(a) for each positive integer $m$ put $J_m=\{ (t_1,t_2,...,t_m)\in
\mathbb{N}^m:t_1<t_2<...<t_m \}$
\(b) given $m \in \mathbb{N}, a \in S^{m+1}, t \in J_m$, and $t \in \tau $, put $x(m,a,t,f)=\prod^m_{i=1}(a(i)f(t_i))a(m+1) $ where $\tau=\mathbb{N_S}$
\(c) We call a subset $A\subseteq S$, a $C$-set if and only if there exists functions $m:P_f(\tau)\rightarrow \mathbb{N}$,$\alpha \in X^{S^{m(F)+1}}_{F \in P_f(\tau)}$, and $\tau \in X^{J_{m(F)}}_{F \in P_f(\tau)}$ such that the following two statements are satisfied:
\(1) If $F,G \in P_f(\tau)$ and $F\subsetneq G$ then $\tau(F)(m(F))<\tau(G)(1)$.
\(2) Whenever $m \in \mathbb{N}, G_1, G_2,...,G_m$ is a finite sequence in $P_f(\tau)$ with $G_1 \subsetneq G_2 \subsetneq ...\subsetneq G_m$ and for each $i \in \{ 1,2,...,m \}, f_i \in G_i$ then we have\
$\prod^m_{i=1}x(m(G_i), \alpha (G_i), \tau (G_i),f_i)\in A$
\(d) We call a subset $A\subseteq S$, a $J$-set if and only if for every $F \in P_f(\tau)$, there exist $m \in \mathbb{N},a \in S^{m+1}$ and $t \in J_m$ such that for all $f \in F$, $x(m,a,t,f)\in A$\
**Definition 1.11.** Let $(X,\langle T_s\rangle _{s \in S})$ be a dynamical system and let $x,y \in X$.
\(a) The pair $(x,y)$ is jointly intermittently uniformly recurrent (abbreviated JIUR) if and only if for every neighborhood $U$ of $y$, $\{ s \in S: T_s(x)\in U$ and $T_s(y)\in U \}$ is piecewise syndetic.
\(b) the pair $(x,y)$ is jointly intermittently almost uniform recurrent (abbreviated as JIAUR) if and only if for every neighborhood $U$ of $y$, $\{ s \in S: T_s(x)\in U$ and $T_s(y)\in U \}$ is a $J$-set.\
Using theorem 1.8, we have dynamical characterizations of quasi-central set and $C$-set in terms of JIUR and JIAUR respectively.\
**Theorem 1.12.** Let S be a semigroup and let $C\subseteq S$. The set $C$ is quasi-central if and only if there exists a dynamical system $(X,\langle T_s\rangle _{s \in S})$, points $x$ and $y$ of $X$ such that $x$ and $y$ are JIUR, and a neighborhood $U$ of $y$ such that $C=\{ s \in S: T_s(x)\in U\}$.\
**Theorem 1.13.** Let S be a semigroup and let $C\subseteq S$. The set $C$ is $C$-set if and only if there exists a dynamical system $(X,\langle T_s\rangle _{s \in S})$, points $x$ and $y$ of $X$ such that $x$ and $y$ are JIAUR, and a neighborhood $U$ of $y$ such that $C=\{ s \in S: T_s(x)\in U \}$.\
Now we will be considering semigroups which are dense in $((0,\infty),+)$. Here ’dense’ means with respect to the usual topology on $((0,\infty),+)$.\
**Definition: 1.14.** If $S$ be a dense subsemigroup of $((0,\infty),+)$ then $O^{+}(S)=\{ p \in \beta S :$ for an $ \epsilon >0, S\cup(0,\epsilon) \in p\}$\
It was proved in \[5, Lemma 2.5\] that $O^{+}(S)$ is a compact right topological subsemigroup of $(\beta S,+)$. It was also noted there in $O^{+}(S)$ is disjoint from $K (\beta S)$ and hence gives some new information which is not available from $K (\beta S)$.\
Being a compact right topological semigroup , $O^{+}(S)$ has a minimal ideal $K (O^{+}(S)$.\
Like discrete case we can define central set, quasi-central set, $C$-set near zero. Dynamical characterization of central set near zero, quasi central set near zero and $C$-set near zero is established in section 2, section 3 and section 4 respectively..\
dynamical characterization of central set near zero
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Let us start this section with the following well-known definition of central set near zero\[5, Definition 4.1(a)\]\
**Definition 2.1.** Let $S$ be a dense subsemigroup of $((0,\infty),+)$. A set $C\subseteq S$ is a central near zero if and only if there is some idempotent $p\in K(O^{+}(S))$ with $C \in p$\
Following definition is \[5, Definition 3.2(b)\].\
**Definition 2.2.** Let $S$ be a dense subsemigroup of $((0,\infty),+)$. A subset $B$ of $S$ is syndetic near zero if and only if for every $\epsilon>0$, there exists some $F \in P_f((0,\epsilon)\cap S)$ and some $\delta>0$ such that $S\cap (0,\delta)\subseteq \cup_{t \in F}(-t+B)$.\
We shall now introduce the the notion of uniform recurrence and proximality near zero.\
**Definition 2.3.** Let S be a dense subsemigroup of $((0,\infty),+)$ and $(X,\langle T_s\rangle _{s\in S})$ be a topological dynamical system.
\(a) A point $x \in X$ is a uniformly recurrent point near zero if and only if for each neighborhood $W$ of $x$, $\{s \in S: T_s \in W \}$ is syndetic near zero.
\(b) Points $x$ and $y$ of $X$ are proximal near zero if and only if for every neighborhood $U$ of the diagonal in $X\times X$, for each $\epsilon>0$ there exists $s\in S \cap (0, \epsilon)$ such that $(T_s(x), T_s(y)) \in U$.\
We now recall \[7, Theorem 19.11\].\
**Theorem 2.4.** Let $(X,\langle T_s \rangle_{s\in S})$ be a dynamical system and define $\theta : S \rightarrow X^X$ by $\theta(s)=T_s$. Then $\tilde{\theta}$ is a continuous homomorphism from $\beta S$ onto the enveloping semigroup of $(X,\langle T_s\rangle _{s\in S})$. ($\tilde{\theta}$ is the continuous extension of $\theta$)\
The following notation will be convenient in the next section.\
**Definition 2.5.**(\[7, Definition 19.12\]) Let $(X,\langle T_s\rangle _{s\in S})$ be a dynamical system and define $\theta : S\rightarrow X^X$ by $\theta(s)=T_s$. For each $p \in \beta S$, let $T_p= \tilde{\theta}(p)$.\
As a immediate consequence of theorem 2.4 we have the following remark \[7, Remark 19.13\]\
**Remark 2.6.** Let $(X,\langle T_s\rangle _{s\in S})$ be a dynamical system and let $p,q \in \beta S$. Then $T_p\circ T_q=T_{pq}$ and for each $x \in X$, $T_p(x)=p-lim_{s \in S}T_s(x)$.\
Clearly it is easy to see that, points $x$ and $y$ of $X$ are proximal near zero if and only if there is some $p \in O^{+}(S)$ such that $T_p(x)=T_p(y)$\
**Lemma 2.7.** Let S be a dense subsemigroup of $((0,\infty),+)$. Let $(X,\langle T_s\rangle _{s\in S})$ be a topological dynamical system and $L$ be a minimal left ideal of $O^{+}(S)$ and $x \in X$.
The following statements are equivalent :
\(a) The point $x$ is a uniformly recurrent point near zero of $(X,\langle T_s\rangle _{s\in S})$.
\(b) There exists $u \in L$ such that $T_u(x)=x$.
\(c) There exists $y \in X$ and an idempotent $u \in L$ such that $T_u(y)=x$.
(d)there exists an idempotent $u \in L$ such that $T_u(x)=x$.\
**Proof.** (a)$\Rightarrow$(b) Choose any $v \in L$. Let $N$ be a set of neighborhoods of $x$ in $X$. For each $U \in N$ let $B_U=\{ s \in S:T_s(x) \in U \}$. Since $x$ is uniformly recurrent point near zero, each $B_U$ is syndetic near zero, for every $\epsilon >0$ there is some $F_{\epsilon} \in P_f((0,\epsilon)\cap S)$ and some $\delta>0$ such that $S\cup(0,\delta)\subseteq \cup_t \in F_{\epsilon}(-t+B)$. So, for each $U \in N$ and $\epsilon>0$, pick $t_{(U,\epsilon)}\in F_{(U,\epsilon)}$ such that
$-t_{(U,\epsilon)}+B_U\in v$ Given $U \in N$ and $\epsilon > 0$, let
$C_{(U,\epsilon)}=\{ t_{(v,\epsilon)}: v\in N$ and $V\subseteq U \}$ and $C_U= \cup_{\epsilon >0}C_{(U,\epsilon)}$, then
$\{ C_U:U \in N \} \cup \{ S \cap (0,\epsilon): \epsilon>0 \}$ has the finite intersection property.
Now pick $w \in O^{+}(S)$ such that $\{C_U: U \in N \}\subseteq w$ and let $u=w+v$. Then $u \in L$ since $L$ is a left ideal of $O^{+}(S)$.
To see that $T_U(x)=x$, let $U \in N$. We need to show that $B_U \in u $, suppose instead that $B_U \not\in u$.
Then $\{ t \in S: -t+B_U \not\in v \}$ and $C_U \in w$ and so pick $t \in C_U$ such that, $-t+B_U\not\in v$. Pick $V\in N$ with $V\subseteq U$ such that $t=t_{(V,\epsilon)}$ for some $\epsilon >0$. Then $-t+B_v \in v$ and $-t+B_V \subseteq -t+B_v$, a contradiction.\
(b)$ \Rightarrow (c)$ Let $K=\{ v \in L:T_v(x)=x \}$. It suffices to show that $K$ is a compact subsemigroup of $L$, since then $K$ has an idempotent. By assumption, $K\neq \phi$ . Further if $v \in L \setminus K$, then there is some neighborhood $U$ of $x$ such that $B=\{ s \in S:T_s(x)\in U \}\not\in v$. Then Cl$B$ is a neighborhood of $v$ in $\beta S$ which misses $K$. Finally, to see that $K$ is a semigroup, let $v,w\in K$. Then by Remark 2.6, $T_{v+w}(x)=T_v(T_w(x))=T_v(x)=x$.\
(c)$\Rightarrow$(d)\
Again we use remark 2.6: $T_u(x)=T_u(T_u(y))=T_{u+u}(y)=T_u(y)=x$\
(d)$\Rightarrow$(a)\
Let $U$ be a neighborhood of $x$ and let $B=\{ s \in S:T_s(x)\in U\}$ and suppose that $B$ is not syndetic near zero. Then there exists $\epsilon >0$ such that $\{ S\setminus \cup_{t \in F}(-t+B)$: F is a finite nonempty subset of $S \cap (0,\epsilon) \}$ $\cup$ $ \{S \cap (0,\delta): 0<\delta < \infty\} $ has the finite intersection property. So pick some $w \in O^{+}(S)$ such that $\{ S\setminus \cup_{t \in F}(-t+B)$: F is a finite nonempty subset of $S \cup (0,\epsilon)\}\subseteq w$.\
Then $(O^{+}(S)+w)\cap ClB=\phi $(For suppose instead one had some $v \in O^{+}(S)$ with $B \in v+w$. Then pick some $t \in P_f(S \cap (0, \epsilon)$ with $-t+B \in w$ ).\
Let $L^{'}= O^{+}(S)+w$. Then $L^{'}$ is a left ideal of $O^{+}(S)$, so $L^{'}+u$ is a left ideal of $O^{+}(S)$ which is contained in $L$ , and hence $L^{'}+u=L$. Thus we may pick some $v \in L^{'}$ such that $v+u=u$. Again using Remark 2.6, $T_v(x)=T_v(T_u(x))=T_{v+u}(x)=T_u(x)=x$, so in particular $B \in v$. But, $v \in L^{'}$ and $L^{'}\cap ClB=\phi$, a contradiction.\
**Lemma 2.8.** Let $S$ be a dense a subsemigroup of $((0,\infty),+)$. Let, $(X,\langle T_s\rangle _{s \in S})$ be a topological dynamical system and let $x \in X$. Then for each $\epsilon > 0$ there is a uniformly recurrent point near zero $y \in Cl\{ T_s(x): s \in S \cap (0, \infty) \}$ such that $x$ and $y$ are proximal near zero.\
**Proof.** Let $L$ be any minimal left ideal of $O^{+}(S)$ and pick an idempotent $u \in L$. Let $y=T_u(x)$. For each $\epsilon > 0$, clearly $y \in Cl\{ T_s(x): s \in S \cap (0, \infty) \}$. By Lemma 2.7, $y$ is a uniformly recurrent point near zero of $(X,\langle T_s\rangle _{s \in S})$. By Remark 2.6 we have $T_u(y)=T_u(T_u(x))=T_{u+u}(x)=T_u(x)$ . So $x$ and $y$ are proximal near zero.\
**Lemma 2.9.** Let $S$ be a dense subsemigroup of $((0, \infty),+)$. Let $(X,\langle T_s\rangle _{s \in S})$ be a topological dynamical system and let $x,y \in X$. If $x$ and $y$ are proximal near zero, then there is a minimal left ideal $L$ of $O^{+}(S)$ such that $T_u(x)=T_u(y)$ for all $u \in L$\
**Proof.** Pick $v \in O^{+}(S)$ such that $T_v(x)=T_v(y)$ and pick a minimal left ideal $L$ of $O^{+}(S)$ such that $L \subseteq O^{+}(S)+v$. to see that $L$ is as required , let $u \in L$ and choose $w \in O^{+}(S)$ such that $u=w+v$. Then again using Remark 2.6, we have $T_u(x)=T_{w+v}(x)=T_w(T_v(x)=T_w(T_v(y))=T_{w+v}(y)=T_u(y)$\
**Lemma 2.10.** Let $S$ be a dense subsemigroup. Let $(X,\langle T_s\rangle {s \in S})$ be a topological dynamical system and let $x,y \in X$. There is an idempotent $u$ in $K(O^(S))$ such that $T_u(x)=y$ if and only if both $y$ is uniformly recurrent near zero and $x$ and $y$ are proximal near zero.\
**Proof.** $(\Rightarrow)$. Since $u$ is a minimal idempotent of $O^{+}(S)$, there is a minimal left ideal $L$ of $O^{+}(S)$ such that $u \in L$. Thus by Lemma 2.7, $y$ is uniformly recurrent near zero. By Remark 2.6, $T_u(y)=T_u(T_u(x))=T_{u+u}(x)=T_u(x)$. so $x$ and $y$ are proximal near zero.\
$(\Leftarrow)$ Pick by Lemma 2.9 a minimal ideal $L$ of $O^{+}(S)$ such that $T_u(x)=T_u(y)$ for all $u \in L$. Pick by Lemma 2.7 an idempotent $u \in L$ such that $T_u(y)=y$.\
We now give a dynamical Characterization of Central sets near zero in the following theorem.\
**Theorem 2.11.** Let $S$ be a dense subsemigroup of $((0, \infty),+)$ and let $B \subseteq S$. Then $B$ is central near zero if and only if there exists a topological dynamical system $(X,\langle T_s\rangle {s \in S})$ and there exists $x,y \in X$ and a neighborhood $U$ of $y$ such that $x$ and $y$ are proximal near zero , $y$ is uniformly recurrent near zero, and $B=\{ s \in S: T_s(x) \in U \}$.\
**Proof.** $(\Rightarrow)$ Let $G=S\cup \{0\}$ and $X=\prod_{s \in G}\{0,1\}$ and for $s \in S$ define $T_s:X\rightarrow X$ by $T_s(x)(t)=x(x+t)$ for all $t \in G$. It is easy to see that $T_s$ is continuous. Now let $x=\chi _{B}$, the characteristic function of $B$. That is, $x(t)=1$ if and only if $t \in B$. Pick a minimal idempotent in $O^{+}(S)$ such that $B \in u$ and let $y=T_u(x)$. Then by Lemma 2.10, $y$ is uniformly recurrent near zero and $x$ and $y$ are proximal near zero.\
Now let $U=\{ x \in X:z(0)=y(0) \}$. Then $U$ is a neighborhood of $y$ in $X$. We note that $y(0)=1$. Indeed, $y=T_u(x)$ so, $\{ s \in S:T_s(x) \in U \}\in u$ and we may choose some $s \in B$ such that $T_s(x) \in U$. Then $y(0)=T_s(x)(0)=x(s+0)=1$ Thus given any $s \in S$,\
$s \in B\Leftrightarrow x(s)=1 \Leftrightarrow T_s(x)(0)=1 \Leftrightarrow T_s(x)\in v$\
$(\Leftarrow)$ Choose a topological dynamical system $(X,\langle T_s\rangle {s \in S})$, points $x,y \in X$ and a neighborhood $U$ of $y$ such that $x$ and $y$ are proximal, $y$ is uniformly recurrent and $B= \{ s \in S : T_s(x) \in U \}$. Choose by Lemma 2.10 a minimal idempotent $u$ in $O^{+}(S)$ such that $T_u(x)=y$. Then $B \in u$\
Dynamical characterization of quasi-central set near zero
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**Definition 3.1.** Let $S$ be a dense subsemigroup of $((0, \infty), +)$. Then $C$ is said to be quasi central near zero if and only if there is an idempotent $p$ in $Cl\text{ }K(O^+(S))$ such that $C \in p$.\
Quasi-central sets near zero have some significant virtues. In the first place it satisfies conclusion of central sets theorem near zero. Secondly, in \[5\] and \[9\] combinatorial characterization of central set near zero and quasi-central set near zero are obtained respectively. The characterization of quasi-central set near zero is much simpler than the characterization of Central sets near zero.\
In this section we shall deduce the dynamical characterization of quasi-central set near zero.\
For this we need the following two definitions.\
**Definition 3.2.**(\[5, Definition $3.4$\]) Let $S$ be a dense subsemigroup of $((0,\infty), +)$. A subset $A$ of $S$ is piecewise syndetic near zero if and only if there exists sequences $\langle F_n\rangle_{n=1}^{\infty}$ and $\langle \delta_n\rangle_{n=1}^{\infty}$ such that\
(1) for each $n \in \mathbb{N}$, $F_n \in \mathcal{P}_f((0,\frac{1}{n}) \cap S)$ and $\delta_n \in (0, \frac{1}{n})$ and\
(2) for all $G \in \mathcal{P}_f(S)$ and $\mu > 0$ there is some $x \in (0,\mu) \cap S$ such that for all $n \in \mathbb{N}$, we have $$(G \cap (0,\delta_n)) + x \subseteq \cup_{t \in F_n}(t+A).$$\
As in the discrete case let us now introduce the notion of jointly interminittently uniform recurrent near zero.\
**Definition 3.3.** Let $(X, \langle T_s\rangle_{s\in S})$ be a dynamical system and let $x,y \in X$. The pair $(x,y)$ is jointly interminittently uniformly recurrent near zero (abbriviated as $JIUR_0$) if and only if for every neighbourhood $U$ of $y$, the set $\{s \in S : T_s(x) \in U \text{ and } T_s(y)\in U\}$ is piecewise syndetic near zero.\
For our purpose, we state the following theorem\[5, Theorem $3.5$\]\
**Theorem 3.4.** Let $S$ be a dense subsemigroup of $((0,\infty), +)$ and let $A \subseteq S$. Then $K(O^+(S)) \cap Cl \text{ }A \neq \phi$ if and only if $A$ is piecewise syndetic near zero.\
**Lemma 3.5.** Let $S$ be a dense subsemigroup of $((0, \infty), +)$ and $$\mathcal{K} = \{A \subseteq S : S \setminus A \text{ is not piecewise syndetic near zero}\}.$$ Then $\mathcal{K}$ is a filter on $S$ with $Cl \text{ } K(O^+(S)) = \overline{\mathcal{K}}$, which is a compact subsemigroup of $\beta S$.\
**Proof.** It is a routine exercise to show that $\mathcal{K}$ is non-empty, does not contain the empty set and is closed under super set. To show that $\mathcal{K}$ is a filter it is enough to prove that $\mathcal{K}$ is closed under finite intersection. Let $A, B \in \mathcal{K}$, then both $S \setminus A$ and $S \setminus B$ are piecewise syndetic near zero. Now we shall show that $A \cap B \in \mathcal{K}$ i.e. $S \setminus (A \cap B)$ is not piecewise syndetic near zero. If possible let $S \setminus (A \cap B)$ is piecewise syndetic near zero. So by Theorem $3.4$, there exists $p \in Cl \text{ }K(O^+(S))$ such that $S \setminus (A \cap B) \in p$. This $(S \setminus A) \cup (S \setminus B) \in p$ and therefore $S \setminus A \in p$ or $S \setminus B \in p$, which is a contradiction. So our claim is proved.
Observe that under the assumption that $\mathcal{K}$ is a filter, $$\mathcal{L}(\mathcal{K}) = \{A \subseteq S : A \text{ is piecewise syndetic near zero}\}$$ So by Theorem 1.6 and Theorem 3.4, we have $Cl \text{ }K(O^+(S)) = \overline{\mathcal{K}}$, which is a compact subsemigroup of $\beta S$.\
In the following theorem we shall give the dynamical characterization of quasi-central set near zero.\
**Theorem 3.6.** Let $S$ be a dense subsemigroup of $((0, \infty), +)$ and let $A \subseteq S$. The set $A$ is quasi-central near zero if and only if there exists a dynamical system $(X,\langle T_s \rangle_{s \in S})$, points $x$ and $y$ of $X$ such that $x, y$ are $JIUR_0$ and a neighbourhood $U$ of $y$ such that $$A = \{s \in S : T_s(x) \in U\}.$$\
**Proof.** Let $$\mathcal{K} = \{B \subseteq S : B \text{ is not a piecewise syndetic set near zero}\},$$ and note that $$\mathcal{L}(\mathcal{K}) = \{A \subseteq S : A \text{ is piecewise syndetic near zero}\}.$$ By Lemma 3.5, we have $\mathcal{K}$ is a filter and $\overline{\mathcal{K}} = Cl \text{ } K(O^+(S))$ which is a compact subsemigroup of $\beta S$. Now choose an idempotent $p$ in $\overline{\mathcal{K}} = Cl \text{ } K(O^+(S))$ such that $c \in p$. Now we can apply Theorem $1.8$ to prove our required statement.\
Dynamical characterization of $C$-set near zero
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We start by giving the combinatorial definitions of $C$-set near zero. As this combinatorial definition is rather complicated, we shall soon state an algebraic characterization showing that $C$-sets are members of idempotents in a certain compact subsemigroup.\
**Definition 4.1**(\[1, Definition 3.1\]) Let $S$ be a dense subsemigroup of $((0,\infty), +)$. The set of sequences in $S$ converging to $0$ is denoted by $\tau_0$.\
**Definition 4.2.**(\[1, Definition $3.6(a)$\]) Let $S$ be a dense subsemigroup of $((0,\infty),+)$ and let $A \subseteq S$. We say $A$ is a $C$-set near zero if and only if for each $\delta \in (0,1)$, there exists functions $a_{\delta} : \mathcal{P}_f(\tau_0) \to \mathcal{P}_f(\mathbb{N})$, such that\
(1) $\alpha_{\delta}(F) < \delta$ for each $F \in \mathcal{P}_f(\tau_0)$, (2) if $F, G \in \mathcal{P}_f(\tau_0)$ and $F \leq G$, then $\max H_{\delta}(F) < \min H_{\delta}(G)$ and (3) whenever $m \in \mathbb{N}$, $G_1,G_2, \cdots , G_m \in \mathcal{P}_f(\tau_0)$, $G_1 \subseteq G_2\subseteq \cdots \subseteq G_m$ and for each $i \in \{1,2, \cdots ,m\}$, $f_i \in G_i$, one has $$\sum_{i=1}^{m}\big(\alpha_{\delta}(G_i)+ \sum_{t \in H_{\delta}(G_i)}f_i(t)\big) \in A.$$\
We now recall \[1, Definition $3.2$\] and \[1, Definition $3.6(b)$\].\
**Definition 4.3.** Let $S$ be a dense subsemigroup of $((0, \infty),+)$ and let $A \subseteq S$. (1) $A$ is said to be $J$-set near zero if and only if whenever $F \in \mathcal{P}_f(\tau_0)$ and $\delta > 0$, there exists $a S \cap (0, \delta)$ and $H \in \mathcal{P}_f(\mathbb{N})$ such that for each $f \in F$, $a + \sum_{t \in H}f(t) \in A$. (2) $J_0(S) = \{p \in O^+(S) : \text{ for all } A \in p, \text{ is a J-set near zero}\}$.\
**Lemma 4.4.** Let $S$ be a dense subsemigroup of $((0, \infty),+)$ and $A_1$, $A_2$ are subsets of $S$. If $A_1 \cup A_2$ is $J$-set near zero then either $A_1$ or $A_2$ is a $J$-set near zero.\
**proof** See \[1, Lemma $3.8$\].\
**Definition 4.5.** Let $(X, \langle T_s\rangle_{s \in S})$ be a dynamical system and $x,y \in X$. The pair $(x,y)$ is jointly almost uniform recurrent (abbreviated $JIAUR_0$) if and only if for every neighbourhood $U$ of $y$, $\{s \in S : T_s(x) \in U \text{ and } T_s(y) \in U\}$ is a $J$-set near zero.\
**Lemma 4.6.** Let $S$ be a dense subsemigroup of $((0,\infty),+)$ and $$\mathcal{K}= \{A \subseteq S : S \setminus A \text{ is not a }J\text{-set near zero}\}.$$ Then $\mathcal{K}$ is a filter on $S$ with $J_0(S) = \overline{\mathcal{K}}$ and $J_0(S)$ is a compact subsemigroup of $\beta S$.\
**Proof.** It is easy to see that $\mathcal{K}$ is non-empty, does not contain the empty set and is closed under super sets. By Lemma 4.4, it follows that $\mathcal{K}$ is closed under finite intersection.
Under the assumption that $\mathcal{K}$ is a filter, we have $$\mathcal{L}(\mathcal{K}) = \{ A \subseteq S : A \text{ is a }J\text{-set}\}.$$ From Theorem $1.6$ it follows that $J_0(S) = \overline{\mathcal{K}}$. Finally, the fact that $J_0(S)$ is a subsemigroup of $\beta S$ follows from \[1, Theorem $3.9$\].\
**Lemma 4.7.** Let $S$ be a dense subsemigroup of $((0,\infty),+)$ and $A \subseteq S$. Then $A$ is a $C$-set near zero if and only if there exists an idempotent $p \in J_0(S)$ such that $A \in p$.\
**Proof** This is proved in \[1, Theorem $3.14$\].\
The following theorem gives a dynamical characterization of $C$-set near zero.\
**Theorem4.8** Let $S$ be a dense subsemigroup of $((0,\infty),+)$ and $A \subseteq S$. Then $A$ is a $C$-set near zero if and only if there exists a dynamical system $(X, \langle T_s \rangle_{s \in S})$ with points $x$ and $y$ in $X$ such that $x$, $y$ are $JIAUR_0$ and a neighbourhood $U$ of $y$ such that $A = \{s \in S : T_s(x) \in U\}$.\
**Proof** Let $\mathcal{K} = \{B \subseteq S : S \setminus B \text{ is not a }J \text{-set near zero}\}$. Since Lemma 4.7, characterizes $C$-set near zero in terms of idempotents in $\overline{\mathcal{K}}$, we can apply Theorem $1.8$ to prove our statement.\
**Acknowledgement.** The author is grateful to Prof. Swapan Kumar Ghosh of Ramakrishna Mission Vidyamandira for continuous inspiration and a number of valuable suggestions towards the improvement of the paper.
[3]{}
. E. Bayatmanesh and M. A. Tootkaboni, Central sets theorem near zero. Topology and its Applications, 210:70-80, 2016.\
. V. Bergelson and N. Hindman, Nonmetrizable topological dynamics and Ramsey theory, Trans. Amer. Math. Soc. 320 (1990), 293-320.\
. S. Burns and N. Hindman, Quasi-central sets and their dynamical characterization, Topology Proc. 31 (2007), 445-455.\
. H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, 1981.\
. N. Hindman and I. Leader, The semigroup of ultrafilters near 0, Semigroup Forum 59 (1999), 33-55.\
. N. Hindman, A. Maleki, and D. Strauss, Central sets and their combinatorial characterization, J. Comb. Theory (Series A) 74 (1996), 188-208.\
. N. Hindman and D. Strauss, Algebra in the Stone-$\breve{C}$ech compactification - theory and application, W.de Gruyter and Co.,Berlin, 1998.\
. John H.Johnson, A dynamical characterization of $C$-sets, https: //arxiv.org/abs/ 1112.0715\
. T. Bhattacharya, S.Chakraborty and S. K. Patra, Combined Algebraic Properties of $C^*$-sets near zero, Semigroup Forum (2019), 1-15.\
. H. Shi and H. Yang, Nonmetrizable topological dynamical characterization of central sets, Fund. Math. 150 (1996), 1-9.\
|
---
abstract: 'Counting the number of ones in a binary stream is a common operation in database, information-retrieval, cryptographic and machine-learning applications. Most processors have dedicated instructions to count the number of ones in a word (e.g., `popcnt` on x64 processors). Maybe surprisingly, we show that a vectorized approach using SIMD instructions can be twice as fast as using the dedicated instructions on recent Intel processors. The benefits can be even greater for applications such as similarity measures (e.g., the Jaccard index) that require additional Boolean operations. Our approach has been adopted by LLVM: it is used by its popular C compiler (Clang).'
address: '5800 Saint-Denis, Montreal (Quebec) H2S 3L5 Canada'
author:
- Wojciech Muła
- Nathan Kurz
- Daniel Lemire
bibliography:
- 'hamming.bib'
title: Faster Population Counts Using AVX2 Instructions
---
=1
Introduction
============
We can represent all sets of integers in $\{0,1,\ldots, 63\}$ using a single 64-bit word. For example, the word `0xAA` (`0b10101010`) represents the set $\{1,3,5,7\}$. Intersections and unions between such sets can be computed using a single bitwise logical operation on each pair of words (AND, OR). We can generalize this idea to sets of integers in $\{0,1,\ldots, n-1 \}$ using $\lceil n/64 \rceil $ 64-bit words. We call such data structures *bitsets*; they are also known as a bit vectors, bit arrays or bitmaps. Bitsets are ubiquitous in software, found in databases [@SPE:SPE2402], version control systems [@gitewah], search engines [@SPE:SPE2325; @SPE:SPE2326; @RoaringDocIdSetBlogPost], and so forth. Languages such as Java and C++ come with their own bitset classes (`java.util.BitSet` and `std::bitset` respectively).
The cardinality of a bitset (the number of one bits, each representing an element in the set) is commonly called a population count, a popcount, a Hamming weight, a sideways addition, or sideways sum. For example, the population counts of the words `0xFFFF`, `0xAA` and `0x00` are 16, 4 and 0 respectively. A frequent purpose for the population count is to determine the size of the intersection or union between two bitsets. In such cases, we must first apply a logical operation on pairs of words (AND, OR) and then compute the population count of the resulting words. For example, the cardinality of the intersection of the sets $A=\{4,5,6,7\}$ and $B=\{1,3,5,7\}$ represented by the words `0xF0` and `0xAA` can be computed as $\vert A \cap B \vert = \operatorname{popcount}(\texttt{0xF0} \operatorname{AND}\texttt{0xAA})=\operatorname{popcount}(\texttt{0xA0})=2$.
Population-count functions are used in cryptography [@hilewitz2004comparing], e.g., as part of randomness tests [@suciu2011never] or to generate pseudo-random permutations [@stefanov2012fastprp]. They can help find duplicated web pages [@Manku:2007:DNW:1242572.1242592]. They are frequently used in bioinformatics [@prokopenko2016utilizing; @Lacour2015; @li2012wham], ecology [@dambros2015effects], chemistry [@zhang2015design], and so forth. Gueron and Krasnov use population-count instructions as part of a fast sorting algorithm [@Gueron01012016].
The computation of the population count is so important that commodity processors have dedicated instructions: `popcnt` for x64 processors and `cnt` for the 64-bit ARM architecture.[^1] The x64 `popcnt` instruction is fast: on recent Intel processors, it has a throughput of one instruction per cycle [@fog2016instruction] (it can execute once per cycle) and a latency of 3 cycles (meaning that the result is available for use on the third cycle after execution). It is available in common C and C++ compilers as the intrinsic `_mm_popcnt_u64`. In Java, it is available as the `Long.bitCount` intrinsic.
Commodity PC processors also support Single-Instruction-Multiple-Data (SIMD) instructions. Starting with the Haswell microarchitecture (2013), Intel processors support the AVX2 instruction set which offers rich support for 256-bit vector registers. The contest between a dedicated instruction operating on 64-bits at a time (`popcnt`) and a series of vector instructions operating on 256-bits at a time (AVX2) turns out to be interesting. In fact, we show that we can achieve twice the speed of the an optimized `popcnt`-based function using AVX2: 0.52 versus 1.02 cycles per 8 bytes on large arrays. Our claim has been thoroughly validated: at least one major C compiler (LLVM’s Clang) uses our technique [@clangMula2015].
Thus, in several instances, SIMD instructions might be preferable to dedicated non-SIMD instructions if we are just interested in the population count of a bitset. But what if we seek the cardinality of the intersection or union, or simply the Jaccard index between two bitsets? Again, the AVX2 instructions prove useful, more than doubling the speed ($2.4\times$) of the computation against an optimized function using the `popcnt` instruction.
Existing Algorithms and Related Work
====================================
Conceptually, one could compute the population count by checking the value of each bit individually by calling `count += (word >> i) & 1` for `i` ranging from 0 to 63, given that `word` is a 64-bit word. While this approach scales linearly in the number of input words, we expect it to be slow since it requires multiple operations for each bit in each word. It is $O(n)$—the best that we can do—but with a high constant factor.
child [node\[fill=blue!10,draw\] [sum of bits 1 & 2]{} child [node\[fill=yellow!30,draw\] [1$^{\text{st}}$ bit]{} edge from parent\[<-,thick,>=latex\]]{} child [node\[fill=yellow!30,draw\] [2$^{\text{nd}}$ bit]{} edge from parent\[<-,thick,>=latex\]]{} edge from parent\[<-,thick,>=latex\] ]{} child [node\[fill=blue!10,draw\] [sum of bits 3 & 4]{} child [node\[fill=yellow!30,draw\] [3$^{\text{rd}}$ bit]{} edge from parent\[<-,thick,>=latex\]]{} child [node\[fill=yellow!30,draw\] [4$^{\text{th}}$ bit]{} edge from parent\[<-,thick,>=latex\]]{} edge from parent\[<-,thick,>=latex\] ]{};
Instead, we should prefer approaches with fewer operations per word. We can achieve the desired result with a tree of adders and bit-level parallelism. In Fig. \[fig:tree-of-adders\], we illustrate the idea over words of 4 bits (for simplicity). We implement this approach with two lines of code.
1. We can sum the individual bits to 2-bit subwords with the line of C code: `( x & 0b0101 ) + ( ( x >> 1 ) & 0b0101 )`. This takes us from the bottom of the tree to the second level. We say to this step exhibits *bit-level parallelism* since two sums are executed at once, within the same 4-bit word.
2. We can then sum the values stored in the 2-bit subwords into a single 4-bit subword with another line of C code: `( x & 0b0011 ) + ( ( x >> 2 ) & 0b0011 )`.
Fig. \[fig:tradd\] illustrates a non-optimized (naive) function that computes the population count of a 64-bit word in this manner.
[c]{}
uint64_t c1 = UINT64_C(0x5555555555555555);
uint64_t c2 = UINT64_C(0x3333333333333333);
uint64_t c4 = UINT64_C(0x0F0F0F0F0F0F0F0F);
uint64_t c8 = UINT64_C(0x00FF00FF00FF00FF);
uint64_t c16 = UINT64_C(0x0000FFFF0000FFFF);
uint64_t c32 = UINT64_C(0x00000000FFFFFFFF);
uint64_t count(uint64_t x) {
x = (x & c1) + ((x >> 1) & c1);
x = (x & c2) + ((x >> 2) & c2);
x = (x & c4) + ((x >> 4) & c4);
x = (x & c8) + ((x >> 8) & c8);
x = (x & c16)+ ((x >> 16)) & c16);
return (x & c32) + ((x >> 32) & c32);
}
[c]{}
``` {style="customc"}
uint64_t c1 = UINT64_C(0x5555555555555555);
uint64_t c2 = UINT64_C(0x3333333333333333);
uint64_t c4 = UINT64_C(0x0F0F0F0F0F0F0F0F);
uint64_t count(uint64_t x) {
x -= (x >> 1) & c1;
x = (( x >> 2) & c2) + (x & c2);
x = ( x + (x >> 4) ) & c4;
x *= UINT64_C(0x0101010101010101);
return x >> 56;
}
```
A fast and widely used tree-of-adder function to compute the population count has been attributed by Knuth [@KnuthV4] to a 1957 textbook by Wilkes, Wheeler and Gill [@Wilkes]: see Fig. \[fig:wwg\]. It involves far fewer than 64 instructions and we expect it to be several times faster than a naive function checking the values of each bit and faster than the naive tree-of-adder approach on processor with a sufficiently fast 64-bit integer multiplication (which includes all x64 processors).
- The first two lines in the `count` function correspond to the first two levels of our simplified tree-of-adders `count` function from Fig. \[fig:tree-of-adders\]. The first line has been optimized. We can verify the optimization by checking that for each possible 2-bit word, we get the sum of the bit values:
- `0b11 - 0b01 = 0b10 = 2`,
- `0b10 - 0b01 = 0b01 = 1`,
- `0b01 - 0b00 = 0b01 = 1`,
- `0b00 - 0b00 = 0b00 = 0`.
- After the first two lines, we have 4-bit population counts (in $\{0b0000,0b0001,0b0010,0b0011,0b0100\}$) stored in 4-bit subwords. The next line sums consecutive 4-bit subwords to bytes. We use the fact that the most significant bit of each 4-bit subword is zero.
- The multiplication and final shift sum all bytes in an efficient way. Multiplying `x` by `0x0101010101010101` is equivalent to summing up `x`, `x << 8`, `x << 16`, …, `x << 56`. The total population count is less than 64, so that the sum of all bytes from `x` fits in a single byte value (in $[0,256)$). In that case, the most significant 8 bits from the product is the sum of all eight byte values.
Knuth also attributes another common technique to Wegner [@Wegner] (see Fig. \[fig:wegner\]) that could be competitive when the population count is relatively low (e.g., less than 4 one bit per 64-bit word). When the population count is expected to be high (e.g., more than 60 one bit per 64-bit words), one could simply negate the words prior to using the function so as to count the number of zeros instead. The core insight behind the Wegner function is that the line of C code `x &= x - 1` sets to zero the least significant bit of `x`, as one can readily check. On an x64 processor, the expression `x &= x - 1` might be compiled to the `blsr` (reset lowest set bit) instruction. On current generation processors, this instruction achieves a throughput of two instructions per cycle with a latency of one cycle [@fog2016instruction]. The downside of the Wegner approach for modern processors is that the unpredictable loop termination adds a mispredicted branch penalty of at least 10 cycles [@Rohou:2015:BPP:2738600.2738614], which for short loops can be more expensive than the operations performed by the loop.
[c]{}
int count(uint64_t x) {
int v = 0;
while(x != 0) {
x &= x - 1;
v++;
}
return v;
}
Another simple and common technique is based on tabulation. For example, one might create a table that contains the corresponding population count for each possible byte value, and then look up and sum the count for each byte. Such a table would require only 256 bytes. A population count for a 64-bit word would require only eight table look-ups and seven additions. On more powerful processor, with more cache, it might be beneficial to create a larger table, such as one that has a population count for each possible `short` value (2 bytes) using 64KB. Each doubling of the bit-width covered by the table halves the number of table lookups, but squares the memory required for the table.
We can improve the efficiency of tree-of-adders techniques by *merging* the trees across words [@Lauradoux]. To gain an intuition for this approach, consider that in the Wilkes-Wheeler-Gill approach, we use 4-bit subwords to store the population count of four consecutive bits. Such a population count takes a value in $\{0,1,2,3,4\}$, yet a 4-bit integer can represent all integers in $[0,16)$. Thus, as a simple optimization, we could accumulate the 4-bit counts across three different words instead of a single one. Next consider that if you sum two 4-bit subwords (representing integers in $[0,16)$) the result is in $[0,32)$ whereas an 8-bit subword (a byte) can represent all integers in $[0,256)$, a range that is four times larger. Hence, we can accumulate the counts over four triple of words. These two optimizations combined lead to a function to compute the population count of twelve words at once (see Fig. \[fig:lauradoux\]) faster than would be possible if we processed each word individually.
[c]{}
uint64_t count(uint64_t *input) {
uint64_t m1 = UINT64_C(0x5555555555555555);
uint64_t m2 = UINT64_C(0x3333333333333333);
uint64_t m4 = UINT64_C(0x0F0F0F0F0F0F0F0F);
uint64_t m8 = UINT64_C(0x00FF00FF00FF00FF);
uint64_t m16= UINT64_C(0x0000FFFF0000FFFF);
uint64_t acc = 0;
for (int j = 0; j < 12; j += 3) {
uint64_t count1 = input[j + 0];
uint64_t count2 = input[j + 1];
uint64_t half1 = input[j + 2];
uint64_t half2 = input[j + 2];
half1 &= m1;
half2 = (half2 >> 1) & m1;
count1 -= (count1 >> 1) & m1;
count2 -= (count2 >> 1) & m1;
count1 += half1;
count2 += half2;
count1 = (count1 & m2)
+ ((count1 >> 2) & m2);
count1 += (count2 & m2)
+ ((count2 >> 2) & m2);
acc += (count1 & m4)
+ ((count1 >> 4) & m4);
}
acc = (acc & m8) + ((acc >> 8) & m8);
acc = (acc + (acc >> 16)) & m16;
acc = acc + (acc >> 32);
return acc;
}
However, even before Lauradoux proposed this improved function, Warren [@warren2007] had presented a superior alternative attributed to a newsgroup posting from 1997 by Seal, inspired from earlier work by Harley. This approach, henceforth called Harley-Seal, is based on a carry-save adder (CSA). Suppose you are given three bit values ($a,b,c\in \{0,1\}$) and you want to compute their sum ($a+b+c\in \{0,1,2,3\}$). Such a sum fits in a 2-bit word. The value of the least significant bit is given by $(a\operatorname{XOR}b) \operatorname{XOR}c$ whereas the most significant bit is given by $(a \operatorname{AND}b) \operatorname{OR}( (a \operatorname{XOR}b) \operatorname{AND}c )$. Table \[table:sum\] illustrates these expressions: the least significant bit ($(a\operatorname{XOR}b) \operatorname{XOR}c$) takes value 1 only when $a+b+c$ is odd and the most significant bit takes value 1 only when two or three of the input bits ($a,b,c$) are set to 1. There are many possible expressions to compute the most significant bit, but the chosen expression is convenient because it reuses the $a \operatorname{XOR}b$ expression from the computation of the least significant bit. Thus, we can sum three bit values to a 2-bit counter using 5 logical operations. We can generalize this approach to work on all 64-bits in parallel. Starting with three 64-bit input words, we can generate two new output words: $h$, which holds the 64 most significant bits, and $l$, which contains the corresponding 64 least significant bits. We effectively compute 64 sums in parallel using bit-level parallelism. Fig. \[fig:csa\] presents an efficient implementation in C of this idea. The function uses 5 bitwise logical operations (two XORs, two ANDs and one OR): it is optimal with respect to the number of such operations [@KnuthV4A 7.1.2]. However, it requires at least three cycles to complete due to data dependencies.
[c]{}
void CSA(uint64_t* h, uint64_t* l,
uint64_t a, uint64_t b, uint64_t c) {
uint64_t u = a ^ b;
*h = (a & b) | (u & c);
*l = u ^ c;
}
[ccc|c|x[1.5cm]{}x[1.5cm]{}]{} $a$ & $b$ & $c$ & $a+b+c$ & $(a\oplus b) \oplus c$ & $(a \land b) \lor \allowbreak ( (a \oplus b) \land c )$\
0 & 0 & 0 & 0 & 0 & 0\
0 & 0 & 1 & 1 & 1 & 0\
0 & 1 & 0 & 1 & 1 & 0\
1 & 0 & 0 & 1 & 1 & 0\
0 & 1 & 1 & 2 & 0 & 1\
1 & 0 & 1 & 2 & 0 & 1\
1 & 1 & 0 & 2 & 0 & 1\
1 & 1 & 1 & 3 & 1 & 1\
=\[minimum height=0.7cm,minimum width=1cm\]
(A3) \[on chain=9,mytape\] […]{}; (A4) \[on chain=9,mytape\] ; (A5) \[on chain=9,mytape\] ; (A6) \[on chain=9,mytape\] ; (A7) \[on chain=9,mytape\] ; (csa1)\[on chain=10,fill=gray!30,minimum width=3cm,below=0.5cm of A6\] [CSA]{}; (A8) \[on chain=10,mytape\] ; (A9) \[on chain=10,mytape\] ; (A5.south) to (A5.south |- csa1.north); (A6.south) to (A6.south |- csa1.north); (A7.south) to (A7.south |- csa1.north); (csa2)\[on chain=11,fill=gray!30,minimum width=3cm,below=0.5cm of A8\] [CSA]{}; (A10) \[on chain=11,mytape\] […]{};
($(csa1.south east)!0.22!(csa1.south)$) to\[out=270,in=90\] ($(csa2.north west)!0.35!(csa2.north)$); (A8.south) to (A8.south |- csa2.north); (A9.south) to (A9.south |- csa2.north);
(csa3)\[on chain=12,below=1cm of A10,minimum width=4cm\] ; (csa12)\[fill=gray!30,minimum width=3cm,left=0.5cm of csa3\] [CSA]{};
(A4.south) to\[out=270,in=90\] ($(csa12.north west)!0.25!(csa12.north)$);
(foursoutput) \[on chain=13,mytape,below=1cm of $(csa12.south west)!0.35!(csa12.south)$,minimum width=2cm\] ;
($(csa12.south west)!0.35!(csa12.south)$) to\[out=270,in=90\] (foursoutput.north);
(twosoutput) \[minimum width=2cm,below=1cm of $(csa12.south east)!0.35!(csa12.south)$\] ;
($(csa12.south east)!0.35!(csa12.south)$) to\[out=270,in=90\] (twosoutput.north);
(onesoutput) \[mytape,minimum width=2cm,below=1.5cm of $(csa2.south east)!0.35!(csa2.south)$\] ;
($(csa2.south east)!0.35!(csa2.south)$) to\[out=270,in=90\] (onesoutput.north);
($(csa2.south west)!0.35!(csa2.south)$) to\[out=270,in=90\] ($(csa12.north east)!0.35!(csa12.north)$);
($(csa1.south west)!0.35!(csa1.south)$) to\[out=270,in=90\] (csa12.north);
From such a CSA function, we can derive an efficient population count. Suppose we start with three words serving as counters (initialized at zero): one for the least significant bits (henceforth `ones`), another one for the second least significant bits (`twos`, so named because each bit set represents 2 input bits), and another for the third least significant bits (`fours`, representing 4 input bits). We can proceed as follows; the first few steps are illustrated in Fig. \[fig:hsillustration\]. We start with a word serving as a population counter $c$ (initialized at zero). Assume with we have a number of words $d_1,d_2,\ldots$ divisible by 8. Start with $i=0$.
- Load two new words ($d_i,d_{i+1}$). Use the CSA function to sum `ones`, $d_i$ and $d_{i+1}$, write the least significant bit of the sum to `ones` and store the carry bits in a temporary register (noted `twosA`). We repeat with the next two input words. Load $d_{i+2},d_{i+3}$, use the CSA function to sum `ones`, $d_i$ and $d_{a+i}$, write the least significant bit of the sum to `ones` and store the carry bits in a temporary register (noted `twosB`).
- At this point, we have three words containing second least significant bits (`twos`, `twosA`, `twosB`). We sum them up using a CSA, writing back the result to `twos` and the carry bits to a temporary register `foursA`.
- We do with $d_{i+4},d_{i+5}$ and $d_{i+6},d_{i+7}$ as we did with $d_i,d_{i+1}$ and $d_{i+2},d_{i+3}$. Again we have three words containing second least significant bits (`twos`, `twosA`, `twosB`). We sum them up with CSA, writing the result to `twos` and to a carry-bit temporary register `foursB`.
- At this point, we have three words containing third least significant bits (`fours`, `foursA`, `foursB`). We can sum them up with a CSA, write the result back to `fours`, storing the carry bits in a temporary register `eights`.
- We compute the population count of the word `eights` (e.g, using the Wilkes-Wheeler-Gill population count) and increment the counter $c$ by the population count.
- Increment $i$ by 8 and continue for as long as we have new words.
When the algorithm terminates, multiply $c$ by 8. Compute the population count of `fours`, multiply the result by 4 and add to $c$. Do similarly with `twos` and `ones`. The counter $c$ contains the population count. If the number of input words is not divisible by 8, adjust accordingly with the leftover words (e.g, using the Wilkes-Wheeler-Gill population count).
In that particular implementation of this idea, we used blocks of eight words. More generally, the Harley-Seal approach works with blocks of $2^n$ words for $n=3,4,5, \ldots$ (8, 16, 32, …). We need $2^n-1$ CSA function calls when using $2^n$ words, and one call to an auxiliary function (e.g., Wilkes-Wheeler-Gill). If we expect the auxiliary function to be significantly more expensive than the CSA function calls, then larger blocks should lead to higher performance, as long as we have enough input data and many available registers. In practice, we found that using blocks of sixteen words works well on current processors (see Fig. \[fig:harleyseal16\]). This approach is only worthwhile if we have at least 16 input words (64-bits/word $\times$ 16 words $=128$ bytes).
[c]{}
uint64_t harley_seal(uint64_t * d,
size_t size) {
uint64_t total = 0, ones = 0, twos = 0,
fours = 0, eights = 0, sixteens = 0;
uint64_t twosA, twosB, foursA, foursB, eightsA, eightsB;
for(size_t i = 0; i < size - size % 16;
i += 16) {
CSA(&twosA, &ones, ones, d[i+0], d[i+1]);
CSA(&twosB, &ones, ones, d[i+2], d[i+3]);
CSA(&foursA, &twos, twos, twosA, twosB);
CSA(&twosA, &ones, ones, d[i+4], d[i+5]);
CSA(&twosB, &ones, ones, d[i+6], d[i+7]);
CSA(&foursB, &twos, twos, twosA, twosB);
CSA(&eightsA, &fours, fours, foursA, foursB);
CSA(&twosA, &ones, ones, d[i+8], d[i+9]);
CSA(&twosB, &ones, ones, d[i+10],d[i+11]);
CSA(&foursA, &twos, twos, twosA, twosB);
CSA(&twosA, &ones, ones, d[i+12],d[i+13]);
CSA(&twosB, &ones, ones, d[i+14],d[i+15]);
CSA(&foursB, &twos, twos, twosA, twosB);
CSA(&eightsB, &fours, fours, foursA,
foursB);
CSA(&sixteens, &eights, eights, eightsA,
eightsB);
total += count(sixteens);
}
total = 16 * total + 8 * count(eights)
+ 4 * count(fours) + 2 * count(twos)
+ count(ones);
for(size_t i = size - size % 16 ; i < size; i++)
total += count(d[i]);
return total;
}
The functions we presented thus far still have their uses when programming with high-level languages without convenient access to dedicated functions (e.g., JavaScript, Go) or on limited hardware. However, they are otherwise obsolete when a sufficiently fast instruction is available, as is the case on recent x64 processors with `popcnt`. The `popcnt` instruction has a reciprocal throughput[^2] of one instruction per cycle. With a properly constructed loop, the load-`popcnt`-add sequence can be executed in a single cycle, allowing for a population count function that processes 64-bits per cycle.
Existing Vectorized Algorithms
------------------------------
To our knowledge, the first published vectorized population count on Intel processor was proposed by Muła in 2008 [@Mula2008]. It is a vectorized form of tabulation on 4-bit subwords. Its key ingredient is the SSSE3 vector instruction `pshufb` (see Table \[ref:sseinstructions\]). The `pshufb` instruction shuffles the input bytes into a new vector containing the same byte values in a (potentially) different order. It takes an input register $v$ and a control mask $m$, treating both as vectors of sixteen bytes. Starting from $v_0, v_1, \ldots, v_{16}$, it outputs a new vector $(v_{m_0},v_{m_1},v_{m_2},v_{m_3}, \ldots, v_{m_{15}})$ (assuming that $0 \leq m_i < 16$ for $i=0,1,\ldots, 15$). Thus, for example, if the mask $m$ is $0,1,2,\ldots, 15$, then we have the identify function. If the mask $m$ is $15, 14, \ldots, 0$, then the byte order is reversed. Bytes are allowed to be repeated in the output vector, thus the mask $0,0,\ldots, 0$ would produce a vector containing only the first input byte, repeated sixteen times. It is a fast instruction with a reciprocal throughput and latency of one cycle on current Intel processors, yet it effectively “looks up” 16 values at once. In our case, we use a fixed input register made of the input bytes $0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4$ corresponding to the population counts of all possible 4-bit integers $0,1,2,3,\ldots, 15$. Given an array of sixteen bytes, we can call `pshufb` once, after selecting the least significant 4 bits of each byte (using a bitwise AND) to gather sixteen population counts on sixteen 4-bit subwords. Next, we right shift by four bits each byte value, and call `pshufb` again to gather sixteen counts of the most significant 4 bits of each byte. We can sum the two results to obtain sixteen population counts, each corresponding to one of the sixteen initial byte values. See Fig. \[fig:mula\] for a C implementation. If we ignore loads and stores as well as control instructions, Muła’s approach requires two `pshufb`, two `pand`, one `paddb`, and one `psrlw` instruction, so six inexpensive instructions to compute the population counts of sixteen bytes. The Muła algorithm requires fewer instructions than the part of Wilkes-Wheel-Gill that does the same work (see Fig. \[fig:wwg\]), but works on twice as many input bytes per iteration.
[c]{}
__m128i count_bytes(__m128i v) {
__m128i lookup = _mm_setr_epi8(0,1,1,2,1,2,2,3,1,2,2,3,2,3,3,4);
__m128i low_mask = _mm_set1_epi8(0x0f);
__m128i lo = _mm_and_si128(v, low_mask);
__m128i hi = _mm_and_si128(
_mm_srli_epi16(v, 4), low_mask);
__m128i cnt1 =
_mm_shuffle_epi8(lookup, lo);
__m128i cnt2 =
_mm_shuffle_epi8(lookup, hi);
return _mm_add_epi8(cnt1, cnt2);
}
The `count_bytes` function from Fig. \[fig:mula\] separately computes the population count for each of the sixteen input bytes, storing each in a separate byte of the result. As each of these bytes will be in $[0,8]$, we can sum the result vectors from up to 31 calls to `count_bytes` using the `_mm_add_epi8` intrinsic without risk of overflow before using the `psadbw` instruction (using the `_mm_sad_epu8` intrinsic) to horizontally sum the individual bytes into two 64-bit counters. In our implementation, we found it adequate to call the `count_bytes function` eight times between each call to `psadbw`.
Morancho observed that we can use both a vector approach, like Muła’s, and the `popcnt` in a hybrid approach [@6787256]. Morancho proposed a family of hybrid schemes that could be up to 22% faster than an implementation based on `popcnt` for sufficiently large input arrays.
instruction C intrinsic description latency
------------- -------------------- ------------------------------------------------------------------------------------------- --------- ------
`paddb` `_mm_add_epi8` add sixteen pairs of 8-bit integers 1 0.5
`pshufb` `_mm_shuffle_epi8` *shuffle* sixteen bytes 1 1
`psrlw` `_mm_srli_epi16` shift right eight 16-bit integers 1 1
`pand` `_mm_and_si128` 128-bit AND 1 0.33
`psadbw` `_mm_sad_epu8` sum of the absolute differences of the byte values to the low 16 bits of each 64-bit word 5 1
Novel Vectorized Algorithms
===========================
Starting with the Haswell microarchiture released in 2013, Intel processors support the AVX2 instruction set with 256-bit vectors, instead of the shorter 128-bit vectors. It supports instructions and intrinsics that are analogous to the SSE intrinsics (see Table \[ref:sseinstructions\]).[^3]
The Muła function provides an effective approach to compute population counts at a speed close to an x64 processor’s `popcnt` instruction when using 128-bit vectors, but after upgrading to AVX2’s 256-bit vectors, it becomes faster than functions using the `popcnt` instruction. We present the basis of such a function in Fig. \[fig:avxmula\] using AVX2 intrinsics; the AVX2 intrinsics are analogous to the SSE intrinsics (see Fig. \[ref:sseinstructions\]). It returns a 256-bit word that can be interpreted as four 64-bit counts (each having value in $[0,64]$). We can then add the result of repeated calls with the `_mm256_add_epi64` intrinsic to sum 64-bit counts.
[c]{}
__m256i count(__m256i v) {
__m256i lookup =
_mm256_setr_epi8(0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4);
__m256i low_mask = _mm256_set1_epi8(0x0f);
__m256i lo = = _mm256_and_si256(v, low_mask);
__m256i hi = _mm256_and_si256(_mm256_srli_epi32(v, 4), low_mask);
__m256i popcnt1 = _mm256_shuffle_epi8(lookup, lo);
__m256i popcnt2 = _mm256_shuffle_epi8(lookup, hi);
__m256i total = _mm256_add_epi8(popcnt1,popcnt2);
return _mm256_sad_epu8(total, _mm256_setzero_si256());
}
For a slight gain in performance, we can call the Muła function several times while skipping the call to `_mm256_sad_epu8`, adding the byte values with `_mm256_add_epi8` before calling `_mm256_sad_epu8` once. Each time we call the Muła function, we process 32 input bytes and get 32 byte values in $[0,8]$. We can add sixteen totals before calling `_mm256_sad_epu8` to sum the results into four 64-bit words (since $8\times 16 =128 < 2^8$), thus processing a block of 512 bytes per call.[^4]
Of all the non-vectorized (or *scalar*) functions, Harley-Seal approaches are fastest. Thus we were motivated to port the approach to AVX2. The carry-save adder that worked on 64-bit words (see Fig. \[fig:csa\]) can be adapted in a straight-forward manner to work with AVX2 intrinsics (see Fig. \[fig:avxcsa\]).
[c]{}
void CSA(__m256i* h, __m256i* l, __m256i a, __m256i b, __m256i c) {
__m256i u = _mm256_xor_si256(a , b);
*h = _mm256_or_si256(_mm256_and_si256(a , b) , _mm256_and_si256(u , c) );
*l = _mm256_xor_si256(u , c);
}
Fig. \[fig:avxhs\] presents an efficient Harley-Seal function using an AVX2 carry-save adder. The function processes the data in blocks of sixteen 256-bit vectors (512B). It calls Muła’s AVX2 function (see Fig. \[fig:avxmula\]).
[c]{}
uint64_t avx_hs(__m256i* d, uint64_t size) {
__m256i total = _mm256_setzero_si256();
__m256i ones = _mm256_setzero_si256();
__m256i twos = _mm256_setzero_si256();
__m256i fours = _mm256_setzero_si256();
__m256i eights = _mm256_setzero_si256();
__m256i sixteens = _mm256_setzero_si256();
__m256i twosA, twosB, foursA, foursB,
eightsA, eightsB;
for(uint64_t i = 0; i < size; i += 16) {
CSA(&twosA, &ones, ones, d[i], d[i+1]);
CSA(&twosB, &ones, ones, d[i+2], d[i+3]);
CSA(&foursA, &twos, twos, twosA, twosB);
CSA(&twosA, &ones, ones, d[i+4], d[i+5]);
CSA(&twosB, &ones, ones, d[i+6], d[i+7]);
CSA(&foursB,& twos, twos, twosA, twosB);
CSA(&eightsA,&fours, fours, foursA,foursB);
CSA(&twosA, &ones, ones, d[i+8], d[i+9]);
CSA(&twosB, &ones, ones, d[i+10],d[i+11]);
CSA(&foursA, &twos, twos, twosA, twosB);
CSA(&twosA, &ones, ones, d[i+12],d[i+13]);
CSA(&twosB, &ones, ones, d[i+14],d[i+15]);
CSA(&foursB, &twos, twos, twosA, twosB);
CSA(&eightsB, &fours, fours, foursA, foursB);
CSA(&sixteens, &eights, eights, eightsA, eightsB);
total=_mm256_add_epi64(total, count(sixteens));
}
total = _mm256_slli_epi64(total, 4);
total = _mm256_add_epi64(total,
_mm256_slli_epi64(count(eights), 3));
total = _mm256_add_epi64(total,
_mm256_slli_epi64(count(fours), 2));
total = _mm256_add_epi64(total,
_mm256_slli_epi64(count(twos), 1));
total =_mm256_add_epi64(total,count(ones));
return _mm256_extract_epi64(total, 0)
+ _mm256_extract_epi64(total, 1)
+ _mm256_extract_epi64(total, 2)
+ _mm256_extract_epi64(total, 3);
}
Processors execute complex machine instructions using low-level instructions called [[$\mu$op]{}s]{}.
- Using the dedicated `popcnt` instruction for the population of an array of words requires loading the word (`movq`), counting the bits (`popcnt`), and then adding the result to the total (`addq`). The load and the `popcnt` can be combined into a single assembly instruction, but internally they are executed as separate µops, and thus each 64-bit word requires three [[$\mu$op]{}s]{}. Apart from minimal loop overhead, these three operations can be executed in a single cycle on a modern x64 superscalar processor, for a throughput of just over one cycle per 8B word.
- The AVX2 Harley-Seal function processes sixteen 256-bit vectors (512B) with 98 [[$\mu$op]{}s]{}: 16 loads (`vpmov`), 32 bitwise ANDs (`vpand`), 15 bitwise ORs (`vpor`), and 30 bitwise XORs (`vpxor`). Each 64-bit word (8B) thus takes just over 1.5 [[$\mu$op]{}s]{}—about half as many as required to use the builtin `popcnt` instruction on the same input.
While fewer [[$\mu$op]{}s]{} does does not guarantee faster execution, for computationally intensive tasks such as this it often proves to be a significant advantage. In this case, we find that it does in fact result in approximately twice the speed.
Beyond population counts
========================
In practice, we often want to compute population counts on the result of some operations. For example, given two bitsets, we might want to determine the cardinality of their intersection (computed as the bit-wise logical AND) or the cardinality of their union (computed as the bit-wise logical OR). In such instances, we need to load input bytes from the two bitsets, generate a temporary word, process it to determine its population count, and so forth. When computing the Jaccard index, given that we have no prior knowledge of the population counts, we need to compute both the intersection and the union, and then we need to compute the two corresponding population counts (see Fig. \[fig:popcntjaccard\]).
[c]{}
void popcnt_jaccard_index(uint64_t* A, uint64_t* B, size_t n) {
double s = 0;
double i = 0;
for(size_t k = 0; k < n; k++) {
s += _mm_popcnt_u64(A[k] | B[k]);
i += _mm_popcnt_u64(A[k] & B[k]);
}
return i / s;
}
Both loads and logical operations benefit greatly from vectorization, and hybrid scalar/vector approaches can be difficult because inserting and extracting elements into and from vectors adds overhead. With AVX2, in one operation, we can load a 256-bit register or compute the logical AND between two 256-bit registers. This is four times the performance of the corresponding 64-bit operations. Thus we can expect good results from fast population count functions based on AVX2 adapted for the computation of the Jaccard index, the cardinality of the intersection or union, or similar operations.
Experimental Results
====================
We implemented our software in C. We use a Linux server with an Intel i7-4770 processor running at . This Haswell processor has of L1 cache and of L2 cache per core with of L3 cache. The machine has of RAM (DDR3-1600 with double-channel). We disabled Turbo Boost and set the processor to run at its highest clock speed. Our software is freely available (<https://github.com/CountOnes/hamming_weight>) and was compiled using the GNU GCC 5.3 compiler with the “`-O3 -march=native`” flags.
For all experiments, we use randomized input bits. However, we find that the performance results are insensitive to the data values.
Table \[tab:speed\] presents our results in number of cycles per word, for single-threaded execution. To make sure our results are reliable, we repeat each test 500 times and check that the minimum and the average cycle counts are within 1% of each other. We report the minimum cycle count divided by the number of words in the input. All the scalar methods (WWG, Laradoux, and HS) are significantly slower than the native `popcnt`-based function. We omit tabulation-based approaches from Table \[tab:speed\] because they are not competitive: 16-bit tabulation uses over 5 cycles even for large arrays. We can see that for inputs larger than 4kB, the AVX2-based Harley-Seal approach is twice as fast as our optimized `popcnt`-based function, while for small arrays (fewer 64 words) the `popcnt`-based function is fastest.
[c|cccccc]{} array size & WWG & Lauradoux & HS & `popcnt` & AVX2 Muła & AVX2 HS\
\
256B& 6.00 & 4.50 &3.25 & **1.12** & 1.38 & —\
512B& 5.56 & 2.88 &2.88 & 1.06 & **0.94** & —\
1kB& 5.38 & 3.62 & 2.66 & 1.03 & 0.81 & **0.69**\
2kB& 5.30 & 3.45 & 2.55 & 1.01 & 0.73 & **0.61**\
4kB& 5.24 & 3.41 & 2.53 & 1.01 & 0.70 & **0.54**\
8kB& 5.24 & 3.36 & 2.42 & 1.01 & 0.69 & **0.52**\
16kB& 5.22 & 3.36 & 2.40 & 1.01 & 0.69 & **0.52**\
32kB& 5.23 & 3.34 & 2.40 & 1.01 & 0.69 & **0.52**\
64kB& 5.22 & 3.34 & 2.40 & 1.01 & 0.69 & **0.52**\
We present the results for Jaccard index computations in Table \[tab:speedji\]. Contrary to straight population counts, the Jaccard-index AVX2 Muła remains faster than the `popcnt`-based function even for small blocks (256B). AVX2 HS provides the best speed, requiring only 1.15 cycles to calculate the Jaccard similarity between each pair of 64-bit inputs. This is more than twice as fast ($2.4\times$) as the `popcnt`-based function. Since the population count is done for each input of the pair, the speed of the similarity is only slightly greater than the speed of calculating the two population counts individually. That is, using AVX2 for both the Boolean operation and both population counts gives us the Boolean operation almost for free.
[c|ccc]{} array size & `popcnt` & AVX2 Muła & AVX2 HS\
\
256B & 3.00 & **2.50** & —\
512B & 2.88 & **2.00** & —\
1kB& 2.94 & 2.00 & **1.53**\
2kB& 2.83 & 1.84 & **1.33**\
4kB& 2.80 & 1.76 & **1.22**\
8kB& 2.78 & 1.75 & **1.16**\
16kB& 2.77 & 1.75 & **1.15**\
32kB& 2.76 & 1.75 & **1.15**\
64kB& 2.76 & 1.74 & **1.15**\
Conclusion
==========
On recent Intel processors, the fastest approach to compute the population count on moderately large arrays (e.g., 4kB) relies on a vectorized version of the Harley-Seal function. It is twice as fast as functions based on the dedicated instruction (`popcnt`). For the computation of similarity functions between two bitsets, a vectorized approach based on the Harley-Seal function is more than twice as fast ($2.4\times$) as an optimized approach based on the `popcnt` instruction.
Future work should consider other computer architectures. Building on our work [@Mula2008], Sun and del Mundo tested various population-count functions on an Nvidia GPU and found that its popcount intrinsic gave the best results [@sunrevisiting].
We are grateful to S. Utcke for pointing out a typographical error in one of the figures.
Funding {#funding .unnumbered}
=======
This work was supported by the Natural Sciences and Engineering Research Council of Canada \[261437\].
Population Counts in AVX-512
============================
Though Intel processors through the current Kaby Lake generation do not yet support the AVX-512 instruction set, it is straight-forward to generalize our vectorized algorithms to 512-bit vectors. However, even beyond the increase in vector width, it should be possible to implement the carry-save adder more efficiently with AVX-512, which also adds the `vpternlogd` instruction. Available through C intrinsics as `_mm512_ternarylogic_epi64`, this instruction allows us to compute arbitrary three-input binary functions in a single operation. Utilizing this, we can replace the 5 logical instructions we needed for AVX2 with just two instructions. The `vpternlogd` instruction relies on an integer parameter $i$ that serves as a look-up table. Given the input bits $x,y,z$, the value given that the $(x+2y+4z)^{\mathrm{th}}$ bit of the parameter $i$ is returned. For example, to compute XOR of the inputs, the $i$ parameter needs to have a 1-bit at indexes 1, 2, 4 and 7 (i.e., $i=2^1+2^2+2^4+2^7=150$ or `0x96` in hexadecimal). Similarly, to compute the most significant bit of the carry-save adder, the $i$ parameter needs to have a 1-bit at indexes 3, 5, 6 and 7 (`0xe8` in hexadecimal). Fig. \[fig:avx512csa\] presents a C function implementing a carry-save adder (CSA) using AVX-512 intrinsics.
[c]{}
void CSA(__m512i* h, __m512i* l, __m512i a, __m512i b, __m512i c) {
*l = _mm512_ternarylogic_epi32(c, b, a, 0x96);
*h = _mm512_ternarylogic_epi32(c, b, a, 0xe8);
}
Intel has also announced that future processors might support the AVX-512 `vpopcnt` instruction [@intel2016] which computes the population count of each word in a vector. To our knowledge, no available processor supports `vpopcnt` at this time.
[^1]: The x64 `popcnt` instruction was first available in the Nehalem microarchitecture, announced in 2007 and released in November 2008. The ARM `cnt` instruction was released as part of the Cortex-A8 microarchitecture, published in 2006 [@7477864].
[^2]: The reciprocal throughput is the number of processor clocks it takes for an instruction to execute.
[^3]: To our knowledge, Muła was first to document the benefits of AVX2 for the population count problem in March 2016 [@Mula2008].
[^4]: We could call the Muła function up to 31 times, since $8 \times 31 = 248 < 2^{8}$.
|
---
abstract: 'We present five epochs of near IR observations of the protoplanetary disk around MWC 480 (HD 31648) obtained with the SpeX spectrograph on NASA’s Infrared Telescope Facility (IRTF) between 2007 and 2013, inclusive. Using the measured line fluxes in the Pa $\beta$ and Br $\gamma$ lines, we found the mass accretion rates to be (1.43 - 2.61)$\times$10$^{-8}$M$_{\astrosun}$y$^{-1}$ and (1.81 - 2.41)$\times$10$^{-8}$M$_{\astrosun}$y$^{-1}$ respectively, but which varied by more than 50% from epoch to epoch. The spectral energy distribution (SED) reveals a variability of about 30% between 1.5 and 10 microns during this same period of time. We investigated the variability using of the continuum emission of the disk in using the Monte-Carlo Radiative Transfer Code (MCRT) HOCHUNK3D. We find that varying the height of the inner rim successfully produces a change in the NIR flux, but lowers the far IR emission to levels below all measured fluxes. Because the star exhibits bipolar flows, we utilized a structure that simulates an inner disk wind to model the variability in the near IR, without producing flux levels in the far IR that are inconsistent with existing data. For this object, variable near IR emission due to such an outflow is more consistent with the data than changing the scale height of the inner rim of the disk.'
author:
- 'Rachel B. Fernandes, Zachary C. Long, Monika Pikhartova, Michael L. Sitko, Carol A. Grady, Ray W. Russell David M. Luria, Dakotah B. Tyler, Ammar Bayyari,William Danchi, John P. Wisniewski'
title: 'VARIABILITY OF DISK EMISSION IN PRE-MAIN SEQUENCE AND RELATED STARS. IV. INVESTIGATING THE STRUCTURAL CHANGES IN THE INNER DISK REGION OF MWC 480'
---
Introduction
============
MWC 480 is a Herbig Ae star with a circumstellar disk has been the focus of numerous investigations during the past decade. The thermal emission of the dust disk has been detected at millimeter and sub-millimeter wavelengths [@mks97; @hughes13; @andrews13]. It was also detected in scattered light at 1.6 $\mu$m by @Grady10 using HST (Near Infrared Camera and Multi-Object Spectrometer (NICMOS) coronographic imagery. Outflowing jets of material were seenin the NICMOS images, and at visible wavelengths using the Goddard Fabry-Perot Interferometer at the Apache Point Observatory. They demonstrated that the star drives parsec-scale bipolar jets with condensations or knots. Such jets might be launched near the inner rim of the dust disk, similar to what is suggested for HD 163296 by @Ellerbroek14 and more generally by @Bans12.
The SED of MWC 480 is consistent with it being a Meeus Group II objects [@me01], which have disks that were postulated to be significantly shadowed, a likely sign of dust grain growth and settling. As such behavior is a sign of disk evolution and aging, it is surprising that it exhibits many indicators of the active accretion generally associated with flared unshadowed Meeus Group I objects such as hot, non-stellar gas and jets. Both imaging by @Kusakabe12 and interferometry combined with the system’s spectral energy distribution (SED) by @Millan-Gabet16 suggest a disk whose ratio of scale height to radial distance is small. The imaging suggested a scale height for the dust at a radial distance of 100 AU of only 30% that of the CO gas scale height of 10$\pm$1.1 AU deduced from modeling CO interferometric data by @Pietu07. Interferometric /SED data by also indicated a thinner disk than most of the objects in their sample.\
The NIR emission of MWC 480 is known to vary by $\sim$30 percent [@Sitko08; @Grady10; @Kusakabe12]. Such NIR variability is not uncommon in pre-main sequence stars, and it has been suggested that such variability is due to changes in the scale height of the “puffed up” inner rim of the dust disk [@Muzerolle09; @Flaherty10; @Espaillat11]. Changing the scale height of this optically thick structure is thought to be the source of the “see-saw” variability exhibited by many of these disk systems, with the near IR and mid IR fluxes changing in opposite directions. In this scenario, as the scale height of the inner rim increases, it produces an increase in near-IR flux (as the solid angle subtended by the disk rim “seen” by the star increased). This simultaneously decreases the emission at longer wavelengths, as more distant portions of the disk experience increased shadowing by the inner disk wall. It should be noted that this behavior is not universal, however, as not all such objects clearly follow this behavior - only about half of the variable sources observed by @Espaillat11 using the Infrared Spectrograph (IRS) of the *Spitzer Space Telescope* (5-38 $\mu$m) actually exhibit this behavior. But if this paradigm applies to MWC 480, the historically low NIR flux seen in 2010 by @Kusakabe12 implied the smallest inner rim height in many years, minimizing the shadow it casts, which coincided with the first significant detection of the outer disk in scattered light.\
A variable inner rim scenario is only one way that variability in NIR fluxes might arise. Numerous studies have suggested that much of the near-IR emission seen in young stellar disks might have been caused by a wind launched from the inner region of the disks which include material from beyond the dust sublimation radius. @Bans12 suggested a bi-layer wind that includes a dusty wind originating in the disk beyond the dust sublimation radius, and a dust-free gaseous wind launched closer to the star. Here, the dusty wind produces emission from material at temperatures of 1500 Kelvins (a “canonical” sublimation temperature often used for silicate grains) and cooler. Dust at these temperatures tends to radiate at wavelengths longer than $\sim$2 $\mu$m and so, in order to increase the flux needed to match the SEDs in these systems required a hotter component - the gas. Interferometric measurements of some young stellar disks, such as that of HD 163296, require that some material be present closer to the star than the canonical dust sublimation radius, but whether this is gaseous or a population of super-refractory grains, is a matter of debate [@Tannirkulam08a; @Tannirkulam08b; @Benisty10].\
A disk wind that is associated with the Herbig-Haro (HH) knots in the observed jets in HD 163296 [@wassell06] has been detected in ALMA observations in the CO gas [@klaassen13]. An example of disk wind that included a significant dust component was suggested by @Ellerbroek14 for producing the drops in brightness at visible wavelengths in HD 163296 during the past 2 decades. In HD 163296, a $\sim$1 magnitude drop in the light at visible wavelengths was observed in 2001, about the time that one ejection event was to have occurred. Such material, illuminated by the star, could also produce changes in the near-IR emission also seen within a year of that event, and has been confirmed in a radiative transfer model by @Pikhartova18. In the case of HD 163296, it was necessary to have a substantial optical depth in the jet to reproduce the observed variability. But the observed presence of a jet in MWC 480 suggested that a similar model for it.\
In this paper, we investigate whether or not the variable inner-rim scenario is consistent with both the existing observed variable NIR emission, the thinness of the disk, and of other existing mid- and far-IR data. The variable inner-rim scenario predicts that there should be significant time-dependent changes in the far-IR flux. Although numerous epochs of simultaneous observations covering the relevant wavelength range are lacking, they should record noticeable scatter in their values, measuring higher and lower flux states. We will also determine whether whether an alternative model, one where changes in the outflowing jet material can produce the observed variable NIR emission, and its effect on the mid- and far-IR flux, provides an equally good or better fit to the observations.\
Observations
============
SpeX
----
Near-IR spectra of MWC 480 were obtained on 5 nights between 2007 and 2013, using the SpeX spectrograph [@rayner03] on NASA’s Infrared Telescope Facility. The SpeX observations were made using the cross-dispersed (“XD”) echelle gratings between 0.8-5.4 $\mu$m, using an entrance slit width of 0.8 arcsec. The spectra were processed using a Spextool reduction package [@cushing04] running under IDL. In Spextool, telluric corrections and flux calibrations are performed using A0V stars [@vacca03] as “Vega analogs". For MWC 480, the A0V stars HD 25152, HD 31592, HD 31295, and HD 31039 were used for the calibration. The observing and data reduction procedures were the same as described by @Sitko12. Beginning with the December 2007 observations, and continuing through 3 of the remaining 4 epochs, we also obtained spectra using the Prism disperser and a 3.0$\arcsec$ wide slit. This technique provides absolute fluxes to an accuracy of $\sim$5% when the seeing was 1.0$\arcsec$ or better.\
BASS Spectrophotometry
----------------------
We observed MWC 480 on five epochs between 2007 and 2011 using The Aerospace Corporation’s Broad-band Array Spectrograph System (BASS). BASS uses a cold beamsplitter to separate the light into two separate wavelength regimes. The short-wavelength beam includes light from 2.9-6 $\mu$m, while the long-wavelength beam covers 6-13.5 $\mu$m. Each beam is dispersed onto a 58-element Blocked Impurity Band (BIB) linear array, thus allowing for simultaneous coverage of the spectrum from 2.9-13.5 $\mu$m. The spectral resolution $R = \lambda$/$\Delta\lambda$ is wavelength-dependent, ranging from about 30 to 125 over each of the two wavelength regions [@hackwell90]. The entrance aperture of BASS is a 1-mm circular hole, whose effective projected diameter on the sky was $\sim$3.5 ”.The observations are calibrated against spectral standard stars located close to the same airmass. Due to its proximity in the sky, $\alpha$ Tau usually serves as the flux calibration star.\
[lcccc]{} 2007.12.09 & SXD & 1.03 & 1.06 & HD 25152\
2007.12.10 & LXD &1.28 & 1.11 & HD 25152\
& Prism & 1.05 & 1.06 & HD 25152\
\
2008.10.04 & SXD & 1.12 & 1.21 & HD 31069\
& LXD & 1.09 & 1.23 & HD 31069\
& Prism & 1.04 & 1.01 & HD 31069\
\
2009.12.01 & SXD & 1.20 & 1.16 & HD 31069\
& LXD & 1.16 & 1.19 & HD 31069\
& Prism & 1.70 & 1.66 & HD 31069\
\
2011.10.016 & SXD &1.02 & 1.07 & HD 25152\
& LXD & 1.02 & 1.14 & HD 25152\
& Prism & 1.02 & 1.05 & HD 25152\
\
2013.09.11 & SXD & 1.18 & 1.23 & HD 31069\
& LXD & 1.10 & 1.15 & HD 31069\
& Prism & 1.05 & 1.13 & HD 31069\
[lccc]{} 1996.10.14 & 1.51-1.66 & 1.59-1.61 & $\alpha$ Tau\
2004.08.05 & 1.28-1.32 & 1.17-1.18 & $\alpha$ Tau\
2006.12.11 & 1.08-1.10 & 1.06-1.07 & $\alpha$ Tau\
2007.08.20 & 1.13-1.15 & 1.13-1.14 & $\alpha$ Tau\
2008.09.03 & 1.35-1.39 & 1.33-1.35 & $\alpha$ Tau\
2009.11.29 & 1.49-1.56 & 1.56-1.60 & $\alpha$ Tau\
2010.10.23 & 1.05-1.06 & 1.05-1.06 & $\alpha$ Tau\
2011.10.15 & 1.03-1.04 & 1.03-1.04 & $\alpha$ Tau\
Archival Data
-------------
In order to construct a complete spectral energy distribution for constraining models of the MWC 480 system, we incorporate a variety of archival data, both via the Simbad/Vizier SED tool and other sources in the literature (the Spitzer Heritage Archive, Mikulski Archive for Space Telescopes, Infrared Space Observatory, various optical-infrared photometric surveys [@Oudmaijer01; @Eiroa01; @dewinter01], 2MASS, AKARI, Herschel PACS, Hubble NICMOS [@Grady10], millimeter surveys [@Pietu06; @ms97], the University of Wisconsin’s Low Resolution Spectrophotometer (0.34-0.58 $\mu$m), the Kitt Peak National Observatory infrared photometers [@Sitko81], and the HiCIAO photometric point from @Kusakabe12 . These data were obtained over a span of many decades and are not contemporaneous with the SpeX observations. Nevertheless, some are uniquely suited for placing useful limits on the nature of the SED from ultraviolet to millimeter wavelengths., in particular, if there is evidence of scatter due to variability at mid-IR wavelength, such as those expected for “see-saw” behavior predicted by the variable inner disk rim model.\
Near-IR Variability of MWC 480
------------------------------
Figure 1 shows the new and archival flux data on MWC 480. In most cases where both SpeX and BASS data were obtained within a few days to a few weeks of one another, the two independent data sets appear to be in agreement with one another, although variability on time scales of days cannot be entirely ruled out, without a more extensive set of observations with a short (days) cadence.\
We also show the difference in the “high state” and “low state” of the SpeX-derived fluxes. It clearly shows that the bulk of the difference is consistent with a change in emission of a component whose temperature is $\sim$ 1600 K, indicating that the changes are occurring close to the sublimation temperature of silicate grains, and hence close to the inner edge of the dust disk. Additional flux at shorter wavelengths is likely coming from gas emission, much of which may occur inside the dust sublimation radius.\
Some of the flux levels we observed in the near-IR would seem to be lower than the bulk of historic data obtained from archival data. It has been suggested that the HiCIAO detection of the disk in scattered light in 2010, compared to more marginal detections at other epochs, might have been due to a minimum in the shadowing of the outer disk by the inner disk wall, often assumed to have a slightly “puffed-up” geometry [@Kusakabe12]. However, we will show that another scenario may be implicated in the flux variability of MWC 480.\
Mass Accretion Rates
====================
Derivation of line Luminosities and mass accretion rates by Epoch
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The mass accretion rates were derived using the same procedures as those described by @Sitko12. The absolute flux-calibrated spectra were modeled as a combination of a stellar photosphere matched to the spectral type of MWC 480, plus a thermal component due to the hottest dust component in the system. For the spectral type, we adopted A3V [@Grady10], and used SAO 206463, a pre-main sequence A0V star [@Houk82] star exhibiting neither significant gas accretion nor thermal dust emission for which we had comparable SpeX data. To this, we added a modified blackbody to approximate the thermal emission of the innermost dust.\
\
Because the models were not able to produce a pseudocontinuum that matched the data in every spectral order at every line to be extracted, the model was adjusted locally using a vertical scaling until the $\chi^2$ of the difference in the continuum nearby (but outside) the line was minimized. For the majority of the lines on all nights, these corrections to the scaling were less than 2$\%$ of the initial model continuum level.\
\
[ccccc]{}
071210 & 1.42 $\pm$ 0.07 & 7.22 $\pm$ 0.36 & 1.84 $\pm$ 0.21 & 2.04$\pm$ 0.24\
081004 & 1.79 $\pm$ 0.10 & 7.67 $\pm$ 0.46 & 2.48 $\pm$ 0.34 & 2.21 $\pm$ 0.31\
091201 & 1.47$\pm$ 0.08 & 6.79 $\pm$ 0.38 & 1.92 $\pm$ 0.25 & 1.89 $\pm$ 0.25\
111016 & 1.77 $\pm$ 0.09 & 7.83 $\pm$ 0.40 & 2.45 $\pm$ 0.29 & 2.77 $\pm$ 0.27\
130911 & 1.14 $\pm$ 0.06 & 7.22 $\pm$ 0.41 & 1.39 $\pm$ 0.19 & 2.044 $\pm$ 0.27\
The line strengths of the Pa$\beta$ and Br$\gamma$ lines were then extracted by subtraction of the adjusted model from the flux-calibrated data, and the net line flux calculated by integrating over the flux-calibrated continuum-subtracted line profile. These were converted to line luminosities using a distance of 142 pc. The line luminosities were transformed to mass accretion luminosities using the calibrations of @Fairlamb17 for Pa$\beta$ and Br$\gamma$, respectively. These were then converted to mass accretion rates using M= [L]{}$_{acc}$R$_{\ast}$/GM$_{\ast}$, with adopted values of R$_{\ast}$ = 1.93 R$_{\astrosun}$, and the stellar mass M$_{\ast}$ =2.15 M$_{\astrosun}$. While there are many different calibrations of the mass accretion rates based on these line, we selected that of @Fairlamb17, as it was derived specifically for Herbig AeBe stars, and has among the smallest calibration uncertainties in the literature. The mass accretion rates for MWC 480 were time-dependent, being (1.43 - 2.61)$\times$10$^{-8}$M$_{\astrosun}$y$^{-1}$ and (1.81 - 2.41)$\times$10$^{-8}$M$_{\astrosun}$y$^{-1}$ derived from Br$\gamma$ and Pa$\beta$ lines, respectively.
Line versus Continuum Emission
------------------------------
One aspect of the continuum and line variability that has generally received little attention is whether these two different sources of flux are in any way related. @Sitko12 illustrated the lime emission strengths of Pa $\beta$, Br $\gamma)$, and the O I line at 0.8446 $\mu$m versus the flux in the K band in SAO 206462. At the start of the increase in K band flux in 2009, the strengths of all three lines were observed to increase, and continued to do so after the continuum flux receded in strength. This might indicate that whatever caused the warm dust emission to increase triggered some sort of response in the gas accretion. However, in the case of HD 163296, @Sitko08 found little change in the Paschen lines between the two epochs with the largest change in NIR continuum yet seen in that object. The Ca II lines actually became stronger as the continuum decreased.\
For MWC 480, the SpeX data were obtained at intervals of 1 to 2 years apart, not close enough in time to make a meaningful light curve. In this case we simply examined whether the gas and dust emission were in any way related to one another. The result is shown in Figure 12 where the Pa $\beta$ and K band fluxes are shown. It is clear that a more concerted effort to monitor MWC 480 will be required to investigate the possible connection between line and continuum emission.
![The flux in Pa $\beta$ versus the flux in the K band. While, unlike SAO 206462 [@Sitko12] no strong correlation is apparent, and a larger sample of observations obtained closer in time are needed to investigate any possible relation.](lines_continuum.pdf)
Radiative Transfer Modeling
===========================
In order to model the SED of MWC 480, we used the three-dimensional Monte Carlo Radiative Transfer (MCRT) code HOCHUNK3D of @Whitney13. HOCHUNK3D allows for two independent, constituent coplanar disks, each having its own set of input parameters which dictate the structure and composition of that disk. These include different vertical scale heights to simulate grain growth and settling [@Dullemond04a; @Dullemond04b] as well as different radial extents. We assigned one disk to contain “large" grains with a small scale height (“settled disk”) and the other disk to contain smaller grains with a larger scale height that overlaps the first disk. Our model includes a third component that was originally developed for an envelope of material infalling onto the disk, but that we employed to create an inner disk “fan”, as an approximation of a disk wind.\
The actual radial dimensions of the disk surrounding MWC 480 can be characterized by the size of the disk observed in scattered light from the dust, the size determined through the thermal emission of the dust, and through the emission by the molecular gas. @Pietu07 traced CO emission to distances of 700-800 au. However @Pietu06, using a 0.73 x 0.53 arcsec beam, traced the millimeter continuum emission by the dust to only 185-190 au, indicating that either the dust does not extend as far as the gas (e.g. @birnstiel14), or that it was simply not detectable. @huang17, using the Atacama Large Millimeter/submillinmeter Array (ALMA), found the outer radius of the dust disk to be 200 au, in agreement with the value of @Pietu06. @Kusakabe12 traced the scattered light out to only 137 au, using the Hight Contrast Instrument for Subaru Next Generation Adaptive Optics (HiCIAO) instrument on the Subaru telescope (the pixel scale was 9.53 mas/pixel, and the PSF FWHM was measured to be 0.07 arcsec.). Given the faintness of the disk (it has largely eluded “strong” detection in the past), it is possible that the instrument was not sufficiently sensitive to detect the disk further out. It might also be the case that the disk is becoming self-shadowed at larger distances. Since our study deals primarily with the thermal emission between 1 $\mu$m and 100 $\mu$m (the latter has characteristic temperatures $\sim$ 30-40 K and for an A3V star, distances $\sim$60 au), and of the scattered light, the 200 au radial size for the disk measured with ALMA was adopted.\
The inner radius of the disk is somewhat difficult to establish. Using the Keck Interferometer Nuller operating in the N band (8.0-13.0 $\mu$m), combined with SED data from 0.5-13.0 $\mu$m, @Millan-Gabet16 modeled the interior region of MWC 480 two ways - one model with a single inner rim and a second model with two inner rims. In the model consisting of a single inner rim, the inner radius was found to be 0.10 au, and had a characteristic temperature of 2500 K. As this greatly exceeds typical sublimation temperatures for silicates ($\sim$1500 K), such a system would require either super-refractory grains or hot gas at those distances (see, for example, @Benisty10 for a discussion for the case of HD 163296). In the two-rim model, they derived distances of 0.44 au and 2.3 au (both with inner rim temperatures below 1500 K). @Lazareff17 observed MWC 480 using the PIONIER instrument on the VLT in the H band (1.55-1.80 $\mu$m). Aided by SED information, they found that an ellipsoid with a Half Width Half Maximum (HWHM) of 1.9 au provided the best fit. While a ring model was determined for many objects in their study, none was found for MWC 480. Due to the uncertainties presented by the inteferometrically-determined dimensions, we placed an emphasis on using the SED in 1-5 $\mu$m as a guide. Nevertheless, we recognize that 1-2 $\mu$m gas emission, which is not included in the HOCHUNK3D code, is likely to cause additional emission in this wavelength.\
Here we investigate two different scenarios for the observed variability of the NIR fluxes derived from the SpeX data. In the first one, we use disk with an inner rim, the scale height of which changes in a manner as to produce the changing NIR fluxes. In the second case, we change the structure and density of an inner wind or jet.
Changing the puffed up inner rim
--------------------------------
We began this investigation by setting the “low” NIR photometric state as the starting point of the modeling. Fitting the low state required setting the inner rim height to 0.05 au, with an added 50% rim “puff”, increasing it to 0.075 au, which reproduced the observed SED. A plot of the temperature and density in this model are shown in Figure 1, and the resulting SED in Figure 2. As is apparent in Figure 2, this model reproduces the NIR low flux state, as well as the remainder of the available data at longer wavelengths.\
To fit the high state, we used the same model, but raised the value of the inner rim height just enough to reproduce the higher NIR flux levels in the SpeX data (increasing the rim “puff" to 50%). The result of this change is shown in Figure 3. As is evident in the resulting SED, the “see-saw” behavior is reproduced, with the model fluxes at longer wavelengths dropping accordingly. The $\sim$30 % change in the NIR flux produces a more substantially visible change at longer wavelengths, since the total fluxes are originally more than an order of magnitude lower to begin with. Figure 4 shows a side-by-side comparison of the inner disk region for the two models .\
More importantly, while raising the height of the inner wall fits the NIR region well, it produced far-IR emission that was substantially lower than all observed flux levels. While there are no far-IR observations that are nearly simultaneous with any of the NIR data, the fact than *none* of the data at longer wavelengths can be fit by this second model suggests that simply increasing the rim height is unlikely to be the source of the observed NIR variations. We checked the reasonableness of this result by determining the change in surface brightness in the outer disk produced by this change in inner rim scale height. We measured the azimuthally-averaged surface brightness in the H band, using a face-on geometry to minimize the effects of disk inclination and disk shape, to provide an indication of just how much change there was in the light actually reaching the outer disk. In these models, the surface brightness, shown in Figure 5, dropped as $\sim r^{-2.0}$, but was roughly only one-half the brightness in the high-rim state as the low-rim state. The relative change is less than what the observer sees from the inner rim, and the disk “sees” the inner regions at higher inclination, and in such a thin disk, even a small change in inner rim height can create a substantial deficit of light from the star reaching the outer disk. One way such differences in light reaching the outer disk could be masked is the delay in any “thermal pulse” from propagating deep onto the disk, effectively smoothing thermal emission response. While it is possible that the time scale for such a “thermal pulse” to propagate into the disk mid-plane will be years (see the discussion in @Sitko12), it is evident that out to at least 37 $\mu$m many disk system show measurable variability, with “see-saw” response [@Espaillat11], which seems to be absent in MWC 480. This, and the presence of a jet in imaging observations of MWC 480, suggest that an alternative model be explored.\
![The horizontal and vertical temperature (upper three panels) and density (lower three panels) distribution in the circumstellar material of the high NIR state, modeled with just a slightly puffed up inner rim. The horizontal and vertical scales are in au, with, from left to right, the the structure from 0-1.5 ua, 0-50 au, and 0-300 au. The temperature scale color bar is the log of the temperature in Kelvins, while the density scale is the log of the mass density in g cm$^{-3}$.]("trho_rz_disk_low".pdf)
![The surface brightness versus radial distance for the variable disk rim height models. Here, rather than deriving the surface brightness of the curved upper disk surface seen by the observer at the inclination of the disk of MWC 480, which will suffer significant distortions, we show the effective scattering of the disk, as seen face, in order to remove the distortion produced by the observed geometry, and allowing azimuthal averaging of the signal. In this manner. Also shown in the upper right corner of the figure is the radial dependence of the scattering. The upper curve represents the “low” inner rim state, which allows more stellar light to reach the outer disk, while the lower curve represents a higher inner rim “puff” that reduces the amount of light reaching the outer disk. A “low” rim state might explain the greater ease of detection compared to previous attempts, but such a change should also change the mid-IR portion of the SED, for which there is no evidence yet.]("sb_puffed_up_inner_rim".pdf)
In this pair of models, any increase in the size of the inner rim, which is generally envisioned to be optically thick (in order to reproduce the “see-saw” behavior observed in some other disks) significantly reduces any direct illumination by the star, plunging the outer disk into darkness, and dropping its emission below all observed levels. Even with non-simultaneous mid- and far-IR archival data sets we would expect to see scatter in the data comparable to the difference between the observed and model flux levels shown in Figure 3. This suggests that this scenario is not likely to explain most of the near-IR variability seen in MWC 480.\
The emission of the inner wall requires a combination of both optical depth and physical scale. An optically thick inner wall can produce the observed NIR emission with a minimum solid angle, as “seen” by the illuminating star. Lower optical depth requires a larger solid angle structure. Because MWC 480 is known to drive bipolar jets, we investigated an alternative geometry where the NIR is dominated by a more optically thin structure.\
Changing the inner disk wind structure
--------------------------------------
For the jet/wind models, we began again with a goal of first reproducing the low flux state in the NIR, and the remaining observations at longer wavelengths. The temperature and density structures of this model is shown in Figure 6, while the resulting model SED is in Figure 7. Changing only parameters relevant to the inner few au, that is by increasing the “ffducial” density of the wind component at 1 au from 2.5$\times$10$^{-16}$ g/cm$^{3}$ to 5$\times$10$^{-16}$ g/cm$^{3}$ (this is a basic model parameter in the code, and is independent of the *actual* inner radius of the model) , increasing the exponent of power law mass density dropoff with distance $r$ from $\rho \sim r^{-1.2}$ to $\rho \sim r^{-1.8}$, and pushing the actual inner envelope gap radius from 1 au to 2.5 au, we reproduced the high NIR flux state, as shown in Figure 8. A closeup of the density in the inner regions of the two models is shown in Figure 9. As is apparent in the model SEDs of the wind/jet models, the agreement between both models and the flux levels at longer wavelengths is better than that of the high flux NIR state of the changing inner rim models. Thus, this scenario would seem to be consistent with the ensemble of data sets that are available.
![The above figure shows the temperature and density distribution of the low NIR state, modeled with just a slightly puffed up inner rim but with material above and below the disk, representing the denser grains of a disk wind.]("trho_rz_jet_low".pdf)
By contrast, in the variable jet/wind models, little change in the longer-wavelength SED occurred. This is more consistent with the ensemble of data at those wavelengths without the need to invoke the possible smearing of a variable response due to the time scale of the propagation of thermal response in the disk. In Figure 14, we calculated the surface brightness of these “jet” models in the same manner as for the variable inner rim models. The minimal response to the light received & scattered by the disk seems to be more consistent with both the mid-IR data and lack of comparable response in the model SED at those wavelengths.\
![The surface brightness for the two “jet” models calculated in the same manner as for the variable inner rim models. Here the change in the wind that fits the higher and lower near-IR observations produces little difference in the scattered light (as well as little difference on the mid-IR SED .]("sb_jet_models".pdf)
Discussion
==========
We have investigated how changing the structure of the inner disk in MWC 480 could explain the observed variability seen at near-IR wavelengths. We find that changing the scale height of the inner rim of the dust disk, which has been successful in describing the variability in other objects, does not seem to apply in the case of MWC 480. Such changes should induce a drop in the flux at longer wavelengths (“see-saw” behavior) which is inconsistent with existing data. And while the thermal response sue to changes in stellar illumination could potentially wash out the response at longer wavelengths, the presence of such a smoothed-out response in many disks systems indicates that this may not explain the “stable” fluxes seen at longer wavelengths, Instead, a more optically thin inner disk wind-like structure is able to produce changes in the NIR without producing far IR changes. Since it is known that MWC 480 possesses outflowing bipolar material, such a model would be required to be consistent with its presence. The principle difference between the models explored here is that the wind model trades optical depth for solid angle as seem by the star. The wind model is much more extended vertically, but has a lower overall mean optical depth than the inner disk rim description. It would be useful to search for the presence of jet-like structures in other object with near-IR variability where evidence for “see-saw” behavior is weak or absent.\
@Kusakabe12 detected the scattered light in the disk of MWC 480 at a time when the near-IR flux was at an historic minimum. This would seem to *favor* the variable inner rim models. However, this could be due to a couple of factors. The first is that the historic low state at the time of the HiCIAO observations is even lower that those we are modeling, based primarily on the SpeX and BASS observations. It would be important to obtain multi-epoch scattered light images, as has been done for HD 163296 (e.g. [@Grady10; @monnier17]). Likewise, multi-epoch interferometry in the near-IR would be important to obtain. [@Tannirkulam08a; @Tannirkulam08b] detected changes in the inner regions of HD 163296. However, comprehensive temporal coverage of MWC 480 is lacking. @jamialahmadi17 present 3 epochs of VLTI and Keck Interferometer data using 2-telescope beam combiners, for a total of 6 points in the uv plane. @Lazareff17 have a single epoch using the 4-telescope PIONIER instrument on the VLTI. A more comprehensive study will be required to obtain a better understanding of the structural changes in the inner regions of MWC 480.
Conclusion
==========
While changes in the scale height of the inner rim of young stellar disks has been implicated in their observed near-IR variability, such structural changes predict measurable variations in the disk flux at wavelengths dominated by more distant regions of the disk - the mid- to far-IR. Despite the $\sim$30% changes observed in the near-IR, flux of MWC 480, there exists little evidence that the corresponding changes at longer wavelengths is occurring. Rather, changes in a more vertically extended, but optically thinner, structure seems capable of explaining the large variability in the near-IR, but without a similar change at longer wavelengths. Such a structure is already known to exist in MWC 480, as evidenced by the bipolar jets seen in deep imaging, and should be included in any model. Such structures have also been implicated in the variability in other similar stars, such as HD 163296. It is ironic that these two objects, that were part of the “inspiration” for the development of the variable inner rim scenario [@Sitko08], should be better-explained by a different mechanism - changes in a wind-like structure - that are also consistent with the observed inference (we do not see their base directly) of the existence of such structures.\
A more rigorous test of these models for MWC 480 can be made with truly contemporaneous NIR and MIR observations. Such an investigation could be done with a combination of ground-based NIR observations, with MIR data obtained with facilities such as NASA’s Wide-Field Infrared Survey Telescope - Astrophysics-Focused Telescope Assets (WFIRST-AFTA) or James Webb Space Telescope (JWST). Despite the relative brightness brightness of the star, coronagraphic imaging of the disk will be possible with JWST.\
MLS is supported by NASA Exoplanet Research Program grants NNX16AJ75G and NNX17AF88G. CAG is supported under NASA Origins of Solar Systems Funding via NNG16PX39P. This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts.\
The results reported herein benefitted from collaborations and/or information exchange within NASA’s Nexus for Exoplanet System Science (NExSS) research coordination network sponsored by NASA’s Science Mission Directorate.\
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Model Parameters
================
[ |p[8cm]{}||p[4cm]{}|p[4cm]{}| ]{}\
Parameter & Value & Source\
RA(J2000) & $04^{h} 58^{m} 46.271^{s}$ & (1)\
DEC(J2000) & $+29\degr 50' 376.61"$ & (1)\
Distance & 142$\pm$7 pc & (2)\
V & 7.62 & (3)\
B-V & 0.16 & (3)\
Luminosity & 15.1$\pm$1.5 L$_{\astrosun}$ & . . .\
Mass & 2.15$\pm$0.05 M$_{\astrosun}$ & (4)\
Age & 7.5$\pm$ 1.5 Myr & (4)\
Disk Inclination & 37.5$\degr$ & (5)\
[ |p[8cm]{}||p[4cm]{}|p[4cm]{}| ]{}\
Parameter & Value & Source\
Disk mass (total)& 0.07 M$_{\astrosun}$ & . . .\
Fraction of mass in large grains disk & 0.2 & . . .\
Settled disk grain file & www003.par & (6)\
Settled disk scale height normalization & 0.005 R$\star$ & . . .\
Rin-Rout & 0.15-200 AU & (5)\
Settled disk radial density $r^{-A}$ & A = 2.0 & . . .\
Settled disk scale height $r{^B}$ & B = 1.0 & . . .\
Envelope grain file & kmh & (7)\
Flared disk grain file & mrn77 & (8)\
Non-settled disk scale height normalization & 0.028 R$\star$ & . ..\
Non-settled Rin-Rout & 0.15-200AU & (5)\
Non-settled radial density $r^{-A}$ & A = 2.15 & . . .\
Non-settledscale height $r^B$ & B = 1.15 & . . .\
Mass and Age of MWC 480
=======================
Using the unweighted mean V magnitude of MWC 480 from @Oudmaijer01 (the range in V here is 0.06 mag based on 5 observations, that in @dewinter01 was 0.04 mag, based on 3 observations, while a range of 0.11 was derived from the Hipparcos Hp observations [@ESA97] that covers both B and V filter wavelengths), the distance from the first-release Gaia archive, and the bolometric correction and effective temperature for an A3 PMS stars from @pecaut13 we derived a bolometric luminosity of 15.1$\pm$1.5 L$_{\astrosun}$ and effective temperature of 8550 K. Figure 9 shows the location of MWC relative to the PMS evolution tracks and isochrones of the non-gray models of @tognelli11, for z=0.02, helium mass fraction of 0.27, convective parameter $\alpha$ = 1.68, and a deuterium abundance of 2 x 10$^{-5}$. The on-line versions of these tracks and isochrones use the abundances of @asplund05 but are nearly indistinguishable from those based on the revised abundances of @asplund09 as seen in the figures in Tognelli et al. From these we derived a mass of 2.15$\pm$0.05 M$_{\astrosun}$ and an age of 7.5$\pm$1.5 Myr.
|
---
abstract: |
Motivated by recent applications of the Lyapunov’s method in artificial neural networks, which could be considered as dynamical systems for which the convergence of the system trajectories to equilibrium states is a necessity. We re-look at a well-known Krasovskii’s stability criterion pertaining to a non linear autonomous system. Instead, we consider the components of the same autonomous system with the help of the elements of Jacobian matrix [**[J]{}**]{}([**[x]{}**]{}), thus proposing much simpler convergence criteria via the method of Lyapunov. We then apply our results to artificial neural networks and discuss our results with respect to recent ones in the field.\
\
[*Keywords and Phrases*]{}: Lyapunov Stability, Hopfield-Tank Neural Networks\
author:
- |
Raveen Goundar[^1]\
and\
Jito Vanualailai\
[*Department of Mathematics and Computing Science,*]{}\
[*University of the South Pacific, Suva, Fiji.*]{}\
\
title: '**Convergence Criteria for a Hopfield-type Neural Network** '
---
Introduction
============
The [*Direct Method of Lyapunov*]{}, which utilizes energy-like functions called [*Lyapunov functions*]{}, is now a well-entrenched technique in the qualitative analysis of mathematical systems governed by differential equations. A flurry of activities by mathematicians, particularly within the period of early 1940s and the late 1960s, extended the work of Lyapunov to produce results that are now indispensable in many applications. (A good modern review of the Lyapunov method and its many applications is by Sastry [@sastry].) This paper is motivated to a large extent by modern applications of the Lyapunov method, especially in the field of artificial neural networks.
We start by considering the autonomous system of the form $${ \mbox{\boldmath $x$} }'(t) = { \mbox{\boldmath $g$} }({ \mbox{\boldmath $x$} }) \,, { \mbox{\boldmath $x$} }(t_{0}) =
{ \mbox{\boldmath $x$} }_{0} \,.
\label{auto1}$$ Throughout the paper, guided by a well-known result of Krasovskii, we will strive to portray a simple and flexible method of proposing a stability criterion for system (\[auto1\]). We conclude by considering an application in artificial neural networks.
Throughout the article, we suppose that, in system (\[auto1\]), ${ \mbox{\boldmath $g$} } \in C[ { \mbox{\boldmath $R$} }^{n}, { \mbox{\boldmath $R$} }^{n}]$, and is smooth enough to guarantee existence, uniqueness and continuous dependence of solutions ${ \mbox{\boldmath $x$} }(t)={ \mbox{\boldmath $x$} }(t; { \mbox{\boldmath $x$} }_{0})$, with ${ \mbox{\boldmath $x$} }=(x_{1}, \ldots, x_{n})^{T}$. The following definition and theorems of Lyapunov will be used in this article. (We will use those in Glendenning [@Glendenning]).
Suppose that the origin, ${{ \mbox{\boldmath $x$} }} = {{ \mbox{\boldmath $0$} }}$, is an equilibrium point for system (\[auto1\]). Let $D$ be an open neighborhood of ${{ \mbox{\boldmath $0$} }}$ and $ V : D \rightarrow {{ \mbox{\boldmath $R$} }}$ be a continuously differentiable function. Then we can define the derivative of $ V $ along trajectories by differentiating $V$ with respect to time using the chain rule, so $$V'({{ \mbox{\boldmath $x$} }}) = \frac{dV({{ \mbox{\boldmath $x$} }})}{dt} = {{ \mbox{\boldmath $x$} }}'\cdot\nabla V({{ \mbox{\boldmath $x$} }}) = {{ \mbox{\boldmath $g$} }}({{ \mbox{\boldmath $x$} }})\cdot\nabla
V({{ \mbox{\boldmath $x$} }}) = \sum_{i=1}^{n} g_{i}({{ \mbox{\boldmath $x$} }}) \frac{\partial
V({{ \mbox{\boldmath $x$} }})}{\partial x_{i}}\,,$$ where the subscripts denote the components of ${{ \mbox{\boldmath $g$} }}$ and ${{ \mbox{\boldmath $x$} }}$. Then $ V $ is a Lyapunov function on $ D $ iff
[(i)]{}
: $V$ is continuously differentiable on $D$;
[(ii)]{}
: $V({{ \mbox{\boldmath $0$} }}) = 0$ and $V({{ \mbox{\boldmath $x$} }}) > 0$ for all ${{ \mbox{\boldmath $x$} }}\in D \setminus \{ {{ \mbox{\boldmath $0$} }}\}$;
[(iii)]{}
: $V'({{ \mbox{\boldmath $x$} }}) \leq 0$ for all ${{ \mbox{\boldmath $x$} }} \in D$.
\[c1d1\]
Let ${{ \mbox{\boldmath $x$} }} = {{ \mbox{\boldmath $0$} }}$ be an equilibrium point for system (\[auto1\]) and $D \subset {{ \mbox{\boldmath $R$} }}^{n}$ be a domain containing ${{ \mbox{\boldmath $x$} }} = {{ \mbox{\boldmath $0$} }}$. Let $V({{ \mbox{\boldmath $x$} }})$ be a Lyapunov function on an open neighborhood of $D$, then ${{ \mbox{\boldmath $x$} }}
= {{ \mbox{\boldmath $0$} }}$ is stable. \[c1t1\]
Let ${{ \mbox{\boldmath $x$} }} = {{ \mbox{\boldmath $0$} }}$ be an equilibrium point for system (\[auto1\]) and $D \subset {{ \mbox{\boldmath $R$} }}^{n}$ be a domain containing ${{ \mbox{\boldmath $x$} }} = {{ \mbox{\boldmath $0$} }}$. Let $V({{ \mbox{\boldmath $x$} }})$ be a Lyapunov function on an open neighborhood of $D$. If $V'({{ \mbox{\boldmath $0$} }}) = 0$ and $V'({{ \mbox{\boldmath $x$} }}) < {{ \mbox{\boldmath $0$} }}$ for all ${{ \mbox{\boldmath $x$} }} \in
D \setminus \{{{ \mbox{\boldmath $0$} }}\}$, then ${{ \mbox{\boldmath $x$} }} = {{ \mbox{\boldmath $0$} }}$ is asymptotically stable. \[c1t2\]
Let ${{ \mbox{\boldmath $x$} }} = {{ \mbox{\boldmath $0$} }}$ be a equilibrium point for system (\[auto1\]) and let $V({{ \mbox{\boldmath $x$} }})$ be a Lyapunov function for all ${{ \mbox{\boldmath $x$} }} \in {{ \mbox{\boldmath $R$} }}^{n}$. If ${{ \mbox{\boldmath $x$} }} =
{{ \mbox{\boldmath $0$} }}$ is asymptotically stable and $V({{ \mbox{\boldmath $x$} }})$ is radially unbounded, then ${{ \mbox{\boldmath $x$} }} = {{ \mbox{\boldmath $0$} }}$ is globally asymptotically stable. \[c1t3\]
We carry the assumption that ${ \mbox{\boldmath $g$} }({\bf 0}) \equiv {\bf 0}$ so that ${\bf 0}$ is the zero solution of (\[auto1\]).
Convergence Criteria
====================
In 1954, Krasovskii [@kra] established an asymptotic stability criterion that avoided the linearization principle, and in the process established a method of estimating the extent of asymptotic stability region for a nonlinear systems. He assumed that ${ \mbox{\boldmath $g$} } \in C^\prime[{ \mbox{\boldmath $R$} }^{n}, { \mbox{\boldmath $R$} }^{n}]$ and ${ \mbox{\boldmath $g$} }({\bf 0}) = {\bf 0}$. Then system (\[auto1\]) can be written as $${ \mbox{\boldmath $x$} }'(t) = \int_{0}^{1} { \mbox{\boldmath $J$} }(s{ \mbox{\boldmath $x$} }) { \mbox{\boldmath $x$} } ds$$ where ${ \mbox{\boldmath $J$} }$ is the Jacobian matrix $${ \mbox{\boldmath $J$} }({ \mbox{\boldmath $x$} }) = \frac{\partial
{ \mbox{\boldmath $g$} }({ \mbox{\boldmath $x$} })}{\partial { \mbox{\boldmath $x$} }} \,.$$ The following result by Krasovskii is a fundamental one in control theory.
Let ${ \mbox{\boldmath $g$} } \in C^{\prime}[{ \mbox{\boldmath $R$} }^{n}, { \mbox{\boldmath $R$} }^{n}]$ and ${ \mbox{\boldmath $g$} }({\bf 0}) = {\bf 0}$. If there exists a constant positive definite symmetric matrix ${\bf P}$ such that $${ \mbox{\boldmath $x$} }^{T} [ { \mbox{\boldmath $P$} } { \mbox{\boldmath $J$} }({ \mbox{\boldmath $x$} }) +
{ \mbox{\boldmath $J$} }^{T}({ \mbox{\boldmath $x$} }) { \mbox{\boldmath $P$} } ] { \mbox{\boldmath $x$} }$$ is a negative definite function, then the zero solution of system (\[auto1\]) is globally asymptotically stable.
For our purpose, we need a criterion that explicitly uses each component of system (\[auto1\]). Thus, using the elements of Jacobian matrix; $J_{ij}({{ \mbox{\boldmath $x$} }})$, we define $${ \mbox{\boldmath $D$} }({ \mbox{\boldmath $x$} }) = \left[ d_{ij}({ \mbox{\boldmath $x$} }) \right]_{ n \times
n }
\label{c1e2}$$\
where $$d_{ij}({ \mbox{\boldmath $x$} }) =\int_{0}^{1}J_{ij}(s{ \mbox{\boldmath $x$} })ds =
\int_{0}^{1} \frac{ \partial g_{i}(s { \mbox{\boldmath $x$} }) }{ \partial
(sx_{j}) } ds \,,$$\
such that system (\[auto1\]) can be rewritten as $${ \mbox{\boldmath $x$} }'(t) = { \mbox{\boldmath $D$} }({ \mbox{\boldmath $x$} }) { \mbox{\boldmath $x$} } \,.
\label{form}$$ A decoupled form for the $i$-th component of system (\[form\]) is $$x_{i}'(t) = d_{ii}({ \mbox{\boldmath $x$} }) x_{i} + \sum_{{j=1} \atop {j \neq
i}}^{n} d_{ij}({ \mbox{\boldmath $x$} }) x_{j} \,.
\label{c2e2}$$
[*Note that in (\[c2e2\]), the term $d_{ij}({ \mbox{\boldmath $x$} })x_{j}$, for $i, j = 1, \ldots, n$, is continuously differentiable with respect to ${ \mbox{\boldmath $x$} } \in { \mbox{\boldmath $R$} }^{n}$ for the simple reason that ${ \mbox{\boldmath $D(x)x=g(x)$} }$ and ${ \mbox{\boldmath $g$} } \in
C^{\prime}[{ \mbox{\boldmath $R$} }^{n}, { \mbox{\boldmath $R$} }^{n}]$. \[condiff\]* ]{}
The following result of ours, guarantees the convergence criteria for autonomous system (\[auto1\]).
Let ${ \mbox{\boldmath $g$} } \in C^{\prime}[{ \mbox{\boldmath $R$} }^{n}, { \mbox{\boldmath $R$} }^{n}]$ and ${ \mbox{\boldmath $g$} }({\bf 0}) = {\bf 0}$. Let $$\beta_{i}({ \mbox{\boldmath $x$} }) =
d_{ii}({ \mbox{\boldmath $x$} }) + \frac{1}{2}
\sum_{{j=1}
\atop {j \neq i}}^{n}
\left( | d_{ij}({ \mbox{\boldmath $x$} }) | + | d_{ji}({ \mbox{\boldmath $x$} }) | \right) \,.$$ Define $D = \{{{ \mbox{\boldmath $x$} }} \in {{ \mbox{\boldmath $R$} }}^{n} : \|{{ \mbox{\boldmath $x$} }}\| \leq
M \}$ for some $M > 0$ and assume that $d_{ij}({ \mbox{\boldmath $x$} })x_{i}$ are continuous on ${ \mbox{\boldmath $R$} }^{n}$ for $i,j=1, \ldots, n$, such that $i \neq j$. Then the zero solution of (\[auto1\]) is
1. stable if $-\infty < \beta_{i}({{ \mbox{\boldmath $x$} }})
\leq 0$ for $i = 1,2, \ldots ,n$ and ${{ \mbox{\boldmath $x$} }} \in D$.
2. asymptotically stable if $-\infty <
\beta_{i}({{ \mbox{\boldmath $x$} }}) < 0$ for $i = 1,2,\ldots,n$ and ${{ \mbox{\boldmath $x$} }}
\in D$.
3. globally asymptotically stable if $-\infty <
\beta_{i}({{ \mbox{\boldmath $x$} }}) < 0$ for all ${{ \mbox{\boldmath $x$} }} \in {{ \mbox{\boldmath $R$} }}^{n}$.
\[raveen\]
Consider $$V({ \mbox{\boldmath $x$} }) = \frac{1}{2} \sum_{i=1}^{n} x_{i}^{2}$$ as a tentative Lyapunov function for system (\[auto1\]). We have, along a solution of (\[auto1\]), $$\begin{aligned}
\frac{d}{dt} \left[ V \right]_{(\ref{auto1})}
& = & \frac{1}{2} \sum_{i=1}^{n} \frac{d}{dt}
\left[ x_{i}^{2} \right]
=
\sum_{i=1}^{n} x_{i} x_{i}'(t)\nonumber \\
& = & \sum_{i=1}^{n} x_{i} \left[
d_{ii}({ \mbox{\boldmath $x$} })x_{i} +
\sum_{{j=1} \atop {j \neq i}}^{n} d_{ij}({ \mbox{\boldmath $x$} }) x_{j} \right]\nonumber \\
& = & \sum_{i=1}^{n} \left[
d_{ii}({ \mbox{\boldmath $x$} }) x_{i}^{2}
+ \sum_{{j=1}
\atop {j \neq i}}^{n} d_{ij}({ \mbox{\boldmath $x$} }) x_{j} x_{i} \right]\nonumber \\
& = & \sum_{i=1}^{n} \left[
d_{ii}({ \mbox{\boldmath $x$} }) x_{i}^{2}
+ \frac{1}{2} \sum_{{j=1}
\atop {j \neq i}}^{n} [ d_{ij}({ \mbox{\boldmath $x$} }) + d_{ji}({ \mbox{\boldmath $x$} }) ] x_{j} x_{i} \right] \nonumber\\
& \leq & \sum_{i=1}^{n} \left[
d_{ii}({ \mbox{\boldmath $x$} }) x_{i}^{2}
+ \frac{1}{4} \sum_{{j=1}
\atop {j \neq i}}^{n} [ | d_{ij}({ \mbox{\boldmath $x$} })| + |d_{ji}({ \mbox{\boldmath $x$} })| ]
(x_{j}^{2} + x_{i}^{2}) \right]\nonumber \\
& = & \sum_{i=1}^{n} \left[
d_{ii}({ \mbox{\boldmath $x$} })
+
\frac{1}{2} \sum_{{j=1}
\atop {j \neq i}}^{n}
[ | d_{ij}({ \mbox{\boldmath $x$} })| + |d_{ji}({ \mbox{\boldmath $x$} })| ]
\right] x_{i}^{2} \label{rav1}\\
& = & \sum_{i=1}^{n} \beta_{i}({ \mbox{\boldmath $x$} }) x_{i}^{2} \,.
\label{c1e3}\end{aligned}$$ Expanded form of system (\[rav1\]) is $$\frac{dV}{dt} \leq \sum_{i=1}^{n} \left[ d_{ii}({ \mbox{\boldmath $x$} }) x_{i} x_{i}
+
\frac{1}{2} \sum_{{j=1}
\atop {j \neq i}}^{n}
[ | d_{ij}({ \mbox{\boldmath $x$} })x_{i}x_{i}| + |d_{ji}({ \mbox{\boldmath $x$} })x_{i}x_{i}| ]
\right] \,.$$ By Remark \[condiff\], the first and third terms of system (\[rav1\]) are continuous on ${ \mbox{\boldmath $R$} }^{n}$, and by assumption of Theorem \[raveen\], the second term is also continuous on ${ \mbox{\boldmath $R$} }^{n}$. Hence $V({{ \mbox{\boldmath $x$} }})$ is continuous on ${{ \mbox{\boldmath $R$} }}^{n}$. Since $$V({{ \mbox{\boldmath $x$} }}) = \frac{1}{2} \sum_{i=1}^{n}{x_{i}}^{2}\,,
\label{c1e4}$$ we have therefore, $V({{ \mbox{\boldmath $0$} }}) = 0$ and $V({{ \mbox{\boldmath $x$} }}) > 0$ for all ${{ \mbox{\boldmath $x$} }} \in {{ \mbox{\boldmath $R$} }}^{n}\setminus\{{ \mbox{\boldmath $0$} }\}$. From equation (\[c1e3\]), $$V'({{ \mbox{\boldmath $x$} }}) \leq \sum_{i=1}^{n} \beta_{i}({{ \mbox{\boldmath $x$} }}){x_{i}}^{2} \label{c1e5}$$ and by condition (a) of Theorem \[raveen\], we have $V'({{ \mbox{\boldmath $x$} }}) \leq 0$ for all ${{ \mbox{\boldmath $x$} }} \in D$. Hence by Theorem \[c1t1\], the zero solution of system (\[auto1\]) is stable. Moreover, by condition (b) of Theorem \[raveen\], equation (\[c1e5\]) implies $V'({{ \mbox{\boldmath $0$} }}) = 0$ and $V'({{ \mbox{\boldmath $x$} }}) < 0$ for all ${{ \mbox{\boldmath $x$} }} \in D \setminus \{ {{ \mbox{\boldmath $0$} }}\}$. Hence by Theorem \[c1t2\], the zero solution of system (\[auto1\]) is asymptotically stable. Furthermore, by condition (c) of Theorem \[raveen\], equation (\[c1e5\]) implies $V'({{ \mbox{\boldmath $0$} }}) = 0$ and $V'({{ \mbox{\boldmath $x$} }}) < 0$ for all ${{ \mbox{\boldmath $x$} }} \in {{ \mbox{\boldmath $R$} }}^{n}$. Note that (\[c1e4\]) implies $V({{ \mbox{\boldmath $x$} }}) \rightarrow \infty$ as $\|{{ \mbox{\boldmath $x$} }}\| \rightarrow
\infty$, thus $V({{ \mbox{\boldmath $x$} }})$ is radially unbounded. Hence by Theorem \[c1t3\], the zero solution of system (\[auto1\]) is globally asymptotically stable.
Let us consider some examples to show the applicability of Theorem \[raveen\].
[*We consider the following two-dimensional system $$\begin{aligned}
\left[
\begin{array}{c}
x_{1}'(t)\\
x_{2}'(t)
\end{array}
\right]
=
\left[
\begin{array}{c}
-2 x_{1} + x_{2}^{2}\\
x_{1}^{2} - 2 x_{2}
\end{array}
\right]\,,
\label{c2e8}\end{aligned}$$ with $x_{1}(t_{0}) = x_{10}$ and $x_{2}(t_{0}) = x_{20}$. In the form of system (\[form\]), system (\[c2e8\]) can be written as $$\begin{aligned}
\left[
\begin{array}{c}
x_{1}'(t)\\
x_{2}'(t)
\end{array}
\right] = \left[
\begin{array}{cc}
-2 & x_{2}\\
x_{1} & -2
\end{array}
\right] \left[
\begin{array}{c}
x_{1}\\
x_{2}
\end{array}
\right]\,.\end{aligned}$$* ]{} \[c2ex1\]
The assumption of Theorem \[raveen\] is satisfied since $$d_{12}({{ \mbox{\boldmath $x$} }})x_{1} = d_{21}({{ \mbox{\boldmath $x$} }})x_{2} = x_{1}x_{2}\,.
\nonumber$$ Next we shall check condition (a) of Theorem \[raveen\]. We have $$\begin{aligned}
\beta_{1}({{ \mbox{\boldmath $x$} }}) & = & d_{11}({{ \mbox{\boldmath $x$} }}) +
\frac{1}{2}\left(|d_{12}({{ \mbox{\boldmath $x$} }})| + |d_{21}({{ \mbox{\boldmath $x$} }})|\right)
\\
& = & -2 + \frac{1}{2}\left(|x_{2}| + |x_{1}|\right)\nonumber\,.\end{aligned}$$ Solving the inequality $\beta _{1}({{ \mbox{\boldmath $x$} }}) < 0$, we have $$|x_{1}| + |x_{2}| < 4\,,$$ and ‘squaring’ both sides gives $${x_{1}}^{2} + {x_{2}}^{2} + 2|x_{1}||x_{2}| < 16\,.$$ Now $${x_{1}}^{2} + {x_{2}}^{2} + 2|x_{1}||x_{2}| < {x_{1}}^{2} +
{x_{2}}^{2} + 2 \times \frac{1}{2}\left({x_{1}}^{2} +
{x_{2}}^{2}\right) = 2{x_{1}}^{2} + 2{x_{2}}^{2}\,.$$ Then let $$2{x_{1}}^{2} + 2{x_{2}}^{2} < 16$$ from which $${x_{1}}^{2} + {x_{2}}^{2} < 8\,.
\nonumber$$ Similarly solving $\beta _{2}({{ \mbox{\boldmath $x$} }}) < 0$, we have $$\beta_{2}({{ \mbox{\boldmath $x$} }}) = d_{22}({{ \mbox{\boldmath $x$} }}) +
\frac{1}{2}\left(|d_{21}({{ \mbox{\boldmath $x$} }})| + |d_{12}({{ \mbox{\boldmath $x$} }})|\right) < 0\,,
\nonumber$$ which gives $$-2 + \frac{1}{2} \left(|x_{1}| + |x_{2}|\right) < 0.
\label{rav2}$$ Further simplification of (\[rav2\]) gives us $${x_{1}}^{2} + {x_{2}}^{2} < 8\,.
\nonumber$$ Therefore, let $$D = \{{{ \mbox{\boldmath $x$} }} \in {{ \mbox{\boldmath $R$} }}^{2} : \|{{ \mbox{\boldmath $x$} }}\| <
\sqrt{8}\}\,.
\nonumber$$ Hence by condition (a) of Theorem \[raveen\], the zero solution of system (\[c2e8\]) is asymptotically stable.
[*We consider the following two-dimensional system $$\begin{aligned}
\left[
\begin{array}{c}
x_{1}'(t)\\
\\
x_{2}'(t)
\end{array}
\right] = \left[
\begin{array}{c}
\displaystyle -4 x_{1} + x_{1}\mbox{sech}(x_{1}) + 4 x_{2}\\
\\\displaystyle -x_{1} - 6 x_{2} - x_{2}\cos{(x_{2})}
\end{array}
\right]\,,
\label{c2e12}\end{aligned}$$ which can be written in the form of system (\[form\]) as $$\begin{aligned}
\left[
\begin{array}{c}
x_{1}'(t)\\
\\
x_{2}'(t)
\end{array}
\right] = \left[
\begin{array}{cc}
\displaystyle -4 + \mbox{sech}(x_{1}) & \displaystyle 4\\
\\
\displaystyle -1 & \displaystyle -6 - \cos(x_{2})
\end{array}
\right] \left[
\begin{array}{c}
x_{1}\\
\\
x_{2}
\end{array}
\right]\,.\end{aligned}$$ The assumption of Theorem \[raveen\] is satisfied since $d_{12}({{ \mbox{\boldmath $x$} }}) x_{1} = 4x_{1}$ and $d_{21}({{ \mbox{\boldmath $x$} }})x_{2} =
-x_{2}$. Next we shall check condition (c) of Theorem \[raveen\]. We have $$\begin{aligned}
\beta_{1}({{ \mbox{\boldmath $x$} }}) & = & d_{11}({{ \mbox{\boldmath $x$} }}) + \frac{1}{2}\left(
|d_{12}({{ \mbox{\boldmath $x$} }})| + |d_{21}({{ \mbox{\boldmath $x$} }})|\right)\\
& = &
-4 + \mbox{sech}(x_{1}) + \frac{1}{2}\left(|4| +
|-1|\right)\\
& = & \mbox{sech}(x_{1}) - \frac{3}{2} \leq 1 - \frac{3}{2} =
-\frac{1}{2} < 0\,.\end{aligned}$$ Similarly, we have $$\begin{aligned}
\beta_{2}({{ \mbox{\boldmath $x$} }}) & = & d_{22}({{ \mbox{\boldmath $x$} }}) +
\frac{1}{2}\left(|d_{21}({{ \mbox{\boldmath $x$} }})| + |d_{12}({{ \mbox{\boldmath $x$} }})|\right)\\
& = & -6 - \cos(x_{2}) + \frac{1}{2}\left(|-1| + |4|\right)\\
& = & -\frac{7}{2} - \cos(x_{2}) \leq -\frac{7}{2} + 1 =
-\frac{5}{2} < 0\,.\end{aligned}$$ Since both $\beta_{1}({{ \mbox{\boldmath $x$} }}) < 0$ and $\beta_{2}({{ \mbox{\boldmath $x$} }})
< 0$ for all ${{ \mbox{\boldmath $x$} }} \in {{ \mbox{\boldmath $R$} }}^{2}$ hence by condition (c) of Theorem \[raveen\], the zero solution of system (\[c2e12\]) is globally asymptotically stable.* ]{}
Application in Artificial Neural Networks
=========================================
Artificial neural networks (ANNs) can be considered as dynamical systems with several equilibrium states. An essential operating condition for a neural network is that all system trajectories must converge to the equilibrium states. (A good overview of the concepts associated with biological neural networks is given in [@arbib]).
We will consider an ANN that is described thoroughly in Lakshmikantham et al. [@laks1], and provide a stability criteria using Theorem \[raveen\]. The ANN in question has $n$ units. To the $i$th unit, we associate its [*activation state*]{} at time $t$, a real number $x_{i}=x_{i}(t)$; an [*output function*]{} $\mu_{i}$; a fixed [*bias*]{} $\theta_{i}$; and an [*output signal*]{} $R_{i}=\mu_{i}(x_{i}+\theta_{i})$. The [*weight*]{} or connection strength on the line from unit $j$ to unit $i$ is a fixed real number $W_{ij}$. When $W_{ij}=0$, there is no transmission from unit $j$ to unit $i$. The [*incoming signal*]{} from unit $j$ to unit $i$ is $S_{ij}=W_{ij}R_{j}$. In addition, there can be a vector ${ \mbox{\boldmath $I$} }$ of any number of [*external inputs*]{} feeding into some or all units, so that we may write ${ \mbox{\boldmath $I$} }=(I_{1}, \ldots,
I_{m})^{T}$.
An ANN with fixed weights is a dynamical system: given initial values of the activation of all units, the future activations can be computed. The future activation states are assumed to be determined by a system of $n$ differential equations, the $i$th equation of which is $$\begin{aligned}
x_{i}'(t) & = &
G_{i}(x_{i}, S_{i1}, \ldots, S_{in}, { \mbox{\boldmath $I$} }) =
G_{i}(x_{i}, W_{i1}R_{1}, \ldots, W_{in}R_{n}, { \mbox{\boldmath $I$} })
\nonumber \\
& = &
G_{i}(x_{i}; W_{i1} \mu_{1}(x_{1}+\theta_{1}), \ldots,
W_{in} \mu_{n}(x_{n}+\theta_{n});
I_{1}, \ldots, I_{m} ) \,.
\label{net1}\end{aligned}$$ With $W_{ij}$, $\theta_{i}$ and $I_{k}$ assumed known, we can write (\[net1\]) as $$x_{i}'(t)= g_{i}(x_{1}, \ldots, x_{n}) \,,
\label{net2}$$ or in vector notation $${ \mbox{\boldmath $x$} }'(t) = { \mbox{\boldmath $g$} }({ \mbox{\boldmath $x$} }) \,,
\label{net3}$$ where ${ \mbox{\boldmath $g$} }$ is a vector on Euclidean space ${ \mbox{\boldmath $R$} }^{n}$ whose $i$th element is $g_{i}$ given in (\[net2\]). We assume that ${ \mbox{\boldmath $g$} }$ is continuously differentiable and satisfies the usual theorems on existence, continuity and uniqueness of solutions. Thus, since ${ \mbox{\boldmath $g$} } \in C^{\prime}[{ \mbox{\boldmath $R$} }^{n},
{ \mbox{\boldmath $R$} }^{n}]$, we can define ${ \mbox{\boldmath $D$} }({ \mbox{\boldmath $x$} })$ as in (\[c1e2\]) but using ${ \mbox{\boldmath $g$} }$ in (\[net3\]). Hence, if ${ \mbox{\boldmath $g$} }(\bf 0) \equiv {\bf 0}$, then system (\[net3\]) can be written as $${ \mbox{\boldmath $x$} }'(t) = { \mbox{\boldmath $D$} }({ \mbox{\boldmath $x$} }) { \mbox{\boldmath $x$} } \,, \;\;
{ \mbox{\boldmath $x$} }(t_{0}) = { \mbox{\boldmath $x$} }_{0} \,,$$ the $i$th component of which in a decoupled form is $$x_{i}'(t) = d_{ii}({ \mbox{\boldmath $x$} }) x_{i}
+
\sum_{{j=1}
\atop {j \neq i}}^{n}
d_{ij}({ \mbox{\boldmath $x$} }) x_{j} \,.$$
First, we state a comparable result by Lakshmikantham et al. [@laks1], page 152, who used the concept of [*vector Lyapunov functions*]{}.
Let ${ \mbox{\boldmath $g$} } \in C^{\prime}[{ \mbox{\boldmath $R$} }^{n}, { \mbox{\boldmath $R$} }^{n}]$ and ${ \mbox{\boldmath $g$} }({\bf 0}) = {\bf 0}$. Let $$\beta_{i}({ \mbox{\boldmath $x$} }) =
d_{ii}({ \mbox{\boldmath $x$} }) +
\sum_{{j=1}
\atop {j \neq i}}^{n}
| d_{ij}({ \mbox{\boldmath $x$} }) | \,.
\label{laks1}$$ Suppose that $$\beta_{i}({ \mbox{\boldmath $x$} }) < 0 \;\; \mbox{ if }
\;\; x_{i}^{2} \geq x_{j}^{2} \,,
\label{laks}$$ for $i, j =1, \ldots, n$ and ${ \mbox{\boldmath $x$} } \in { \mbox{\boldmath $R$} }^{n}$, ${ \mbox{\boldmath $x$} } \neq {\bf 0}$. Then the zero solution of (\[net3\]) is globally asymptotically stable. \[LaksThm1\]
If we apply condition (b) of Theorem \[raveen\], then we obtain a simpler convergence criteria.
Let ${ \mbox{\boldmath $g$} } \in C^{\prime}[{ \mbox{\boldmath $R$} }^{n}, { \mbox{\boldmath $R$} }^{n}]$ and ${ \mbox{\boldmath $g$} }({\bf 0}) = {\bf 0}$. Let $$\beta_{i}({ \mbox{\boldmath $x$} }) = d_{ii}({ \mbox{\boldmath $x$} }) + \frac{1}{2} \sum_{{j=1}
\atop {j \neq i}}^{n} \left( | d_{ij}({ \mbox{\boldmath $x$} }) | + |
d_{ji}({ \mbox{\boldmath $x$} }) | \right) \,.$$ Define $D = \{{{ \mbox{\boldmath $x$} }} \in {{ \mbox{\boldmath $R$} }}^{n} : \|{{ \mbox{\boldmath $x$} }}\| \leq
M \}$ for some $M > 0$ and assume that $d_{ij}({ \mbox{\boldmath $x$} })x_{i}$ are continuous on ${ \mbox{\boldmath $R$} }^{n}$ for $i,j=1, \ldots, n$, such that $i \neq j$. Then the zero solution of (\[net3\]) is asymptotically stable if $-\infty < \beta_{i}({{ \mbox{\boldmath $x$} }}) < 0$ for $i = 1,2,\ldots,n$ and ${{ \mbox{\boldmath $x$} }} \in D$. \[LaksThm11\]
Thus, the application of Theorem \[raveen\] to artificial neural network, considering system (\[net3\]), gives us a simpler criterion guaranteeing asymptotic stability as showed by Theorem \[LaksThm1\]. Hence the strong condition $ x_{i}^{2}
\geq x_{j}^{2}$ that appears in Theorem \[LaksThm1\] is not necessary.
Next, we look at a specific case of (\[net3\]). The specific ANN is of the additive type and is often referred to as the Hopfield-Tank ANN, a much studied class of network dynamics [@hopfield]. It is described by the nonlinear differential equation $$\begin{aligned}
x_{i}'(t)
& = & -a_{i} x_{i}(t) + \sum_{j=1}^{n} W_{ij} \; \mu_{j}(x_{j}(t)
+ \theta_{j}) +
I_{i}(t) \nonumber \\
& = & -a_{i} x_{i}(t) + \sum_{j=1}^{n} W_{ij} \; \nu_{j}(x_{j}(t)) +
I_{i}(t) \,,
\label{NN}\end{aligned}$$ where $a_{i}>0$ is the constant [*decay rate*]{}, $I_{i}(t)$ is the external input (to the $i$th neuron) defined almost everywhere on $[0, \infty)$ and $\nu_{i}$ is the suppressed notation for the fixed $\theta_{i}$ by having $\theta_{i}$ incorporated into $\mu_{i}$. The function $\nu_{i}$ is called the [*neuron activation function*]{}.
Now, define ${ \mbox{\boldmath $A$} }=\mbox{diag}(-a_{1}, \ldots, -a_{n})$, ${ \mbox{\boldmath $x$} }=
(x_{1}, \ldots , x_{n})^{T}$, $$h_{i}({ \mbox{\boldmath $x$} })= \sum_{j=1}^{n} W_{ij} \nu_{j}(x_{j}) \mbox{
with } { \mbox{\boldmath $h$} }({ \mbox{\boldmath $x$} })=(h_{1}({ \mbox{\boldmath $x$} }), \ldots,
h_{n}({ \mbox{\boldmath $x$} }))^{T} \,,$$ and ${ \mbox{\boldmath $u$} }(t) = (I_{i}(t), \ldots, I_{n}(t))^{T}$. Then (\[NN\]) is the $i$th component of the system $${ \mbox{\boldmath $x$} }'(t) = { \mbox{\boldmath $A$} } { \mbox{\boldmath $x$} } + { \mbox{\boldmath $h$} }({ \mbox{\boldmath $x$} }) +
{ \mbox{\boldmath $u$} }(t) \,, \;\; { \mbox{\boldmath $x$} }(t_{0}) = { \mbox{\boldmath $x$} }_{0} \,.
\label{input}$$ When the external input vector, ${ \mbox{\boldmath $u$} }$, is zero, the nonautonomous system (\[input\]) reduces to the autonomous system $${ \mbox{\boldmath $x$} }'(t) = { \mbox{\boldmath $A$} } { \mbox{\boldmath $x$} } + { \mbox{\boldmath $h$} }({ \mbox{\boldmath $x$} }) \,,
\;\; { \mbox{\boldmath $x$} }(t_{0}) = { \mbox{\boldmath $x$} }_{0} \,. \label{noinput}$$ For this, we assume that ${ \mbox{\boldmath $x$} }^{*}=(x_{1}^{*}, \ldots,
x_{n}^{*})^{T}$ is an equilibrium point, so that ${ \mbox{\boldmath $A$} }
{ \mbox{\boldmath $x$} }^{*} + { \mbox{\boldmath $h$} }({ \mbox{\boldmath $x$} }^{*}) = {\bf 0}$. By translating the origin, ${\bf 0}$, to this equilibrium point, we can make ${\bf 0}$ an equilibrium point. In this case, ${ \mbox{\boldmath $h$} }({\bf 0}) \equiv {\bf 0}$. Since this is of great notational help, we will henceforth consider ${\bf 0}$ as an equilibrium point or zero solution of (\[noinput\]).
Let us next assumed that ${ \mbox{\boldmath $h$} } \in C^{\prime}[{ \mbox{\boldmath $R$} }^{n},
{ \mbox{\boldmath $R$} }^{n}]$. Then using the elements of Jacobian matrix, $J_{ij}({ \mbox{\boldmath $x$} })$, we define $${ \mbox{\boldmath $F$} }({ \mbox{\boldmath $x$} }) = \left[ f_{ij}({ \mbox{\boldmath $x$} }) \right]_{ n \times
n } \;\; \mbox{ where } \;\; f_{ij}({ \mbox{\boldmath $x$} }) =
\int_{0}^{1}J_{ij}({ \mbox{\boldmath $x$} })ds = \int_{0}^{1} \frac{
\partial h_{i}(s { \mbox{\boldmath $x$} }) }{
\partial (sx_{j}) } ds ,$$ hence system (\[noinput\]) can be rewritten as $${ \mbox{\boldmath $x$} }'(t) = { \mbox{\boldmath $A$} }{ \mbox{\boldmath $x$} } + { \mbox{\boldmath $F$} }({ \mbox{\boldmath $x$} })
{ \mbox{\boldmath $x$} } = \left[ { \mbox{\boldmath $A$} } + { \mbox{\boldmath $F$} }({ \mbox{\boldmath $x$} }) \right]
{ \mbox{\boldmath $x$} } \,.
\label{noinput2}$$ The $i$th component of (\[noinput2\]) in a decoupled form is $$x_{i}'(t) = [ -a_{i} + f_{ii}({ \mbox{\boldmath $x$} })] x_{i}(t) + \sum_{{j=1}
\atop {j \neq i}}^{n} f_{ij}({ \mbox{\boldmath $x$} }) x_{j} \,.$$ Thus the following theorem is an application of our result; Theorem \[raveen\].
Let ${ \mbox{\boldmath $h$} } \in C^{\prime}[{ \mbox{\boldmath $R$} }^{n}, { \mbox{\boldmath $R$} }^{n}]$ and ${ \mbox{\boldmath $h$} }({\bf 0}) = {\bf 0}$. Let $$\beta_{i}({ \mbox{\boldmath $x$} }) = -a_{ii} + f_{ii}({ \mbox{\boldmath $x$} }) + \frac{1}{2}
\sum_{{j=1} \atop {j \neq i}}^{n} \left( | f_{ij}({ \mbox{\boldmath $x$} }) | +
| f_{ji}({ \mbox{\boldmath $x$} }) | \right) \,.$$ Define $D = \{{{ \mbox{\boldmath $x$} }} \in {{ \mbox{\boldmath $R$} }}^{n} : \|{{ \mbox{\boldmath $x$} }}\| \leq
M \}$ for some $M > 0$ and assume that $f_{ij}({ \mbox{\boldmath $x$} })x_{i}$ are continuous on ${ \mbox{\boldmath $R$} }^{n}$ for $i,j=1, \ldots, n$, such that $i \neq j$. Then the zero solution of (\[noinput\]) is
1. stable if $-\infty < \beta_{i}({{ \mbox{\boldmath $x$} }})
\leq 0$ for $i = 1,2, \ldots ,n$ and ${{ \mbox{\boldmath $x$} }} \in D$.
2. asymptotically stable if $-\infty <
\beta_{i}({{ \mbox{\boldmath $x$} }}) < 0$ for $i = 1,2,\ldots,n$ and ${{ \mbox{\boldmath $x$} }}
\in D$.
3. globally asymptotically stable if $-\infty <
\beta_{i}({{ \mbox{\boldmath $x$} }}) < 0$ for all ${{ \mbox{\boldmath $x$} }} \in {{ \mbox{\boldmath $R$} }}^{n}$.
\[raveen-2\]
Applying Theorem \[raveen\] to system (\[noinput\]), and hence to system (\[noinput2\]), with ${ \mbox{\boldmath $D$} }({ \mbox{\boldmath $x$} }) = { \mbox{\boldmath $A$} }
+ { \mbox{\boldmath $F$} }({ \mbox{\boldmath $x$} })$, $d_{ii}({ \mbox{\boldmath $x$} })=-a_{i} +
f_{ii}({ \mbox{\boldmath $x$} })$ and $d_{ij}({ \mbox{\boldmath $x$} }) = f_{ij}({ \mbox{\boldmath $x$} })$, we easily obtain the conclusion of Theorem \[raveen-2\].
Let us consider one example of Theorem \[raveen-2\].
[*Let us consider two-neural autonomous system. $$\begin{aligned}
\left[
\begin{array}{c}
x_{1}'(t)\\
x_{2}'(t)
\end{array}
\right] = \left[
\begin{array}{cc}
-a_{1} & 0\\
0 & -a_{2}
\end{array}
\right] \left[
\begin{array}{c}
x_{1}\\
x_{2}
\end{array}
\right] + \left[
\begin{array}{c}
h_{1}({{ \mbox{\boldmath $x$} }})\\
h_{2}({{ \mbox{\boldmath $x$} }})
\end{array}\right]
\label{c4e6}\end{aligned}$$* ]{}
with $x_{1}(t_{0}) = x_{10},\,\,\,x_{2}(t_{0}) = x_{20},\,\,\,0
\leq t_{0} \leq t$, where, $$\begin{aligned}
a_{1} & = & 10,\,\,\,a_{2} = 10\,,\\
h_{1}({{ \mbox{\boldmath $x$} }}) & = & B_{11}\nu_{1}(x_{1}) +
B_{12}\nu_{2}(x_{2}) =-3x_{1} + x_{2} - \tanh(3x_{1})\,,\\
h_{2}({{ \mbox{\boldmath $x$} }}) & = & B_{21}\nu_{1}(x_{1}) +
B_{22}\nu_{2}(x_{2}) = x_{1} - x_{2} + \frac{1}{5}\tanh(3x_{2})\,.\end{aligned}$$ In the form of system (\[noinput2\]), system (\[c4e6\]) can be written as $$\begin{aligned}
\left[\begin{array}{c} x_{1}'(t)\\
x_{2}'(t)
\end{array}
\right] = \left(\left[
\begin{array}{cc}
-10 & 0 \\
0 & -10
\end{array}
\right] +\ \left[
\begin{array}{cc}
-3 - \tau(x_{1}(t)) & 1\\
1 & -1 + \displaystyle\frac{1}{5}\tau(x_{2}(t))
\end{array}\right]\right)\left[
\begin{array}{c}
x_{1}\\
x_{2}
\end{array}
\right]\,,\end{aligned}$$ where for $i = 1,2,$ we define $$\tau(x_{i}(t)) = \left\{
\begin{array}{ll}
\displaystyle \frac{\tanh(3x_{i})}{x_{i}} & \mbox{$x_{i} \neq 0$}\,,\\
3 & \mbox{$x_{i} = 0$}\,,
\end{array}
\right.$$ noting that $ 0 < \tau(x_{i}) \leq 3$ for all $x_{i} \in {{ \mbox{\boldmath $R$} }}^{2}$. The assumption of Theorem \[raveen-2\] is satisfied since $f_{12}({{ \mbox{\boldmath $x$} }})x_{1} = x_{1}$ and $ f_{21}({{ \mbox{\boldmath $x$} }})x_{2} = x_{2}$. Now we shall check condition (c) of Theorem \[raveen-2\]. We have $$\begin{aligned}
\beta_{1}({{ \mbox{\boldmath $x$} }}) & = & -a_{1} + f_{11}({{ \mbox{\boldmath $x$} }}) +
\frac{1}{2}(|f_{12}({{ \mbox{\boldmath $x$} }})| + |f_{21}({{ \mbox{\boldmath $x$} }})|)\nonumber\\
& = &
-10 - 3 - \tau(x_{1}(t)) + \frac{1}{2}(|1| + |1|)\nonumber\\
& = & -12 - \tau(x_{1}(t))
\label{rat1}\\
& < & -12 \nonumber\end{aligned}$$ for all ${{ \mbox{\boldmath $x$} }}\in{{ \mbox{\boldmath $R$} }}^{2}\setminus\{{{ \mbox{\boldmath $0$} }}\}$ and $$\begin{aligned}
\beta_{2}({{ \mbox{\boldmath $x$} }}) & = & -a_{2} + f_{22}({{ \mbox{\boldmath $x$} }}) +
\frac{1}{2}(|f_{21}({{ \mbox{\boldmath $x$} }})| + |f_{12}({{ \mbox{\boldmath $x$} }})|)\nonumber\\
& = & -10 - 1 + \frac{1}{5}\tau(x_{2}(t)) + \frac{1}{2}(|1| +
|1|)\nonumber\\
& = & -10 + \frac{1}{5}\tau(x_{2}(t))
\label{rat2}\\
& < & -10 + \frac{3}{5} = -\frac{47}{5} \nonumber\end{aligned}$$ for all ${{ \mbox{\boldmath $x$} }} \in {{ \mbox{\boldmath $R$} }}^{2}\setminus\{{{ \mbox{\boldmath $0$} }}\}$. Clearly, both $\beta_{1}({{ \mbox{\boldmath $x$} }}) < 0$ and $\beta_{2}({{ \mbox{\boldmath $x$} }}) < 0$ for all ${{ \mbox{\boldmath $x$} }}\in{{ \mbox{\boldmath $R$} }}^{2}\setminus\{{{ \mbox{\boldmath $0$} }}\}$.
Next, we shall check the condition on $\beta_{i}({{ \mbox{\boldmath $x$} }})$ for ${{ \mbox{\boldmath $x$} }} = {{ \mbox{\boldmath $0$} }}$, where $ i = 1,2$. From (\[rat1\]), we have $$\beta_{1}({{ \mbox{\boldmath $x$} }}) = -12 - \tau(x_{1}(t))\,.$$ Therefore, $$\beta_{1}({{ \mbox{\boldmath $0$} }}) = -12 - 3 = -15 \,.$$ Similarly, from (\[rat2\]), we have $$\beta_{2}({{ \mbox{\boldmath $x$} }}) = -10 + \frac{1}{5} \tau(x_{2}(t))\,.$$ Therefore, $$\beta_{2}({{ \mbox{\boldmath $0$} }}) = -10 + \frac{3}{5} = -\frac{47}{5} \,.$$ Since $\beta_{1}({{ \mbox{\boldmath $x$} }}) < 0$ and $\beta_{2}({{ \mbox{\boldmath $x$} }}) < 0$ for all ${{ \mbox{\boldmath $x$} }} \in {{ \mbox{\boldmath $R$} }}^{2}$, therefore, by condition (c) of Theorem \[raveen-2\], the zero solution of system (\[c4e6\]) is globally asymptotically stable.
Conclusion
==========
We have established the criteria for stability, asymptotic stability and global asymptotic stability for a non linear autonomous system via the method of Lyapunov. We have also considered the usefulness of our main results by application of it to artificial neural networks.
Further research in this direction is being carried out, considering a non autonomous system, wherein the external input source is not assumed to be zero. Determining the convergence criteria for a non autonomous system and to measure its rate of convergence will be of grandness in applications to artificial neural networks.
[99]{} S. Sastry, [*Nonlinear Systems: Analysis, Stability and Control*]{}, New York: Springer-Verlag, 1999. P. Glendenning, [*Stability, instability and chaos: an introduction to the theory of nonlinear differential equations*]{}, Cambridge University Press, INC., New York. N. N. Krasovskii, “On the stability in the large of a system of nonlinear differential equations", [*Prikl. Mat. Meh.*]{}, vol 18, pp. 735–737, 1954. J. Vanualailai, T. Soma and S. Nakagiri, “Convergence of Solutions and Practical Stability of Hopfield-type Neural Networks with Time-Varying External Inputs", [*Nonlinear Studies*]{}, vol 9, pp. 109–122, 2002. M. A. Arbib (editor), [*The Handbook of Brain Theory and Neural Networks*]{}, London: MIT Press, 1995. V. Lakshmikantham, V. M. Matrosov and S. Sivasundaram, [*Vector Lyapunov Functions and the Stability Analysis of Nonlinear Systems*]{}, The Netherlands: Kluwer Academic Publishers, 1991. J. J. Hopfield and D. W. Tank, “Computing with neural circuits: A Model,” [*Science*]{}, vol. 233, pp. 625–632, 1986.
[^1]: [Corresponding Author. Email: raveeng@hotmail.com]{}
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abstract: 'We consider the 1D Schrödinger operator $Hy=-y''''+(p+q)y$ with a periodic potential $p$ plus compactly supported potential $q$ on the real line. The spectrum of $H$ consists of an absolutely continuous part plus a finite number of simple eigenvalues in each spectral gap $\g_n\ne \es, n\geq 0$, where $\g_0$ is unbounded gap. We prove the following results: 1) we determine the distribution of resonances in the disk with large radius, 2) a forbidden domain for the resonances is specified, 3) the asymptotics of eigenvalues and antibound states are determined, 4) if $q_0=\int_\R qdx=0$, then roughly speaking in each nondegenerate gap $\g_n$ for $n$ large enough there are two eigenvalues and zero antibound state or zero eigenvalues and two antibound states, 5) if $H$ has infinitely many gaps in the continuous spectrum, then for any sequence $\s=(\s)_1^\iy, \s_n\in \{0,2\}$, there exists a compactly supported potential $q$ such that $H$ has $\s_n$ bound states and $2-\s_n$ antibound states in each gap $\g_n$ for $n$ large enough. 6) For any $q$ (with $q_0=0$), $\s=(\s_n)_{1}^\iy$, where $\s_n\in \{0,2\}$ and for any sequence $\d=(\d_n)_1^\iy\in \ell^2, \d_n>0$ there exists a potential $p\in L^2(0,1)$ such that each gap length $|\g_n|=\d_n, n\ge 1$ and $H$ has exactly $\s_n$ eigenvalues and $2-\s_n$ antibound states in each gap $\g_n\ne \es$ for $n$ large enough.'
address: 'School of Math., Cardiff University. Senghennydd Road, CF24 4AG Cardiff, Wales, UK. email KorotyaevE@cf.ac.uk, [Partially supported by EPSRC grant EP/D054621.]{}'
author:
- Evgeny Korotyaev
title: 1D Schrödinger operator with periodic plus compactly supported potentials
---
\[section\] \[theorem\][**Lemma**]{} \[theorem\][**Corollary**]{} \[theorem\][**Proposition**]{} \[theorem\][**Definition**]{} \[theorem\][*Remark*]{} c ł Ł § ¶ ø Ø
ß 2[\^[2]{}]{}
Introduction
============
Consider the Schrödinger operator $H=H_0+q$ acting in $L^2(\R)$, where $H_0=-{d^2\/dx^2}+p(x)$ and $p\in L^2(0,1)$ is the real 1-periodic potential. The real compactly supported potential $q$ belongs to the class $\cQ_{t}=\{q\in L^2(\R ): \ [0,t] \ is\ the\ convex\ hull\ of\ the\ support\ of\ q\}$ for some $t>0$. The spectrum of $H_0$ consists of spectral bands ${\mathfrak{S}}_n$ and is given by (see Fig. 1) $$\s(H_0)=\s_{ac}(H_0)=\cup {\mathfrak{S}}_n,\qq {\mathfrak{S}}_n=[E^+_{n-1},E^-_n],n\ge 1,\qq
E_0^+<..\le E^+_{n-1}< E^-_n \le E^+_{n}<...$$ We assume that $E_0^+=0$. The sequence $E_0^+<E_1^-\le E_1^+\ <\dots$ is the spectrum of the equation $$\lb{1}
-y''+p(x)y=\l y, \ \ \ \ \l\in \C ,$$ with the 2-periodic boundary conditions, i.e. $y(x+2)=y(x), x\in \R$. The bands ${\mathfrak{S}}_n, {\mathfrak{S}}_{n+1}$ are separated by a gap $\g_{n}=(E^-_{n},E^+_n)$ and let $\g_0=(-\iy,E_0^+)$. If a gap degenerates, that is $\g_n=\es $, then the corresponding bands ${\mathfrak{S}}_{n} $ and ${\mathfrak{S}}_{n+1}$ merge. If $E_n^-=E_n^+$ for some $n$, then this number $E_n^{\pm}$ is the double eigenvalue of equation with the 2-periodic boundary conditions. The lowest eigenvalue $E_0^+=0$ is always simple and the corresponding eigenfunction is 1-periodic. The eigenfunctions, corresponding to the eigenvalue $E_{2n}^{\pm}$, are 1-periodic, and for the case $E_{2n+1}^{\pm}$ they are anti-periodic, i.e., $y(x+1)=-y(x),\ \ x\in\R$.
It is well known, that the spectrum of $H$ consists of an absolutely continuous part $\s_{ac}(H)=\s_{ac}(H_0)$ plus a finite number of simple eigenvalues in each gap $\g_n\ne \es, n\geq 0$, see [@Rb], [@F1]. Moreover, in every open gap $\g_n$ for $n$ large enough the operator $H$ has at most two bound states [@Rb] and precisely one bound state in the case $q_0=\int_\R q(x)dx\ne 0$ [@Zh], [@F2]. Note that the potential $q$ in [@Rb],[@Zh] belongs to the more general class, see also [@GS], [@So].
=1.00mm
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Let $\vp(x,z), \vt(x,z)$ be the solutions of the equation $-y''+py=z^2y$ satisfying $\vp'(0,z)=\vt(0,z)=1$ and $\vp(0,z)=\vt'(0,z)=0$, where $u'=\pa_x u$. The Lyapunov function is defined by $\D(z)={1\/2}(\vp'(1,z)+\vt(1,z))$. The function $\D^2(\sqrt \l)$ is entire, where $\sqrt \l$ is defined by $\sqrt 1=1$. Introduce the function $\O(\l)=(1-\D^2(\sqrt \l))^{1\/2}, \l\in \ol\C_+$ and we fix the branch $\O(\l)=(1-\D^2(\sqrt \l))^{1\/2}$ by the condition $\O(\l+i0)>0$ for $\l\in {\mathfrak{S}}_1=[E^+_{0},E^-_1]$. Introduce the two-sheeted Riemann surface $\L$ of $\O(\l)=(1-\D^2(\sqrt \l))^{1\/2}$ obtained by joining the upper and lower rims of two copies of the cut plane $\C\sm\s_{ac}(H_0)$ in the usual (crosswise) way. The n-th gap on the first physical sheet $\L_1$ we will denote by $\g_n^{(1)}$ and the same gap but on the second nonphysical sheet $\L_2$ we will denote by $\g_n^{(2)}$ and let $\g_n^{(0)}$ be the union of $\ol\g_n^{(1)}$ and $\ol\g_n^{(2)}$, i.e., $$\g_n^{(0)}=\ol\g_n^{(1)}\cup \ol\g_n^{(2)}.$$
Introduce the function $D(\l)=\det (I+q(H_0-\l)^{-1})$, which is meromorphic in $\L$, see [@F1]. Recall that $1/D$ is the transmission coefficient in the S-matrix for the operators $H,H_0$, see Sect. 2. If $D$ has some poles, then they coincide with some $E_n^\pm$. It is well known that if $D(\l)=0$ for some zero $\l\in \L_1$, then $\l$ is an eigenvalue of $H$ and $\l\in\cup \g_n^{(1)}$. Note that there are no eigenvalues on the spectrum $\s_{ac}(H_0)\ss \L_1$, since $|D(\l)|\ge 1$ on $\s_{ac}(H_0)\ss \L_1$ (see and all these facts in [@F1]).
Define the functions $A, J$ by $$J(\l)= 2\O(\l+i0)\Im D(\l+i0),\qqq \ A(\l)=\Re D(\l+i0)-1, \qq for \qq \l\in \s(H_0)\ss\L_1.$$ In Lemma \[T32\] we will show that $A, J$ are entire functions on $\C$ and they are real on the real line. Instead of the function $D$ we will study the [**modified function**]{} $\Xi=2i\O D$ on $\L$. We will show that $\Xi$ satisfies $$\lb{T1-1}
\Xi=2i\O D=2i \O (1+A)-J\qqq on \qqq \L,$$ see Theorem \[T1\]. Recall that $\O$ is analytic in $\L$ and $\O(\l)=0$ for some $\l\in \L$ iff $\l=E_n^-$ or $\l=E_n^+$ for some $n\ge 0$. Then the function $\Xi$ is analytic on $\L$ and has branch points $E_n^\pm, \g_n\ne \es$. The zeros of $\Xi$ define the eigenvalues and resonances, similar to the case $p=0$. Define the set $\L_0 =\{\l\in \L: \l=E_n^+\in \L_1 $ and $\l=E_n^+\in \L_2, \g_n=\es, n\ge 1\}\ss \L$. In fact with each $\g_n=\es$ we associate two points $E_n^+\in \L_1$ and $E_n^+\in \L_2$ from the set $\L_0$. If each gap of $H_0$ is empty, then $\L_0$ is a union of two sets $\{(\pi n)^2, n\in \N\}\ss \L_1$ and $\{(\pi n)^2, n\in \N\}\ss \L_2$. If each gap of $H_0$ is not empty, then $\L_0=\es$.
It is known that the gaps $\g_n=\es$ do not give contribution to the states. Recall that S-matrix for $H,H_0$ is meromorphic on $\L$, but it is analytic at the points from $\L_0$ (see [@F1]). Roughly speaking there is no difference between the points from $\L_0$ and other points inside the spectrum of $H_0$.
The gap $\g_n^{(1)}\ss\L_1$ is so-called physical gap and the gap $\g_n^{(2)}\ss\L_2$ is so-called non-physical gap. If $q_0=\int_\R q(x)dx\ne 0$, then $H$ has precisely one bound state on each physical gap $\ne \es$ and odd number $\ge 1$ of resonances on each non-physical gap $\ne \es$ for $n$ large enough [@F1].
We explain roughly why antibound states are important. Consider the operator $H_\t=H_0+\t q$, where $\t\in \R$ is the coupling constant. If $\t=0$, then $H=H_0$ and there are no eigenvalues, complex resonances and $H$ has only virtual states, which coincide with each $E_n^\pm , , \g_n\ne \es$, since $\X=2i\O$ on $\L$ at $q=0$ and $\O(E_n^\pm)=0$. If $\t$ is increasing, then there are eigenvalues, antibound states (close to the end of gaps) and complex resonances. If $\t$ is increasing again, then there are no new eigenvalues but some complex resonances ($\z\in \C_+\ss\L_2$ and $\ol \z\in \C_-\ss\L_2$) reach some non physical gap and transform into new antibound states. If $\t$ is increasing again, then some new antibound states will be virtual states, and then later they will be bound states. Thus if $\t$ runs through $\R_+$, then there is a following transformation: resonances $\to $ antibound states $\to $ virtual states $\to $ bound states $\to $ virtual states...
For each $ n\ge 1$ there exists an unique point $E_n\in [E_n^-,E_n^+]$ such $$\lb{dzn}
(-1)^n\D(\sqrt E_n)=\max_{z\in [E_n^-,E_n^+]} |\D(\sqrt\l)|=\cosh h_n, \qq \ {\rm for \ some} \ h_n\ge 0.$$ Note that if $\g_n=\es$, then $E_n=E_n^\pm$ and if $\g_n\ne \es$, then $E_n\in \g_n$.
Let potentials $(p,q)\in L^2(0,1)\ts \cQ_t, t>0$. Then $\Xi$ satisfies and
i\) There exist even number $\ge 0$ of states (counted with multiplicity) on each set $\g_n^{(0)}\ne \es,n\ge 1$, where $\g_n^{(0)}$ is a union of the physical $\ol \g_n^{(1)}\ss \L_1$ and non-physical gap $\ol \g_n^{(2)}\ss\L_2$ (here $\g_n\ne \es$).
ii\) Let $\|q\|_t=\!\int_0^t\!\!|q(x)|dx$. There are no states in the “forbidden” domain $\mD_F\sm \cup \ol \g_n^{(2)}\ss\L_2$, where $$\begin{gathered}
\lb{T1-3}
\mD_F=\{\l\in \L_2: |\l|^{1\/2}>\max \{180e^{2\|p\|_1},C_Fe^{2t|\Im \sqrt \l|}\}\},\ \ C_F=12\|q\|_te^{\|p\|_1+\|q\|_t+2\|p\|_t}.\end{gathered}$$
iii\) In each $\g_n^{(0)}\ne \es, n\ge 1+{e^{t\pi /2}\/\pi}C_F$ there exists exactly two simple real states $\l_n^\pm\in \g_n^{(0)}$ such that $E_n^-\le\l_n^-<E_n<\l_n^+\le E_n^+$.
iv\) Let $\l\in \g_n^{(1)}$ be an eigenvalue of $H$ for some $n\ge 0$, i.e., $\Xi(\l)=0$ and let $\l^{(2)}\in \g_n^{(2)}\ss \L_2$ be the same number but on the second sheet $\L_2$. Then $\l^{(2)}$ is not an anti-bound state, i.e., $\Xi(\l^{(2)})\ne 0$.
1\) The forbidden domain $\mD_F\cap \C_-$ is similar to the case $p=0$, see [@K2].
2\) In Theorem \[T2\] we show that $\l_n^\pm\to E_n^\pm$ as $n\to \iy$.
Let $\m_n^2, n\ge 1$ be the Dirichlet spectrum of the equation $-y''+py=\m^2y$ on the interval $[0,1]$ with the boundary condition $y(0)=y(1)=0$. It is well known that each $\m_n^2\in [E^-_n,E^+_n ], n\ge 1$. Define the coefficients $q_0=\int_\R q(x)dx$ and $$\lb{fco}
\wh q_n=\wh q_{cn}+i\wh q_{sn},\qq
\wh q_{cn}=\int_\R q(x)\cos 2\pi nxdx,\qqq
\wh q_{sn}=\int_\R q(x)\sin 2\pi nxdx, \ n\ge 1.$$ In order to formulate Theorem \[T2\] we define the functions $\P_{cn}, \P_{sn},
c_n=\cos \f_n, s_n=\sin \f_n, n\ge 1$ (depending from $p\in L^2(0,1)$) by $$\lb{ip1}
\P_{cn}={E_n^-+E_n^+\/2}-\m_n^2={|\g_n|\/2}c_n,\qq
\P_{sn}={|\g_n|\/2}s_n,\
\qq
\sign s_n=\sign |\vp'(1,\m_n)|.$$ The identity ${E_n^-+E_n^+\/2}-\m_n^2={|\g_n|\/2}c_n$ defines $c_n=\cos \f_n\in [-1,1]$ and the identity $\sign s_n=\sign |\vp'(1,\m_n)|$ defines $s_n=\sin \f_n$ and the angle $\f_n\in [0,2\pi)$. Recall the results from [@K5]:
[*The mapping $\P: \cH\to \ell^2\os \ell^2$ given by $\P=((\P_{cn})_1^\iy,(\P_{sn})_1^\iy)$ is a real analytic isomorphism between real Hilbert spaces $\cH=\{p\in L^2(0,1): \int_0^1p(x)dx=0\}$ and $\ell^2\os \ell^2$.*]{}
Let $\#(H,r, X)$ be the total number of state of $H$ in the set $X\subseteq \L$ having modulus $\le r^2$, each state being counted according to its multiplicity.
Let $(p,q)\in L^2(0,1)\ts \cQ_t$ and let $q_0=\int_\R q(x)dx$. Then
i\) Let $\g_n\ne \es$ for some $n$ large enough. Then $H$ has exactly two simple states $\l_n^-,\l_n^+\in \g_n^{(0)}$. Moreover, if $\l$ is one of $\l_n^-,\l_n^+$ and satisfies $(-1)^{n+1}J(\l)>0$ (or $(-1)^{n+1}J(\l)<0$ or $J(\l)=0$), then $\l$ is a bound state (or an anti-bound state or a virtual state).
ii\) The following asymptotics hold true: $$\begin{gathered}
\lb{T2-1}
\sqrt{\l_n^\pm}=\sqrt{E_n^\pm}\mp {2|\g_n|\/(4\pi n)^3}(\mp q_0-c_n\wh q_{cn}+s_n\wh q_{sn}+ O(1/n))^2,\\
\qqq (-1)^{n+1}J(\l_n^\pm)={|\g_n|\/(2\pi n)^2}(\mp q_0-c_n\wh q_{cn}+s_n\wh q_{sn}+ O(1/n)),
\\
if \qq q_0>0 \ \Rightarrow \qq \l_n^-\in \L_1\ is \ bound \ state, \qqq
\l_n^+\in \L_2\ is \ anti \ bound \ state, \\
if \qq q_0<0 \ \Rightarrow \ \qq
\l_n^-\in \L_2 \ is \ anti \ bound \ state, \qqq \l_n^+\in \L_1\ is \ bound \ state.\end{gathered}$$
iii\) Let $\wh q_n=\wh q_{cn}+i\wh q_{sn}=|\wh q_n|e^{i\t_n}, n\ge 1$ and $q_0=0$. Assume that $|\cos (\f_n+\t_n)|>\ve>0$ and $|\wh q_n|>n^{-\a}$ for $n$ large enough and for some $\ve, \a\in (0,1)$. Then the operator $H$ has $\s_n=1-\sign \cos (\f_n+\t_n)$ bound states in the physical gap $\g_n^{(1)}\ne \es$ and $2-\s_n$ resonances inside the nonphysical gap $\g_n^{(2)}\ne \es$ for $n$ large enough.
iv\) For some integer $N_S\in \Z$ the following asymptotics hold true: $$\lb{T2-2}
\#(H,r,\L_2\sm\cup\g_n^{(2)})=r{2t+o(1)\/\pi}\qq as \qq r\to \iy,$$ $$\lb{T2-3}
\#(H,r,\cup \G_n)=\#(H_0,r,\cup \G_n)+2N_S \qq as \qq r\to \iy, \qq r^2\notin \cup\ol\g_n.$$
. 1) First term in the asymptotics does not depend on the periodic potential $p$. Recall that asymptotics for the case $p=0$ was obtained by Zworski [@Z].
2\) The main difference between the distribution of the resonances for the case $p\ne \const$ and $p=\const$ is the bound states and anti bound states in high energy gaps, see (iii) and .
3\) Assume that a potential $u\in L^2(\R)$ is compactly supported, $\supp u\ss (0,t)$ and satisfies $|\wh u_{n}|=o(n^{-\a})$ as $n\to \iy$. Then in the case (iii) the operator $H+u$ has also $1-\sign \cos (\f_n+\t_n)$ bound states in each gap $g_n\ne \es$ for $n$ large enough.
4\) In the proof of we use the Paley Wiener type Theorem from [@Fr], the Levinson Theorem (see Sect. 3) and a priori estimates from [@KK], [@M].
5\) Assume that $\m_n^2=E_n^-$ or $\m_n^2=E_n^+$ with $|\g_n|>0$ for all $n\in \N_0$, where $ \N_0\ss \N$ is some infinite subset. Then for all large $n\in \N_0$ the following asymptotics hold true: $$\lb{T4-1}
\sqrt{\l_n^\pm}=\sqrt{E_n^\pm}\mp {2|\g_n|\/(4\pi n)^2}\rt(\mp q_0-c_n\wh q_{cn}+ {O(1)\/n}\rt)^2
\qqq \as \qq n\to \iy.$$ There is no $\wh q_{sn}$ in asymptotics . Let in addition $q_0=0$ and $|\wh q_{cn}|>n^{-\a}$ for all $n\in \N_0$ and some $\a\in (0,1)$. Then $H$ has exactly $\s_n=1-\sign c_n\wh q_{cn}$ eigenvalues in each gap $\g_n^{(1)}$ and $2-\s_n$ resonances inside each gap $\g_n^{(2)}\ne \es$ for $n\in \N_0$ large enough.
6\) Recall that if $p$ is even i.e., $p\in L_{even}^2(0,1)=\{p\in L^2(0,1),
p(x)=p(1-x), x\in (0,1)\}$, iff $s_n=0$ ( or $\m_n^2\in \{E_n^-,E_n^+\}$, or $c_n\in \{\pm 1\}$) for all $n\ge 1$, see [@GT], [@KK1].
Consider some inverse problems for the operator $H$.
i\) Let the operator $H_0$ have infinitely many gaps $\g_n\ne \es$ for some $p\in L^2(0,1)$. Then for any sequence $\s=(\s_n)_{1}^\iy$, where $\s_n\in \{0,2\}$, there exists some potential $q\in \cQ^t,t>0$ such that $H$ has exactly $\s_n$ eigenvalues in each gap $\g_n^{(1)}\ne \es$ and $2-\s_n$ resonances inside each gap $\g_n^{(2)}\ne \es$ for $n$ large enough.
ii\) Let $q\in \cQ_t,t>0$ satisfy $q_0=0$ and let $|\wh q_{cn}|>n^{-\a}$ for all $n$ large enough and some $\a\in (0,1)$. Let $\s=(\s_n)_{1}^\iy$, where $\s_n\in \{0,2\}$ and infinitely many $\d_n>0$. Then for any sequence $\d=(\d_n)_1^\iy\in \ell^2$ where each $\d_n\ge 0, n\ge 1$ there exists a potential $p\in L^2(0,1)$ such that each gap length $|\g_n|=\d_n, n\ge 1$. Moreover, $H$ has exactly $\s_n$ eigenvalues in each physical gap $\g_n^{(1)}\ne \es$ and $2-\s_n$ resonances inside each non-physical gap $\g_n^{(2)}\ne \es$ for $n$ large enough.
[**Remark .**]{} 1) In the proof we use results from the inverse spectral theory from [@K5].
2\) We have $\a\in ({1\/2},1)$, if all gaps are open.
A lot of papers are devoted to the resonances for the Schrödinger operator with $p=0$, see [@Fr], [@H], [@K1], [@K2], [@S], [@Z] and references therein. Although resonances have been studied in many settings, but there are relatively few cases where the asymptotics of the resonance counting function are known, mainly one dimensional case [@Fr], [@K1], [@K2], [@S], and [@Z]. We recall that Zworski \[Z\] obtained the first results about the distribution of resonances for the Schrödinger operator with compactly supported potentials on the real line. The author obtained the characterization (plus uniqueness and recovering) of $S$-matrix for the Schrödinger operator with a compactly supported potential on the real line [@K2] and the half-line [@K1], see also [@Z1], [@BKW] about uniqueness.
For the Schrödinger operator on the half line the author [@K3] obtained the stability results:
\(i) If $\vk^0=(\vk^0)_1^\iy$ is a sequence of zeros (eigenvalues and resonances) of the Jost function for some real compactly supported potential $q_0$ and $\vk-\vk^0\in\ell_\ve^2$ for some $\ve>1$, then $\vk$ is the sequence of zeroes of the Jost function for some unique real compactly supported potential.
\(ii) The measure associated with the zeros of the Jost function is the Carleson measure, the sum $\sum
(1+|\vk_n^0|)^{-\a}, \a>1$ is estimated in terms of the $L^1$-norm of the potential $q^0$.
Brown and Weikard [@BW] considered the Schrödinger operator $-y''+(p_A+q)y$ on the half-line, where $p_A$ is an algebro-geometric potentials and $q$ is a compactly supported potential. They proved that the zeros of the Jost function determine $q$ uniquely.
Christiansen [@Ch] considered resonances associated to the Schrödinger operator $-y''+(p_{S}+q)y$ on the real line, where $p_S$ is a step potential. She determined asymptotics of the resonance-counting function. Moreover, she obtained that the resonances determine $q$ uniquely.
In the proof of theorems we use properties of the quasimomentum from [@KK], [@M], a priori estimates from [@KK], [@M], and results from the inverse theory for the Hill operator from [@K5], see .
The plan of the paper is as follows. In Section 2 we describe the preliminary results about fundamental solutions. In Sections 3 we study the function $\x$ and prove Theorem \[T1\]. In Sections 4 we prove the main Theorem \[T2\]-\[T3\], which are crucial for Section 3 and 4.
Preliminaries
==============
We will work with the momentum $z=\sqrt \l$, where $\l$ is an energy and recall that $\sqrt 1=1$. If $\l\in \g_n, n\ge 1$, then $z\in g_{\pm n}$ and if $\l\in \g_0=(-\iy,0)$, then $z\in g_0^\pm=i\R_\pm$, where the momentum gaps $g_n$ are given by $$\lb{2}
g_n=(e_n^-,e_n^+)=-g_{-n},\qq e_n^\pm=\sqrt{E_n^\pm}>0,\qq
n\ge 1,\qq and \qq \D(e_{n}^{\pm})=(-1)^n.$$ Introduce the cut domain (see Fig.2) $$\cZ_0=\C\sm \cup \ol g_n.$$
(120.67,34.33) (20.33,21.33)[(1,0)[100.33]{}]{} (70.33,10.00)[(0,1)[24.33]{}]{} (69.00,19.00)[(0,0)\[cc\][$0$]{}]{} (120.33,19.00)[(0,0)\[cc\][$\Re z$]{}]{} (67.00,33.67)[(0,0)\[cc\][$\Im z$]{}]{} (81.33,21.33)[(1,0)[9.67]{}]{} (100.33,21.33)[(1,0)[4.67]{}]{} (116.67,21.33)[(1,0)[2.67]{}]{} (60.00,21.33)[(-1,0)[9.33]{}]{} (40.00,21.33)[(-1,0)[4.67]{}]{} (24.33,21.33)[(-1,0)[2.33]{}]{} (81.67,24.00)[(0,0)\[cc\][$e_1^-$]{}]{} (91.00,24.00)[(0,0)\[cc\][$e_1^+$]{}]{} (100.33,24.00)[(0,0)\[cc\][$e_2^-$]{}]{} (105.00,24.00)[(0,0)\[cc\][$e_2^+$]{}]{} (115.33,24.00)[(0,0)\[cc\][$e_3^-$]{}]{} (120.00,24.00)[(0,0)\[cc\][$e_3^+$]{}]{} (59.33,24.00)[(0,0)\[cc\][$-e_1^-$]{}]{} (50.67,24.00)[(0,0)\[cc\][$-e_1^+$]{}]{} (40.33,24.00)[(0,0)\[cc\][$-e_2^-$]{}]{} (34.67,24.00)[(0,0)\[cc\][$-e_2^+$]{}]{} (26.00,24.00)[(0,0)\[cc\][$-e_3^-$]{}]{} (19.50,24.00)[(0,0)\[cc\][$-e_3^+$]{}]{}
Below we will use the momentum variable $z=\sqrt \l$ and the corresponding the Riemann surface $\cM$, which is more convenient for us, than the Riemann surface $\L$. Slitting the n-th momentum gap $g_n$ (suppose it is nontrivial) we obtain a cut $G_n$ with an upper $g_n^+$ and lower rim $g_n^-$. Below we will identify this cut $G_n$ and the union of of the upper rim (gap) $\ol g_{n}^+$ and the lower rim (gap) $\ol g_{n}^{\ -}$, i.e., $$G_{n}=\ol g_{n}^+\cup \ol g_{n}^-,\ \ where \ \ g_{n}^\pm =g_n\pm i0.
$$ Also we will write $z\pm i0\in g_n^\pm$, if $z\in g_n$.
In order to construct the Riemann surface $\cM$ we take the cut domain $\cZ_0=\C\sm \cup \ol g_n$ and identify (i.e. we glue) the upper rim $g_{n}^+$ of the slit $G_n$ with the upper rim $g_{-n}^+$ of the slit $G_{-n}$ and correspondingly the lower rim $g_{n}^-$ of the slit $G_n$ with the lower rim $g_{-n}^-$ of the slit $G_{-n}$ for all nontrivial gaps.
The mapping $z=\sqrt \l$ from $\L$ onto $\cM$ is one-to-one and onto. The gap $\g_n^{(1)}\ss \L_1$ maps onto $g_n^+\ss \cM_1$ and the gap $\g_n^{(2)}\ss \L_2$ maps onto $g_n^-\ss \cM_2$. From a physical point of view, the upper rim $g_{n}^+$ is a physical gap and the lower rim $g_{n}^-$ is a non physical gap. Moreover, $\cM\cap\C _+=\cZ_0\cap\C _+$ plus all physical gaps $g_{n}^+$ is a so-called physical “sheet” $\cM_1$ and $\cM\cap\C _-=\cZ_0\cap\C _-$ plus all non physical gaps $g_{n}^-$ is a so-called non physical “sheet” $\cM_2$. The set (the spectrum) $\R\sm \cup g_n$ joints the first and second sheets.
We introduce the quasimomentum $k(\cdot )$ for $H_0$ by $k(z)=\arccos \D(z),\ z \in \cZ_0=\C\sm \cup \ol g_n$. The function $k(z)$ is analytic in $z\in\cZ_0$ and satisfies: $$\lb{pk}
(i)\qq k(z)=z+O(1/z)\qq as \ \ |z|\to \iy, \qq and \qq (ii)\qq \Re k(z\pm i0) |_{[e_n^-,e_n^+]}=\pi n,\qq \ n\in \Z,$$ and $\pm \Im k(z)>0$ for any $z\in \C_\pm$, see [@M], [@KK]. The function $k(\cdot)$ is analytic on $\cM$ and satisfies $\sin k(z)=\O(z^2), z\in \cM$. Moreover, the quasimomentum $k(\cdot)$ is a conformal mapping from $\cZ_0$ onto the quasimomentum domain $\cK=\C\sm \cup \G_n$, where the slit $\G_n=(\pi n-ih_n,\pi n+ih_n)$. The height $h_n$ is defined by the equation $\cosh h_n=(-1)^n\D(e_n)$, where $e_n\in [e_n^-,e_n^+]$ and $\D'(e_n)=0$. The Floquet solutions $\p_{\pm}(x,z), z \in \cZ_0$ of $H_0$ is given by $$\lb{3}
\p_\pm(x,z)=\vt(x,z)+m_\pm(z)\vp(x,z),\ \
m_\pm={\b\pm i\sin k\/ \vp(1,\cdot)},\ \
\b={\vp'(1,\cdot)-\vt(1,\cdot)\/2},$$ where $\vp(1,z)\p^+(\cdot,z)\in L^2(\R_+)$ for all $z\in\C_+\cup\cup_{} g_n$. If $p=0$, then $k=z$ and $\p_\pm(x,z)=e^{\pm izx}$.
(120.67,34.33) (20.33,20.00)[(1,0)[102.33]{}]{} (71.00,7.00)[(0,1)[27.00]{}]{} (70.00,18.67)[(0,0)\[cc\][$0$]{}]{} (124.00,18.00)[(0,0)\[cc\][$\Re k$]{}]{} (67.00,33.67)[(0,0)\[cc\][$\Im k$]{}]{} (87.00,15.00)[(0,1)[10.]{}]{} (103.00,17.00)[(0,1)[6.]{}]{} (119.00,18.00)[(0,1)[4.]{}]{} (56.00,15.00)[(0,1)[10.]{}]{} (39.00,17.00)[(0,1)[6.]{}]{} (23.00,18.00)[(0,1)[4.]{}]{} (85.50,18.50)[(0,0)\[cc\][$\pi$]{}]{} (54.00,18.50)[(0,0)\[cc\][$-\pi$]{}]{} (101.00,18.50)[(0,0)\[cc\][$2\pi$]{}]{} (36.00,18.50)[(0,0)\[cc\][$-2\pi$]{}]{} (117.00,18.50)[(0,0)\[cc\][$3\pi$]{}]{} (20.00,18.50)[(0,0)\[cc\][$-3\pi$]{}]{} (87.00,26.00)[(0,0)\[cc\][$\pi+ih_1$]{}]{} (56.00,26.00)[(0,0)\[cc\][$-\pi+ih_1$]{}]{} (103.00,24.00)[(0,0)\[cc\][$2\pi+ih_2$]{}]{} (39.00,24.00)[(0,0)\[cc\][$-2\pi+ih_2$]{}]{} (119.00,23.00)[(0,0)\[cc\][$3\pi+ih_3$]{}]{} (23.00,23.00)[(0,0)\[cc\][$-3\pi+ih_3$]{}]{}
The function $\sin k$ and each function $\vp(1,\cdot)\p_{\pm}(x,\cdot), x\in \R$ are analytic on the Riemann surface $\cM$. Recall that the Floquet solutions $\p_\pm(x,z), (x,z)\in \R\ts \cM$ satisfy (see [@T]) $$\lb{f1}
\p_\pm(0,z)=1, \qq \p_\pm(0,z)'=m_\pm(z),
\qq \p_\pm(1,z)=e^{\pm ik(z)}, \qq \p_\pm(1,z)'=e^{\pm ik(z)}m_\pm(z),$$ $$\lb{f2}
\p_\pm(x,z)=e^{\pm ik(z)x}(1+O(1/z)) \qq as \qq |z|\to \iy, \qq (x,z)\in \R\ts \cZ_\ve,$$ where the set $\cZ_\ve =\{z\in \cZ_0: \dist \{z,g_n\}>\ve, g_n\ne \es, n\in \Z\},\ \ve>0$. Below we need the simple identities $$\lb{LD0}
\b^2+1-\D^2=1-\vp'(1,\cdot)\vt(1,\cdot)= -\vp(1,\cdot)\vt'(1,\cdot).$$ Let $\cD_r(z_0)=\{|z-z_0|<r\}$ be a disk for some $r>0$. Recall the well known properties of the function $m_\pm$ (see [@T]).
i\) $\Im m_+ (z)>0$ for all $(z,n)\in (z_{n-1}^+,z_{n}^-)\ts \N$ and the following asymptotics hold true: $$\lb{Tm-1}
m_\pm (z)=\pm iz+O(1) \qq as \qq |z|\to \iy, \qq z\in \cZ_\ve,\ve >0.$$
ii\) Let $g_n=\es$ for some $n\in \Z$. Then the functions $\sin k(\cdot), m_\pm$ are analytic in the disk $B(\m_n,\ve)\ss\cZ_0$ for some $\ve>0$ and the functions $\sin k(z)$ and $\vp(1,z)$ have the simple zero at $\m_n$. Moreover, $m_\pm$ satisfies $$\lb{Tm-2}
m_\pm (\m_n)={\b'(\m_n)\pm i(-1)^nk'(\m_n)\/\pa_z\vp(1,\m_n)},
\qq \Im m_\pm (\m_n)\ne 0.$$
iii\) If the function $m_+$ has a pole at $\m_n+i0$ for some $n\in \N$, then $k(\m_n+i0)=\pi n+ih_{sn}$ and $$\begin{gathered}
\lb{Tm-31}
\qq h_{sn}>0,\qq
\b(\m_n)=i\sin k(\m_n+i0)= -(-1)^n\sinh h_{sn},\qq m_+\in \mA(\m_n-i0),\\
\qq m_+(\m_n+z)={\r_n+O(z)\/z} \qq as \ z\to 0, \ z\in \C_+,\
\qq \r_n={-2\sinh |h_{sn}|\/(-1)^n\pa_z\vp(1,\m_n)}<0.\end{gathered}$$
iv\) If the function $m_+$ has a pole at $\m_n-i0$ for some $n\in \N$, then $k(\m_n-i0)=\pi n+ih_{sn}$ and $$\begin{gathered}
\lb{Tm-32}
h_{sn}<0,\qq
\b(\m_n)=-i\sin k(\m_n-i0)=(-1)^n\sinh h_{sn},\ m_+\in \mA(\m_n^+),\\
\qq m_+(\m_n+z)={\r_n+O(z)\/z}\qq as \ z\to 0, \ z\in \C_-.\end{gathered}$$
iv\) $\m_n=e_n^-$ or $\m_n=e_n^+$ (here $e_n^-\ne e_n^+$) for some $n\ne 0$ iff $$m_+(\m_n+z)={\r_n^\pm+O(z)\/\sqrt z}\qq as \ z\to 0, \ z\in \C_+,\qq\ {\rm some} \ \const \
\r_n^\pm\ne 0.$$
The following asymptotics hold true as $n\to \iy$ (see [@PT], [@K5]): $$\lb{sde}
\m_n=\pi n+\ve_n(p_{c0}-p_{cn}+O(\ve_n)),
\qqq
\ve_n={1/2\pi n},$$ $$\lb{ape}
e_n^\pm=\pi n+\ve_n(p_0\pm |p_n|+O(\ve_n)), \qq \qq
p_n=\int_0^1p(x)e^{-i2\pi nx}dx=p_{cn}-ip_{sn}.$$ The equation $-f''+(p+q)f=z^2 f,\ z\ne e_n^\pm$ has unique solutions $f_\pm (x,z)$ such that $$\lb{bcf}
f_+(x,z)=\p_+(x,z), \ x\ge t, \qqq {\rm and} \ \ \ \
f_-(x,z)=\p_-(x,z), \ x\le 0,$$ and $f_+(x,z)=\ol f_+(x,-z), z^2\in\s(H_0)\sm \{e_n^\pm, n\in \Z\}$. This yields $$f_+(x,z)=b(z)f_-(x,z)+a(z)f_-(x,-z),\qq a={w\/w_0},\qq b={s\/w_0},$$ where $z^2\in\s(H_0)\sm \{e_n^\pm, n\in \Z\}$ and $$\begin{gathered}
\ w=\{f_-,f_+\}, \ \ \ \
s=\{f_+(x,z),f_-(x,-z)\},\
w_0=\{\p_-,\p_+\}={2i\sin k\/\vp(1,\cdot)}\end{gathered}$$ and $\{f, g\}=fg'-f'g$ is the Wronskian. The scattering matrix $\cS_M$ for $H, H_0$ is given by $$\cS_M (z)\ev {\begin{pmatrix}}a(z)^{-1}& r_-(z)\\ r_+(z)&a(z)^{-1}{\end{pmatrix}},
\ \ \ \ r_{\pm}={s(\mp z)\/ w(z)}=\mp {b(\mp z)\/ a(z)}, \ \ \ \ z^2\in \s_{ac}(H),
\lb{1.7}$$ where $1/a$ is the transmission coefficient and $r_{\pm}$ is the reflection coefficient. We have the following identities from [@F1], [@F3]: $$\lb{iab}
|a(z)|^2=1+|b(z)|^2,\qqq z^2\in \s_{ac}(H),$$ $$\lb{iab1}
a(z)=D(z^2), \qqq z\in \cZ_0.$$ The functions $a,b, s,w $ are analytic in $\cZ_0$ and real on $i\R$. Then the following identities hold true: $$a(-z)=\ol a(\ol z), \qq w(-z)=\ol w(\ol z), \qq s(-z)=\ol s(\ol z),\qq w_0(-z)=\ol w_0(\ol z),\qq z\in \cZ_0.$$
Let $\vp(x,z,\t), \ (z,\t)\in \C\ts \R$ be the solutions of the equation $$\lb{x+t}
-\vp''+p(x+\t)\vp=z^2 \vp, \qq \ \vp(0,z,\t)=0,\qq \vp'(0,z,\t)=1.$$ The function $\vp(1,z,x)$ for all $(x,z)\in R\ts \C$ satisfies the following identity (see [@Tr]) $$\lb{if}
\vp(1,\cdot,\cdot)=\vp(1,\cdot)\vt^2-\vt'(1,\cdot)\vp^2+2\b\vp\vt=
\vp(1,\cdot)\p_-\p_+.$$ Let $\wt\vt, \wt\vp$ be the solutions of the equations $-y''+(p+q)y=z^2y, z\in \C$ and satisfying $$\lb{wtc}
\wt\vp(x,z)=\vp(x,z),\qqq \wt\vt(x,z)=\vt(x,z) \ \ for \ \ all \ x\ge t.$$ A solution of the equation $-y''+(p-z^2)y=f, y(0)=y'(0)=0$ has the form $
y=\int_0^x\vp(x-\t,z,\t)f(\t)d\t.
$ Hence the solutions $\wt\vt, \wt\vp$ and $f_+$ of the equation $-y''+(p+q)y=z^2y$ satisfy the equation $$\lb{ep}
y(x,z)=y_0(x,z)-\int_x^t\vp(x-\t,z,\t)q(\t)y(\t,z)d\t, \qqq x\le t,$$ where $y$ is one from $\wt\vt, \wt\vp$ and $f_+$; $y_0$ is the corresponding function from $\vt, \vp$ and $\p_+$. For each $x\in \R$ the functions $\wt\vt (x,z), \wt\vp (x,z)$ and $\vt (x,z), \vp (x,z)$ are entire in $z\in\C$ and satisfy $$\begin{gathered}
\lb{efs}
\max \{||z|_1\wt\vp(x,z)|, \ |\wt\vp'(x,z)| , |\wt\vt(x,z)|,
{1\/|z|_1}|\wt\vt'(x,z)| \} \le X_1=e^{|\Im z||2t-x|+\|q\|_t+\|p\|_t+\int_x^t|p(\t)|d\t},\\
|z|_1=\max\{1, |z|\},\qqq
|\wt\vt(x,z)-\vt(x,z)|\le {X_1\/|z|}\|q\|_t,\qq
|\wt\vp(x,z)-\vp(x,z)|\le {X_1\/|z|^2}\|q\|_t,\end{gathered}$$ where $\|p\|_t=\int_0^t|p(s)|ds$ and $$\begin{gathered}
\lb{efs1}
\max \{|z|_1|\vp(x,z)|, \ |\vp'(x,z)| , |\vt(x,z)|,
{1\/|z|_1}|\vt'(x,z)| \} \le X=e^{|\Im z|x+\|p\|_x},\\
|\vp(x,z)-{\sin zx\/z}|\le {X\/|z|^2}\|p\|_x,
\qq |\vt(x,z)-{\cos zx}|\le {X\/|z|}\|p\|_x,\end{gathered}$$ for $(p,x,z)\in L_{loc}^1(\R)\ts \R\ts \C$, see \[PT\]. These estimates yield $$\begin{gathered}
\lb{asb} \b(z)=\int_0^1{\sin z(2x-1)\/z}p(x)dx+{O(e^{|\Im
z|})\/z^2},\\ \b'(z)=\int_0^1{\cos z(2x-1)\/z} p(x)(2x-1)
dx+{O(e^{|\Im z|})\/z^2}\qq as \qq |z|\to \iy.\end{gathered}$$
For all $z\in \cZ_0$ the following identities and asymptotics hold true: $$\lb{T22-1}
f_+(\cdot,z)=\wt\vt (\cdot,z)+m_+(z)\wt\vp (\cdot,z),$$ $$\begin{gathered}
\lb{T22-2}
f_+(0,z)\!\!=1\!+\!\int_0^t\!\vp(x,z)q(x)f_+(x,z)dx,\
f_+'(0,z)\!\!=\! m_+(z)-\!\int_0^t\! \vt(x,z)q(x)f_+(x,z)dx,\end{gathered}$$ $$\lb{T22-3} f_+(x,z)=e^{ik(z)x}+e^{(2t-x){\mathfrak{J}}}O(1/z),\qqq \qq
{\mathfrak{J}}={|\Im z|- \Im z\/2}, \qq x\in [0,t],$$ $$\lb{T22-4}
f_+(0,z)=1+e^{2t{\mathfrak{J}}}O(1/z), \qq f_+'(0,z)=iz+O(1)+e^{2t{\mathfrak{J}}}o(1)$$ as $|z|\to \iy, z\in \cZ_\ve, \ve>0$, where $\wh q(z)=\int_0^tq(x)e^{2izx}dx, \ z\in \C$.
Using , we obtain . Using the identity $\vp(x,\cdot,s)=\vt(t,\cdot)\vp(x+s,\cdot)-\vp(s,,\cdot)\vt(x+s,\cdot)$, we obtain $\vp(-s,\cdot,s)=-\vp(s,\cdot)$ and $\vp'(-s,\cdot,s)=\vt(s,\cdot)$. Substituting the last identities into we get .
Standard iteration arguments (see [@PT]) for the equation give .
Substituting into we obtain .
i\) For each $z\in \cZ_0$ the following identities hold true: $$\lb{T23-1}
w(z)=f_+'(0,z)-m_-(z)f_+(0,z)=w_0(z)-\int_0^tq(x)\p_-(x,z)f_+(x,z)dx,$$ $$\lb{T23-2}
s(z)=f_+(0,z)m_+(z)-f_+'(0,z)=\int_0^tq(x)\p_+(x,z)f_+(x,z)dx.$$ The functions $\x(z)=2i\sin k(z)a(z)$ and $s(\cdot)$ have the following asymptotics: $$\lb{T23-3}
\x(z)=2i\sin z(1+O(e^{2t{\mathfrak{J}}}/z)) ,\qqq s(z)=O(e^{2t{\mathfrak{J}}}),$$ as $|z|\to \iy, z\in \cZ_\ve, \ve>0$.
ii\) The function $s(\cdot)$ has exponential type $\r_\pm$ in the half plane $\C_\pm$, where $\r_+=0, \r_-=2t$.
We have $w=\{f^-,f^+\}=\p^-{f^+}'-m^-f^+|_{x=0}$, which yields the identity in . Substituting into we obtain . Asymptotics from Lemma \[T22\] and imply .
ii\) We show $\r_-=2t$. Due to , $s$ has exponential type $\r_-\le 2t$. The decompositions $f_+=e^{ixz}(1+h)$ and $\p_+=e^{ixz} (1+\e)$ give $(1+h)(1+\e)=1+T, T=h+\e +\e h$ and $$\lb{esff}
s(z)=\int_0^tq(x)\p^+(x,z)f^+(x,z)dx=\int_0^tq(x)e^{i2xz}(1+T(x,z))dx,
\qqq \qq z\in\cZ_\ve.$$
Asymptotics , , and $k(z)=z+O(1/z)$ as $|z|\to \iy$ (see [@KK]) yield $$\lb{esff1}
\e(x,z)=O(1/z),\qq \qqq h(x,z)=e^{2(t-x)|\Im z|}O(1/z)\qq as \ |z|\to \iy,\qq z\in \cZ_\ve.$$ We need the following variant of the Paley Wiener type Theorem from [@Fr]:
We can not apply this result to the function $T(x,z), z\in\C_-$, since $m_+(z)$ may have a singularity at $\m_n-i0\in \ol g_n^-$ if $g_n\ne \es$. But we can use this result for the function $T(x,z-i), z\in\C_-$, since , imply $\sup _{x\in [0,1]}|T(x,-i+\t)|=O(1/\t)$ as $\t\to \pm\iy$. Then the function $s(z)$ has exponential type $2t$ in the half plane $\C_-$. The proof for $\r_+=0$ is similar.
Analyze of the function $\x$
============================
Below we need the identities and the asymptotics as $n\to \iy$ from [@KK]: $$\lb{35} (-1)^{n+1}i\sin k(z)=\sinh v(z)=\pm |\D^2(z)-1|^{1\/2}>0\qq
\ all \qq z\in g_n^\pm,$$ $$\lb{pav}
v(z)=\pm |(z-e_n^-)(e_n^+-z)|^{1\/2}(1+O(n^{-2})),\qq
\sinh v(z)=v(z)(1+O(|g_n|^2),\qq z\in \ol g_n^\pm.$$ Let $\n_n^2, n\ge 1$ be the Neumann spectrum of the equation $-y''+py=\n^2y$ on the interval $[0,1]$ with the boundary condition $y(0)=y(1)=0$. It is well known that each $\n_n^2\in [E^-_n,E^+_n ], n\ge 1$.
Let $p\in L^1(0,1)$.
i\) Then the following asymptotics hold true uniformly for $z\in [e_n^-,e_n^+]$ as $n\to \iy$: $$\lb{T31-1}
\vp(1,z)=(-1)^n{(z-\m_n)\/\pi n}(1+O(1/n)),\qq$$ $$\lb{T31-2}
-{\vt'(1,z)\/z^2}=(-1)^n{(z-\n_n)\/\pi n}(1+O(1/n)).$$ ii) Let $z\in g_n$ and $e_n^-\ge R_p=8e^{\|p\|_1}$. Then the following estimates hold true (here $\dot u=\pa_z u$) $$\lb{T31-3} |g_n|^2\le {8e^{\|p\|_1}\/|z_n|}<1,$$ $$\lb{T31-4}
|\dot \vp(1,z)|\le {3e^{\|p\|_1}\/2|z|},
\qqq |\vp(1,z)|_{z\in g_n} \le |g_n|{3e^{\|p\|_1}\/2|z|},\qq$$ $$\lb{T31-5}
|\dot \vt'(1,z)|\le |z|{3e^{\|p\|_1}\/2},\qqq
|\vt'(1,z)|\le |g_n||z|{3e^{\|p\|_1}\/2},$$ $$\lb{T31-6}
|\dot\b(z)|\le {3e^{\|p\|_1}\/2|z|}, \qqq |\b(e_n^\pm)|\le |g_n|{3e^{\|p\|_1}\/2},\qqq
|\b(z)|\le |g_n|{9e^{\|p\|_1}\/4|z|}.$$ iii) In each disk $\cD_{\pi\/4}(\pi n)\ss \cD=\{|z|>32e^{2\|p\|_1}\}$ there exists exactly one momentum gap $g_n$ of the operator $H_0$. Moreover, if $g_n,g_{n+1}\ss\cD$, then $e_{n+1}^--e_{n}^+\ge \pi$.
i\) We have the Taylor series $\vp(1,z)=\dot\vp(1,\m_n)\t+\ddot\vp(1,\m_n+\a \t){\t^2\/2}$ for any $z\in [e_n^-, e_n^+]$ and some $\a\in [0,1]$, where $\t=z-\m_n$ and $\dot \vp=\pa_z \vp$. Asymptotics give $\dot\vp(1,\m_n)=2(-1)^n(1+O(1/n))/(2\pi n)$ and $\ddot\vp(1,\m_n+\a \t)\t=O(n^{-2})$, which yields . Similar arguments imply .
ii\) Using $|\D(z)-\cos z|\le {e^{\|p\|_1}\/|z|}$, for all $|z|\ge 2$, we obtain $${h_n^2\/2}\le \cosh h_n-1=|\D(z_n)|-1\le {e^{\|p\|_1}\/|z_n|}.$$ Then the estimate $|g_n|\le 2h_n$ (see [@KK]) gives .
Due to , the function $f=z\vp(1,z)$ has the estimate $|f(z)|\le C_0=e^{\|p\|_1}, z\in \R$. Then the Bernstein inequality gives $|\dot f(z)|=|\vp(1,z)+z\dot \vp(1,z)|\le C_0, z\in \R$, which yields $|\dot \vp(1,z)|\le {C_0\/|z|}(1+{1\/|z|}), z\in \R$. Moreover, we obtain $|\vp(1,z)|\le |g_n|\max_{z\in g_n} |\dot \vp(1,z)|\le |g_n|{3C_0\/2|z|}$.
The proof of and the estimate $|\dot\b(z)|\le {3e^{\|p\|_1}\/2|z|}$ is similar. Identity gives
$\b^2(e_n^\pm)=-\vp(1,e_n^\pm)\vt'(1,e_n^\pm)$. Then , imply $|\b(e_n^\pm)|\le |g_n|{3e^{\|p\|_1}\/2}$. Using these estimates and $\b(z)=\b(e_n^-)+\b'(z_*)(z-e_n^-)$ for some $z_*\in g_n$ we obtain .
iii\) Using we obtain $$|(\D^2(z)-1)-(\cos^2z-1)|\le 2X|\D(z)-\cos z|\le 2X^2/|z|,\qq X=e^{|\Im |+\|p\|_1}.$$ After this the standard arguments (due to Rouche’s theorem) give the proof of iii).
The function $a$ is not convenient, since $a$ is not analytic on $\cM$. The modified function $\x$ in the momentum variable $z$ is given by $$\lb{ix}
\x(z)=2i\sin k(z)a(z)=\Xi(z^2), \qqq z\in \cZ_0.$$
The number $e_n=\sqrt{E_n}\in [e_n^-,e_n^+]$ satisfies $|\D(e_n)|=\max_{z\in g_n} |\D(z)|=\cosh h_n\ge 1$ for some $h_n\ge 0$.
i\) The following identity and asymptotics hold true: $$\begin{gathered}
\lb{T32-1}
\x(z)=2i\sin k(z)(1-A(z^2))-J(z^2),\qq z\in \cM,\qqq J(z^2)=\int_\R q(x)Y_1(x,z)dx
\\
A(z^2)=\int_\R q(x)Y_2(x,z)dx,\qqq
Y_1=\vp_1\vt \wt\vt-\vt_1'\vp \wt\vp+\b(\vp \wt\vt+\vt \wt\vp),
\qqq Y_2={1\/2}(\vp \wt\vt-\vt \wt\vp),\\
\x(z)=2(-1)^{n+1}(1+A(z^2))\sinh v(z)-J(z^2),\qqq z\in g_n^\pm\ne \es,\end{gathered}$$ where $v=\Im k$ and $\pm v(z)>0$ for $z\in g_n^\pm$, $$\begin{gathered}
\lb{T32-2}
Y_1=\vp(1,z,x)+Y_{11}(z,x),\ \
Y_{11}=\vp_1\vt \wt\vt_*-\vt_1'\vp \wt\vp_*+\b(\vp \wt\vt_*+\vt \wt\vp_*),
\ \
\wt\vt_*=\wt\vt-\vt, \wt\vp_*=\wt\vp-\vp,
\\
Y_{11}(z,x)=O(|g_n|/n^2)\ \ as \ \ z\in g_n, n\to \iy.\end{gathered}$$ Moreover, the functions $J, A$ are entire and $\x$ is analytic on $\cM$.
ii\) Let $|z|\ge 2$. Then the following estimates hold true $$\begin{gathered}
\lb{T32-3}
|J(z^2)|\le C_{p,q}\|q\|_t \rt(|\vp(1,z)|+{|\vt'(1,z)|\/|z|^2}+{|\b(z)|\/|z|}\rt)e^{2t|\Im z|}
\le {3C_*\/|z|}e^{(2t+1)|\Im z|},
\\
|A(z^2)|\le {\|q\|_t^2C_{p,q}\/|z|^2}e^{2t|\Im z|},\qq where \qq C_*=\|q\|_te^{\|p\|_1+\|q\|_t+2\|p\|_t}, \qq
C_{p,q}=e^{2\|p\|_t+\|q\|_t},\end{gathered}$$ $$\lb{T32-4}
|J(e_n^2)|\le {R_1\/|z_n|}\sinh h_n,\qqq if \qq e_n^-\ge R_p=8e^{\|p\|_1}, \qqq R_1=11C_*,$$
i\) We rewrite the identity in the form $$\lb{318}
\x(z)=2i\sin k(z)-\int_\R q(x)Y(x,z)dx,\qq
Y=\vp_1\p_-(x,z)f_+(x,z),$$ for $z\in \C_+$. Using , we rewrite $Y$ in the form $$Y=\vp_1(\vt+m_-\vp)(\wt\vt+m_+\wt\vp)=\vp_1\rt(\vt \wt\vt
+m_+\vt \wt\vp+m_-\vp \wt\vt-{\vt_1'\/\vp_1}\vp \wt\vp\rt)
=Y_1-i2Y_2\sin k.$$ Substituting the last identity into $\int_\R q(x)Y(x,z)dx$ and using , we obtain .
Substituting asymptotics from , Lemma \[T31\] into $Y_{11}$ we obtain $Y_{11}(z,x)=O(|g_n|/n^2)$ as $z\in g_n, n\to \iy.$
ii\) Using , and and Lemma \[T31\], we obtain . Estimates and Lemma \[T31\] give $|J(e_n^2)|\le {21C_*\/4|z_n|} |g_n|$; and the estimate $|g_n|\le 2|h_n|\le 2\sinh h_n$ from [@KK] yields .
This lemma gives that the function $\x$ is analytic on $\cM$ and the function $\Xi$ is analytic on $\L$. We define the bound states, resonances in terms of momentum variable $z\in \cM$. Recall that there are bound states on the physical gaps and resonances on the non physical gaps. Define the set $\cM_0 =\{z\in \cM: z=e_{\pm n}^+\in \cM_1 $ and $z=e_{\pm n}^+\in \cM_2, \g_n=\es, n\ge 1\}\ss \cM$. The set $\cM_0$ is the image of $\L_0$ (see before Definition of states, Section 1) under the mapping $z=\sqrt \l$.
Of course, $z^2$ is really the energy, but since the momentum $z$ is the natural parameter, we will abuse the terminology.
The kernel of the resolvent $R=(H-z^2)^{-1}, z\in \C_+,$ has the form $$R(x,x',z)={f_- (x,z )f_+(x',z)\/-w(z)}={R_1(x,x',z)\/-\x(z)},\ \ \ x<x',\ \
R_1=\vp(1,z)f_- (x,z )f_+(x',z),$$ and $R(x,x',z )=R(x',x,z ),\ x>x'$. Identity yields $f_\pm=\wt\vt+m_\pm \wt\vp,\qq \wt\vt=\wt\vt(x),\qq \wt\vp=\wt\vp(x)$. Let $ \wt \vt_*=\wt\vt(x'),\qq \wt \vp_*=\wt\vp(x')$. Then using we obtain $$R_1(x,x',z)=\vp(1,\cdot)\wt \vt \wt \vt_*+(\b-i\sin k)\wt \vp \wt \vt_*+
(\b+i\sin k)\wt\vt \wt \vp_*-\vt'(1,\cdot)\wt \vp_*\wt \vp.$$ Thus if $\x(z)=\vp(1,z)w(z)=0$ at some $z\in\cM$, then $(H-z^2)^{-1}$ has singularity at $z$. The poles of $R(x,x',z)$ define the bound states and resonances. The zeros of $\x$ define the bound states and resonances, since the function $R_1=\vp_1(z)f^-(x,z )f^+(x',z)$ is locally bounded.
If $q=0$, then $R_0=(H_0-z^2)^{-1}$ has the form $$R_0(x,x',z)={R_{10}(x,x',z)\/-\x_0(z)},\qq \x_0=\vp(1,z)w_0(z)=2i\sin k(z),$$$$R_{10}=\vp_1(z)\p_- (x,z )\p_+(x',z)=
\vp(1,\cdot)\vt\vt_*+(\b-i\sin k)\vp\vt_*+
(\b+i\sin k)\vt\vp_*-\vt'(1,\cdot)\vp\vp_*,$$ where $\vp=\vp(x,z ),..$ and $\vp_*=\vp(x',z ),..$ Thus $R_0(x,x',z)$ has singularity at some $z\in\cM$ iff $\sin k(z)=0$, i.e., $k(z)=\pi n $ and then $z=e_n^\pm$.
Define the functions $F, S$ by $$F=\x(z)\x(-z), \qqq S(z)=\vp^2(1,z)s(z)s(-z), \qq z\in\cZ_0.$$
i\) The functions $F(z), S(z), z\in\cZ_0$ have analytic continuations into the whole complex plane $\C$ and satisfy $$\lb{T33-1}
F(z)=4(1-\D^2(z))(1+A(z^2))^2+J^2(z^2)=4(1-\D^2(z^2))+S(z), \qqq z\in \C.$$ Moreover, $F(z)>0$ and $S(z)\ge 0$ on each interval $(e_{n-1}^+,e_n^-), n\ge 1$ and $F$ has even number of zeros on each interval $[e_n^-,e_n^+], n\ge 1$. The function $F$ has only simple zeros at $e_n^\pm, g_n\ne \es$.
ii\) If $g_n=(e_n^-,e_n^+)= \es$ for some $n\ne 0$, then each $f_\pm(x,\cdot ), x\in \R$ is analytic in some disk $\cD(e_n^+,\ve),\ve >0$. Moreover, $\m_n=e_n^\pm$ is a double zero of $F$ and $e_n^+$ is not a state of $H$.
iii\) Let $z\in g_n^+$ be a bound state for some $n\ge 1$, i.e., $\x(z+i0)=0$. Then $z-i0\in g_n^-$ is not an anti-bound state and $\x(z-i0)\ne 0$.
iv\) Let $z\in i\R_+$ be a bound state, i.e., $\x(z)=0$. Then $-z\in i\R_-$ is not an anti-bound state and $\x(-z)\ne 0$.
v\) $z\in \C_-\sm i\R$ is a zero of $F$ iff $z\in \C_-\sm i\R$ is a zero of $\x$ (with the same multiplicity).
vi\) $z\in i\R_-$ is a zero of $F$ iff $z\in i\R_-$ or $-z\in i\R_-$ is a zero of $\x$.
vii\) Let $g_n\ne \es, n\ge 1$. The point $z\in g_n$ is a zero of $F$ iff $z+i0\in g_n^+$ or $z-i0\in g_n^-$ is a zero of $\x$ (with the same multiplicity).
viii\) Let $e_n^-> \max\{8e^{\|p\|_1}, 11C_*\}, C_*=\|q\|_te^{\|p\|_1+\|q\|_t+2\|p\|_t}$ and $|g_n|>0$. Then $$\lb{T33-2}
F(e_n)\le - \rt(1-{R_1^2\/|e_n|^2} \rt)\sinh^2 h_n<0, \qqq \where
\qqq e_n=\sqrt {E_n}>0,$$ and $E_n,h_n$ are defined by . Moreover, $F$ has at list two zeros in the segment $[e_n^-,e_n^+]$.
i\) Using we deduce that $F=\x(z)\x(-z)$ is entire and satisfies $F(z)=(1-\D^2(z))(2-A(z^2))^2+J^2(z^2)$. Using we obtain that $S$ is entire and satisfies .
Recall that $F\ge 0$ and $S\ge0$ inside each $(e_{n-1}^+,e_n^-)$. Moreover, $F>0$, since $a\ne 0$ inside each $(e_{n-1}^+,e_n^-)$. Due to $F(e_n^\pm)\ge 0$, we get that $F$ has even number of zeros on each interval $[e_n^-,e_n^+], n\ge 1$. We have $F=F_0+S, F_0=4(1-\D^2)$. Consider the case $z=e_n^+$, the proof for $z=e_n^-$ is similar. Thus if $F(z)=0$, then we get $S(z)=0$, since $\D^2(e_n^+)=1$. Moreover, $F_0'(e_n^+)=-2\D(e_n^+)\D'(e_n^+)>0$ and $S'(e_n^+)\ge 0$, which gives that $z=e_n^+$ is a simple zero of $F$.
ii\) The function $m_\pm$ is analytic in $\cZ_0$, then each $f^\pm(x,\cdot), x\in \R$ is analytic in $\cZ_0$. Moreover, yields $\b(\m_n)=\vp(1,\m_n)=\vt'(1,\m_n)=0$ and then $J(\m_n^2)=0$. Thus by , the function $F$ has a double zero at $\m_n$ at list.
iii\) If $z\in g_n, n\ge 1$ is a bound state. Then yields $$0=\x(z+i0)=2(-1)^{n+1}\sinh v(z+i0)(1+A(z^2))-J(z^2),$$ $$2\sinh v(z+i0)={(-1)^{n+1}J(z^2)|\/(1+A(z^2))}>0.$$ If we assume that $z-i0$ is a anti-bound state, then $2\sinh v(z-i0)={(-1)^{n+1}J(z^2)\/(1+A(z^2))}<0$, which gives contradiction. The proof of iv) is similar.
v\) The function $\x$ has not zeros in $\C_+\sm iR$, see [@F1]. This yields v).
vi\) The property $F(z)=F(-z), z\in \C$ gives vi).
vii\) The statement vii) follows from iii).
viii\) Estimate gives $|A(e_n^2)|\le {1\/2}$ and imply . The function $F(e_n^\pm)\ge 0$ and due to , we deduce that $F$ has at list two zeros in the segment $[e_n^-,e_n^+]$.
[**Proof of Theorem \[T1\]**]{}. Identity have been proved in Lemma \[T33\].
i\) The function $\x$ is analytic in $\cZ_0$ and is real on $i\R$. Then the set of zeros of $\x$ is symmetric with respect to the imaginary line and satisfies .
The statement ii) has been proved in Lemma \[T33\].
iii\) Using Lemma \[T32\], and we obtain $$|A(z^2)|\le {\|q\|_t^2\/|z|^2}X_2,\qqq |J(z^2)|\le {3\|q\|_t\/|z|}
X_2X\qqq |\D(z)-\cos z|\le {X\/|z|},\qq |z|\ge 2,$$ where $ X_2=e^{|\Im z|2t+\|q\|_t+2\|p\|_t},\qq X=e^{|\Im z|+\|p\|_1} $. Substituting these estimates into the identity $$F-4\sin^2 z=4(\cos^2z-\D^2)+J^2+4(1-\D^2)A(A+2)$$ we obtain $$|F(z)-4\sin^2 z|\le 9X^2C_0,\qqq
C_0={1\/|z|}+{\|q\|_t^2X_2^2\/|z|^2}+{\|q\|_t^2X_2\/|z|^2}\rt(2+{\|q\|_t^2X_2\/|z|^2}\rt),\qq |z|\ge 2.$$ Using $e^{|{\Im}z|}\le 4|\sin z|$ for all $|z-\pi n|\ge {\pi \/4}, n\in \Z$, (see p. 27 \[PT\]), we obtain $$9X^2=9e^{2|\Im z|+2\|p\|_1}\le |4\sin^2 z|r_0^2
\qq all \ |z-\pi n|\ge{\pi\/4},\qq n\in \Z, \qqq r_0=6e^{\|p\|_1},$$ which yields $$|F(z)-4\sin^2 z|\le 4|\sin^2 z|{r_0^2C_0 \/|z|}< 4|\sin^2 z|,\qq all
\qq |z|\ge 2, z\notin \cup \cD(\pi n,{\pi\/4}),$$ since $${r_0^2C_0\/|z|}\le
{r_0^2\/|z|}+{r_0^2\|q\|_t^2X_2^2\/|z|^2}+{r_0^2\|q\|_t^2X_2\/|z|^2}\rt(2+{\|q\|_t^2X_2\/|z|^2}\rt)<{1\/5}
+{1\/4}+{1\/2}+{1\/(24)^2}<{19\/20},$$ where we have used: $$if \ z\in \cD_F=\{z\in \C: |z|>\max \{180e^{2\|p\|_1},C_Fe^{2t|\Im z|}\}\}\qq \Rightarrow \qq
{r_0^2\/|z|}<{1\/5},\qq {r_0\|q\|_tX_2\/|z|}<{1\/2},$$ and recall that $$\mD_F=\{\l\in \L_2: |\l|^{1\/2}>\max \{180e^{2\|p\|_1},C_Fe^{2t|\Im \sqrt \l|}\}\},\ \ C_F=12\|q\|_te^{\|p\|_1+\|q\|_t+2\|p\|_t}.$$ Thus by Rouche’s theorem, $F$ has as many roots, counted with multiplicities, as $\sin^2 z$ in each disk $\cD_{\pi\/4}(\pi n)\ss\cD_F$. Since $\sin z$ has only the roots $\pi n, n\ge 1$, then $F$ has two zeros in each disk $\cD_{\pi\/4}(\pi n)\ss\cD_F$ and $F$ has not zeros in $\cD_F\sm \cup \cD_{\pi\/4}(\pi n)$.
$F(e_n^\pm)\ge 0$ for all $n$ and Lemma \[T33\] yields $F(e_n)<0$ for all $n$, where $e_n^-> \max\{8e^{\|p\|_1}, 11C_*\}$, see . Then there exists exactly two simple real zeros $\vk_n^\pm=\sqrt{\l_n^\pm}>0$ of $F$ such that $e_n^-\le \vk_n^-<e_n<\vk_n^+\le e_n^+$. Moreover, $F$ has not no states in the “logarithmic” domain $\cD_F\cap \C_-$.
The statements iv) and v) have been proved in Lemma \[T33\].
Proof of Theorem 1.2-1.4
========================
. i) Theorem \[T1\] gives $\vk_n^\pm=e_n^\pm\mp \d_n^\pm=\sqrt{\l_n^\pm}$. Let $\vk=\vk_n^\pm, \d=\d_n^\pm$. Then the equation $0=\x(\vk)=(-1)^{n+1}2(1+A(\vk^2))\sinh v(\vk)-J(\vk^2),\ \ \vk\in \ol g_n^\pm\ne \es$ and imply $$\sinh |v(\vk)|=O(J(\vk^2))=\ve O( |\vp(1,\vk)|+{|\vt'(1,\vk)|\/|\vk|^2}+{|\b(\vk)|\/\vk}) =\ve O(|g_n|), \qq \ve={1\/2\pi n}$$ as $n\to \iy$. Moreover, using the estimate $|(z-e_n^-)(z-e_n^+)|^{1\/2}\le |v(z)|$ for each $z\in g_n$ (see [@KK]) we obtain $|\d(|g_n|-\d)|^{1\/2}\le |v(\vk)|=\ve O(|g_n|)$, which yields $\d=\ve^2 O(|g_n|)$. Thus the points $\vk_n^\pm$ are close to $e_n^\pm$ and satisfy: $$\lb{asd}
\d_n^-=\vk_n^--e_n^-=\ve^2 O(|g_n|),\qqq and \qqq
\d_n^+=e_n^+-\vk_n^+=\ve^2 O(|g_n|).$$ Recall that $\vk_n^-$ and $\vk_n^+$ are simple. Consider the first case $\d=\d_n^-=|g_n|O(\ve^2)$.
Using we obtain $$J=J_{10}+J_{10},\qqq J_{10}(z)=\int_\R\vp(1,z,x)q(x)dx.$$ Let $\m_n^2(\t), \t\in \R$ be the Dirichlet eigenvalue for the problem $-y''+q(x+\t)y=z^2 y, y(0)=y(1)=0$. In this case $\cos \f_n$ in is a function from $\t\in \R$. Below we need facts from [@K5]: $$\lb{mz}
{E_n^-+E_n^+\/2}-\m_n^2(\t)={|\g_n|\/2}\cos \f_n(\t),\qq
\f_n(\t)=\f_n(0)+2\pi n\t+O(\ve),$$ as $n\to \iy$ uniformly with respect to $\t\in [0,1]$. Asymptotics yield $$\vp(1,\vk,x)={(-1)^n\/\pi n}(1+O(\ve))(\vk-\m_n(x))={(-1)^n\/\pi n}(1+O(\ve))(e_n^--\m_n(x)+\ve^2 O(|g_n|)).$$ Thus we rewrite $e_n^--\m_n(x)$ in the form $$e_n^--\m_n(x)={{E_n^-+E_n^+-|\g_n|\/2}-\m_n^2(x)\/e_n^-+\m_n(x)}=
{|\g_n|\/2} {\cos \f_n(x)-1\/e_n^-+\m_n(x)}.$$ This gives $$J_{10}(\vk)=\int_\R\vp(1,\vk,x)q(x)dx={(-1)^n\/\pi n}(1+O(\ve))
\int_\R \rt[{|\g_n|\/2} {\cos \f_n(x)-1\/e_n^-+\m_n(x)}+\ve^2 O(|g_n|)\rt] q(x)dx.$$ $$={(-1)^n|g_n|\/2\pi n}
\int_\R \rt[\cos y_n(x)-1+ O(\ve)\rt]q(x)dx
={(-1)^n|g_n|\/2\pi n}(-q_0+c_n\wh q_{cn}-s_n\wh q_{sn}+ O(\ve)).$$ where $c_n=\cos y_n(0), s_n=\sin y_n(0)$, and thus $$(-1)^{n+1}J(\vk^2)={|g_n|\/2\pi n}I_n^-, \qqq I_n^-=q_0-c_n\wh q_{cn}+s_n\wh q_{sn}+ O(\ve).$$ Using we obtain $$\lb{as2}
\sinh v(\vk)={(-1)^{n+1}J(\vk^2)\/2+2A(\vk^2)}={\ve |g_n|I_n^-\/2+O(\ve^2)},\qq
\sign v(\vk)=\sign (-1)^{n+1}J(\vk^2)=\sign I_n^-.$$ Note that if $v(\vk)>0$ (or $v(\vk)<0$), then $\vk\in g_n^+$ (or $\vk\in g_n^-$) and $\vk$ is a bound state (or a resonance). Moreover, if $v(\vk)=0$, then $\vk=e_n^-$ or $\vk=e_n^+$ is a virtual state. Then gives $\sinh v(\vk)=v(\vk)(1+O(|g_n|^2\ve^2))$ and using asymptotics , we obtain $$v(\vk)=\sqrt{\d(|g_n|-\d)}(1+O(\ve^2))=\sqrt{\d|g_n|}(1+O(\ve^2)).$$ Thus yields $\d_n^-={|g_n|\ve^2\/4}(I_n^-)^2$ and gives $|\g_n|=(2\pi n)|g_n|(1+O(\ve^2))$, which yields $\d_n^-={2|\g_n|\/(4\pi n)^3}(I_n^-)^2(1+O(\ve^2))$.
If $q_0>0$, then $I_n^->0$ and above arguments yield that $\vk_n^-$ is a bound state and $\vk_n^+$ is an anti bound state. Conversely, if $q_0<0$, then $I_n^-<0$ and we deduce that $\vk_n^-$ is an anti bound state and $\vk_n^+$ is a bound state.
Similar arguments imply the proof for the case $\d_n^+=e_n^+-\vk_n^+=|g_n|O(\ve^2)$.
iii\) We have $-c_n\wh q_{cn}+s_n\wh q_{sn}=-|\wh q_n|\cos (\f_n+\t_n)$. Then (i) yields the statement (ii).
iv\) An entire function $f(z)$ is said to be of $exponential$ $ type$ if there is a constant $\a$ such that $|f(z)|\leq $ const. $e^{\a |z|}$ everywhere. The function $f$ is said to belong to the Cartwright class $\mE_\r,$ if $f$ is entire, of exponential type, and the following conditions are fulfilled: $$\int _{\R}{\log ^+|f(x)|dx\/ 1+x^2}<\iy ,\ \
\r_\pm(f)=\r,\ \ \ {\rm where}\ \ \
\r_{\pm}(f)\ev \lim \sup_{y\to \iy} {\log |f(\pm iy)|\/y}.$$
Denote by $\cN^+(r,f)$ the number of zeros of $f$ with real part $\geq 0$ having modulus $\leq r$, and by $\cN^-(r,f)$ the number of its zeros with real part $< 0$ having modulus $\leq r$, each zero being counted according to its multiplicity. We recall the well known result (see \[Koo\]).
Let $\cN (r,f)$ be the total number of zeros of $f$ with modulus $\le r$. Denote by $\cN_+(r,f)$ (or $\cN_-(r,f)$) the number of zeros of $f$ with imaginary part $>0$ (or $<0$) having modulus $\le r$, each zero being counted according to its multiplicity.
Let $\pm \z_n>0, n\in \N$ be all real zeros $\ne 0$ of $F$ and let the zero $\z_0=0$ has the multiplicity $n_0\le 2$. Let $F_1=z^{n_0}\lim_{r\to \iy}\prod_{|\z_n|\le r}(1-{z\/\z_n})$. The Levinson Theorem and Lemma \[T23\] imply $$\cN(r,F)=\cN(r,F_1)+\cN(r,F/F_1)=2r{1+2t+o(1)\/\pi},\qq
\cN(r,F_1)=2r{1+o(1)\/\pi} \ \ \as \ r\to\iy.$$ Then Lemma \[T33\] gives the identities $\cN_-(r,F)=\cN_+(r,F)=\cN_-(r,\x)+N_*$ for some integer $N_*\ge 0$. Thus we obtain $$\cN(r,F)=\cN(r,F_1)+2\cN_-(r,\x)+2N_*=2r{1+2t+o(1)\/\pi},$$ which yields $\cN_-(r,\x)={2rt+o(r)\/\pi}$ as $r\to \iy$ and .
Due to , the high energy states of $H$ and $H_0$ are very close. This gives .
. i) Let the operator $H_0$ have infinitely many gaps $\g_n\ne \es$ for some $p\in L^2(0,1)$ and let $\s=(\s_n)_{1}^\iy$ be any sequence, where $\s_n\in \{0,2\}$.
We take a potential $q$ and let $\wh q_n=\wh q_{cn}+i\wh q_{sn}=|\wh q_n|e^{i\t_n}, n\ge 1$ and $q_0=0$. We also assume that $|\wh q_n|>n^{-\a}$ for $n$ large enough and for some $\a\in (0,1)$. For each $\f_n$ we take $\t_n$ such that $|\cos (\f_n+\t_n)|>\ve>0$. Then due to Theorem \[T2\] iii), the operator $H$ has $\s_n=1-\sign \cos (\f_n+\t_n)$ bound states in the physical gap $\g_n^{(1)}\ne \es$ and $2-\s_n$ resonances inside the nonphysical gap $\g_n^{(2)}\ne \es$ for $n$ large enough. Thus changing $\t_n$ we obtain $\cos (\f_n+\t_n)<0$, which yields $\s_n=2$ or we obtain $\sign \cos (\f_n+\t_n)>0$, which yields $\s_n=0$.
ii\) Let $q\in \cQ_t,t>0$ satisfy $q_0=0$ and let $|\wh q_{cn}|>n^{-\a}$ for all $n$ large enough and some $\a\in (0,1)$. Let $\s=(\s_n)_{1}^\iy$ be any sequence, where $\s_n\in \{0,2\}$. Let $\d=(\d_n)_1^\iy\in \ell^2$ be a sequence of nonnegative numbers $\d_n\ge 0, n\ge 1$ and infinitely many $\d_n>0$.
Recall the result from [@K5]:
[*The mapping $\P: \cH\to \ell^2\os \ell^2$ given by $\P=((\P_{cn})_1^\iy,(\P_{sn})_1^\iy)$ is a real analytic isomorphism between real Hilbert spaces $\cH=\{p\in L^2(0,1): \int_0^1p(x)dx=0\}$ and $\ell^2\os \ell^2$.*]{}
Then for the sequence $\d=(\d_n)_1^\iy\in \ell^2$ there exists a potential $p\in L^2(0,1)$ such that each gap length $|\g_n|=\d_n$. Assume that $E_0^+=0$. Moreover, using Theorem \[T2\] we deduce that $H$ has exactly two simple states $\l_n^-,\l_n^+\in \g_n^{(0)}$ for $n$ large enough and $\l_n^-,\l_n^+$ have asymptotics $$\sqrt{\l_n^\pm}=\sqrt{E_n^\pm}\mp {2|\g_n|\/(4\pi n)^3}(R_n+ O(1/n))^2,\qq
\qqq (-1)^{n+1}J(\l_n^\pm)={|\g_n|\/(2\pi n)^2}(R_n+ O(1/n)),$$ where $R_n=-c_n\wh q_{cn}+s_n\wh q_{sn}=-|\wh q_n|\cos (\f_n+\t_n)$ and $\wh q_n=\wh q_{cn}+i\wh q_{sn}=|\wh q_n|e^{i\t_n}$ and $\f_n$ is defined in . The parameters $\t_n$ are fixed, but due to above results from [@K5] the angles $\f_n$ can be any numbers. If we take $\f_n$ such that $|R_n|>\ve>0$ for all $\g_n\ne \es$ and some $\ve>0$. Moreover, we take $\f_n$ such that $\s_n=1-\sign \cos (\f_n+\t_n)$. Then Theorem \[T2\] we obtain that the operator $H$ has $\s_n=1$ bound states in the physical gap $\g_n^{(1)}\ne \es$ and $2-\s_n$ resonances inside the nonphysical gap $\g_n^{(2)}\ne \es$ for $n$ large enough.
The various parts of this paper were written at ESI, Vienna, and Mathematical Institute of the Tsukuba Univ., Japan and Ecole Polytechnique, France. The author is grateful to the Institutes for the hospitality.
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author:
- 'L. Wang, P. Norberg, M. Bethermin, N. Bourne, A. Cooray, W. Cowley, L. Dunne, S. Dye, S. Eales, D. Farrah, C. Lacey, J. Loveday, S. Maddox, S. Oliver, M. Viero'
date: 'Received / Accepted'
title: 'The faint end of the 250 $\mu$m luminosity function at $z<0.5$'
---
Introduction
============
Luminosity functions (LF) are fundamental properties of the observed galaxy populations that provide important constraints on models of galaxy formation and evolution (e.g. Lacey et al. 2015; Schaye et al. 2015). Studying the LF at far-infrared (FIR) and sub-millimetre (sub-mm) wavelengths is critical. Half of the energy ever emitted by galaxies has been absorbed by dust and re-radiated in the FIR and sub-mm (Hauser & Dwek 2001; Dole et al. 2006). The spectra of most IR luminous galaxies peak in the FIR and sub-mm (Symeonidis et al. 2013; Casey et al. 2014). Finally, our knowledge of the FIR and sub-mm LF is relatively poor.
The first 250 $\mu$m LF measurement was made by Eales et al. (2009) with observations conducted using the Balloon-borne Large Aperture Submm Telescope (BLAST; Devlin et al. 2009). [*Herschel*]{} (Pilbratt et al. 2010) significantly improved over BLAST with increased sensitivity, higher resolution, and larger areal coverage. Dye et al. (2010) detected strong evolution in the 250 $\mu$m LF out to $z\sim0.5$, using the [*Herschel*]{}-Astrophysical Terahertz Large Area Survey (H-ATLAS; Eales et al. 2010). Using the [*Herschel*]{} Multi-tiered Extragalactic Survey (HerMES; Oliver et al. 2012), Vaccari et al. (2010) presented the first constraints on the 250, 350, and 500 $\mu$m as well as the infrared bolometric (8-1000 $\mu$m) LF at $z<0.2$. More recently, combining [*Herschel*]{} data with multi-wavelength datasets, Marchetti et al. (2016) derived the LF at 250, 350, and 500 $\mu$m as well as the bolometric LF over $0.02<z<0.5$. Evolution in luminosity ($L^*_{250}\propto(1+z)^{5.3\pm0.2}$) and density ($\Phi^*_{250}\propto(1+z)^{-0.6\pm0.4}$) are found at $z<0.2$. Marchetti et al. (2016), however, were unable to constrain evolution beyond $z\sim0.2,$ as only the brightest galaxies can be individually detected at higher redshifts. Despite the significant progress made, the determination of the LF is still hampered by many difficulties. Large samples over large areas are required for accuracy. We need to focus on smaller areas with increased sensitivity, however, to probe the faint end. At the [*Herschel*]{}-SPIRE (Griffin et al. 2010) wavelengths, confusion (related to the relatively poor angular resolution) is a serious challenge for source extraction, flux estimation, and cross-identification with sources detected at other wavelengths. In addition, issues such as completeness and selection effects due to the combination of several surveys are extremely difficult to quantify (e.g. Casey et al. 2012).
In this paper, we present a new analysis of the 250 $\mu$m LF by stacking deep optically selected galaxy catalogues from the Sloan Digital Sky Survey (SDSS) on the SPIRE 250 $\mu$m images. We bypass some major difficulties in previous measurements (e.g. complicated selection effects, reliability of the cross-identification). The paper is organised as follows. In Section 2, we describe the relevant data products from the SDSS and [*Herschel*]{} surveys. In Section 3, we explain our stacking method, which recovers the mean properties and underlying distribution functions. In Section 4, we present our results and compare with previous measurements. Finally, we give conclusions in Section 5. We assume $\Omega_m=0.25$, $\Omega_{\Lambda}=0.75$, and $H_0=73$ km s$^{-1}$ Mpc$^{-1}$. Flux densities are corrected for Galactic extinction (Schlegel, Finkbeiner & Davis 1998).
Data
====
Optical galaxy samples from SDSS
--------------------------------
The SDSS Data Release 12 (DR 12) contains observations from 1998 to 2014 over a third of the sky (Alam et al. 2015) in $ugriz$. The DR 12 includes photometric redshift ($z_{\rm phot}$) using an empirical method known as a kd-tree nearest neighbour fit (KF) (Csabai et al. 2007), which is extended with a template-fitting method to derive parameters, such as $k$ corrections and absolute magnitudes, using spectral templates from Dobos et al. (2012). The DR 12 features an expanded training set (extending to $z=0.8$), an updated method of template-fitting, and a more detailed approach to errors (Beck et al. 2016). Following recommendations on the SDSS website, we selected galaxies (located in the three Galaxy And Mass Assembly (GAMA) equatorial fields with [*Herschel*]{} coverage) with photoErrorClass equal to 1, -1, 2, and 3, which have an average RMS error in ($1+z$) of 0.02, 0.03, 0.03 and 0.03, respectively. We constructed volume-limited samples in five redshift bins, $z1=[0.02, 0.1]$, $z2=[0.1, 0.2]$, $z3=[0.2, 0.3]$, $z4=[0.3, 0.4]$ and $z5=[0.4, 0.5]$. In each bin, we only selected galaxies that were bright enough to be seen throughout the corresponding volume, given the apparent magnitude limit is $r=20.4$ which corresponds to the 90% completeness limit for single pass images (Annis et al. 2014). We also take the most adverse $k$ correction in a given redshift bin into account in deriving the luminosity limit owing to the nature of flux-limited surveys.
The SDSS stripe along the celestial equator in the south Galactic cap, known as “Stripe 82”, was the subject of repeated imaging. The resulting depths are roughly 2 magnitudes deeper than the single-epoch imaging. We used the Stripe 82 Coadd photometric redshift catalogue constructed using artificial neural network (Reis et al. 2012). The median photo-$z$ error is $\sigma_z=0.031$ and the photo-$z$ is well measured up to $z\sim0.8$. Following the procedure applied to the DR 12, we also constructed volume-limited samples in five redshift bins.We performed $k$ corrections in the optical bands to $z=0.1$ using KCORRECT v4\_2 (Blanton et al. 2002; Blanton & Roweis 2007). The luminosity limit as a function of redshift is calculated using an apparent magnitude limit of $r=22.4,$ which corresponds to the 90% completeness limit for the Coadd data (Annis et al. 2014) . This deeper catalogue allows us to probe 250 $\mu$m LF down to even fainter luminosities than the DR 12 catalogue.
Fig. \[samples\] shows the rest-frame $r$-band absolute magnitude $M_r$ ($k-$corrected to $z=0.1$) as a function of $z_{\rm phot}$ for galaxies with $r<20.4$ in the GAMA fields and for galaxies with $r<22.4$ in the Stripe 82 area with [*Herschel*]{} coverage. The red boxes indicate the redshift boundaries and $M_r$ limits used to define volume-limited samples. When carrying out the stacking procedure, we further bin galaxies in each redshift slice along the $M_r$ axis. The minimum bin width along $M_r$ is 0.15 mag but can be increased to ensure that the minimum number of galaxies in a given redshift and $M_r$ bin is 1000.
![Rest-frame r-band absolute magnitude $M_r$ vs. photometric redshift $z_{\rm phot}$ for DR12 galaxies with $r<20.4$ (black dots) and Stripe 82 galaxies with $r<22.4$ (green dots), in areas with [*Herschel*]{}-SPIRE coverage. For clarity, only 20% of the DR12 sample and $10\%$ of the Stripe 82 sample are plotted. The red boxes indicate the volume-limited subsamples in five redshift slices (solid: DR12; dashed: Stripe 82).[]{data-label="samples"}](Mr_vs_z_r20p4_and_s82_r22p4_crop.png){height="2.5in" width="3.3in"}
[*Herschel*]{} survey 250 $\mu$m maps
---------------------------------------
The H-ATLAS survey conducted observations at 100, 160, 250, 350, and 500 $\mu$m of the three equatorial fields also observed in the GAMA spectroscopic survey (Driver et al. 2011); these equatorial fields are G09, G12, and G15 centred at a right ascension of $\sim$9, 12, and 15 hours, respectively. For this study, we cut out a rectangle inside each of the GAMA fields with a total area of 95.6 deg$^2$. The version of the data used in this paper is the Phase 1 version 3 internal data release. The SPIRE maps, which have unit of Jy/beam, were made using the methods described by Valiante et al. (2016, in prep). Large-scale structures and artefacts are removed by running the NEBULISER routine developed by Irwin (2010). We estimated the local background by fitting a Gaussian to the peak of the histogram of pixel values in $30\times30$ pixel boxes and subtracted this background from the raw map.
As the deeper SDSS Coadd catalogue is located in Stripe 82, we also used maps from the two [*Herschel*]{} surveys in the Stripe 82 region, i.e. the [*Herschel*]{} Stripe 82 Survey (HerS; Viero et al. 2014) and the HerMES Large-Mode Survey (HeLMS; Oliver et al. 2012). The joint HeRS and HeLMS areal coverage between -10$^{\circ}$ and 37$^{\circ}$ (RA) covers the subset of Stripe 82 that has the lowest level of Galactic dust emission (or cirrus) foregrounds. For this study, we combined 39.1 deg$^2$ in HeRS and 47.6 deg$^2$ in HeLMS, which are covered by the SDSS Coadd data. The SPIRE data, obtained from the [*Herschel*]{} Science Archive, were reduced using the standard ESA software and the custom-made software package, SMAP (Levenson et al. 2010; Viero et al. 2014). Maps were made using an updated version of SMAP/SHIM, which is an iterative map-maker designed to optimally separate large-scale noise from signal. Viero et al. (2013) provide greater detail on these maps.
Method
======
![Top: estimated mean $L250$ as a function of the intrinsic population mean ($\mu$) and standard deviation ($\sigma$) of $\log{L250}$. For each set of ($\mu$, $\sigma$), we generate $\sim$2000 random numbers representing the 250 $\mu$m luminosities drawn from the log-normal distribution specified by ($\mu$, $\sigma$). The estimates of the mean 250 $\mu$m luminosity $\bar{m}$ are derived from these specific realisations of log-normal distributions. Bottom: the estimated standard deviation of $L250$ as a function of $\mu$ and $\sigma$.[]{data-label="method"}](MEAN_true_crop.png "fig:"){height="2.in" width="3.4in"} ![Top: estimated mean $L250$ as a function of the intrinsic population mean ($\mu$) and standard deviation ($\sigma$) of $\log{L250}$. For each set of ($\mu$, $\sigma$), we generate $\sim$2000 random numbers representing the 250 $\mu$m luminosities drawn from the log-normal distribution specified by ($\mu$, $\sigma$). The estimates of the mean 250 $\mu$m luminosity $\bar{m}$ are derived from these specific realisations of log-normal distributions. Bottom: the estimated standard deviation of $L250$ as a function of $\mu$ and $\sigma$.[]{data-label="method"}](NRMS_true_crop.png "fig:"){height="2.in" width="3.4in"}
Stacking was used for determining the mean properties of sources detected at another wavelength that are individually too dim to be detected at the working wavelength. For a given galaxy sample, we can stack[^1] the 250 $\mu$m images centred at the positions of the galaxies weighted by luminosity distance squared ($D_L^2$) and $k$ correction to derive the mean rest-frame 250 $\mu$m luminosity. To apply the $k$ correction at rest-frame 250 $\mu$m, we used $$K(z) = \left(\frac{\nu_o}{\nu_e}\right)^{3+\beta} \frac{e^{h\nu_e/kT_{\rm dust}} - 1}{e^{h\nu_o/kT_{\rm dust} } - 1},$$ where $\nu_o$ is the observed frequency and $\nu_e=(1+z)\nu_o$ is the emitted frequency in the rest frame. We assumed a mean dust temperature of $T_{\rm dust} = 18.5K$ and emissivity index $\beta=2$, following Bourne et al. (2012).
In this paper, we extend the traditional stacking method to reconstruct the LF. The key assumption is that the rest-frame 250 $\mu$m luminosities $L_{250}$ of galaxies in a narrow bin of $z$ and $M_r$ follow a log-normal distribution, i.e. the logarithm of the luminosities, $\log L_{250}$, follow a normal distribution with mean $\mu$ and standard deviation $\sigma$. In contrast, we used to $m$ denote the mean of $L_{250}$ and $s$ to denote the standard deviation of $L_{250}$. The two sets of parameters can be related to each other as, $$\mu = \ln{ (m/\sqrt{1+s^2/m^2} )}, \sigma=\sqrt{\ln{(1 + s^2/m^2)}}.$$ With stacking, we can estimate the mean of $L_{250}$ ($\bar{m}$) and the standard deviation of $L_{250}$ ($\bar{s}$). We use $m$ and $s$ to denote the intrinsic population mean and standard deviation parameters, and $\bar{m}$ and $\bar{s}$ to denote estimates[^2] of the intrinsic parameters.
To recover the LF in a given redshift bin, we need to infer $\mu$ and $\sigma$ as a function of $M_r$, using combinations of $\bar{m}$ and $\bar{s}$. In Fig. \[method\], we plot the estimated mean and standard deviation of $L_{250}$, i.e. $\bar{m}$ and $\bar{s}$ as a function of the intrinsic population mean and standard deviation of $\log L_{250}$, i.e. $\mu$ and $\sigma$. To make this plot, we generated $\sim$2000 random numbers (representing the 250 $\mu$m luminosities) drawn from a log-normal distribution for each set of ($\mu$, $\sigma$) values. The estimates $\bar{m}$ and $\bar{s}$ were derived from these specific samples (i.e. realisations) of log-normal distributions. The estimates $\bar{m}$ and $\bar{s}$ become noisy when $\sigma$ is large (even in the absence of noise), even though $m$ and $s$ can be related to $\mu$ and $\sigma$ analytically (Eq. 2). This is because $\bar{m}$ and $\bar{s}$ are sensitive to the large values in the tail of the distribution. To take the effect of realistic noise into account, we injected synthetic galaxies with log-normally distributed $L_{250}$ (drawn from distributions of known $\mu$ and $\sigma$) at random locations in the map. We can then measure the mean and standard deviation of $L_{250}$ from the stacks of synthetic galaxies in the presence of realistic noise and compare with the estimated mean and standard deviation of $L_{250}$ from the stacks of real galaxies. We summarise the main steps of recovering the 250 $\mu$m LF using our modified stacking method in Appendix A.
![Mean rest-frame 250 $\mu$m luminosity $L250$ vs. $Mr$ for the DR12 (solid squares) and Stripe 82 galaxies (open stars) in five redshift bins. Error bars correspond to the error on the mean.[]{data-label="mean_property"}](magr_vs_meanL250_s82_r22p4_gama_r20p4_crop.eps){height="2.6in" width="3.4in"}
Results
=======
Fig. \[mean\_property\] shows the mean rest-frame 250 $\mu$m luminosity $L_{\rm 250}$ as a function of $M_r$ for the DR 12 and Stripe 82 galaxies. There is good agreement in the overlapping $M_r$ range; this agreement is generally below 0.1 dex difference. At the faint end, galaxies exhibit a steep correlation between $L_{\rm 250}$ and $M_r$ without significant evolution with redshift. At the bright end, the mean $L_{\rm 250}$ as a function of $M_r$ begins to flatten with significant redshift evolution. As optically red galaxies dominate at the bright end, the redshift evolution can be explained by the evolution in the red galaxy population, which was first observed in Bourne et al. (2012). In the two highest redshift bins, $z4$ and $z5$, the depth of DR 12 means that we are only able to probe the bright galaxies with a flattened relation between the mean $L_{\rm 250}$ and $M_r$. As explained in Appendix A, our method only works if there is a roughly monotonic relation between the mean $L_{\rm 250}$ and $M_r$. Therefore, we do not use DR 12 at $z>0.3$. Fig. \[LF\_zbins\] shows our reconstructed rest-frame 250 $\mu$m LF, using DR 12 in the GAMA fields and the deeper Coadd data in Stripe 82. The luminosity limit reached by our method corresponds to the mean $L_{250}$ of the galaxies in the faintest $M_r$ bin in each redshift slice. Good agreement can be found between our results and previous determinations in the overlapping luminosity range. The dashed line in each panel is a modified Schechter function (Saunders et al. 1990) fit to our results (in the GAMA fields and Stripe 82) and measurements from Marchetti et al. (2015), $$\phi(L) = \frac{dn}{dL}=\phi^* \left(\frac{L}{L^*}\right)^{1-\alpha} \exp{\left[-\frac{1}{2\sigma^2} \log^2_{10}\left(1+\frac{L}{L^*}\right)\right]},$$ where $\phi^*$ is the characteristic density, $L^*$ is the characteristic luminosity, $\alpha$ describes the faint-end slope, and $\sigma$ controls the shape of the cut-off at the bright end. We assume $\sigma$ and $\alpha$ do not change with redshift. Table 1 lists the best-fit and marginalised error for the parameters in the modified Schechter function. We find strong positive luminosity evolution $L^*_{250}(z)\propto(1+z)^{4.89\pm1.07}$ and moderate negative density evolution $\Phi^*_{250}(z)\propto(1+z)^{-1.02\pm0.54}$ over $0.02 < z< 0.5$.
![ 250 $\mu$m LF in five redshift bins. Our results are plotted as filled stars (black: Stripe 82; red: GAMA fields), which agree well with previous measurements (green circles: Dye et al. 2010; blue circles: Marchetti et al. 2015). The dashed line is the best fit to our measurements (GAMA fields and Stripe 82) and Marchetti et al. (2015).[]{data-label="LF_zbins"}](IRLF_r20p4_and_s82_r22p4_crop.png){height="3.9in" width="3.4in"}
Parameter Best value error
------------------------------------ ------------ -------
$\log L_1^*$ ($z1=[0.02, 0.1]$) 9.17 0.11
$\log L_2^*$ ($z2=[0.1, 0.2]$) 9.37 0.11
$\log L_3^*$ ($z3=[0.2, 0.3]$) 9.50 0.12
$\log L_4^*$ ($z4=[0.3, 0.4]$) 9.66 0.12
$\log L_5^*$ ($z5=[0.4, 0.5]$) 9.87 0.13
$\log \phi_1^*$ ($z1=[0.02, 0.1]$) -1.60 0.02
$\log \phi_2^*$ ($z2=[0.1, 0.2]$) -1.60 0.03
$\log \phi_3^*$ ($z3=[0.2, 0.3]$) -1.70 0.06
$\log \phi_4^*$ ($z4=[0.3, 0.4]$) -1.59 0.10
$\log \phi_5^*$ ($z5=[0.4, 0.5]$) -1.92 0.13
$\sigma$ 0.35 0.01
$\alpha$ 1.03 0.02
: Best-fit values and marginalised errors of the parameters in the modified Schechter functions.[]{data-label="table:selection"}
Conclusion
==========
We study the low-redshift, rest-frame 250 $\mu$m LF using stacking of deep optically selected galaxies from the SDSS survey on the [*Herschel*]{}-SPIRE maps of the GAMA fields and the Stripe 82 area. Our method not only recovers the mean 250 $\mu$m luminosities $L_{250}$ of galaxies that are too faint to be individually detected, but also their underlying distribution functions.We find very good agreement with previous measurements. More importantly, our stacking method probes the LF down to much fainter luminosities ($\sim25$ times fainter) than achieved by previous efforts. We find strong positive luminosity evolution $L^*_{250}(z)\propto(1+z)^{4.89\pm1.07}$ and moderate negative density evolution $\Phi^*_{250}(z)\propto(1+z)^{-1.02\pm0.54}$ at $z< 0.5$. Our method bypasses some major difficulties in previous studies, however, it critically relies on the input photometric redshift catalogue. Therefore, issues such as photometric redshift bias and accuracy would have an impact. Over the coming years, our stacking method of reconstructing the LF will deliver even more accurate results and also extend to even fainter luminosities and higher redshifts. This is because, although we are probably not going to have any FIR/sub-mm imaging facility that will surpass [*Herschel*]{} in terms of areal coverage, sensitivity, and resolution in the near future, our knowledge of the optical and near-IR Universe will increase dramatically with ongoing and planned surveys such as DES and LSST. In addition, large and deep spectroscopic surveys such as EUCLID and DESI will further improve the quality of photometric redshift.
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The modified stacking method
============================
Below we summarise the main steps of recovering the 250 $\mu$m LF in a given redshift bin using our modified stacking method:
1\. Stack the 250 $\mu$m images centred on the galaxies in a given $M_r$ bin, weighted by luminosity distance squared ($D_L^2$) and $k-$correction. Measure the mean and standard deviation of the rest-frame 250 $\mu$m luminosity $L_{\rm 250}$, i.e. $\bar{m}$ and $\bar{s}$. Note that the estimates $\bar{m}$ and $\bar{s}$ are affected by instrument noise in the 250 $\mu$m images.
2\. Generate $n$ bootstrap realisations for each sample (i.e. the set of galaxies in a given $M_r$ bin) and repeat Step 1 for all realisations. Form an estimate of the error on $\bar{m}$ and $\bar{s}$, using the $n$ bootstrap realisations.
3\. Generate synthetic galaxies[^3] with random $L_{\rm 250}$ values drawn from log-normal distributions set by known $\mu$ and $\sigma$ values and add them to random locations in the map. The $\sigma$ values (i.e. the standard deviation of $\log L_{\rm IR}$) are chosen to sample linearly between 0.027 and 2.17 with a width of 0.027. The $\mu$ values (i.e. the mean of $\log L_{\rm IR}$) are sampled linearly between 6.478 and 12.088 with a width of 0.035. Measure the mean and standard deviation of $L_{\rm 250}$ of the synthetic galaxies, taking into account the effect of instrument noise in the 250 $\mu$m images.
4\. Repeat Step 3 $n$ times. Each time sampling different random locations in the maps.
5\. By comparing the measured mean and standard deviation of $L_{\rm 250}$ of the real galaxies with the mean and standard deviation estimates of the synthetic galaxies (for all $n$ repetitions), select all sets of $\mu$ and $\sigma$ values that give reasonably close mean and standard deviation to the real values using $\chi^2$ statistics.
6\. For each set from the accepted $\mu$ and $\sigma$ values, generate log-normally distributed $L_{\rm 250}$ and assign them randomly to galaxies in a given $M_r$ bin. Calculate the resulting distribution function of $L_{\rm 250}$.
7\. Repeat Step 6 for all accepted values of $\mu$ and $\sigma$, so we have multiple realisations of the distribution function of $L_{\rm 250}$ for galaxies in a single $M_r$ bin.
8\. Repeat Step 1 to 7 for all $M_r$ bins. The 250 $\mu$m LF is derived by adding up contributions to a given $L_{\rm 250}$ bin from galaxies in all $M_r$ bins. Using the multiple realisations, form a median estimate of the final 250 $\mu$m LF and its confidence range.
For this method to work properly, it is important that the mean r-band luminosity $M_r$ and the mean 250 $\mu$m luminosity has a more or less monotonic relation. Otherwise, one could have situations where some sources in a given bin in $L_{250}$ have fainter $M_r$ values than are included in the optical prior list. Our method is similar to the stacking approach in Bethermin et al. (2012) which was used to derive the SPIRE number counts. The main difference is that, in Bethermin et al. (2012), the aim was to recover the mean and standard deviation of the logarithm of the 250 $\mu$m flux rather than luminosity.
PN acknowledges the support of the Royal Society through the award of a University Research Fellowship, the European Research Council, through receipt of a Starting Grant (DEGAS-259586) and the support of the Science and Technology Facilities Council (ST/L00075X/1). NB acknowledges funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 312725. LD and SJM acknowledge support from the European Research Council Advanced Investigator grant, COSMICISM and Consolidator grant, cosmic dust.
The H-ATLAS is a project with Herschel, which is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. The H-ATLAS web site is <http://www.h-atlas.org/>.
Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.
SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
[^1]: We use the IAS library (<http://www.ias.u-psud.fr/irgalaxies/files/ias_stacking_lib.tgz>) (Bavouzet 2008; B[é]{}thermin et al. 2010) to perform stacking. To avoid introducing bias, we did not clean the image of any detected sources.
[^2]: An estimator is a statistic, which is a function of the values in a given sample, used to estimate a population parameter. An estimate is a specific value of the estimator calculated from a particular sample.
[^3]: The number of synthetic galaxies is equal to the number of real galaxies in a given $z$ and $M_r$ bin.
|
---
abstract: 'By evaporating a drop of lipid dispersion we generate the myelin morphology often seen in dissolving surfactant powders. We explain these puzzling nonequilibrium structures using a geometric argument: The bilayer repeat spacing increases and thus the repulsion between bilayers decreases when a multilamellar disk is converted into a myelin without gain or loss of material and with number of bilayers unchanged. Sufficient reduction in bilayer repulsion can compensate for the cost in curvature energy, leading to a net stability of the myelin structure. A numerical estimate predicts the degree of dehydration required to favor myelin structures over flat lamellae.'
author:
- |
Jung-Ren Huang[^1], Ling-Nan Zou, Thomas A. Witten\
[*James Franck Institute and Department of Physics, University of Chicago*]{}\
[*5640 S. Ellis Avenue, Chicago, Illinois 60637, USA*]{}
title: ' [Confined Multilamellae Prefer Cylindrical Morphology]{} '
---
Introduction
============
The phospholipid molecules that constitute the chief component of cellular plasma membranes spontaneously self-assemble to form bilayers when dissolved in water. These bilayers tend to stack to form multilayers, also known as [*multilamellae*]{}. At high lipid concentrations, the stable morphology consists of flat, stacked bilayers with quasi-long-range order [@Laughlin; @Roux-1994; @deGennes-1993-1]. Yet, under certain nonequilibrium conditions, the bilayers bend to form long-lived spherical multilamellae known as onions[@Diat-1993; @Buchanan-2000] or nested cylindrical tubes called myelin figures or simply, myelins (see Fig.\[fig:myelins\_exp\])[@Buchanan-2000; @Virchow-1854; @Sakurai-1990; @Haran-2002; @Buchanan-2004], despite the additional penalty in curvature energy. The formation of myelins has been a mystery since their discovery more than 150 years ago[@Virchow-1854; @Buchanan-2004]. Myelins offer a variety of potential applications such as encapsulation and controlled delivery of drugs[@Lasic-1993]. Thus a better understanding of their formation and structure is desired.
In this paper we identify a type of constrained equilibrium that leads naturally to myelin formation. The simple geometrical mechanism proposed here offers a plausible and general explanation for the longstanding puzzle of myelin formation and has explicit implications about their internal makeup. We note that in contrast to the $H_{II}$-to-$L_\alpha$-to-$H_{II}$ reentrant phase transition[@Kozlov-1994] that involves topological changes, the formation of myelins in our system involves only geometrical changes.
This paper is organized as follows: In Section \[sec:exp\] we outline our experimental results of myelin formation. Based on the experimental observations, we propose a model in Sections \[sec:model\] and \[sec:instability\] to explain the formation of myelins in our experiment. In Section \[sec:permeation\] we investigate the effect of water permeation on our model. In Section \[sec:discussion\] we discuss the implications and limitations of our model. Finally Section \[sec:conclusions\] concludes our work.
Experiment\[sec:exp\]
=====================
The main goal of this paper is to present our model of myelin formation. Therefore we only summarize our experimental results that are pertinent to the model. More complete description of the experiment will be given elsewhere[@Zou-2005].
In our experiment we observe a drop of dilute suspension of dimyristoyl phosphatidylcholine (DMPC) bilayer vesicles in water as it evaporates[@Zou-2005] (Fig.\[fig:exp\]). The drop is heated to $27-30^\circ$C, above the chain-melting temperature of DMPC ($\approx 24^\circ$C)[@Mabrey-1976]. Thus the bilayers are in the fluid state ($L_\alpha$ phase)[@Laughlin].
The evaporative flow creates a ring deposit of concentrated lipid around the drop’s perimeter[@Deegan-1997]. From this deposit many pancake-like multilamellar disks develop and grow inward. As the disks grow, they undergo a shape transition to form myelins. Experimental evidence suggests that materials, i.e., lipid and water enter the disk-myelin complex mainly via its root embedded in the dry deposit region (Fig.\[fig:exp\])[@Buchanan-2000; @Haran-2002]. If evaporation is halted, growth stops and the myelin is resorbed into its parent disk; but the myelin’s cylindrical morphology is retained during the resorption. This suggests that the disk-myelin complex is in quasi-equilibrium[@Warren-2001], and hence free energy analysis is applicable to the disk-to-myelin shape transition.
Model Definition\[sec:model\]
=============================
In this section and the next one, we present a theory to account for the myelin formation, i.e., the disk-to-myelin shape transition, observed in our experiment (Fig.\[fig:exp\]). We will show that the formation of myelins can be favorable because their cylindrical form results in a larger bilayer repeat spacing (i.e., separation between bilayers) and hence lower inter-bilayer repulsion than flat multilamellae.
We postulate that ([**a**]{}) the number of bilayers is unchanged during the shape transition, and ([**b**]{}) the bilayers can exchange materials, i.e., lipid and water, freely to achieve quasi-equilibrium[@Warren-2001]. The number of bilayers, denoted by $N$, is unlikely to change on the time scale of the shape transition, which is about 1 second. The bilayers may exchange materials via defects such as screw dislocations[@Kleman-1983; @Benton-1986; @Sein-1996].
The free energy includes a curvature energy favoring flat bilayers[@TW-2004] and a repulsion between bilayers[@Lis-1982; @Israelachvili-1992; @Israelachvili-1993]. The curvature energy of a bilayer of area $A$ takes the form of (Fig.\[fig:D-to-M\]) $$\frac{\kappa_c}{2} \int_{A} dA (c_1+c_2-c_0)^2,
\label{eqn:F^c}$$ where $\kappa_c$ is the bending stiffness, $c_1$ and $c_2$ are the principal curvatures, and $c_0$ is the spontaneous curvature[@TW-2004].
The bending stiffness $$\kappa_c=1.2\times 10^{-19}\mbox{J}
\label{eqn:kappa_c}$$ for DMPC bilayers[@Hackl-1997]. (However, Ref.[@Pabst-2003] gives a smaller value of $\kappa_c$.) In our experiment we expect $c_0 \simeq 0$. Postulate ([**a**]{}) implies that the topology of the disk-myelin complex does not change in the shape transition. Therefore we can neglect the Gaussian curvature contributions according to the Gauss-Bonnet theorem[@Struik-1950]. With (\[eqn:F\^c\]) the curvature energy of a myelin consisting of $N$ concentric, uniformly spaced bilayer cylinders is given by $$\kappa_c \frac{\pi L}{D_m} \ln\left(\frac{R_o}{R_i}\right),
\label{eqn:F^c_m}$$ where $L$ is the myelin length and $D_m$ the bilayer repeat spacing (Fig.\[fig:D-to-M\]); $R_o$ and $R_i$ are the radii of the outermost and innermost cylinders respectively. The total bilayer area $$A = \pi N (R_i + R_o) L.
\label{eqn:A_m}$$ By construction $$R_o=R_i + (N-1)D_m.
\label{eqn:R_o}$$ In this paper we set (see Section \[sec:discussion\]) $$R_i=\frac{D_m}{2}
\label{eqn:R_i}$$ for convenience, and we only consider myelins of large aspect ratios, i.e., $L\gg R_o$, so that the end caps are negligible.
The repulsion between bilayers is determined experimentally: The authors of [@Lis-1982] found that their pressure data were well represented as the sum of an exponentially falling hydration force plus a van der Waals attraction (Table 1 of [@Lis-1982]). Our estimates of inter-bilayer interaction energy use this functional form. Specifically, given water layer thickness $d_w$ and bilayer repeat spacing $d$ (Fig.\[fig:D-to-M\]), the inter-bilayer pressure $$P = P_1 + P_2,
\label{eqn:P}$$ where the hydration pressure $$P_1(d_w) = P_0 \exp\left[-\frac{d_w}{\lambda}\right],
\label{eqn:P_1}$$ and the van der Waals attraction $$P_2(d_w) = -\frac{H}{6\pi} \left[ \frac{1}{d_w^3}-
\frac{2}{d^3}+\frac{1}{(d+d_w)^3} \right].
\label{eqn:P_2}$$ The inter-bilayer interaction energy per unit bilayer area is therefore equal to $$-\int_\infty^{d_w} d(\tilde{d}_w) P(\tilde{d}_w).
\label{eqn:repulsion}$$ In this paper the bilayer thickness $d_l$ is taken to be constant (see Fig.\[fig:D-to-M\]). The pressure $P_0=10^{9.94}$dyne/cm$^2$ and $\lambda=0.26$nm for DMPC bilayers, and the Hamaker coefficient $H$ is set to $10^{-20}$J in this paper[@Lis-1982; @Evans-1986]. When the bilayers are curved, this energy is in principle altered. In Section \[sec:discussion\] we argue that such corrections are negligible in the case of interest.
Postulate ([**b**]{}) proposed at the beginning of the section implies that the important external parameters for determining the free energies are the amount of lipid and that of water in the disk-myelin complex. Under moderate external pressure, the bilayer thickness $d_l$ is approximately constant and, moreover, both lipid bilayers and water are virtually incompressible (Fig.\[fig:D-to-M\])[@Lis-1982]. This means we can describe the amount of lipid by the total area $A$ of all the bilayers, which is measured at the bilayer’s mid-surface (S in Fig.\[fig:D-to-M\]). Similarly, the total amount of lipid plus water is equivalent to the total volume $V$ of the complex.
Given the total bilayer area $A$, volume $V$, and the number of bilayers $N$, the disk-myelin complex is internally confined. In this case the lipid concentration $A/V$ determines the bilayer repeat spacing and thus the inter-bilayer repulsion, as shown in the next section.
Instability of the Disk\[sec:instability\]
==========================================
We now illustrate how a sufficiently large lipid concentration, i.e., area-to-volume ratio $A/V$, can make a multi-lamellar disk unstable to myelin formation.
We consider a large but thin disk composed of $N$ nested bilayer disks, with uniform bilayer repeat spacing $D_d$, volume $V$, and total bilayer area $A$. Because the disk is large and thin, its rim part is negligible. Figure \[fig:D-to-M\] shows the example with $N=3$. If the area of each bilayer is denoted by $A_1$, then the total area $A = 2N A_1$ and the total volume $V=(2N-1) D_d A_1$. Thus the repeat spacing $$D_d = \left(1+\frac{1}{2N-1}\right)\frac{V}{A}.
\label{eqn:D_d}$$ This $D_d$ determines the inter-bilayer interaction energy (see (\[eqn:repulsion\])). For these flat bilayers, the curvature energy (\[eqn:F\^c\]) is arbitrarily small.
To show the instability of this disk, we convert it into a myelin composed of $N$ concentric and uniformly spaced cylindrical bilayers, keeping both $V$ and $A$ fixed (Fig.\[fig:D-to-M\]). Using (\[eqn:A\_m\]), (\[eqn:R\_o\]) and (\[eqn:R\_i\]) the bilayer area $A = \pi L N^2 D_m$ while the myelin volume $V =\pi R_o^2 L = \pi L (N-1/2)^2 D_m^2$. Thus $V/A = (1-1/2N)^2 D_m$ so that the myelin repeat spacing $$\begin{aligned}
D_m &=& \left(1 + \frac{1}{2N-1}\right)^2 \frac{V}{A}
\nonumber \\
&=&\left(1+\frac{1}{2N-1}\right) D_d > D_d.
\label{eqn:D_m}\end{aligned}$$ This $D_m$ determines the inter-bilayer interaction energy of the myelin (see (\[eqn:repulsion\])). With (\[eqn:R\_o\]) and (\[eqn:R\_i\]) its curvature energy (\[eqn:F\^c\_m\]) can be written as $$\kappa_c \frac{\pi L}{D_m} \ln (2N-1)
\label{eqn:F^c_m1}$$ Because the myelin has a larger repeat spacing, it has a lower repulsion energy than the disk. If the decrease in the repulsion energy can compensate for the myelin’s curvature energy (see (\[eqn:repulsion\]) and (\[eqn:F\^c\_m1\])), the myelin, rather than the disk, becomes the thermodynamically more stable morphology. Here we define a threshold inter-bilayer pressure $P_{th}$, above which the disk becomes unstable against myelin formation. Namely, $P_{th}$ is the pressure at which the decrease in the inter-bilayer repulsion energy due to the disk-to-myelin transition is equal to the curvature energy of the myelin.
Although the fractional change in spacing $$\frac{D_m-D_d}{D_d} = \frac{1}{2N-1} \approx \frac{1}{2N}
\label{eqn:frac_D}$$ is tiny when $N\gg 1$, the reduction in the total repulsion energy is proportional to $N$, and therefore the effect of spacing increase can still be significant, as demonstrated by the example below.
Since the inter-bilayer pressure increases rapidly with decreased repeat spacing (\[eqn:P\]), a multilamellar disk with modest dehydration can easily have enough repulsion to become unstable. A typical disk observed in the experiment should be somewhat dehydrated because part of it is in the dry deposit region (Fig.\[fig:exp\]). The disk thickness is observed to be about 2.5 $\mu$m[@Zou-2005]. Given the equilibrium spacing[@Lis-1982], we infer the number $N$ of nested bilayer disks to be approximately 200.
Taking $N=200$ and using the measured bending stiffness (\[eqn:kappa\_c\]) and inter-bilayer repulsion (\[eqn:P\]) for these DMPC bilayers[@Lis-1982], our model predicts that the energy of the disk becomes larger than that of the myelin when the dehydration exceeds 2.1%, i.e., a 2.1% decrease in the repeat spacing from the equilibrium value of 6.22nm[@Lis-1982]. The inter-bilayer pressure $P = P_{th} \approx$ 0.32atm at such dehydration (\[eqn:P\]). The myelin converted from this disk has a repeat spacing about 1.9% less than the equilibrium value and an inter-bilayer pressure $P$ of about 0.3atm, with the van der Waals attraction $P_2\simeq $ 0.013atm (\[eqn:P\_2\]). Since the difference between the repeat spacings, $D_d$ and $D_m$, is tiny, the myelin diameter is nearly identical with the thickness of the parent disk; this is consistent with our experimental observations[@Zou-2005]. Here we assume that the threshold pressure $P_{th}$ is sufficiently low so that the bilayer thickness $d_l$ is unchanged during the shape transformation (Fig.\[fig:D-to-M\]). With the experimental data shown in Fig.1 of [@Lis-1982], our result of $P_{th}\approx $ 0.32atm suggests that a constant $d_l$ is indeed a fair approximation.
Based on our model we propose the following scenario to explain the myelin formation in our experiment (Fig.\[fig:exp\]): As water evaporates, lipid and water enter the disk, and hence both $A$ and $V$ increase. Because the contact line gradually moves inward due to water evaporation, the disk becomes more and more dehydrated, i.e., the lipid concentration $A/V$ increases. An increase in $A/V$ leads to a decrease in the repeat spacing $D_d$ (\[eqn:D\_d\]) and thus an increase in the inter-bilayer repulsion (\[eqn:P\]). The repeat spacing keeps decreasing until a threshold, below which the disk is unstable and a myelin with a larger spacing grows out of it in order to lower the free energy.
Effect of Water Permeation\[sec:permeation\]
============================================
The above simple example has a large pressure difference $\approx$ 0.3atm between the inside and outside of the disk-myelin complex. This pressure might induce sufficient water permeation through the bilayers to reduce the dehydration and restore the stability of the disk morphology, thus invalidating our model. In the following we perform a self-consistency check to show this is not the case: In practice the system should be in a steady state, where the pressure drop occurs across a significant fraction of the complex. This suggests that the pressure difference across a bilayer in the complex of Figure \[fig:exp\] is about $1\mbox{atm}/N \approx 1\mbox{atm}/200=5\times 10^{-3}$atm. The water permeability coefficients $p_w$ for typical phosphatidylcholine bilayers are known to be 30-100 $\mu$m/s[@Marrink-1994]. Assuming the validity of the solubility-diffusion mechanism[@Finkelstein-1987], the volume flux of water across a bilayer of area $A_2$ is given by $$J_v = p_w \frac{\bar{V}_w A_2}{RT} \Delta P,$$ where $\Delta P$ is the pressure difference across the bilayer, $\bar{V}_w=18$cm$^3$/mole, the partial molar volume of water, $R$ the gas constant and $T$ the temperature. Using permeability coefficient $p_w=100\mu$m/s and assuming that water enters the complex only via permeation and spreads evenly into all water layers, we estimate that permeation causes each water layer to swell at the rate of $\dot{d}_w \approx 2\times 10^{-3}$nm/s, or equivalently, 1.8% of the equilibrium spacing per minute (Fig.\[fig:D-to-M\]). Given the disk diameter $=15\mu$m and the swelling rate $\dot{d}_w$, the flux of water permeating into the disk part of the complex is $$J_d \approx 0.06\mu\mbox{m$^3$/s},$$ which only accounts for a small portion of the total water influx $$J_T \approx 1\mu\mbox{m$^3$/s}$$ inferred from the observed myelin growth rate $\approx 0.5\mu$m/s (Fig.\[fig:exp\]). The growth of myelins implies that both the bilayer area $A$ and volume $V$ of the complex increase with time. Our model, nevertheless, requires that the concentration $A/V$ is maintained at a sufficiently high level so that the disk is always unstable against the formation of myelins. Since $J_d \ll J_T$, the effect of water permeation is too weak to cause any significant decrease in $A/V$ and thus to compromise the proposed mechanism for myelin formation. By the same token, the flux of water permeating into a myelin in Figure \[fig:exp\] is given by $$J_m \approx 2.9\times 10^{-3} L\,\mu\mbox{m$^3$/s},$$ where the myelin length $L$ is in $\mu$m and the myelin diameter is set to $2.5\mu$m. Now we can define a critical myelin length $L_c$ using the equation $$J_T=J_d + J_m(L_c).$$ When the myelin length $L > L_c$, our mechanism for myelin formation no longer holds because the flux due to water permeation, $J_d + J_m$ exceeds the total influx $J_T$ and thus the concentration $A/V$ must decrease, restoring the stability of the disk morphology. Given $J_T$, $J_d$ and $J_m$ calculated above, $L_c$ is about $320\mu$m. In Figure \[fig:exp\] the lengths of all the myelins are less than $L_c$, which means our model for myelin formation can be applied to this system with self-consistency.
Discussion\[sec:discussion\]
============================
The model presented in Sections \[sec:model\] and \[sec:instability\] offers a simple geometrical explanation for the formation of myelins in our experiment: The bilayer repeat spacing increases and therefore the inter-bilayer repulsion decreases when a multilamellar disk is transformed into a myelin under the constraints of fixed volume, bilayer area, and number of bilayers (see (\[eqn:D\_d\]) and (\[eqn:D\_m\])). If the lipid concentration is sufficiently high, the decrease in the inter-bilayer repulsion energy is greater than the curvature energy of the myelin, and thus the disk is unstable against myelin formation. Our model can be thought of as a minimal model for myelin formation, from which more sophisticated models can be constructed. For systems with non-negligible disk perimeter and myelin end-caps, the proposed geometrical mechanism still holds, but the expressions for $D_d$ and $D_m$ are no longer as simple as (\[eqn:D\_d\]) and (\[eqn:D\_m\]), respectively[@Huang-2005]. In the following we will discuss the limitations as well as the implications of our model.
In Section \[sec:model\] we impose two artificial (i.e., non-physical) constraints on the myelin geometry in order to illustrate the geometrical mechanism that destabilizes the disk: The constituent bilayer cylinders of a myelin are uniformly spaced, and the radius $R_i=D_m/2$ (equation (\[eqn:R\_i\])). With these two constraints the calculations of the energies (\[eqn:F\^c\_m\]) and (\[eqn:repulsion\]) are greatly simplified. Freeing these two constraints would not weaken our proposed mechanism for myelin formation in that allowing $D_m$ to be non-uniform or $R_i$ to vary can only lower the myelin energy further. We will investigate the myelin structure without these two constraints in another work[@Huang-2005].
In our model we neglect the thermal undulations of bilayers[@Helfrich-1978; @Helfrich-1984] (Fig.\[fig:D-to-M\]). The reasons are twofold: First, the persistence length[@deGennes-1982; @Sornette-1994] of DMPC bilayers is much larger than any lengths of the disk-myelin complexes observed in our experiment (Fig.\[fig:exp\]). This means the bilayers are stiff, and thus their undulations should be negligible. Secondly, although bilayer thermal undulations decrease the bending stiffness $\kappa_c$ (\[eqn:kappa\_c\])[@Sornette-1994; @Peliti-1985; @Helfrich-1985] and increase the inter-bilayer pressure $P$ (\[eqn:P\])[^2], these effects, however, would only help destabilize the disk and therefore would not invalidate our model. Moreover, the threshold pressure $P_{th}$ defined in Section \[sec:instability\] is insensitive to bilayer undulations, as shown below: The decrease in the inter-bilayer repulsion energy due to the disk-to-myelin transition can be approximated with $P_{th} A(D_m-D_d) = P_{th} A D_d/(2N-1)$ (see (\[eqn:repulsion\]) and (\[eqn:frac\_D\])). By definition $P_{th}$ satisfies (see (\[eqn:F\^c\_m\])–(\[eqn:R\_i\]), and (\[eqn:D\_m\])) $$\begin{aligned}
P_{th} \frac{A D_d}{2N-1} &\approx &
\kappa_c \frac{\pi L}{D_m} \ln\left(\frac{R_o}{R_i}\right)
\nonumber \\
&=& \kappa_c \frac{A(N-1/2)^2 \ln\left(2N-1\right)}{D_d^2\cdot N^3(N+1/2)}.
\label{eqn:P_th}\end{aligned}$$ Using $N=200$ and equation (\[eqn:kappa\_c\]), and assuming $D_d$ is about equal to the equilibrium value of 6.22nm for DMPC bilayers (Section \[sec:instability\]), the above equation yields the threshold pressure $P_{th}\approx 0.3$atm regardless of bilayer undulations. This estimate of $P_{th}$ is close to the result, 0.32atm, obtained in Section \[sec:instability\]. In the above calculation we neglect the weak dependence of $\kappa_c$ on bilayer undulations[@Sornette-1994; @Peliti-1985; @Helfrich-1985].
Based on the experimental evidence described in Section \[sec:exp\], we postulate that the bilayers can exchange materials through defects to reach quasi-equilibrium (postulate ([**b**]{}) in Section \[sec:model\]). This postulate is important in our model in that it greatly simplifies the problem. Therefore the formation of defects in systems like ours deserve more detailed investigation[@Kleman-1983; @Benton-1986; @Sein-1996].
Our theory says nothing about how the myelin size is determined. This size appears to be determined by pre-existing structure in the dry lipid from which the disk and myelins grow (Fig.\[fig:exp\]). However, our theory does suggest consequences of changes in size: Since the bilayer repeat spacing is of the order of the equilibrium spacing, the overall diameter of the myelin tube is proportional to the number of bilayers $N$. We see from equation (\[eqn:P\_th\]) that the threshold pressure $P_{th}$ is proportional to $\ln N/N$ for large $N$. Thus an increase in the myelin diameter should produce a roughly proportionate decrease in the internal pressure.
Our analysis of inter-bilayer interaction energy assumes that the bilayers are flat. However, in the myelin structure the bilayers are curved. This in principle induces a correction to the interaction energy per unit area (\[eqn:repulsion\]). In the following we show that this correction is negligible when the number of bilayers $N\gg 1$: Equations (\[eqn:A\_m\])–(\[eqn:R\_i\]), (\[eqn:P\_1\]) and (\[eqn:frac\_D\]) imply that the net change of the interaction energy due to the spacing increase is approximately $$P_1 A (D_m-D_d) \simeq \frac{\pi}{2}L D_m^2 P_1 N
\sim N.
\label{eqn:main}$$ Using equation (\[eqn:P\_1\]) the interaction energy density (\[eqn:repulsion\]) of two adjacent bilayer cylinders of radii $\simeq R$ in a myelin is about $$P_1 \lambda \left[1 + c \left(\frac{D_m}{R}\right)^2\right],
\label{eqn:C_correct}$$ where the dimensionless prefactor $c$ is expected to be of order 1. The correction term $c(D_m/R)^2$ results from the bilayer curvature $1/R$. Terms of order $(D_m/R)^1$ would depend on the direction of curvature, and are thus ruled out by symmetry. With (\[eqn:R\_o\]), (\[eqn:R\_i\]) and (\[eqn:C\_correct\]) the correction to the inter-bilayer interaction energy of a myelin due to nonzero bilayer curvatures is given by $$\begin{aligned}
c P_1\lambda \sum_{n=1}^{N} A_n \left(\frac{D_m}{R_n}\right)^2
&\simeq & 2\pi c P_1\lambda D_m L \ln(2N)
\nonumber \\
&\sim & \ln(2N),
\label{eqn:secondary}\end{aligned}$$ where the decay length $\lambda < D_m$ (\[eqn:P\_1\]), and $R_n = R_i + (n-1)D_m$ and $A_n=2\pi R_n L$ are the radius and area of the $n$-th bilayer cylinder of the myelin, respectively. The above estimates suggest that the curvature effect (\[eqn:secondary\]) is negligible compared to our main effect (\[eqn:main\]) when $N \gg 1$.
In Section \[sec:instability\] we show that the repeat spacing increases when an $N$-bilayer disk is transformed into an $N$-bilayer myelin (see (\[eqn:D\_d\]) and (\[eqn:D\_m\])). The spacing would increase further if the myelin were converted into concentric spheres or onions[@Diat-1993] also composed of $N$ uniformly spaced bilayers. Specifically, the onion spacing $\simeq (1+1/(2N-1)) D_m > D_m$ for $N\gg 1$. This suggests that our model might help understand the formation of onions observed in [@Diat-1993] and [@Buchanan-2000]. In our experiment (Fig.\[fig:exp\]) their formation is, however, kinetically blocked because they cannot form continuously from a disk or myelin. Furthermore, the relative stability of onions is also influenced by the Gaussian bending stiffness[@TW-2004], since the transformation from a disk or myelin to onions changes the topology[@Struik-1950].
Although our model is invented mainly to account for the disk-to-myelin transition shown in Fig.\[fig:exp\], it also sheds some light on the formation of myelins observed in contact experiments (see Fig.\[fig:myelins\_exp\])[@Buchanan-2000; @Sakurai-1990; @Haran-2002]: In a contact experiment water is brought into contact with concentrated surfactant. Immediately after contact, the surfactant molecules self-organize to form many multilamellae at the interface[@Sein-1996]. These multilamellae should have high lipid concentrations, and therefore their inter-bilayer interaction is strongly repulsive. As a result, the bilayers along the contact interface would curve to form myelinic structures.
Conclusions\[sec:conclusions\]
==============================
In this paper we propose a geometrical mechanism to account for the formation of myelins: In short, if a stack of flat bilayers is internally confined and the inter-bilayer interaction is repulsive, geometrical packing alone can lead to myelin formation. We believe this geometric mechanism can help explain why myelins form in a variety of experiments[@Buchanan-2000; @Sakurai-1990; @Haran-2002], where internal confinement and inter-bilayer repulsion also appear to play important roles. Our findings may help develop techniques in controlled growth of myelin or onion structures, which have potential applications in encapsulation and drug delivery[@Lasic-1993]. We emphasize that our model, by its very nature, addresses only the equilibrium aspect of myelin formation. Many questions such as the myelin size distribution and the myelin growth rate still await answers.
[**[Acknowledgements]{}**]{}\
We would like to thank Prof. de Gennes and Prof. Kozlov for useful discussions. This work was supported in part by the MRSEC Program of the National Science Foundation under Award Number DMR-0213745. L.-N. Zou was partially supported by a GAANN fellowship from the U.S. Department of Education.
R.G. Laughlin, [*The Aqueous Phase Behavior of Surfactants*]{} (Academic Press, San Diego, 1996) D. Roux, C.R. Safinya, F. Nallet, in [*Micelles, Membranes, Microemulsions, and Monolayers*]{}, edited by W.M. Gelbart, A. Ben-Shaul, D. Roux (Springer, New York, 1994) P.G. De Gennes, J. Prost, [*The Physics of Liquid Crystals*]{}, 2nd edn. (Oxford University Press, New York, 1993) O. Diat, D. Roux, F. Nallet, [J. Phys. II France **3**]{}, 1427 (1993) M. Buchanan, S.U. Egelhaaf, M.E. Cates, [ Langmuir **16**]{}, 3718 (2000) R. Virchow, [ Virchow’s Arch. **6**]{}, 562 (1854) I. Sakurai, T. Suzuki, S. Sakurai, [ Mol. Cryst. Liq. Cryst. **180B**]{}, 305 (1990) M. Haran, A. Chowdhury, C. Manohar, J. Bellare, [ Colloids Surf. A **205**]{}, 21 (2002) M. Buchanan, in [*Nonlinear Dynamics in Polymeric Systems*]{}, edited by J.A.Pojman, Q. Tran-Cong-Miyata (American Chemical Society, Washington, DC, 2004) D.D. Lasic, [*Liposomes: From Physics to Applications*]{} (Elsevier, Amsterdam, 1993). M.M. Kozlov, S. Leikin, R.P. Rand, [ Biophys. J.]{} [**67**]{}, 1603 (1994) L.-N. Zou, S.R. Nagel (unpublished) S. Mabrey, J.M. Sturtevant, [ Proc. Natl. Acad. Sci. USA]{} [**73**]{}, 3862 (1976) R.D. Deegan [ et al.]{}, [Nature **389**]{}, 827 (1997); [Phys. Rev. E]{} [**62**]{}, 756 (2000) P.B. Warren, M. Buchanan, [ Curr. Opin. Colloid Interface Sci. **6**]{}, 287 (2001) M. Kléman, [*Points, Lines and Walls: In Liquid Crystals, Magnetic Systems and Various Ordered Media*]{} (John Wiley & Sons, New York, 1983) W.J. Benton, K.H. Raney, C.A. Miller, [ J. Colloid Interface Sci.]{} [**110**]{}, 363 (1986) A. Sein, J.B.F.N. Engberts, [ Langmuir **12**]{}, 2924 (1996) T.A. Witten, [*Structured Fluids: Polymers, Colloids, Surfactants*]{} (Oxford University Press, New York, 2004) L.J. Lis [ et al.]{}, [ Biophys. J.]{} [**37**]{}, 657 (1982)
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[^1]: email: jhuang2@uchicago.edu
[^2]: The thermal undulations of bilayers alter the functional form of the hydration pressure $P_1$ (\[eqn:P\_1\])[@Evans-1986; @Rand-1989].
|
---
abstract: 'Equip the edges of the lattice $\ZZ^2$ with i.i.d. random capacities. We prove a law of large numbers for the maximal flow crossing a rectangle in $\RR^2$ when the side lengths of the rectangle go to infinity. The value of the limit depends on the asymptotic behaviour of the ratio of the height of the cylinder over the length of its basis. This law of large numbers extends the law of large numbers obtained in [@GrimmettKesten84] for rectangles of particular orientation.'
title: 'Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation'
---
-1cm [Raphaël Rossignol[^1]]{}\
[*Université Paris Sud, Laboratoire de Mathématiques, bâtiment 425, 91405 Orsay Cedex, France*]{}\
[*E-mail:*]{} raphael.rossignol@math.u-psud.fr\
0.5cm and\
0.5cm [Marie Théret]{}\
[*École Normale Supérieure, Département Mathématiques et Applications, 45 rue d’Ulm, 75230 Paris Cedex 05, France*]{}\
[*E-mail:*]{} marie.theret@ens.fr
[*AMS 2000 subject classifications:*]{} Primary 60K35; secondary 82B43.
[*Keywords :*]{} First passage percolation, maximal flow, law of large numbers.
Introduction
============
The model of maximal flow in a randomly porous medium with independent and identically distributed capacities has been introduced by [@Chayes] and [@Kesten:flows]. The purpose of this model is to understand the behaviour of the maximum amount of flow that can cross the medium from one part to another.
All the precise definitions will be given in section \[chapitre6sec:notations\], but let us draw the general picture in dimension $d$. The random medium is represented by the lattice $\ZZ^d$. We see each edge as a microscopic pipe which the fluid can flow through. To each edge $e$, we attach a non-negative capacity $t(e)$ which represents the amount of fluid (or the amount of fluid per unit of time) that can effectively go through the edge $e$. Capacities are then supposed to be random, identically and independently distributed with common distribution function $F$. Let $A$ be some hyperrectangle in $\RR^d$ and $n$ an integer. The portion of medium that we will look at is a box $B_n$ of basis $nA$ and of height $2h(n)$, which $nA$ splits into two boxes of equal volume. The boundary of $B_n$ is thus split into two parts, $A_n^1$ and $A_n^2$. There are two protagonists in this play, two types of flows through $B_n$: the maximal flow $\tau_n$ for which the fluid can enter the box through $A_n^1$ and leave it through $A_n^2$, and the maximal flow $\phi_n$ for which the fluid enters $B_n$ only through its bottom side and leaves it through its top side. The first quality of $\tau_n$ is that it is (almost) a subadditive quantity, whereas $\phi_n$ is not. The main question now is: “How do $\phi_n$ and $\tau_n$ behave when $n$ is large ?”.
In this paper, we shall understand this question as “Is there a law of large numbers for $\phi_n$ and $\tau_n$ ?”, and let us say that such results do indeed already exist. However, it is important to stress that the orientation of $A$ plays an important role in these results. Indeed, the first ones were obtained for “straight” boxes, i.e., when $A$ is of the form $\prod_{i=1}^{d-1} [0,a_i] \times \{0\}$. Especially concerning the study of $\phi_n$, this simplifies considerably the task. Let us draw a precise state of the art. The law of large numbers for $\tau_n$ were proved under mild hypotheses: in [@Kesten:flows] for straight boxes and in [@RossignolTheret08b] for general boxes. These results follow essentially from the subadditivity property already alluded to. Suppose that $t(e)$ has finite expectation, $\vec{v}$ denotes a unit vector orthogonal to a hyperrectangle $A$ containing the origin of the graph, and $h(n)$ goes to infinity. Then there is a function $\nu$ defined on $S^{d-1}$ such that: $$\nu(\vec{v}) \,=\, \lim_{n\rightarrow \infty} \frac{\tau(nA,
h(n))}{\H^{d-1}(nA)} \qquad \textrm{a.s. and in }L^1\,,$$ where $\H^{d-1}(nA)$ is the $(d-1)$-dimensional Hausdorff measure of $nA$. If the height function $h(n)$ is negligible compared to $n$, $\phi_n$ satisfies the same law of large numbers as $n$ (see for example [@RossignolTheret08b]). Otherwise, the law of large numbers for $\phi_n$ was proved only for straight boxes, with suboptimal assumptions on the height $h$, the moments of $F$ and on $F(\{0\})$, in [@Kesten:flows]. In dimension 2, this was first studied in [@GrimmettKesten84]. The assumption on $F(\{0\})$ was optimized in [@Zhang] and [@Zhang07]. The assumptions on the moments of $F$ and the height $h$ have been improved in [@RossignolTheret08b]. A specificity of the lattice $\ZZ^d$, namely its invariance under reflexions with respect to integer coordinate hyperplanes, implies that the law of large numbers is the same for $\phi_n$ and $\tau_n$ in straight cylinders (provided $\log h(n)$ does not grow too fast).
Summarizing, $\tau_n$ is fairly well studied concerning laws of large numbers, but for $\phi_n$, nothing is known when the boxes are not straight, except when the height is small compared to $n$ (note however a related result by [@Garet2], cf. also Remark \[rem:Garet2\]). This paper aims at filling this gap, although we can do so only in dimension 2. For instance, suppose that $2h(n)/(nl(A))$ goes to $\tan(\alpha)$ when $n$ goes to infinity, with $\alpha\in [0,\frac{\pi}{2}]$ and $l(A)$ denoting the length of the line segment $A$. Our main results imply, under some conditions on $F$ and $A$, that: $$\label{chapitre6eq:introlgn}
\frac{\phi_n}{nl(A)}\xrightarrow[n\rightarrow \infty]{}\inf_{\widetilde{\theta} \in [\theta-\alpha,\theta+\alpha]} \frac{\nu_{\widetilde{\theta}}}{\cos
(\widetilde{\theta} - \theta)} \qquad \textrm{a.s. and in }L^1 \;,$$ where we re-encoded the function $\nu$ as follows: $\nu_{\widetilde{\theta}}:=\nu(\vb )$ when $\vb$ makes an angle $\wt$ with $(1,0)$. Notice that there is no reason for the limit in (\[chapitre6eq:introlgn\]) to be identical to $\nu_\theta$. Thus, something different happens when the boxes are not straight. Notice that this fact can already be observed when $F$ is concentrated on one point. For instance, if $t(e)=1$ deterministically and $2h(n)/(nl(A))$ goes to $\tan(\alpha)$ when $n$ goes to infinity, with $\alpha>\frac{\pi}{4}$, then one may easily compute that $\nu_\theta=|\cos \theta|+|\sin \theta|$, whereas the limit of $\phi_n/(nl(A))$ is $\min\{1/|\cos
\theta |,1/|\sin \theta |\}$. Moreover the moment conditions on $F$ that we need to prove (\[chapitre6eq:introlgn\]) are very weak.
The paper is organized as follows. In section \[chapitre6sec:notations\], we give the precise definitions and state the main result of the paper. Section \[chapitre6subsec:concentration\] is devoted to a deviation result for $\phi_n$. In section \[sec:CVesp\], we prove the convergence of the rescaled expectation of $\phi_n$. Finally, we complete the proof of the law of large numbers for $\phi_n$ in section \[secLLN\].
Notations, background and main results {#chapitre6sec:notations}
======================================
The most important notations are gathered in sections \[chapitre6subsec:maxflow\] to \[chapitre6subsec:duality\], the relevant background is described in section \[chapitre6subsec:mainresults\] while our main results are stated in section \[sec:main\].
Maximal flow on a graph {#chapitre6subsec:maxflow}
-----------------------
First, let us define the notion of a flow on a finite unoriented graph $G=(V,\E)$ with set of vertices $V$ and set of edges $\E$. Let $t=(t(e))_{e\in \E}$ be a collection of non-negative real numbers, which are called *capacities*. It means that $t(e)$ is the maximal amount of fluid that can go through the edge $e$ per unit of time. To each edge $e$, one may associate two oriented edges, and we shall denote by $\smash{\overrightarrow{\E}}$ the set of all these oriented edges. Let $A$ and $Z$ be two finite, disjoint, non-empty sets of vertices of $G$: $A$ denotes the source of the network, and $Z$ the sink. A function $\theta$ on $\smash{\overrightarrow{\E}}$ is called *a flow from $A$ to $Z$ with strength $\|\theta\|$ and capacities $t$* if it is antisymmetric, i.e. $\theta_{\overrightarrow{xy}}=-\theta_{\overrightarrow{yx}}$, if it satisfies the node law at each vertex $x$ of $V\smallsetminus (A\cup Z)$: $$\sum_{y\sim x}\theta_{\overrightarrow{xy}}=0\;,$$ where $y\sim x$ means that $y$ and $x$ are neighbours on $G$, if it satisfies the capacity constraints: $$\forall e\in \E,\;|\theta(e)|\leq t(e)\;,$$ and if the “flow in” at $A$ and the “flow out” at $Z$ equal $\|\theta\|$: $$\|\theta\|=\sum_{a\in A}\sum_{\substack{y\sim a\\ y\not \in
A}}\theta(\overrightarrow{ay})=\sum_{z\in Z}\sum_{\substack{y\sim
z\\ y\not \in Z}}\theta(\overrightarrow{yz})\;.$$ The *maximal flow from $A$ to $Z$*, denoted by $\phi_t(G,A,Z)$, is defined as the maximum strength of all flows from $A$ to $Z$ with capacities $t$. We shall in general omit the subscript $t$ when it is understood from the context. The *max-flow min-cut theorem* (see [@Bollobas] for instance) asserts that the maximal flow from $A$ to $Z$ equals the minimal capacity of a cut between $A$ and $Z$. Precisely, let us say that $E\subset\E$ is a cut between $A$ and $Z$ in $G$ if every path from $A$ to $Z$ borrows at least one edge of $E$. Define $V(E)=\sum_{e\in
E}t(e)$ to be the capacity of a cut $E$. Then, $$\label{chapitre6eq:maxflowmincut}
\phi_t(G,A,Z)=\min\{V(E)\mbox{ s.t. }E\mbox{ is a cut between
}A\mbox{ and }Z \mbox{ in } G\}\;.$$ By convention, if $A$ or $Z$ is empty, we shall define $\phi_t(G,A,Z)$ to be zero.
On the square lattice {#chapitre6subsec:squarelattice}
---------------------
We shall always consider $G$ as a piece of $\ZZ^2$. More precisely, we consider the graph $\LL=(\mathbb{Z}^{2},
\mathbb E ^{2})$ having for vertices $\mathbb Z ^{2}$ and for edges $\mathbb E ^{2}$, the set of pairs of nearest neighbours for the standard $L^{1}$ norm. The notation $\langle x,y\rangle$ corresponds to the edge with endpoints $x$ and $y$. To each edge $e$ in $\mathbb{E}^{2}$ we associate a random variable $t(e)$ with values in $\mathbb{R}^{+}$. *We suppose that the family $(t(e), e \in \mathbb{E}^{2})$ is independent and identically distributed, with a common distribution function $F$*. More formally, we take the product measure $\mathbb {P}=F^{\otimes \Omega}$ on $\Omega= \prod_{e\in \mathbb{E}^{2}} [0, \infty[$, and we write its expectation $\mathbb{E}$. If $G$ is a subgraph of $\LL$, and $A$ and $Z$ are two subsets of vertices of $G$, *we shall denote by $\phi(G,A,Z)$ the maximal flow in $G$ from $A$ to $Z$*, where $G$ is equipped with capacities $t$. When $B$ is a subset of $\RR^2$, and $A$ and $Z$ are subsets of $\ZZ^2\cap B$, we shall denote by $\phi(B,A,Z)$ again the maximal flow $\phi(G,A,Z)$ where $G$ is the induced subgraph of $\ZZ^2$ with set of vertices $\ZZ^2\cap B$.
We denote by $\overrightarrow{e}_1$ (resp. $\overrightarrow{e}_2$) the vector $(1,0)\in\RR^2$ (resp. $(0,1)$). Let $A$ be a non-empty line segment in $\RR^2$. We shall denote by $l(A)$ its (euclidean) length. All line segments will be supposed to be closed in $\mathbb{R}^2$. We denote by $\va$ the vector of unit euclidean norm orthogonal to $\operatorname{hyp}(A)$, the hyperplane spanned by $A$, and such that there is $\theta\in[0,\pi[$ such that $\va=(\cos \theta,\sin\theta)$. Define $\vc=( \sin\theta,-\cos\theta)$ and denote by $a$ and $b$ the end-points of $A$ such that $(b-a).\vc >0$. For $h$ a positive real number, *we denote by $\operatorname{cyl}(A,h)$ the cylinder of basis $A$ and height $2h$*, i.e., the set $$\operatorname{cyl}(A,h) \,=\, \{x+t \va \,|\, x\in A \,,\, t\in
[-h,h] \}\,.$$ We define also *the $r$-neighbourhood $\mathcal{V} (H,r)$ of a subset $H$ of $\mathbb{R}^d$* as $$\mathcal{V}(H,r) \,=\, \{ x \in \mathbb{R}^d \,|\, d(x,H)<r\}\,,$$ where the distance is the euclidean one ($d(x,H) = \inf \{\|x-y\|_2
\,|\, y\in H \}$).
Now, $D(A,h)$ denotes the set of *admissible boundary conditions* on $\operatorname{cyl}(A,h)$ (see Figure \[chapitre6fig:notations\]): $$D(A,h)=\left\{(k,\tilde \theta) \,|\, k\in [0,1]\mbox{ and
}\tilde\theta\in
\left[\theta-\arctan\left(\frac{2hk}{l(A)}\right),\theta+\arctan\left(\frac{2h(1-k)}{l(A)}\right)\right]\right\}\;.$$
(0,0)![An admissible boundary condition $(k,\wt)$.[]{data-label="chapitre6fig:notations"}](notationsthese.eps "fig:")
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The meaning of an element $\kappa=(k,\tilde \theta)$ of $D(A,h)$ is the following. We define $$\vb \,=\, (\cos \widetilde{\theta}, \sin
\widetilde{\theta}) \qquad \textrm{and} \qquad \vd
\,=\, (\sin \widetilde{\theta}, -\cos \widetilde{\theta})\,.$$ In $\operatorname{cyl}(nA,h(n))$, we may define two points $c$ and $d$ such that $c$ is “at height $2kh$ on the left side of $\operatorname{cyl}(A,h)$”, and $d$ is “on the right side of $\operatorname{cyl}(A,h)$” by $$c=a+(2k-1)h\va \,,\quad
(d-c)\mbox{ is orthogonal to }\vb \quad\mbox{ and
} \quad d\mbox{ satisfies }\vec{cd}\cdot \vd >0 \;.$$ Then we see that $D(A,h)$ is exactly the set of parameters so that $c$ and $d$ remain “on the sides of $\operatorname{cyl}(A,h)$”.
We define also $\D(A,h)$, the set of angles $\tilde\theta$ such that there is an admissible boundary condition with angle $\tilde\theta$: $$\D(A,h)=\left[\theta-\arctan\left(\frac{2h}{l(A)}\right),\theta+\arctan\left(\frac{2h}{l(A)}\right)\right]\;.$$ It will be useful to define the *left side (resp. right side) of $\operatorname{cyl}(A,h)$*: let $\operatorname{left}(A)$ (resp. $\operatorname{right}(A)$) be the set of vertices in $\operatorname{cyl}(A,h)\cap\ZZ^2$ such that there exists $y\notin \operatorname{cyl}(A,h)$, $\langle x,y\rangle \in
\mathbb{E}^d$ and $[ x,y [$, the segment that includes $x$ and excludes $y$, intersects $a+[-h,h].\va $ (resp. $b+[-h,h].\va $).
Now, the set $\operatorname{cyl}(A,h) \smallsetminus (c+\RR(d-c))$ has two connected components, which we denote by $\mathcal{C}_1(A,h,k,\tilde\theta)$ and $\mathcal{C}_2(A,h,k,\tilde\theta)$. For $i=1,2$, let $A_i^{h,k,\tilde\theta}$ be the set of the points in $\mathcal{C}_i(A,h,k,\tilde\theta) \cap \mathbb{Z}^2$ which have a nearest neighbour in $\mathbb{Z}^2 \smallsetminus \operatorname{cyl}(A,h)$: $$A_i^{h,k,\tilde\theta}\,=\,\{x\in \mathcal{C}_i(A,h,k,\tilde\theta) \cap
\mathbb{Z}^2 \,|\, \exists y \in \mathbb{Z}^2 \smallsetminus \operatorname{cyl}(A,h) \,,\,
\|x-y\|_{1} =1 \}\,.$$
We define *the flow in $\operatorname{cyl}(A,h)$ constrained by the boundary condition $\kappa=(k,\tilde\theta)$* as: $$\phi^\kappa(A,h):=\phi(\operatorname{cyl}(A,h),A_1^{h,k,\tilde\theta},A_2^{h,k,\tilde\theta})\;.$$ A special role is played by the condition $\kappa = (1/2, \theta)$, and we shall denote: $$\tau(A,h) = \tau (\operatorname{cyl}(A,h), \va )=\phi^{(1/2, \theta)}(A,h)\;.$$ Let $T(A,h)$ (respectively $B(A,h)$) be the top (respectively the bottom) of $\operatorname{cyl}(A,h)$, i.e., $$T(A,h) \,=\, \{ x\in \operatorname{cyl}(A,h) \,|\, \exists y\notin \operatorname{cyl}(A,h)\,,\,\,
\langle x,y\rangle \in \mathbb{E}^d \mbox{ and }\langle x,y\rangle
\mbox{ intersects } A+h\va \}$$ and $$B(A,h) \,=\, \{ x\in \operatorname{cyl}(A,h) \,|\, \exists y\notin \operatorname{cyl}(A,h)\,,\,\,
\langle x,y\rangle \in \mathbb{E}^d \mbox{ and }\langle x,y\rangle
\mbox{ intersects } A-h\va \} \,.$$ We shall denote the flow in $\operatorname{cyl}(A,h)$ from the top to the bottom as: $$\phi(A,h)= \phi (\operatorname{cyl}(A,h), \va)=\phi(\operatorname{cyl}(A,h),T(A,h),B(A,h))\;.$$
Duality {#chapitre6subsec:duality}
-------
The main reason why dimension 2 is easier to deal with than dimension $d\geq 3$ is duality. Planar duality implies that there are only $O(h^2)$ admissible boundary conditions on $\operatorname{cyl}(A,h)$. Let us go a bit into the details.
The dual lattice $\LL^*$ of $\LL$ is constructed as follows: place a vertex in the centre of each face of $\LL$ and join two vertices in $\LL^*$ if and only if the corresponding faces of $\LL$ share an edge. To each edge $e^*$ of $\LL^*$, we assign the time coordinate $t(e)$, where $e$ is the unique edge of $\mathbb{E}^2$ crossed by $e^*$. Now, let $A$ be a line segment in $\RR^2$. Let $G_A$ be the induced subgraph of $\LL$ with set of vertices $\operatorname{cyl}(A,h)\cap \ZZ^2$. Let $G_A^*$ be the planar dual of $G_A$ in the following sense: $G_A^*$ has set of edges $\{e^*\mbox{ s.t. }e\in G_A\}$, and set of vertices those vertices which belong to this set of edges. Now, we define $\operatorname{left}^*(A)$ (resp. $\operatorname{right}^*(A)$) as the set of vertices $v$ of $G_A^*$ which have at least one neighbour in $\LL^*$ which is not in $G_A$ and such that there exists an edge $e^*$ in $G_A^*$ with $v\in e^*$ and $e^*\cap \operatorname{left}(A)\not =\emptyset$ (resp. $e^*\cap \operatorname{right}(A)\not =\emptyset$).
It is well known that the (planar) dual of a cut between the top and the bottom of $\operatorname{cyl}(A,h)$ is a self-avoiding path from “left” to “right”. Furthermore, if the cut is minimal for the inclusion, the dual self-avoiding path has only one vertex on the left boundary of the dual of $A\cap\ZZ^2$ and one vertex on the right boundary. The following lemma is a formulation in our setting of those classical duality results (see for instance [@GrimmettKesten84] p.358 and [@Bollobas], p.47).
Let $A$ be a line segment $\RR^2$ and $h$ be a positive real number. If $E$ is a set of edges, let $$E^*=\{e^* \,|\,e\in E\}\;.$$ If $E$ is a cut between $B(A,h)$ and $T(A,h)$, minimal for the inclusion, then $E^*$ is a self-avoiding path from $\operatorname{left}^*(A)$ to $\operatorname{right}^*(A)$ such that exactly one point of $E^*$ belongs to $\operatorname{left}^*(A)$, exactly one point of $E^*$ belongs to $\operatorname{right}^*(A)$, and these two points are the end-points of the path.
An immediate consequence of this planar duality is the following.
\[chapitre6lem:duality\] Let $A$ be any line segment in $\RR^2$ and $h$ a positive real number. Then, $$\phi(A,h)= \min_{\kappa\in D(A,h)}\phi^\kappa(A,h) \;.$$
Notice that the condition $\kappa$ belongs to the non-countable set $D(A,h)$, but the graph is discrete so $\phi^\kappa(A,h)$ takes only a finite number of values when $\kappa\in D(A,h)$. Precisely, there is a finite subset $\tilde D(A,h)$ of $D(A,h)$, such that: $$\label{chapitre6eq:Oh2}
\operatorname{card}(\tilde D(A,h))\leq C_4 h^2\;,$$ for some universal constant $C_4$, and: $$\phi(A,h)= \min_{\kappa\in \tilde D(A,h)}\phi^\kappa(A,h) \;.$$
Background {#chapitre6subsec:mainresults}
----------
First, let us recall some facts concerning the behaviour of $\tau(nA,h(n))$ when $n$ and $h(n)$ go to infinity. Using a subadditive argument and deviation inequalities, Rossignol and Théret have proved in [@RossignolTheret08b] that $\tau(nA,
h(n))$ satisfies a law of large numbers:
\[chapitre6thm:LGNtau\] We suppose that $$\int_{[0,\infty[} x \, dF (x) \,<\, \infty\, .$$ For every unit vector $\va =(\cos\theta,\sin\theta)$, there exists a constant $\nu_\theta$ depending on $F$, $d$ and $\theta$, such that for every non-empty line-segment $A$ orthogonal to $\va$ and of euclidean length $l(A)$, for every height function $h: \NN \rightarrow \RR^+$ satisfying $\lim_{n\rightarrow \infty} h(n) = +\infty$, we have $$\lim_{n\rightarrow \infty} \frac{\tau(nA, h(n))}{n l(A)}
\,=\, \nu_\theta \qquad \textrm{in } L^1 \,.$$ Moreover, if the origin of the graph belongs to $A$, or if $$\int_{[0,\infty[} x^{2} \, dF (x) \,<\, \infty \,,$$ then $$\lim_{n\rightarrow \infty} \frac{\tau(nA, h(n))}{nl(A)}
\,=\, \nu_\theta \qquad \textrm{a.s.}$$ Under the added assumption that $\lim_{n\rightarrow \infty} h(n) /n =0$, the variable $\phi(nA,h(n))$ satisfies the same law of large numbers as $\tau(nA,h(n))$, under the same conditions.
This law of large numbers holds in fact for every dimension $d\geq 2$. Let us remark that (in dimension two) $\nu_\theta$ is equal to $\mu(\vc ) = \mu(\va)$, where $\mu(.)$ is the time-constant function of first passage percolation as defined in [@Kesten:StFlour], (3.10) p. 158. This equality follows from the duality considerations of section \[chapitre6subsec:duality\] and standard first passage percolation techniques (see also Theorem 5.1 in [@GrimmettKesten84]) that relate cylinder passage times to unrestricted passage times (as in [@HammersleyWelsh], Theorem 4.3.7 for instance). Boivin has also proved a very similar law of large numbers (see Theorem 6.1 in [@Boivin]). Notice that for the definition of $\mu(.)$, Kesten requires only the existence of the first moment of the law $F$ in the proof from [@Kesten:StFlour], and it can also be defined under the weaker condition $\int_0^{\infty}(1
-F(x))^4\;dx<\infty$.
One consequence of this equality between $\nu$ and $\mu$ is that $\theta\mapsto \nu_\theta$ is either constant equal to zero, or always non-zero. In fact the following property holds (cf. [@Kesten:StFlour], Theorem 6.1 and Remark 6.2 p. 218):
\[propnu\] We suppose that $\int_{[0,+\infty[} x \, dF(x) <\infty$. Then $\nu_\theta$ is well defined for all $\theta$, and we have $$\nu_\theta \,>\,0 \iff F(0) \,<\, 1/2\,.$$
There exists a law of large numbers for the variable $\phi(nA, h(n))$ when the rectangle we consider is straight, i.e., $\theta = 0$. It has been proved in [@GrimmettKesten84], Corollary 4.2, that:
\[thmllnphidroit\] Suppose that $A=[0,1]\times\{0\}$, $\int_{[0,+\infty[} x \, dF(x) <\infty$, $$h(n)\xrightarrow[n\rightarrow\infty]{}\infty \quad\mbox{ and
}\quad\frac{\log h(n)}{n}\xrightarrow[n\rightarrow\infty]{}0\;.$$ Then, $$\frac{\phi(nA, h(n))}{n}\xrightarrow[n\rightarrow\infty]{a.s}\nu_0\;.$$
Notice that in [@GrimmettKesten84], the condition on $F$ is in fact weakened to $\int_0^{\infty}(1 -F(x))^4\;dx<\infty$, obtaining the convergence to $\mu((0,1))$. However, our definition of $\nu_\theta$ requires a moment of order 1.
Finally, let us remark that [@Garet2] proved a law of large numbers for the maximal flow between a compact convex set $\Lambda \subset\RR^2$ and infinity. This is somewhat related to our main result, Theorem \[chapitre6thm:lgn\], see Remark \[rem:Garet2\]. Before stating Garet’s result, we need some notations. For every convex bounded set $\Lambda \subset \RR^2$, we denote by $\partial ^* \Lambda$ the set of all the points $x$ of the boundary $\partial \Lambda$ of $\Lambda$ where $\Lambda$ admits a unique outer normal, which is denoted by $\vec{v}_{\Lambda} (x)$. We denote the coordinates of $\vec{v}_{\Lambda}
(x)$ by $(\cos(\theta(\Lambda,x)), \sin(\theta(\Lambda, x)))$ for every $x$ in $\partial ^* \Lambda$. We denote by $\sigma (\Lambda)$ the maximal flow from $\Lambda$ to infinity. Let $\H^1$ be the one-dimensional Hausdorff measure. Theorem 2.1 in [@Garet2] is the following:
\[thmgaret\] We suppose that $F(0)<1/2$ and that $$\exists \gamma>0 \qquad \int_{[0,+\infty[} e^{\gamma t(e)} \,dF(x)
\,<\,\infty \,.$$ Then, for each bounded convex set $\Lambda \subset \RR^2$ with the origin of the graph $0$ in its interior, we have $$\label{eqgaret}
\lim_{n\rightarrow \infty} \frac{\sigma(n\Lambda)}{n} \,=\, \int_{\partial
^* \Lambda} \nu_{\theta(\Lambda, x)} d\H^1(x) \,=\, \mathcal{I} (\Lambda)
\,>\,0\,.$$
Main result {#sec:main}
-----------
We recall that for all $n\in \NN$, we have defined $$\D(nA,h(n))=\left[\theta-\arctan\left(\frac{2h(n)}{nl(A)}\right),\theta+\arctan\left(\frac{2h(n)}{nl(A)}\right)\right]\;.$$ We may now state our main result.
\[chapitre6thm:lgn\] Let $A$ be a non-empty line-segment in $\RR^2$, with euclidean length $l(A)$. Let $\theta\in[0,\pi[$ be such that $(\cos\theta,\sin\theta)$ is orthogonal to $A$ and $(h(n))_{n\geq 0}$ be a sequence of positive real numbers such that: $$\label{chapitre6eq:conditionshn}
\left\lbrace \begin{array}{l}h(n)\xrightarrow[n\rightarrow
\infty]{}+\infty\;,\\
\frac{\log h(n)}{n}\xrightarrow[n\rightarrow
\infty]{}0\;.\end{array}\right.$$ Define: $$\overline{\D}=\limsup_{n\rightarrow
\infty}\D(nA,h(n))=\bigcap_{N\geq 1}\bigcup_{n\geq
N}\D(nA,h(n))\;,$$ and $$\underline{\D}=\liminf_{n\rightarrow \infty}\D(nA,h(n))=\bigcup_{N\geq 1}\bigcap_{n\geq N}\D(nA,h(n))\;.$$ Suppose that $F$ has a finite moment of order 1: $$\label{chapitre6eq:conditionsFLGN}
\int_0^\infty x \, dF(x)<\infty\;.$$ Then, $$\label{eq:liminfmoyenne}
\liminf_{n\rightarrow \infty}\frac{\EE[\phi(nA,h(n))]}{nl(A)}=\inf \left\{
\frac{\nu_{\widetilde{\theta}}}{\cos (\widetilde{\theta} - \theta)}
\,|\, \widetilde{\theta} \in \overline{\D} \right\}$$ and $$\label{eq:limsupmoyenne}\limsup_{n\rightarrow\infty}\frac{\EE[\phi(nA,h(n))]}{nl(A)}= \inf\left\{
\frac{\nu_{\widetilde{\theta}}}{\cos (\widetilde{\theta} - \theta)} \,|\,
\widetilde{\theta} \in \underline{\D} \right\} \,.$$ Moreover, if $$\label{chapitre6eq:conditionsFLGNbis}
\int_0^\infty x^2 \, dF(x)<\infty\,,$$ or if: $$\label{conditionO}
0\mbox{ is the middle of }A\;,$$ then $$\liminf_{n\rightarrow \infty}\frac{\phi(nA,h(n))}{nl(A)}=\inf \left\{
\frac{\nu_{\widetilde{\theta}}}{\cos (\widetilde{\theta} - \theta)}
\,|\, \widetilde{\theta} \in \overline{\D} \right\} \qquad a.s.$$ and $$\limsup_{n\rightarrow\infty}\frac{\phi(nA,h(n))}{nl(A)}= \inf\left\{
\frac{\nu_{\widetilde{\theta}}}{\cos (\widetilde{\theta} - \theta)} \,|\,
\widetilde{\theta} \in \underline{\D} \right\} \qquad a.s.$$
It is likely that condition (\[chapitre6eq:conditionsFLGN\]) can be weakened to $\int_0^\infty
(1-F(x))^4\;dx$, as in Theorem \[thmllnphidroit\]. This would require to define $\nu$ a bit differently.
\[chapitre6corollaire\] We suppose that conditions (\[chapitre6eq:conditionshn\]) on $h$ are satisfied. We suppose also that there is some $\alpha\in \left[0,\frac{\pi}{2}\right]$ such that: $$\frac{2h(n)}{nl(A)}\xrightarrow[n\rightarrow
\infty]{}\tan \alpha \;.$$ Then, if condition (\[chapitre6eq:conditionsFLGN\]) on $F$ is satisfied, we have $$\lim_{n\rightarrow \infty}\frac{\phi(nA,h(n))}{nl(A)}=\inf\left\{
\frac{\nu_{\widetilde{\theta}}}{\cos (\widetilde{\theta} - \theta)} \,|\,
\widetilde{\theta} \in [\theta-\alpha,\theta+\alpha] \right\} \qquad
\textrm{in } L^1\,.$$ Moreover, if condition (\[chapitre6eq:conditionsFLGNbis\]) or (\[conditionO\]) are satisfied, then $$\lim_{n\rightarrow \infty}\frac{\phi(nA,h(n))}{nl(A)}=\inf\left\{
\frac{\nu_{\widetilde{\theta}}}{\cos (\widetilde{\theta} - \theta)} \,|\,
\widetilde{\theta} \in [\theta-\alpha,\theta+\alpha] \right\} \qquad
\textrm{a.s.}$$
It has already been remarked in [@Theret:small] (see the discussion after Theorem 2) that the condition on $h$ is the good one to have positive speed when one allows edge capacities to be null with positive probability.
Notice that Theorem \[chapitre6thm:lgn\] is consistent with Theorem \[thmllnphidroit\], the existing law of large numbers for $\phi(nA, h(n)$ in the straight case. Indeed, it is known that $\nu$ satisfies the weak triangle inequality (see section 4.4 in [@RossignolTheret08b]), and for symmetry reasons, it implies that when $\theta \in
\{0,\pi /2 \}$, the function $\wt\mapsto \nu_{\widetilde{\theta}} / \cos
(\widetilde{\theta} - \theta)$ is minimum for $\wt=\theta$ and thus, Theorem \[chapitre6thm:lgn\] implies that $\phi(nA,h(n))/(nl(A))$ converges to $\nu_0$, the limit of $\tau(nA,h(n))/(nl(A))$, when $\operatorname{cyl}(nA, h(n))$ is a straight cylinder. In fact, the same phenomenon occurs for any $\theta$ such that there is a symmetry axis of direction $\theta$ for the lattice $\ZZ^2$. These directions in $[0,\pi[$ are of course $\{0, \pi /4, \pi/ 2, 3 \pi /4\}$. Also, Corollary \[chapitre6corollaire\] is consistent with the fact that for general boxes, when $h(n)$ is small with respect to $n$, $\phi(nA,h(n))/(nl(A))$ and $\tau(nA,h(n))/(nl(A))$ have the same limit.
\[rem:Garet2\] Theorems \[chapitre6thm:lgn\] and \[thmgaret\] are related. First, they are stated in dimension two only, because both proofs use the duality of the planar graph to define the path which is the dual of a cutset, and then the fact that such paths can be glued together if they have a common endpoint. These properties hold only in dimension two: the dual of an edge in dimension greater than three is a unit surface, and it is much more difficult to study the boundary of a surface. This is the reason why these theorems are not yet generalized in higher dimensions (see also Remark \[remd3\]). Moreover, the expressions of the limits $\mathcal{I}(\Lambda)$ and $\eta_{\theta,h}$ appearing in these theorems are very similar. On one hand, the constant $\eta_{\theta,h}$ is the infimum of the integral of $\nu$ along the segments that cut the top from the bottom of $\operatorname{cyl}(A,h(n)/n)$ for large $n$. Since $\nu$ satisfies the weak triangle inequality, $\eta_{\theta, h}$ is also equal to infimum of the integral of $\nu$ along the polyhedral curves that have the same property of cutting. On the other hand, Garet only has to consider the case of a polyhedral convex set $\Lambda$ during his proof, and he proves the important following property: if $\Lambda \subset
\Lambda '$, where $\Lambda$ and $\Lambda'$ are polyhedral and $\Lambda$ is convex, then $\mathcal{I}(\Lambda) \leq \mathcal{I}(\Lambda ')$. Thus, for a polyhedral convex set $\Lambda$, $\mathcal{I}(\Lambda)$ is the infimum of the integral of $\nu$ along the polyhedral curves that cut $\Lambda$ from infinity.
Sketch of the proof {#chapitre6subsec:sketchLGN}
-------------------
We suppose that $A$ is a non-empty line segment in $\RR^2$. To shorten the notations, we shall write $D_n=D(nA,h(n))$, the set of all admissible conditions for $(nA,h(n))$: $$D_n=\left\{(k,\tilde \theta) \,|\,k\in [0,1]\mbox{ and
}\tilde\theta\in
\left[\theta-\arctan\left(\frac{2h(n)k}{nl(A)}\right),\theta+\arctan\left(\frac{2h(n)(1-k)}{nl(A)}\right)\right]\right\}\;,$$ and $$\D_n=\left[\theta-\arctan\left(\frac{2h(n)}{nl(A)}\right),\theta+\arctan\left(\frac{2h(n)}{nl(A)}\right)\right]\;.$$ Also, we shall use: $$\phi_n=\phi(nA,h(n)),\quad
\phi_n^\kappa=\phi^\kappa(nA,h(n))\quad\mbox{ and }\quad\tau_n=\tau(nA,h(n))\;.$$
First, notice that $0\leq\phi_n\leq\tau_n$. If $F(0)\geq
\frac{1}{2}$, then $\tau_n/n$ converges to zero, and so does $\phi_n$, so Theorem \[chapitre6thm:lgn\] is trivially true. [*We shall therefore make the following hypothesis in the rest of the article:*]{} $$\label{eq:F(0)}
F(0)<\frac{1}{2}\;.$$
Now, let us draw a sketch of the proof of Theorem \[chapitre6thm:lgn\]. Recall that from Lemma \[chapitre6lem:duality\], $$\phi_n=\min_{\kappa\in D_n}\phi_n^\kappa\;.$$ First, we shall study the asymptotics of $\EE(\phi_n)$ (section 4):
[**Step 1.**]{} By a subadditive argument (see Figure \[chapitre6emboitement1\]), we show in section \[chapitre6subsec:upperbound\] that $$\limsup_{n\rightarrow \infty} \frac{\EE[\phi_n]}{nl(A)} \,\leq\, \inf_{\wt \in
\underline{\D}} \frac{\nu_{\wt}}{\cos (\wt - \theta)}$$ and $$\liminf_{n\rightarrow \infty} \frac{\EE[\phi_n]}{nl(A)} \,\leq\, \inf_{\wt \in
\overline{\D}} \frac{\nu_{\wt}}{\cos (\wt - \theta)} \,.$$
[**Step 2.**]{} On the other hand, by a similar subadditive argument (see Figure \[chapitre6emboitement2\]), we show in section \[chapitre6subsec:lowerbound\] that $$\liminf_{n\rightarrow \infty}\inf_{\kappa \in D_n}
\frac{\EE[\phi_n^{\kappa}]}{nl(A)} \,\geq\, \inf_{\wt \in
\overline{\D}} \frac{\nu_{\wt}}{\cos (\wt - \theta)}$$ and $$\limsup_{n\rightarrow \infty}\inf_{\kappa \in D_n}
\frac{\EE[\phi_n^{\kappa}]}{nl(A)} \,\geq\, \inf_{\wt \in
\underline{\D}} \frac{\nu_{\wt}}{\cos (\wt - \theta)} \,.$$
[**Step 3.**]{} Using deviation results for the variables $\phi_n^\kappa$ (section \[chapitre6subsec:concentration\]), we prove in section \[chapitre6subsec:endLGN\] that $\EE[\phi_n]$ is equivalent to $\inf_{\kappa \in D_n} \EE[\phi_n^{\kappa}]$, and this ends the study of the asymptotic behaviour of $\EE[\phi_n]$.
Next, we relate $\phi_n$ and $\EE(\phi_n)$ to show the almost sure asymptotics (section \[secLLN\]):
[**Step 4.**]{} A deviation result for $\phi_n$ obtained in section \[chapitre6subsec:concentration\] shows that almost surely, asymptotically, $\phi_n/n$ is at least as large as $\EE(\phi_n)/n$.
[**Step 5.**]{} Finally, we use again the subadditive argument of the first step of the proof to prove that almost surely, $$\limsup_{n\rightarrow \infty} \frac{\phi_n}{nl(A)} \,\leq\, \inf_{\wt \in
\underline{\D}} \frac{\nu_{\wt}}{\cos (\wt - \theta)}$$ and $$\liminf_{n\rightarrow \infty} \frac{\phi_n}{nl(A)} \,\leq\, \inf_{\wt \in
\overline{\D}} \frac{\nu_{\wt}}{\cos (\wt - \theta)} \,.$$
Deviation properties of the maximal flows {#chapitre6subsec:concentration}
=========================================
The following proposition, due to Kesten, allows to control the size of the minimal cut, and is of fundamental importance in the study of First Passage Percolation.
\[chapitre6prop:5.8Kesten\] Suppose that $F(0)<\frac{1}{2}$. Then, there are constants ${\varepsilon}$, $C_1$ and $C_2$, depending only on $F$, such that: $$\PP\left( \begin{array}{c} \exists\mbox{ a self-avoiding path }\gamma\mbox{ in }\LL^*\mbox{,
starting at }(\frac{1}{2},\frac{1}{2})\mbox{,} \\ \mbox{with }\operatorname{card}(\gamma)\geq m\mbox{ and
}\sum_{e^*\in\gamma}t(e^*)\leq {\varepsilon}m \end{array} \right) \,\leq\, C_1e^{-C_2m}\;.$$
Thanks to Proposition \[chapitre6prop:5.8Kesten\] and general deviation inequalities due to [@Boucheronetal03], we obtain the following deviation result for the maximal flows $\phi_n$ and $\phi_n^\kappa$. The proof is exactly the same as the proof of Proposition 4.3 in [@RossignolTheret08b], using Proposition \[chapitre6prop:5.8Kesten\] instead of Zhang’s result. We reproduce it here for the sake of completeness.
\[prop:deviation\] Suppose that hypotheses (\[chapitre6eq:conditionsFLGN\]) and (\[eq:F(0)\]) hold. Then, for any $\eta\in]0,1]$, there are strictly positive constants $C(\eta,F)$, $K_1(F)$ and $K_2(F)$, such that, for every $n\in\NN^*$, and every non-degenerate line segment $A$, $$\label{chapitre6eq:deviationkappa}
\max_{\kappa\in
D_n}\PP(\phi_n^\kappa<\EE(\phi_n^\kappa)(1-\eta))\leq
K_1e^{-C(\eta,F)\min_\kappa\EE(\phi_n^\kappa)}\;.$$ and: $$\label{eq:dev_phi_ordre1}
\PP\left( \phi_n \leq \EE(\phi_n)(1-\eta) \right) \leq K_2h(n)^2e^{-C(\eta,F)\EE(\phi_n)}\;.$$
Let us fix $A$, $n\in\NN^*$ and $\kappa=(k,\tilde\theta)\in D_n$. First, we prove the result for $\phi_n^\kappa$. We shall denote by $E_{\phi_n^\kappa}$ a cut whose capacity achieves the minimum in the dual definition (\[chapitre6eq:maxflowmincut\]) of $\phi_n^\kappa$. Since $\PP\left( \phi_n^\kappa\leq \EE(\phi_n^\kappa)(1-\eta) \right)$ is a decreasing function of $\eta$, it is enough to prove the result for all $\eta$ less than or equal to some absolute $\eta_0\in]0,1[$. We use this remark to exclude the case $\eta =1$ in our study, thus, from now on, let $\eta$ be a fixed real number in $]0,1[$.
We order the edges in $\operatorname{cyl}(nA,h(n))$ as $e_1,\ldots,e_{m_n}$. For every hyperrectangle $A$, we denote by $\mathcal{N}(A,h)$ the minimal number of edges in $A$ that can disconnect $A_1^h$ from $A_2^h$ in $\operatorname{cyl}(A,h)$. For any real number $r\geq \mathcal{N}(nA,h(n))$, we define: $$\psi_n^r\,=\, \min\left\{ \begin{array}{c} V(E)\mbox{ s.t. }\operatorname{card}(E)\leq
r\mbox{ and } E \mbox{ cuts }\\
(nA)_1^{h(n),k,\tilde\theta}\mbox{
from }(nA)_2^{h(n),k,\tilde\theta}\mbox{ in }\operatorname{cyl}(nA,h(n)) \end{array} \right\}\;.$$ Now, suppose that hypotheses (\[chapitre6eq:conditionsFLGN\]) and (\[eq:F(0)\]) hold, let ${\varepsilon}$, $C_1$ and $C_2$ be as in Proposition \[chapitre6prop:5.8Kesten\], and define $r=(1-\eta)\EE(\phi_n^\kappa)/{\varepsilon}$. Suppose first that $r<\mathcal{N}(nA,h(n))$. Then, $$\begin{aligned}
\PP(\phi_n^\kappa\leq (1-\eta)\EE(\phi_n^\kappa))&=&\PP(\phi_n^\kappa\leq
(1-\eta)\EE(\phi_n^\kappa)\mbox{ and } \operatorname{card}(E_{\phi_n^\kappa}) \geq
(1-\eta)\EE(\phi_n^\kappa)/{\varepsilon})\;,\\
&\leq&C_1e^{-C_2(1-\eta)\EE(\phi_n^\kappa)/{\varepsilon}}\;,\end{aligned}$$ from Proposition \[chapitre6prop:5.8Kesten\], and the desired inequality is obtained. Suppose now that we have $r\geq\mathcal{N}(nA,h(n))$. Then, $$\begin{aligned}
\nonumber \PP(\phi_n^\kappa\leq (1-\eta)\EE(\phi_n^\kappa))&=&\PP(\phi_n^\kappa\leq
(1-\eta)\EE(\phi_n^\kappa)\mbox{ and }\psi_n^r\not=\phi_n^\kappa)+\PP(\psi_n^r\leq (1-\eta)\EE(\phi_n^\kappa))\;,\\
\label{eq:tautaunr} &\leq&C_1e^{-C_2r}+\PP(\psi_n^r\leq (1-\eta)\EE(\psi_n^r))\;,\end{aligned}$$ from Proposition \[chapitre6prop:5.8Kesten\] and the fact that $
\phi_n^\kappa\leq \psi_n^r$. Now, we truncate our variables $t(e)$. Let $a$ be a positive real number to be chosen later, and define $\tilde t(e)=t(e)\land
a$. Let: $$\tilde \psi_n^r\,=\, \min\left\{ \begin{array}{c} \sum_{e\in
E}\tilde t(e)\mbox{ s.t. }\operatorname{card}(E)\leq r\mbox{ and }E \mbox{ cuts }\\
(nA)_1^{h(n),k,\tilde\theta}\mbox{
from }(nA)_2^{h(n),k,\tilde\theta}\mbox{ in }\operatorname{cyl}(nA,h(n)) \end{array} \right\}\;.$$ Notice that $\tilde \psi_n^r\leq \psi_n^r$. We shall denote by $E_{\tilde\psi_n^r}$ a cutset whose capacity achieves the minimum in the definition of $\tilde \psi_n^r$. If there are more than one, we use a deterministic method to select a unique one with the minimal number of edges among these. Then, $$\begin{aligned}
0\leq \EE(\psi_n^r)-\EE(\tilde\psi_n^r)&\leq &\EE\left[\sum_{e\in
E_{\tilde\psi_n^r}}t(e)-\sum_{e\in
E_{\tilde\psi_n^r}}\tilde t(e)\right]\;,\\
&\leq &\EE\left[\sum_{e\in
E_{\tilde\psi_n^r}}t(e)\II_{t(e)\geq a}\right]\;,\\
&=&\sum_{i=1}^{m_n}\EE(t(e_i)\II_{t(e_i)\geq a}\II_{e_i\in
E_{\tilde\psi_n^r}})\;,\\
&=&\sum_{i=1}^{m_n}\EE\left\lbrack \EE\left(t(e_i)\II_{t(e_i)\geq a}\II_{e_i\in
E_{\tilde\psi_n^r}}|(t(e_j))_{j\not= i}\right)\right\rbrack\;.\end{aligned}$$ Now, when $(t(e_j))_{j\not= i}$ is fixed, $t(e_i)\mapsto\II_{e_i\in
E_{\tilde\psi_n^r}}$ is a non-increasing function and $t(e_i)\mapsto
t(e_i)\II_{t(e_i)\geq a}$ is of course non-decreasing. Furthermore, since the variables $(t(e_i))$ are independent, the conditional expectation $\EE\left(.|(t(e_j))_{j\not= i}\right)$ corresponds to expectation over $t(e_i)$, keeping $(t(e_j))_{j\not= i}$ fixed. Thus, Chebyshev’s association inequality (see [@HardyLittlewoodPolya52], p. 43) implies: $$\begin{aligned}
\EE\big(t(e_i)\II_{t(e_i)\geq a}\II_{e_i\in
E_{\tilde\psi_n^r}} & |(t(e_j))_{j\not= i}\big)\\
&\,\leq\, \EE\left(t(e_i)\II_{t(e_i)\geq a}|(t(e_j))_{j\not= i}\right)\EE\left(\II_{e_i\in
E_{\tilde\psi_n^r}}|(t(e_j))_{j\not= i}\right)\;,\\
&\,=\, \EE\left(t(e_1)\II_{t(e_1)\geq a}\right)\EE\left(\II_{e_i\in
E_{\tilde\psi_n^r}}|(t(e_j))_{j\not= i}\right)\;.\end{aligned}$$ Thus, $$\label{eq:nucontinue}0\leq \EE(\psi_n^r)-\EE(\tilde\psi_n^r) \leq
\EE\left(t(e_1)\II_{t(e_1)\geq a}\right)\EE( \operatorname{card}(E_{\tilde\psi_n^r}))\leq
r\EE\left(t(e_1)\II_{t(e_1)\geq a}\right)\;.$$ Now, since $F$ has a finite moment of order 1, we can choose $a=a(\eta,F,d)$ such that: $$\frac{1-\eta}{{\varepsilon}}\EE\left(t(e_1)\II_{t(e_1)\geq
a}\right)\leq\frac{\eta}{2}\;,$$ to get: $$\begin{aligned}
\nonumber
\EE(\psi_n^r)-\EE(\tilde\psi_n^r)\leq\frac{\eta}{2}\EE(\phi_n^\kappa)\leq\frac{\eta}{2}\EE(\psi_n^r)\;,\\
\label{eq:taunrtautilde}
\PP(\psi_n^r\leq (1-\eta)\EE(\psi_n^r))\leq\PP\left(\tilde \psi_n^r\leq\EE(\tilde \psi_n^r)-\frac{\eta}{2}\EE( \psi_n^r)\right)\;.\end{aligned}$$ Now, we shall use Corollary 3 in [@Boucheronetal03]. To this end, we need some notations. We take $\tilde t'$ an independent collection of capacities with the same law as $\tilde t=(\tilde t(e_i))_{i=1\ldots,m_n}$. For each edge $e_i\in\operatorname{cyl}(A,h)$, we denote by $\tilde t^{(i)}$ the collection of capacities obtained from $\tilde t$ by replacing $\tilde t(e_i)$ by $\tilde t'(e_i)$, and leaving all other coordinates unchanged. Define: $$V_-:=\EE\left\lbrack\left.\sum_{i=1}^{m_n}(\tilde\psi_n^r(t)-\tilde\psi_n^r(t^{(i)}))_-^2\right|t\right\rbrack\;,$$ where $\tilde\psi_n^r(t)$ is the maximal flow through $\operatorname{cyl}(nA,h(n))$ when capacities are given by $t$. We shall denote by $R_{\tilde\psi_n^r}$ the intersection of all the cuts whose capacity achieves the minimum in the definition of $\tilde \psi_n^r$. Observe that: $$\tilde\psi_n^r(t^{(i)})-\tilde\psi_n^r(t)\leq (\tilde t'(e_i)-\tilde
t(e_i))\II_{e_i\in R_{\tilde\psi_n^r}}\;,$$ and thus, $$V_-\leq a^2 \EE [\operatorname{card}(R_{\tilde\psi_n^r})]\leq a^2r=a^2(1-\eta)\EE(\phi_n^\kappa)/{\varepsilon}\;.$$ Thus, Corollary 3 in [@Boucheronetal03] implies that, for every $\eta\in]0,1[$, $$\PP \left(\tilde \psi_n^r\leq \EE(\tilde \psi_n^r)-\frac{\eta}{2}\EE( \psi_n^r)\right)\leq
e^{-\frac{\EE(\psi_n^r)^2\eta^2{\varepsilon}}{16a^2(1-\eta)\EE(\phi_n^\kappa)}}\leq e^{-\frac{\EE(\phi_n^\kappa)\eta^2{\varepsilon}}{16a^2(1-\eta)}}\;.$$ Using inequalities (\[eq:taunrtautilde\]) and (\[eq:tautaunr\]) and taking the maximum over $\kappa\in D_n$, this ends the proof of Inequality (\[chapitre6eq:deviationkappa\]).
To see that (\[eq:dev\_phi\_ordre1\]) holds, notice that $\EE(\phi_n)\leq \min_{\kappa\in D_n}\EE(\phi_n^\kappa)$. Thus, (\[eq:dev\_phi\_ordre1\]) is a consequence of inequalities (\[chapitre6eq:deviationkappa\]) and (\[chapitre6eq:Oh2\]).
Asymptotic behaviour of the expectation of the maximal flow {#sec:CVesp}
===========================================================
Upper bound {#chapitre6subsec:upperbound}
-----------
From now on, we suppose that the conditions (\[chapitre6eq:conditionsFLGN\]) on $F$ and (\[chapitre6eq:conditionshn\]) on $h$ are satisfied. We consider a line segment $A$, of orthogonal unit vector $\va = (\cos\theta, \sin\theta)$ for $\theta
\in [0, \pi[$, and a function $h: \NN \rightarrow \RR^+ $ satisfying $\lim_{n\rightarrow
\infty}h(n) = +\infty$. Recall that $\D_n=\D(nA,h(n))$. For all $\wt \in
\D_n $, we define $$k_n \,=\, \frac{1}{2} + \frac{nl(A) \tan (\wt - \theta)}{4 h(n)} \,,$$ and thus $\kappa_n=(k_n,\widetilde{\theta})\in D_n$. We want to compare $\phi_n^{\kappa_n}$ with the maximal flow $\tau$ in a cylinder inside $\operatorname{cyl}(nA, h(n))$ and oriented towards the direction $\widetilde{\theta}$. In fact, we must use the subadditivity of $\tau$ and compare $\phi_n^{\kappa_n}$ with a sum of such variables $\tau$.
We consider $n$ and $N$ in $\NN$, with $N$ a lot bigger than $n$. The following definitions can seem a little bit complicated, but Figure \[chapitre6emboitement1\] is more explicit.
(0,0)![The cylinders $\operatorname{cyl}(NA,h(N))$ and $G_i$, for $i=1,...,\M$.[]{data-label="chapitre6emboitement1"}](emboitement1bis.eps "fig:")
\#1\#2\#3\#4\#5[ @font ]{}
(8295,9399)(1039,-10573) (3301,-2986)[(0,0)\[lb\]]{} (1876,-4486)[(0,0)\[rb\]]{} (3151,-2086)[(0,0)\[rb\]]{} (2701,-2911)[(0,0)\[rb\]]{} (6976,-2536)[(0,0)\[lb\]]{} (6751,-2011)[(0,0)\[rb\]]{} (7801,-7636)[(0,0)\[lb\]]{} (6976,-7561)[(0,0)\[b\]]{} (5851,-4936)[(0,0)\[lb\]]{} (5926,-4561)[(0,0)\[rb\]]{} (5926,-5311)[(0,0)\[lb\]]{} (5326,-6436)[(0,0)\[rb\]]{} (7801,-6886)[(0,0)\[rb\]]{} (5701,-3811)[(0,0)\[lb\]]{} (5251,-10486)[(0,0)\[lb\]]{} (5251,-9886)[(0,0)\[lb\]]{} (2101,-10486)[(0,0)\[lb\]]{} (2101,-9886)[(0,0)\[lb\]]{} (3601,-8311)[(0,0)\[rb\]]{} (3676,-4336)[(0,0)\[lb\]]{}
We choose two functions $h', \zeta :\NN \rightarrow \RR^+$ such that $$\lim_{n\rightarrow \infty} h'(n) \,=\, \lim_{n\rightarrow \infty}
\zeta(n) \,=\, + \infty \,,$$ and $$\label{chapitre6cond1}
\lim_{n\rightarrow \infty}\frac{h'(n)}{\zeta (n)} \,=\, 0 \,.$$ We consider a fixed $\wt \in \D_N$. Let $$\vb \,=\, (\cos \widetilde{\theta}, \sin \widetilde{\theta}) \quad
\textrm{and} \quad \vd \,=\, (\sin
\widetilde{\theta}, -\cos \widetilde{\theta}) \,.$$ In $\operatorname{cyl}(NA,h(N))$, we denote by $x_N$ and $y_N$ the two points corresponding to the boundary conditions $\kappa_N$, such that $\overrightarrow{x_Ny_N} \cdot \vd >0 $. Notice that according to our choice of $k_N$, the segments $[x_N,y_N]$ and $NA$ cut each other in their middle. If we denote by $L(N,\wt)$ the distance between $x_N$ and $y_N$, we have: $$L(N,\wt)\,=\, \frac{Nl(A)}{\cos(\widetilde{\theta}-\theta)} \,.$$ We define $$\operatorname{cyl}'(n) \,=\, \operatorname{cyl}([0 ,n \vd ],h'(n)) \,.$$ We will translate $\operatorname{cyl}'(n)$ numerous times inside $\operatorname{cyl}(NA,h(N))$. We define $$t_i \,=\, x_N + \left(\zeta(n) + (i-1) n \right) \vd \,,$$ for $i=1,..., \M$, where $$\M \,=\, \M(n,N) \,=\, \left\lfloor \frac{L(N,\wt) -2\zeta(n)}{n} \right\rfloor \,.$$ Of course we consider only $N$ large enough to have $\M \geq 2$. For $i=1,...,\M$, we denote by $\widetilde{G_i}$ the image of $\operatorname{cyl}'(n)$ by the translation of vector $\overrightarrow{0 t_i}$. For $n$ (and thus $N$) sufficiently large, thanks to condition (\[chapitre6cond1\]), we know that $\widetilde{G_i} \subset \operatorname{cyl}(NA,h(N))$ for all $i$. We can translate $\widetilde{G_i}$ again by a vector of norm strictly smaller than $1$ to obtain an integer translate of $\operatorname{cyl}'(n)$ (i.e., a translate by a vector whose coordinates are in $\ZZ^2$) that we will call $G_i$. Now we want to glue together cutsets of boundary condition $(1/2, \wt)$ in the cylinders $G_i$. We define: $$\F_1(n,N,\kappa_N) \,=\, \left(\bigcup_{i=1}^{\M} \mathcal{V} (t_i,
\zeta_0) \right) \, \bigcap \,\operatorname{cyl}(NA,h(N)) \,,$$ where $\zeta_0$ is a fixed constant larger than $4$, and: $$\F_2(n,N,\kappa_N) \,=\, \mathcal{V} \left( [x_N,x_N + \zeta(n) \vd ]
\cup [z_\M , y_N ] ,\zeta_0 \right)\, \bigcap \,\operatorname{cyl}(NA,h(N)) \,.$$ Let $F_1(n,N,\kappa_N)$ (respectively $F_2(n,N,\kappa_N)$) be the set of the edges included in $\F_1(n,N,\kappa_N)$ (respectively $\F_2(n,N,\kappa_N)$). If for every $i=1,...,\M$, $\G_i$ is a cutset of boundary condition $(1/2,\wt)$ in $G_i$, then $$\bigcup_{i=1}^{\M} \G_i \cup F_1(n,N,\kappa_N) \cup F_2(n,N,\kappa_N)$$ contains a cutset of boundary conditions $\kappa_N$ in $\operatorname{cyl}(NA,h(N))$. We obtain: $$\label{chapitre6lien_phiF_tau_1} \phi_N^{\kappa_N} \,\leq\, \sum_{i=1}^{\M} \tau(G_i,\vb ) +
V(F_1(n,N,\kappa_N) \cup F_2(n,N,\kappa_N) ) \,,$$ and so, $$\label{chapitre6lien_phi_tau} \forall \widetilde{\theta}\in \D_N \qquad \phi_N \,\leq\,
\phi_{N}^{\kappa_N} \,\leq\, \sum_{i=1}^{\M}\tau(G_i, \vb ) + V( F_1(n,N,\kappa_N) \cup F_2(n,N,\kappa_N) ) \,.$$ There exists a constant $C_5$ such that: $$\operatorname{card}( F_1(n,N,\kappa_N) ) \,\leq\, C_5
\M \quad \textrm{and} \quad \operatorname{card}(F_2(n,N,\kappa_N))\,\leq\, C_5 \left( \zeta(n) + n \right) \,,$$ and since the set of edges $F_1(n,N,\kappa_N) \cup F_2(n,N,\kappa_N)$ is deterministic, $$\EE [V(F_1(n,N,\kappa_N) \cup F_2(n,N,\kappa_N))] \,\leq\, C_5 \EE(t)\left(
\M + \zeta(n) + n \right) \,.$$ So $$\label{eqNn}
\forall \widetilde{\theta}\in \D_N \qquad
\frac{\EE(\phi_N)}{Nl(A)} \,\leq\, \frac{\M n}{N l(A)}
\times \frac{\EE [\tau(\operatorname{cyl}'(n), \vb )]}{n}
+ \frac{C_5 \EE(t)\left( \M + \zeta(n) + n \right) }{Nl(A)} \,.$$ We want to send $N$ to infinity. First, let $\wt \in \underline{\D}$. Then for all $N$ large enough, $\wt \in \D_N$, and thus for all $n$ large enough we have $$\limsup_{N\rightarrow \infty}\frac{\EE(\phi_N)}{Nl(A)} \,\leq\,
\frac{1}{\cos(\wt - \theta)} \frac{\EE [\tau(\operatorname{cyl}'(n), \vb )]}{n} +
\frac{C_5 \EE(t)}{n \cos(\wt - \theta)}\,.$$ Sending $n$ to infinity, thanks to Theorem \[chapitre6thm:LGNtau\], we obtain that $$\label{chapitre6upper1}
\limsup_{N\rightarrow \infty } \frac{\EE(\phi_N)}{Nl(A)}
\,\leq\, \inf_{\widetilde{\theta} \in \underline{\D}}
\frac{\nu_{\widetilde{\theta}}}{\cos (\widetilde{\theta}-\theta)} \,.$$ We now suppose that $\wt \in \overline{\D}$. Let $\psi:\NN \rightarrow
\NN$ be strictly increasing and such that for all $N$, $\wt \in \D_{\psi(N)}$. Then thanks to Equation (\[eqNn\]), sending first $N$ to infinity and then $n$ to infinity, we obtain that $$\label{chapitre6eq:upper1overline}
\liminf_{N\rightarrow \infty } \frac{\EE(\phi_N)}{Nl(A)}
\,\leq\, \limsup_{N\rightarrow \infty } \frac{\EE(\phi_{\psi(N)})}{\psi(N)l(A)}
\,\leq\, \inf_{\widetilde{\theta} \in \overline{\D}}
\frac{\nu_{\widetilde{\theta}}}{\cos (\widetilde{\theta}-\theta)} \,.$$
Lower bound {#chapitre6subsec:lowerbound}
-----------
We do the symmetric construction of the one done in section \[chapitre6subsec:upperbound\]. We consider $n$ and $N$ in $\NN$ and take $N$ a lot bigger than $n$. We choose functions $\zeta', h'': \NN \rightarrow \RR^+$ such that $$\lim_{n\rightarrow \infty} \zeta'(n) \,=\, \lim_{n\rightarrow \infty}
h''(n) \,=\, +\infty \,,$$ and $$\label{chapitre6cond2}
\lim_{n\rightarrow \infty}\frac{ h(n)}{\zeta' (n)} \,=\,0 \,.$$ We consider $\kappa=(k,\widetilde{\theta})\in D_n$. Keeping the same notations as in section \[chapitre6subsec:upperbound\], we define $$\operatorname{cyl}''(N) \,=\, \operatorname{cyl}\left([0,N \vd ] ,
h''(N)\right) \,.$$ We will translate $\operatorname{cyl}(nA, h(n))$ numerous times in $\operatorname{cyl}''(N)$. The figure \[chapitre6emboitement2\] is more explicit than the following definitions.
(0,0)![The cylinders $\operatorname{cyl}''(N)$ and $B_i$, for $i=1,...,\N$.[]{data-label="chapitre6emboitement2"}](emboitement2these.eps "fig:")
\#1\#2\#3\#4\#5[ @font ]{}
(9540,7965)(1168,-8539) (6301,-6226)[(0,0)\[lb\]]{} (10156,-2146)[(0,0)\[lb\]]{} (1456,-5236)[(0,0)\[rb\]]{} (2206,-4621)[(0,0)\[rb\]]{} (3451,-8461)[(0,0)\[lb\]]{} (7651,-7861)[(0,0)\[lb\]]{} (9301,-3286)[(0,0)\[lb\]]{} (7651,-8461)[(0,0)\[lb\]]{} (3451,-7861)[(0,0)\[lb\]]{} (5221,-1351)[(0,0)\[lb\]]{} (5551,-2836)[(0,0)\[rb\]]{} (5326,-1636)[(0,0)\[lb\]]{} (6076,-1936)[(0,0)\[lb\]]{} (5881,-3031)[(0,0)\[lb\]]{} (5026,-4636)[(0,0)\[lb\]]{} (5176,-5386)[(0,0)\[rb\]]{}
The condition $\kappa$ defines two points $x_n$ and $y_n$ on the boundary of $\operatorname{cyl}(nA, h(n))$ (see section \[chapitre6subsec:upperbound\]). As in section \[chapitre6subsec:upperbound\], we denote by $L(n,\wt)$ the distance between $x_n$ and $y_n$, and we have $$L(n,\wt) \,=\, \frac{nl(A)}{cos(\widetilde{\theta}-\theta)}\,.$$ We define $$z_i \,=\, \left(\zeta'(n) + (i-1) L(n,\wt)\right) \vd \,,$$ for $i=1,..., \N$, where $$\N \,=\, \left\lfloor \frac{N-2\zeta'(n)}{L(n,\wt)} \right\rfloor \,.$$ Of course we consider only $N$ large enough to have $\N \geq 2$. For $i=1,...,\N$, we denote by $\widetilde{B_i}$ the image of $\operatorname{cyl}(nA, h(n))$ by the translation of vector $\overrightarrow{x_n z_i}$. For $N$ sufficiently large, thanks to condition (\[chapitre6cond2\]), we know that $\widetilde{B_i} \subset \operatorname{cyl}''(N)$ for all $i$. We can translate $\widetilde{B_i}$ again by a vector of norm strictly smaller than $1$ to obtain an integer translate of $\operatorname{cyl}(nA,
h(n))$ (i.e., a translate by a vector whose coordinates are in $\ZZ^2$) that we will call $B_i$. Now we want to glue together cutsets of boundary condition $\kappa$ in the different $B_i$’s. We define: $$\E_1(n,N,\kappa) \,=\, \left(\bigcup_{i=1}^\N \mathcal{V} (z_i, \zeta) \right)
\, \bigcap \,\operatorname{cyl}''(N) \,,$$ where $\zeta$ is still a fixed constant bigger than $4$, and: $$\E_2(n,N,\kappa) \,=\, \mathcal{V} \left( [0,\zeta'(n)
\vd ] \cup [z_\N, N \vd ]
,\zeta \right)\, \bigcap \,\operatorname{cyl}''(N) \,.$$ Let $E_1(n,N,\kappa)$ (respectively $E_2(n,N,\kappa)$) be the set of the edges included in $\E_1(n,N,\kappa)$ (respectively $\E_2(n,N,\kappa)$). Then, still by gluing cutsets together, we obtain: $$\label{chapitre6lien_phiF_tau_2}
\tau(\operatorname{cyl}''(N), \vb ) \,\leq\, \sum_{i=1}^\N
\phi^{\kappa}(B_i, \va) + V(E_1(n,N,\kappa) \cup E_2(n,N,\kappa)) \,.$$ On one hand, there exists a constant $C_6$ (independent of $\kappa$) such that: $$\operatorname{card}(E_1(n,N,\kappa) \cup E_2(n,N,\kappa)) \,\leq\, C_6 \left( \N + \zeta'(n) +
L(n,\wt) \right) \,,$$ and since the sets $E_1(n,N,\kappa)$ and $E_2(n,N,\kappa)$ are deterministic, we deduce: $$\EE [ V(E_1(n,N,\kappa) \cup E_2(n,N,\kappa))] \,\leq\, C_6 \EE(t) \left( \N + \zeta'(n) +
L(n,\wt) \right)\,.$$ On the other hand, the variables $(\phi^{\kappa}(B_i))_{i=1,...,\N}$ are identically distributed, with the same law as $\phi_{n}^\kappa$ (because we only consider integer translates), so (\[chapitre6lien\_phiF\_tau\_2\]) leads to $$\EE [\tau(\operatorname{cyl}''(N), \vb )] \,\leq\, \N \EE
[\phi_{n}^\kappa] + C_6 \EE(t) \left( \N + \zeta'(n) + L(n,\wt) \right)\,.$$ Dividing by $N$ and sending $N$ to infinity, we get, thanks to Theorem \[chapitre6thm:LGNtau\]: $$\nu_{\widetilde{\theta}} \,\leq\, \frac{\EE[\phi_{n}^\kappa]}{L(n,\wt)} +
\frac{C_6 \EE(t)}{ L(n,\wt)}\,,$$ and so: $$\frac{\EE [\phi_{n}^\kappa]}{nl(A)} \,\geq\,
\frac{\nu_{\widetilde{\theta}}}{\cos (\widetilde{\theta} - \theta)} -
\frac{C_6 \EE(t)}{nl(A)} \,.$$ Since $C_6$ is independent of $\kappa$, $$\inf_{\kappa\in D_n} \frac{\EE [\phi_{n}^\kappa]}{nl(A)} \,\geq\,
\inf_{\widetilde{\theta} \in \D_n} \frac{\nu_{\widetilde{\theta}}}{\cos
(\widetilde{\theta} - \theta)} - \frac{C_6 \EE(t)}{nl(A)} \,.$$ First, we affirm: $$\label{chapitre6liminf_inf}
\liminf_{n\rightarrow \infty}
\inf_{\widetilde{\theta} \in \D_n} \frac{\nu_{\widetilde{\theta}}}{\cos
(\widetilde{\theta} - \theta)} \,\geq\, \inf_{\widetilde{\theta} \in
\overline{\D}}\frac{\nu_{\widetilde{\theta}}}{\cos
(\widetilde{\theta} - \theta)}\,,$$ and thus: $$\label{chapitre6lower1}
\liminf_{n\rightarrow \infty} \inf_{\kappa\in D_n} \frac{\EE
[\phi_{n}^\kappa]}{nl(A)} \,\geq\, \inf_{\widetilde{\theta} \in
\overline{\D}}\frac{\nu_{\widetilde{\theta}}}{\cos
(\widetilde{\theta} - \theta)}\,.$$ We also claim that: $$\label{chapitre6limsup_inf}
\limsup_{n\rightarrow \infty}
\inf_{\widetilde{\theta} \in \D_n} \frac{\nu_{\widetilde{\theta}}}{\cos
(\widetilde{\theta} - \theta)} \,\geq\, \inf_{\widetilde{\theta} \in
\underline{\D}}\frac{\nu_{\widetilde{\theta}}}{\cos
(\widetilde{\theta} - \theta)}\,,$$ and therefore: $$\label{chapitre6eq:lower1underline}
\limsup_{n\rightarrow \infty} \inf_{\kappa\in D_n} \frac{\EE
[\phi_{n}^\kappa]}{nl(A)} \,\geq\, \inf_{\widetilde{\theta} \in
\underline{\D}}\frac{\nu_{\widetilde{\theta}}}{\cos
(\widetilde{\theta} - \theta)}\,.$$
Let us prove Inequality (\[chapitre6liminf\_inf\]). In fact, we will state a more general result:
\[chapitre6inversion\] Let $\theta \in [0,\pi[$, and $f$ be a lower semi-continuous function from $[\theta-\pi/2, \theta+\pi/2]$ to $\RR^+ \cup \{+\infty\}$. Then we have $$\liminf_{n\rightarrow \infty} \inf_{\wt \in \D_n} f(\wt) \,\geq\,
\inf_{\wt \in \\ad (\overline{\D})} f(\wt) \,,$$ where $\operatorname{ad}(\overline{\D})$ is the adherence of $\overline{\D}$.
We consider a positive $\varepsilon$. For all $n$, since $f$ is lower semi-continuous and $\D_n$ is compact, there exists $\wt_n \in \D_n$ such that $f (\wt_n) = \inf_{\wt \in \D_n} f(\wt) $. Up to extracting a subsequence, we can suppose that the sequence $(\inf_{\wt \in
\D_n}f (\wt))_{n\geq 0}$ converges towards $\displaystyle{\liminf_{n\rightarrow \infty} \inf_{\wt\in \D_n}
f(\wt)}$, and so: $$\lim_{n\rightarrow \infty} f( \wt_n ) \,=\,
\liminf_{n\rightarrow \infty} \inf_{\wt\in \D_n} f(\wt) \,.$$ The sequence $(\wt_n)_{n\geq 0}$ (in fact the previous subsequence) takes values in the compact $[\theta-\pi/2, \theta + \pi/2]$, so up to extracting a second subsequence we can suppose that $(\wt_n)_{n\geq 0}$ converges towards a limit $\wt_\infty$ in this compact. Since $f$ is lower semi-continuous, $$f(\wt_\infty) \,\leq\, \lim_{n\rightarrow \infty}f (\wt_n) \,=\,
\liminf_{n\rightarrow \infty} \inf_{\wt\in \D_n} f(\wt) \,,$$ and we just have to prove that $\wt_\infty$ belongs to $\operatorname{ad}(\overline{\D})$. Indeed, for all positive $\varepsilon$, $\wt_n
\in [\wt_\infty - \varepsilon, \wt_\infty + \varepsilon ]$ for an infinite number of $n$. We remember that all the $\D_n$ are closed intervals centered at $\theta$. If $\wt_\infty = \theta$, the result is obvious. We suppose that $\wt_\infty >\theta$ for example, and thus, for $\varepsilon$ small enough, $\wt_\infty - \varepsilon >\theta$. Then $[\theta, \wt_\infty -
\varepsilon]$ is included in an infinite number of $\D_n$, so $\wt_\infty -
\varepsilon$ belongs to $\overline{\D}$, and then $\wt_\infty$ belongs to $\operatorname{ad}(\overline{\D})$. The same holds if $\wt_\infty < \theta$. This ends the proof of Lemma \[chapitre6inversion\].
We use Lemma \[chapitre6inversion\] with $f(\wt) =
\nu_{\widetilde{\theta}} / \cos (\widetilde{\theta} - \theta)$. Here $f$ is lower semi-continuous, because $\wt \rightarrow \nu_{\wt}$ is continuous since it satisfies the weak triangle inequality. Indeed, it is obvious in dimension $2$ because $\nu_{\wt} = \mu(\vb)$ which satisfies the (ordinary) triangle inequality, but it has also been proved in any dimension $d\geq 2$ (see section 4.4 in [@RossignolTheret08b]). Moreover we know that $f$ is finite and continuous on $]\theta - \pi/2, \theta + \pi/2[$, infinite at $\theta + \pi/2$ and $\theta - \pi/2$ and $$\lim_{\wt \rightarrow \theta+\pi/2} f(\wt) \,=\,
\lim_{\wt \rightarrow \theta-\pi/2} f(\wt) \,=\, +\infty \,,$$ so we can even say in this case: $$\inf_{\wt \in \operatorname{ad}(\overline{\D})} f(\wt) = \inf_{\wt \in \overline{\D}}
f(\wt) \,,$$ and we obtain Inequality (\[chapitre6liminf\_inf\]).
Let us now prove Inequality (\[chapitre6limsup\_inf\]). We state again a more general result:
\[chapitre6inversion-bis\] Let $\theta \in [0,\pi[$, and $f$ be a lower semi-continuous function from $[\theta - \pi/2, \theta+\pi/2]$ to $\RR^+ \cup \{+\infty\}$. Then we have $$\limsup_{n\rightarrow \infty} \inf_{\wt \in \D_n} f(\wt) \,\geq\,
\inf_{\wt \in \operatorname{ad}(\underline{\D})} f(\wt) \,,$$ where $\operatorname{ad}(\underline{\D}) $ is the adherence of $\underline{\D}$.
We denote $\operatorname{ad}(\underline{\D})$ by $[\theta -\alpha, \theta + \alpha]$. For all integer $p\geq 1$, there exists $n_p\geq
n_{p-1}$ ($n_0 = 1$) such that: $$\theta + \alpha + 1/p \,\notin\, \D_{n_p} \qquad \mbox{and} \qquad \theta - \alpha -
1/p \,\notin\, \D_{n_p} \,,$$ thus $$\D_{n_p} \,\subset \, ]\theta - \alpha - 1/p , \theta + \alpha + 1/p[
\,,$$ then $$\begin{aligned}
\limsup_{n\rightarrow \infty} \inf_{\wt \in \D_n} f(\wt) & \,\geq\,
\limsup_{p\rightarrow \infty} \inf_{\wt \in \D_{n_p}} f(\wt) \\
& \,\geq\, \limsup_{p\rightarrow \infty} \inf_{\wt \in [\theta - \alpha -
1/p, \theta + \alpha + 1/p]} f(\wt) \,.\end{aligned}$$ The function $f$ is lower semi-continuous and $[\theta - \alpha - 1/p, \theta + \alpha +
1/p]$ is compact, so for all integers $p$ there exists $\wt_p \in [\theta - \alpha -
1/p, \theta + \alpha + 1/p]$ such that $f(\wt_p) = \inf_{\wt \in [\theta
- \alpha - 1/p, \theta + \alpha + 1/p]} f(\wt)$. Up to extraction, we can suppose that $(\wt_p)_{p\geq 1}$ converges towards a limit $\wt_\infty$, that belongs obviously to $[\theta-\alpha, \theta +
\alpha]$. Finally, because $f$ is lower semi-continuous, $$\inf_{\wt \in [\theta-\alpha, \theta + \alpha]} f(\wt) \,\leq\,
f(\wt_\infty) \,\leq\, \limsup_{p\rightarrow \infty} f(\wt_p) \,\leq\,
\limsup_{n\rightarrow \infty} \inf_{\wt \in \D_n} f(\wt) \,,$$ so Lemma \[chapitre6inversion-bis\] is proved.
As previously, we use Lemma \[chapitre6inversion-bis\] with $f(\wt) =
\nu_{\widetilde{\theta}} / \cos (\widetilde{\theta} - \theta)$. Again, we have: $$\inf_{\wt \in \operatorname{ad}(\underline{\D})} f(\wt) = \inf_{\wt \in \underline{\D}}
f(\wt) \,,$$ and Equation (\[chapitre6limsup\_inf\]) is proved.
End of the study of the mean {#chapitre6subsec:endLGN}
----------------------------
Now, we are able to conclude the proof of (\[eq:liminfmoyenne\]) and (\[eq:limsupmoyenne\]). First, we show that $\EE(\phi_n)$ and $\min_\kappa\EE(\phi_n^\kappa)$ are of the same order.
\[chapitre6lem:infmoyenne\] Let $A$ be a line segment in $\RR^2$. Suppose that conditions (\[chapitre6eq:conditionshn\]) and (\[chapitre6eq:conditionsFLGN\]) are satisfied. Then, $$\lim_{n\rightarrow\infty}\frac{\EE(\phi_{n})}{\min_{\kappa\in
D_n}\EE(\phi_{n}^\kappa)}=1\;.$$
Notice that $\EE(\phi_{n})\leq \min_{\kappa\in
D_n}\EE(\phi_{n}^\kappa)$, and thus it is sufficient to show that: $$\liminf_{n\rightarrow\infty}\frac{\EE(\phi_{n})}{\min_{\kappa\in
D_n}\EE(\phi_{n}^\kappa)}\geq 1\;.$$ Recall from (\[chapitre6eq:Oh2\]) and Lemma \[chapitre6lem:duality\] that there is a finite subset $\tilde D_n$ of $D_n$, such that: $$\operatorname{card}(\tilde D_n) \leq C_4 h(n)^2\;,$$ for some constant $C_4$ and every $n$, and $$\label{chapitre6eqphitau}\phi_{n}= \min_{\kappa\in \tilde D_n}\phi^\kappa_{n} \;.$$ Thus, for $\eta$ in $]0,1[$, $$\begin{aligned}
\PP(\min_{\kappa\in \tilde D_n}\phi^\kappa_n\geq \min_{\kappa\in \tilde
D_n}\EE(\phi^\kappa_n)(1-\eta))&=&1-\PP(\exists \kappa\in \tilde D_n,\;\phi^\kappa_n<
\min_{\kappa\in \tilde D_n}\EE(\phi^\kappa_n)(1-\eta))\;,\\
&\geq &1- |\tilde D_n|\max_{\kappa\in \tilde D_n}\PP(\phi^\kappa_n <
\min_{\kappa\in \tilde D_n}\EE(\phi^\kappa_n)(1-\eta))\;,\\
&\geq &1- C_4 h(n)^2\max_{\kappa\in \tilde D_n}\PP(\phi^\kappa_n < \EE(\phi^\kappa_n)(1-\eta))\;.\end{aligned}$$ Now, Proposition \[prop:deviation\] implies that for $\eta$ in $]0,1[$, $$\begin{aligned}
\PP(\min_{\kappa\in D_n}\phi^\kappa_n\geq \min_{\kappa\in
D_n}\EE(\phi^\kappa_n)(1-\eta))&\geq &1- C_4
K_1h(n)^2e^{-C(\eta,F)\min_{\kappa\in D_n}\EE(\phi^\kappa_n)}\;,\end{aligned}$$ where $C(\eta,F)$ is strictly positive. Now, let $\eta_0$ be fixed in $]0,1/2[$. $$\begin{aligned}
\EE(\min_{\kappa\in D_n}\phi_n^\kappa)&=&\int_0^{+\infty}\PP(\min_{\kappa\in D_n}\phi_n^\kappa
\geq t)\;dt\;,\\
&\geq &\int_0^{\min_{\kappa\in D_n}\EE(\phi_n^\kappa)}\PP\left(\min_{\kappa\in D_n}\phi_n^\kappa \geq \min_{\kappa\in D_n}\EE(\phi_n^\kappa)-u\right)\;du\;,\\
&\geq & \min_{\kappa\in D_n}\EE(\phi_n^\kappa)\int_{\eta_0}^{(1-\eta_0)}\PP\left(\min_{\kappa\in
D_n}\phi_n^\kappa\geq \min_{\kappa\in D_n}\EE(\phi_n^\kappa)(1-\eta)\right)\;d\eta\;,\\
&\geq & \min_{\kappa\in D_n}\EE(\phi_n^\kappa)(1-2\eta_0)\left(1-C_4
K_1h(n)^2e^{-C(1-\eta_0,F)\min_{\kappa\in D_n}\EE(\phi^\kappa_n)}\right)\;.\end{aligned}$$ Thanks to Inequality (\[chapitre6lower1\]), we know that there is a strictly positive constant $C(A)$ such that: $$\liminf_{n\rightarrow\infty}\frac{\min_{\kappa \in \tilde
D_n }\EE(\phi_n^\kappa)}{n}\geq C(A)\;.$$ Thus, using assumption (\[chapitre6eq:conditionshn\]), namely the fact that $\log h(n)$ is small compared to $n$, $$\liminf_{n\rightarrow\infty}\frac{\EE(\phi_{n})}{\min_{\kappa\in
D_n}\EE(\phi_{n}^\kappa)}\geq 1-2\eta_0\;.$$ Since this is true for any $\eta_0\in ]0,1/2[$, this finishes the proof of Lemma \[chapitre6lem:infmoyenne\].
Now, inequalities (\[chapitre6upper1\]), (\[chapitre6eq:lower1underline\]) and Lemma \[chapitre6lem:infmoyenne\] give: $$\label{cclesp1}
\limsup_{n\rightarrow \infty} \frac{\EE [\phi_n] }{nl(A)} \,=\,
\inf_{\wt \in \underline{\D}} \frac{\nu_{\wt}}{ \cos(\wt - \theta)}$$ which is (\[eq:limsupmoyenne\]). Similarly, inequalities (\[chapitre6eq:upper1overline\]), (\[chapitre6lower1\]) and Lemma \[chapitre6lem:infmoyenne\] give: $$\label{cclesp2}
\liminf_{n\rightarrow \infty} \frac{\EE [\phi_n] }{nl(A)} \,=\,
\inf_{\wt \in \overline{\D}} \frac{\nu_{\wt}}{ \cos(\wt - \theta)} \,.$$ which is (\[eq:liminfmoyenne\]).
Proof of the law of large numbers {#secLLN}
=================================
Using Borel-Cantelli’s Lemma and Proposition \[prop:deviation\], we obtain that $$\liminf_{n\rightarrow \infty} \frac{\phi_n - \EE[\phi_n]}{nl(A)}
\,\geq\,0 \,,$$ and thus, using Equations (\[cclesp1\]) and (\[cclesp2\]), that $$\label{ajout1}
\liminf_{n\rightarrow \infty} \frac{\phi_n}{nl(A)} \,\geq\, \liminf_{n\rightarrow \infty} \frac{\EE [\phi_n] }{nl(A)} \,=\,
\inf_{\wt \in \overline{\D}} \frac{\nu_{\wt}}{ \cos(\wt - \theta)}$$ and $$\label{ajout2}
\limsup_{n\rightarrow \infty} \frac{\phi_n}{nl(A)} \,\geq\, \limsup_{n\rightarrow \infty} \frac{\EE [\phi_n] }{nl(A)} \,=\,
\inf_{\wt \in \underline{\D}} \frac{\nu_{\wt}}{ \cos(\wt - \theta)}\,.$$ It can seem a bit strange to bound $\displaystyle{\limsup_{n\rightarrow
\infty} \phi_n / (nl(A))}$ from below in the study of the asymptotic behavior of $\phi_n$. The reason is the following: we do not only want to prove the convergence of the rescaled flow $\phi_n$ in some cases, we want to obtain a necessary and sufficient condition for this convergence to hold. Thus we need to know exactly the values of $\displaystyle{\limsup_{n\rightarrow \infty} \phi_n / (nl(A))}$ and $\displaystyle{\liminf_{n\rightarrow \infty} \phi_n / (nl(A))}$. We will prove the converse of Inequalities (\[ajout1\]) and (\[ajout2\]). For that purpose we use again the geometrical construction performed in section \[chapitre6subsec:upperbound\]. Suppose only for the moment that $$\int_{[0,+\infty[} x \,dF(x) \,<\,\infty\,.$$ Let $\wt_1 \in \underline{\D}$ be such that $$\frac{\nu_{\wt_1}}{\cos (\wt_1 - \theta)} \,=\, \inf_{\wt \in
\underline{\D}} \frac{\nu_{\wt}}{\cos(\wt - \theta)} \,.$$ Such a $\wt_1$ exists, since $$\inf_{\wt \in \underline{\D}} \frac{\nu_{\wt}}{\cos(\wt - \theta)}
\,=\, \inf_{\wt \in \operatorname{ad}(\underline{\D}) } \frac{\nu_{\wt}}{\cos(\wt - \theta)}$$ as stated in section \[chapitre6subsec:lowerbound\], $ \operatorname{ad}(\underline{\D})$ is compact and the function $\wt \mapsto \nu_{\wt} / \cos
(\wt - \theta)$ is lower semi-continuous. For all $N$ large enough, $\wt_1
\in \D_N$, and we only consider such large $N$. First suppose that $0$, the origin of the graph, is the middle of $A$. Then consider $\kappa_N = (k_N,
\wt_1)$ as defined in section \[chapitre6subsec:upperbound\]. We performed the geometrical construction of section \[chapitre6subsec:upperbound\]: we consider several integer translates $G_i$, for $i=1,...,\M(n,N)$, of $\operatorname{cyl}'(n)$ inside $\operatorname{cyl}(NA, h(N))$. Since $0$ belongs to $[x_N, y_N]$, we can construct the cylinders $G_i$ and the sets of edges $F_1(n,N,\kappa_N)$ and $F_2(n,N,\kappa_N)$ in such a way that $$\forall N_1 \leq N_2 \quad (G_i)_{i=1,...,\M(n,N_1)} \,\subset\, (G_i)_{i=1,..., \M(n,N_2)} \quad \textrm{and}\quad F_1(n,N_1,\kappa_{N_1}) \,\subset\,
F_1(n,N_2,\kappa_{N_2}) \,.$$ We use again Inequality (\[chapitre6lien\_phi\_tau\]) to obtain that: $$\label{eqfin}
\frac{\phi_N}{Nl(A)} \,\leq\, \frac{n \M}{Nl(A)} \frac{1}{\M} \sum_{i=1}^{\M}
\frac{\tau(G_i,\vec{v} (\wt_1))}{n} + \frac{V(F_1(n,N,\kappa_N) )}{Nl(A)} +
\frac{V( F_2(n,N,\kappa_N))}{Nl(A)} \,.$$ The variables $ (\tau(G_i,\wt_1),i=1,...,\M(n,N))$ are not independent. However, each cylinder $G_i$ can intersect at most the two other cylinders that are its neighbours, thus we can divide the family $ (\tau(G_i,\wt_1),i=1,...,\M(n,N))$ into two families $
(\tau(G_{i},\wt_1),i\in\{1,...,\M(n,N)\} \cap P_j)$ for $j=1,2$, $P_1
= 2\NN$ and $P_2 = 2\NN +1$, such that for each $j\in \{1,2\}$, the family $
(\tau(G_{i},\wt_1),i\in\{1,...,\M(n,N)\} \cap P_j)$ is i.i.d. Since $$\int_{[0,+\infty[} x \,dF(x) \,<\,\infty \,,$$ it is easy to see that the variable $\tau(\operatorname{cyl}'(n), \wt_1)$ is integrable (we can compare this variable with the capacity of a deterministic cutset), and we can apply the strong law of large numbers to each of the two families of variables described above to finally obtain that $$\label{eqfin1}
\lim_{N\rightarrow \infty} \frac{n \M}{Nl(A)} \frac{1}{\M} \sum_{i=1}^{\M}
\frac{\tau(G_i,\vec{v} (\wt_1))}{n} \,=\, \frac{1}{\cos(\wt_1 - \theta)} \frac{\EE[\tau(\operatorname{cyl}'(n) ,
\wt_1)]}{n} \qquad \textrm{a.s.}$$ Up to increasing a little the sets $F_1(n,N,\kappa_N)$, we can suppose that for all $N$, we have $$\operatorname{card}(F_1(n,N,\kappa_N)) \,=\, C_5 \M(n,N)\,,$$ and thus, by the strong law of large numbers, we obtain that $$\label{eqfin2}
\lim_{N \rightarrow \infty} \frac{V(F_1(n,N,\kappa_N) )}{Nl(A)} \,=\,
\frac{C_5 \EE[t(e)]}{n \cos(\wt_1 - \theta)}\qquad \textrm{a.s.}$$ Moreover, we know that $$\operatorname{card}(F_2 (n,N,\kappa_N)) \,\leq\, C_5 (n+\zeta(n))\,,$$ thus for all $\eta > 0$ we have $$\begin{aligned}
\sum_{N\in \NN^*} \PP[V(F_2(n,N,\kappa_N))\geq \eta N l(A)]
& \,\leq\,\sum_{N\in \NN^*} \PP \left[ \sum_{i=1}^{C_5 (n+\zeta(n))} t_i \geq
\eta N l(A) \right] \\
&\,\leq\, \EE\left[1+\frac{1}{\eta l(A)}\sum_{i=1}^{C_5 (n+\zeta(n))} t_i\right]<\infty\;.
$$ where $(t_i, i\in \NN)$ is a family of i.i.d. variables with distribution function $F$. By a simple Borel-Cantelli’s Lemma, we conclude that $$\label{eqfin3}
\lim_{N \rightarrow \infty} \frac{V(F_2(n,N,\kappa_N) )}{Nl(A)} \,=\,
0 \qquad \textrm{a.s.}$$ Combining Equations (\[eqfin\]), (\[eqfin1\]), (\[eqfin2\]) and (\[eqfin3\]), and sending $n$ to infinity, thanks to Theorem \[chapitre6thm:LGNtau\] we obtain that $$\limsup_{N\rightarrow \infty} \frac{\phi_N}{Nl(A)} \,\leq\,
\frac{\nu_{\wt_1}}{\cos(\wt_1 - \theta)} \,=\, \inf_{\wt\in
\underline{\D}} \frac{\nu_{\wt}}{ \cos(\wt - \theta)}\qquad \textrm{a.s.}$$ Similarly, we can choose $\wt_2 \in \overline{\D}$ satisfying $$\frac{\nu_{\wt_2}}{\cos(\wt_2 - \theta)} \,=\,\inf_{\wt \in
\overline{\D}} \frac{\nu_{\wt}}{ \cos(\wt - \theta)} \,.$$ We consider a subsequence $(\psi(N), N\in \NN)$ of $\NN$ such that for all $N$, $\wt_2 \in \D_{\psi(N)}$. If $0$ is the middle of $A$, for every $N$ we consider $k_{\psi(N)}$ as defined in section \[chapitre6subsec:upperbound\], and which is such that $\kappa_{\psi(N)} =(k_{\psi(N)} , \wt_2) \in D_{\psi(N)}$ and $0$ belongs to the segments $[x_N, y_N]$ determined by the boundary condition $\kappa_N$. Then we obtain exactly by the same methods that $$\liminf_{N\rightarrow \infty} \frac{\phi_N}{Nl(A)}\,\leq\,\limsup_{N\rightarrow \infty} \frac{\phi_{\psi(N)}}{\psi(N)l(A)} \,\leq\,
\frac{\nu_{\wt_2}}{\cos(\wt_2 - \theta)} \,=\, \inf_{\wt\in
\overline{\D}} \frac{\nu_{\wt}}{ \cos(\wt - \theta)}\qquad \textrm{a.s.}$$ If the condition on the origin $0$ of the graph is not satisfied, we suppose that $$\int_{[0,+\infty[} x^2 \,dF(x) \,<\,\infty \,.$$ To obtain Equations (\[eqfin1\]) and (\[eqfin2\]) in the case where $0$ is the middle of $A$, we have used the strong law of large numbers. If $0$ is not the middle of $A$ we may not construct the cylinders $(G_i,i\in
\{1,...,\M(n,N)\})$ such that the same $G_i$’s appear for different $N$. Thus we obtain cylinders $(G_i(N), i\in \{1,...,\M(n,N)\})$ that depend on $N$. The sets $(\tau(G_i(N),
\wt_1), i\in \{ 1,...,\M(n,N) \}\cap P_j)$ (resp. $(t(e), e\in
F_1(n,N,\kappa_N))$) are families of i.i.d. random variables for a given $N$, and $\tau(G_i(N))$ (resp. $t(e)$) has the same law whatever the value of $i$ and $N$ (resp. whatever $e$ and $N$), but we are not in the conditions of application of the strong law of large numbers: we consider the behavior of a sequence of the form $$\left( \frac{\sum_{i=1}^{n} X_i^{(n)}}{n}\,,\,\, n\in \NN \right) \,,$$ where $(X_i^{(j)})_{i,j}$ is an array of i.d. random variables such that for each $n$, the variables $(X_1^{(n)},...,X_n^{(n)})$ are independent. Thanks to Theorem 3 in [@HsuRobbins47], we know that such a sequence converges a.s. towards $\EE(X_1^{(1)})$ as soon as $\EE[(X_1^{(1)})^2] <\infty$. This theorem is based on a result of complete convergence (see Theorem 1 in [@HsuRobbins47]) and a Borel-Cantelli’s Lemma. If $t(e)$ admits a moment of order $2$, the same holds for $\tau(G_i(N),\wt_1)$, thus we can use Theorem 3 in [@HsuRobbins47] to get Equations (\[eqfin1\]) and (\[eqfin2\]) again. This ends the proof of Theorem \[chapitre6thm:lgn\].
Obviously, the condition $$\label{chapitre6CS}
\inf_{\widetilde{\theta} \in \underline{\D}} \frac{\nu_{\widetilde{\theta}}}{\cos
(\widetilde{\theta} - \theta)} \,=\, \inf_{\widetilde{\theta} \in
\overline{\D}} \frac{\nu_{\widetilde{\theta}}}{\cos (\widetilde{\theta} -
\theta)} \,:=\, \eta_{\theta,h} \,,$$ necessary and sufficient for the convergence a.s. of $(\phi_n /
(nl(A)))_{n\geq 0}$, is closely linked to the asymptotic behaviour of $h(n)/n$. Indeed we know that $$\D_n \,=\, [\theta - \alpha_n, \theta +\alpha_n] \,,$$ where $\alpha_n = \arctan \left( \frac{2h(n)}{nl(A)} \right)$. If $\lim_{n\rightarrow \infty} 2h(n)/(nl(A))$ exists in $\RR^+ \cup \{+\infty \}$, and we denote it by $\tan \alpha$ ($\alpha \in [0,\pi/2]$), then $\underline{\D}$ and $\overline{\D}$ are equal to $[\theta - \alpha, \theta + \alpha]$ or $]\theta - \alpha,
\theta + \alpha[$, and we obtain that $\eta_{\theta,h}$ exists and $$\eta_{\theta,h} \,=\, \inf_{\widetilde{\theta} \in [\theta - \alpha,
\theta + \alpha]} \frac{\nu_{\widetilde{\theta}}}{\cos (\widetilde{\theta} -
\theta)} \,.$$ As previously, we do not care keeping $\theta + \alpha$ and $\theta -
\alpha$ in the infimum. Then we obtain the a.s. convergence appearing in Corollary \[chapitre6corollaire\]. Obviously, if there exists a $\widetilde{\theta_0}$ such that $$\frac{\nu_{\widetilde{\theta_0}}}{\cos (\widetilde{\theta_0} -
\theta)} \,=\, \inf_{\widetilde{\theta} \in [\theta-\pi/2,\theta +\pi/2]}
\frac{\nu_{\widetilde{\theta}}}{\cos (\widetilde{\theta} - \theta)}$$ and if $$\liminf_{n\rightarrow \infty} \frac{2h(n)}{nl(A)} \,\geq\, |\tan
(\widetilde{\theta_0} -\theta)| \,,$$ then $\eta_{\theta,h}$ also exists (and equals $\nu_{\widetilde{\theta_0}}/
\cos (\widetilde{\theta_0} - \theta) $) and is the limit of $(\phi_n/(nl(A)))_{n\in
\NN}$ almost surely, even if $\lim_{n\rightarrow \infty} h(n)/n$ does not exist.
To complete the proof of Corollary \[chapitre6corollaire\], it remains to prove the convergence of $\phi_n/nl(A)$ in $L^1$. Suppose first that the condition (\[conditionO\]) is satisfied. Then, one can find a sequence of sets of edges $(E(n))_{n\in\NN}$ such that for each $n$, $E(n)$ is a cut between $T(nA, h(n))$ and $B(nA,h(n))$, $E(n)\subset E(n+1)$ and: $$\lim_{n\rightarrow \infty}\frac{\operatorname{card}(E(n)) }{n l(A)} \quad
\textrm{exists}\;,$$ cf. Lemma 4.1 in [@RossignolTheret08b], for instance. Now, define: $$f_n=\frac{\phi_n}{n l(A)}\quad \mbox{ and }\quad g_n= \frac{1}{nl(A)}\sum_{e\in
E(n)}t(e)\;.$$ We know that $(g_n)_{n\in\NN}$ converges almost surely and in $L^1$, thanks to the usual law of large numbers, thus the family $(g_n)_{n\in\NN}$ is equi-integrable. Since $0\leq f_n\leq g_n$ for every $n$, the family $(f_n)_{n\in\NN}$ is equi-integrable too, so its almost sure convergence towards $\eta_{\theta,h}$ implies its convergence in $L^1$ towards the same limit.
It remains to show the convergence in $L^1$ without the condition (\[conditionO\]). Let $A''$ be the translate of $A$ such that $0\in A''$, and $0$ is the center of $A''$, thus condition (\[conditionO\]) holds for $A''$. For any fixed $n$, there exists a segment $A_n'$ which is a translate of $nA$ by an integer vector and such that $d_\infty(0,nA'_n)<1$ and $d_\infty(nA'',A'_n)<1$, where $d_\infty$ denotes the distance induced by $\|.\|_\infty$. We want to compare the maximal flow through $\operatorname{cyl}(nA'',h(n))$ with the maximal flow through $\operatorname{cyl}(A_n',h(n))$. We have to distort a little bit the cylinder $\operatorname{cyl}(nA'',h(n))$. We only consider $n$ large enough so that $h(n) >1$. Thus the following inclusions hold: $$\operatorname{cyl}\left( \left( n- \left\lceil \frac{2}{l(A)} \right\rceil
\right) A'', h(n) -1 \right)\,\subset\, \operatorname{cyl}(A_n',h(n)) \,\subset\, \operatorname{cyl}\left( \left( n+ \left\lceil \frac{2}{l(A)} \right\rceil
\right) A'', h(n) +1\right) \,,$$ where $\lceil x \rceil$ is the smallest integer bigger than or equal to $x$. We get $$\begin{aligned}
\phi \left( \left( n- \left\lceil \frac{2}{l(A)} \right\rceil
\right) A'', h(n) +1\right) \,\leq\, \phi(A_n',h(n))
\,\leq\, \phi \left( \left( n+ \left\lceil
\frac{2}{l(A)} \right\rceil \right) A'', h(n) -1 \right)\,,\end{aligned}$$ (see Figure \[comparaison\]).
(0,0)![The cylinder $\operatorname{cyl}(A_n',h(n))$.[]{data-label="comparaison"}](comparaison.eps "fig:")
\#1\#2\#3\#4\#5[ @font ]{}
(10063,6033)(2539,-6982) (8851,-1411)[(0,0)\[lb\]]{} (8851,-2236)[(0,0)\[lb\]]{} (9151,-5386)[(0,0)\[lb\]]{} (9151,-5986)[(0,0)\[lb\]]{} (8851,-6886)[(0,0)\[lb\]]{} (8851,-2836)[(0,0)\[lb\]]{} (8851,-4336)[(0,0)\[lb\]]{} (5401,-4336)[(0,0)\[b\]]{}
Using the convergence in $L^1$ for $A''$ which satisfies the condition (\[conditionO\]), we see that $$\frac{\phi \left( \left( n- \left\lceil \frac{2}{l(A)} \right\rceil
\right) A'', h(n) +1\right)}{nl(A)} \quad\textrm{and} \quad
\frac{\phi \left( \left( n+ \left\lceil
\frac{2}{l(A)} \right\rceil \right) A'', h(n) -1 \right)}{nl(A)}$$ converge to $\eta_{\theta,h}$ in $L^1$ as $n$ goes to infinity. It is obvious that the small difference in the parameters $n$ and $h(n)$ does not change the value of the limit $\eta_{\theta,h}$. We get the convergence of $\tau(A'_n,h(n))/(nl(A))$ to $\eta_{\theta,h}$ in $L^1$. But since $A_n'$ is an integer translate of $nA$, it implies the convergence of $\tau(nA,h(n))/(nl(A))$ to $\eta_{\theta,h}$ in $L^1$.
\[remd3\] In dimension $d\geq 3$, if we denote by $\vec{v}$ a unit vector orthogonal to a non-degenerate hyperrectangle $A$ and by $\overrightarrow{\D_n(A)}$ the set of all admissible directions for the cylinder $\operatorname{cyl}(nA, h(n))$, i.e., the set of the vectors $\vec{v}' $ in $S^{d-1}$ such that there exists a hyperplane $\mathcal{P}$ orthogonal to $\vec{v}' $ that intersects $\operatorname{cyl}(nA, h(n))$ only on its “vertical faces”, and if $\lim_{n\rightarrow \infty} h(n) / n$ exists (thus $\overrightarrow{\D(A)} =
\operatorname{ad}(\underline{\overrightarrow{\D(A)}}) = \operatorname{ad}(\overline{\overrightarrow{\D(A)}})$ exists), we conjecture that $$\lim_{n\rightarrow \infty} \frac{\phi(nA, h(n))}{n^{d-1}\H^{d-1}(A)}
\,=\, \inf_{\vec{v}' \in \overrightarrow{\D(A)}}
\frac{\nu (\vec{v}' )}{|\vec{v} \cdot \vec{v}' |}
\qquad \textrm{a.s.} \,,$$ under assumptions (\[chapitre6eq:conditionsFLGN\]) on $F$ and if $h(n)$ goes to infinity with $n$ in such a way that we have $\lim_{n\rightarrow \infty} \log h(n) /
n^{d-1} =0$. We could not prove this conjecture, because we are not able to prove that $\phi(nA, h(n))$ behaves asymptotically like $\min_{\kappa
\in K} \phi^{\kappa}(nA, h(n))$, where $K$ is the set of the flat boundary conditions, i.e., the boundary conditions given by the intersection of a hyperplane with the vertical faces of $\operatorname{cyl}(nA, h(n))$.
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[^1]: Raphaël Rossignol was supported by the Swiss National Science Foundation grants 200021-1036251/1 and 200020-112316/1.
|
---
author:
- 'Leonardo Biliotti, Alessandro Ghigi'
title: Homogeneous bundles and the first eigenvalue of symmetric spaces
---
Introduction
============
Let $X$ be a compact complex manifold and let $E$ be a holomorphic vector bundle of rank $r$ over $X$. The [Gieseker point]{} of $E$ is the map $$\begin{gathered}
T_E: \Lambda^r H^0(X,E) \longrightarrow H^0(X, \det E) \\
\end{gathered}$$ that sends an element $s_1\wedge \dots \wedge s_r \in \Lambda^r
H^0(X,E)$ to the section $ x \mapsto s_1(x) \wedge \cdots \wedge
s_r(x)$ of $H^0(X,\det E)$. This map was first considered by Gieseker in his work [@gieseker-vector-surfaces] in order to construct the moduli space of vector bundles on a projective manifold. He proved that for the set of Gieseker stable bundles $E$ with fixed rank and Chern classes on a polarised $(X,H)$ there is a uniform $k_0 $ such that for $k>k_0$, $T_{E(k)} = T_{E \otimes H^{\otimes k}}$ is a stable vector (in the sense of geometric invariant theory) with respect to the action of ${\operatorname{SL}}\bigl ( H^0(X,E(k)) \bigr )$ on ${\operatorname{Hom}}\bigl ( \Lambda^r
H^0(X,E(k)), H^0(X,\det E(k)) \bigr)$.
In this paper we consider the Gieseker point of homogeneous bundles over rational homogeneous spaces. Such bundles are known to be Mumford-Takemoto stable [@ramanan-homogeneous], [@umemura-homogeneous] and this implies they are Gieseker stable [@kobayashi-vector p.191]. So we already know that after twisting with a sufficiently ample line bundle their Gieseker point is stable. The interest here is in the stability of $T_E$ itself, without allowing any twist. Our result is the following.
\[gieseker-intro\] Let $E{\rightarrow}X$ be an irreducible homogeneous vector bundle over a rational homogeneous space $X=G/P$. If $H^0(E) \neq 0$, then $T_E$ is stable.
We give two proofs of this result. The first is algebraic and uses a criterion of Luna for an orbit to be closed. This proof works over any algebraically closed field of characteristic zero. The second proof uses invariant metrics and relies on a result by Xiaowei Wang [@wang-xiaowei-balance].
Our interest for this result is connected with a problem in [Kähler ]{}geometry. Consider a compact [Kähler ]{}manifold $X$ and fix a [Kähler ]{}class $a \in H^2(X)$. Bourguignon, Li and Yau [@bourguignon-li-yau], gave an upper bound for the first eigenvalue of the Laplacian $\Delta_g : C^\infty(X) {\rightarrow}{C^\infty}(X)$ relative to any [Kähler ]{}metric $g$ whose [Kähler ]{}form ${\omega}_g$ lies in the class $a$. The bound depends on the numerical invariants ($h^0$ and degree) of a globally generated line bundle $L$ over $ X$. To get the best estimate one has to choose appropriately the bundle. As is shown in [@bourguignon-li-yau], if $X=\mathbb{P}^n$ and $L={\mathcal{O}}_{\mathbb{P}^n}(1)$ one gets the upper bound $2$, which is optimal since it is achieved by the Fubini-Study metric.
In paper [@arezzo-ghigi-loi] Arezzo, Loi and the second author, generalised this result, by substituting a vector bundle $E$ to the line bundle $L$. In this case one gets the same kind of estimate, but the vector bundle $E$ must satisfy an additional condition, namely its Gieseker $T_E$ point must be stable. By this method one gets an upper bound for ${\lambda}_1$ on the complex Grassmannian [@arezzo-ghigi-loi]. Such a bound is optimal since it is achieved by the symmetric metric. It is important to notice that if one twists the bundle $E$ by a positive line bundle $H$, the estimate gotten from the twisted bundle $E(k)$ is very rough. In fact the estimate blows up as $k {\rightarrow}\infty$ (see below). So it is important to obtain some information on the stability of $T_E$ without twisting the bundle $E$.
The main motivation for the present work was to extend the estimate for ${\lambda}_1$ to other Hermitian symmetric spaces of the compact type using appropriate homogeneous bundles. We are able to prove the following.
\[classical-intro\] Let $X$ be a compact irreducible Hermitian symmetric space of ABCD-type. Then $${\lambda}_1(X, g) \leq 2$$ for any [Kähler ]{}metric $g$ whose [Kähler ]{}class ${\omega}_g$ lies in $2\pi{\operatorname{c}}_1(X)$. This bound is attained by the symmetric metric.
It should be mentioned here that El Soufi and Ilias [@el-soufi-ilias-Pacific Rmk. 1, p.96] have proved that the symmetric metric is a critical point (in suitable sense) for the functional $\lambda_1$ on the set of *all* Riemannian metrics with fixed volume.
Curiously in the two exceptional examples (E-type) the best estimate gotten by this method is strictly larger than 2, which is ${\lambda}_1$ of the symmetric metric.
\[exceptional-intro\] If $X= E_6/P({\alpha}_1)$ (resp. $X= E_7/P({\alpha}_7)$) then $${\lambda}_1(X, g) \leq \frac{36}{17}
\qquad \text{ resp. } \quad
{\lambda}_1(X, g) \leq \frac{133}{53}$$ for any [Kähler ]{}metric $g$ whose [Kähler ]{}class ${\omega}_g$ lies in $2\pi{\operatorname{c}}_1(X)$.
It would be interesting to understand if this is a deficiency of the method, or if these symmetric spaces do in fact support metrics with ${\lambda}_1$ larger than 2.
#### Acknowledgements
We wish to thank Prof. Gian Pietro Pirola and Prof. Andrea Loi for various interesting discussions. The second author is also grateful to Prof. Peter Heinzner for inviting him to Ruhr–Universität Bochum during the preparation of this work. He also acknowledges partial support by MIUR COFIN 2005 “Spazi di moduli e teoria di Lie”.
Stability of the Gieseker point
===============================
Let $X$ be a compact complex manifold and let $E{\rightarrow}X$ be a holomorphic vector bundle of rank $r$. Set $$V=H^0(X,E), \quad V'=H^0(X,\det E) \quad {\mathbb{W}}= {\operatorname{Hom}}(\Lambda^r V, V').$$ The algebraic group ${\operatorname{GL}}(V)$ acts linearly on $V$ hence on $\Lambda^r
V$. It therefore also acts on ${\mathbb{W}}$. For $a\in {\operatorname{GL}}(V)$ let ${{\, \Lambda^ra}} $ be the induced map on $\Lambda^r V$. The action of ${\operatorname{GL}}(V)$ on ${\mathbb{W}}$ is given by $$a.T := T {\circ}({{\, \Lambda^ra}}){^{-1}}.
\label{eq:GLact}$$ Consider the above action restricted to the subgroup ${\operatorname{SL}}(V) \subset
{\operatorname{GL}}(V)$. According to the terminology of geometric invariant theory, a point $T\in {\mathbb{W}}$ is [stable]{} (for this restricted action) if the orbit of ${\operatorname{SL}}(V)$ through $T$ is closed in ${\mathbb{W}}$ and the stabiliser of $T$ inside ${\operatorname{SL}}(V)$ is finite. We denote by ${\widetilde{S}}$ and $S$ the stabilisers of $T_E$ in ${\operatorname{GL}}(V)$ and ${\operatorname{SL}}(V)$ respectively: $$\label{eq:def-S}
{\widetilde{S}}=\{a\in {\operatorname{GL}}(V): a.T_E=T_E\} \qquad S=\{a\in {\operatorname{SL}}(V): a.T_E=T_E\}.$$ For $x\in X$ put $$\begin{gathered}
V_x = \{ s\in V: s(x) = 0\} \qquad V'_x = \{ t\in V': t(x) = 0\} .\end{gathered}$$ The following two simple lemmata will be used in the (algebraic) proof of Thm. 1.
\[genero\] Let $E$ be globally generated of rank $r$. Then a section $s\in V$ belongs to $V_x$ if and only if for any choice of $r-1$ sections $
s_2, \ldots{}, s_r$ of $E$ the section $ T_E(s, s_2, \ldots{} , s_r
)$ of $\det E$ lies in $ V'_x$.
The proof is immediate. Let ${\operatorname{Aut}}(E)$ be the group of holomorphic bundle automorphisms of $E$. If $f\in {\operatorname{Aut}}(E)$ and $x\in X$, the map $f_x : E_x {\rightarrow}E_x$ is a linear isomorphism. The function $x\mapsto \det f_x$ is holomorphic hence a (nonzero) constant and $f\mapsto \det f$ is a character of ${\operatorname{Aut}}(E)$. For $f\in {\operatorname{Aut}}(E)$ and $s \in H^0(X,E)$ put $$ {\varepsilon}(f) (s) := f {\circ}s.$$ This defines a representation ${\varepsilon}: {\operatorname{Aut}}(E) {\rightarrow}{\operatorname{GL}}(V)$.
\[lemma-aut\] Let $E$ be globally generated. Then $
{\varepsilon}( \{ f\in {\operatorname{Aut}}(E) : \det f =1\}) ={\widetilde{S}}$.
[**Proof**. ]{}For $f\in {\operatorname{Aut}}(E)$ $$\begin{gathered}
\bigl ( {\varepsilon}(f). T_E\bigr ) (s_1 , \ldots{}, s_r )(x) =
\bigl( {\varepsilon}(f){^{-1}}s_1\bigr ) (x) \wedge \cdots{} \wedge \bigl(
{\varepsilon}(f){^{-1}}s_r\bigr ) (x) = \\
= \bigl ( f_x{^{-1}}s_1(x) \bigr ) \wedge \cdots{} \wedge \bigl (
f_x{^{-1}}s_r(x) \bigr ) = \bigl ( \det f \bigr ){^{-1}}\cdot s_1(x) \wedge \ldots{} \wedge
s_r(x) = \\
=\bigl ( \det f \bigr ){^{-1}}\cdot T_E(s_1 , \ldots{}, s_r)(x).
\end{gathered}$$ So $$\label{eq:computa-computa}
{\varepsilon}(f). T_E = \bigl ( \det f \bigr ){^{-1}}\cdot T_E.$$ If $\det f=1$ then ${\varepsilon}(f).T_E =T_E$. This proves that $ {\varepsilon}( \{
f\in {\operatorname{Aut}}(E) : \det f =1\}) \subset {\widetilde{S}}$. Conversely, let $a\in {\widetilde{S}}$. We claim that $a ( V_x) = V_x$ for any $x\in X$. Indeed let $s\in
V_x$. Then for any $s_2, \ldots{}, s_r \in V$ $$\begin{gathered}
T_E(as, s_2, \ldots{}, s_r) = T_E(as, aa{^{-1}}s_2 , \ldots{},
aa{^{-1}}s_r) =\\
=(a{^{-1}}.\, T_E) (s, a{^{-1}}s_2, \ldots{}, a{^{-1}}s_r)=\\
= T_E(s, a{^{-1}}s_2, \ldots{}. a{^{-1}}s_r)(x) =0.\end{gathered}$$ So $a s \in V_x$ by Lemma \[genero\], and indeed $a(V_x) =V_x$ as claimed. We get therefore an induced isomorphism $$\begin{gathered}
f_x : E_x \cong V/ V_x {\rightarrow}V/ V_x \cong E_x.\end{gathered}$$ By construction $f_x ( s(x)) = (a\, s)(x)$. Since $E$ is globally generated this ensures that $f$ is holomorphic so $f \in {\operatorname{Aut}}(E)$ and ${\varepsilon}(f)=a$. By $ \bigl ( \det f \bigr
){^{-1}}\cdot T_E = {\varepsilon}(f) .T_E = a.T_E =T_E$. Since $E$ is globally generated, $T_E \neq 0$ and it follows that $\det f =1$. [$\Box$\
]{}We recall two results that will be needed in the following.
\[luna\] Let $H$ be a reductive group and $K \subset H$ a reductive subgroup. Let $X$ be an affine $H$-variety. If $x\in X$ is a fixed point of $K$ the orbit $Hx$ is closed if and only if the orbit $N_H(K)x$ is closed.
This criterion is due to Luna [@luna-inv-75 Cor. 1] and is based on the Slice Theorem. For a complex analytic proof of the Slice Theorem see [@heinzner-schwarz-Cartan].
We recall that a [rational homogeneous space]{} is a projective variety $X$ of the form $G/P$ with $G$ a simply connected complex semisimple Lie group and $P$ a parabolic subgroup without simple factors. (See for example [@akhiezer-libro], [@ottaviani-rat], [@baston-eastwood].) Such spaces are also called [generalised flag manifolds]{}. A homogeneous vector bundle $E$ over $X$ is of the form $E= G \times_P {U}$ where ${U}$ is a representation of $P$. If the representation is irreducible the vector bundle itself is called [irreducible]{}.
\[ramajan\] Let $E{\rightarrow}X$ be an irreducible homogeneous vector bundle over a rational homogeneous space $X$. Then $E$ is simple, i.e. ${\operatorname{Aut}}(E) = {\mathbb{C}}^* \cdot I_E$.
**First proof of Thm. \[gieseker-intro\].** Let $X=G/P$ be as above. By Bott-Borel-Weil theorem (Thm. \[bbw\] below) the hypothesis $H^0(E) \neq 0$ already ensures that $E$ is globally generated, so both $V$ and $V'$ have positive dimension and $T_E \not \equiv 0$. By the same theorem $G$ acts irreducibly on both $V$ and $V'$. Denote by $ \rho : G {\rightarrow}{\operatorname{GL}}(V) $ and $ \sigma : G{\rightarrow}{\operatorname{GL}}(V') $ these representations. Since $G$ is semisimple all the characters of $G$ are trivial. In particular any representation of $G$ on a vector space $U$ has image contained in ${\operatorname{SL}}(U)$. So in fact $ \rho : G {\rightarrow}{\operatorname{SL}}(V) $ and $
\sigma : G{\rightarrow}{\operatorname{SL}}(V')$. The Gieseker point is $G$-equivariant, that is $$\label{equivario} T_E( {{\, \Lambda^r}}\rho(g) (u )) = \sigma(g)
( T(u) ) \qquad u \in \Lambda^r V.$$ Set $H= {\operatorname{SL}}(V) \times G$ and define a representation $ \varpi : H {\rightarrow}{\operatorname{GL}}({\mathbb{W}}) $ by $$\varpi(a,g) T = \sigma(g){\circ}T {\circ}{{\, \Lambda^ra}}{^{-1}}.$$ Let $\tau : G {\rightarrow}H$ be the morphism $\tau(g)=(\rho(g), g)$ and let ${{K}}$ be the image of $\tau$. ${{K}}\subset H$ is a closed reductive subgroup and by $T_E$ is a fixed point of ${{K}}$ acting via $\varpi$. We claim that the normaliser $N_H({{K}})$ is a finite extension of ${{K}}$. In fact, denote by ${\operatorname{Ad}}$ the conjugation on $H$. Given $n\in N_H({{K}})$ put $${\varphi}(n) ={\operatorname{Ad}}(n){\phantom{}_{\text{\raisebox{.4ex}{$|$}}{{K}}}} : {{K}}{\rightarrow}{{K}}.$$ Let ${\operatorname{Aut}}({{K}})$ denote the group of automorphisms of ${{K}}$ and ${\operatorname {Inn}}({{K}})$ the subgroup of inner automorphisms. Then ${\varphi}: N_H({{K}})
{\rightarrow}{\operatorname{Aut}}({{K}})$ is a morphism of groups. Since ${{K}}$ is semisimple ${\operatorname{Aut}}({{K}})$ is a finite extension of ${\operatorname {Inn}}({{K}})$ (see or [@onishchik-vinberg-seminario Thm. 1 p.203]). Put $N'={\varphi}{^{-1}}({\operatorname {Inn}}({{K}}))$. Then $N' \lhd N_H({{K}})$ and $$N_H({{K}})/N' \hookrightarrow {\operatorname{Aut}}({{K}})/{\operatorname {Inn}}({{K}}).$$ Therefore $N_H({{K}})$ is a finite extension of $N'$ and it is enough to prove that $N'$ is a finite extension of ${{K}}$. Indeed if $n\in N'$ there is some $k\in K$ such that $nk'n{^{-1}}= kk'k{^{-1}}$ for any $k'\in K$. So $ k{^{-1}}n $ centralises ${{K}}$. If $k{^{-1}}n =(a,g) $ (with $a\in {\operatorname{SL}}(V)$ and $g\in G$) this means that for any $g'\in G$ we have $$\begin{gathered}
a\rho(g') = \rho(g') a \qquad
\qquad gg'=g'g.\end{gathered}$$ The second formula says that $g\in Z(G)$. The first formula says that $a: V {\rightarrow}V$ commutes with the representation $\rho $. Since this is irreducible Schur lemma implies that $a={\varepsilon}I$ for some ${\varepsilon}\in
{\mathbb{C}}^*$. But $a\in {\operatorname{SL}}(V)$, so ${\varepsilon}^p=1$ where $p=\dim V$. Denote by $U_p$ the group of $p$–roots of unity. Then $k{^{-1}}n= ({\varepsilon},g) \in
U_p \times Z(G)$. This proves that the composition $$U_p\times Z(G) \rightarrow N' {\rightarrow}N'/ {{K}}$$ is onto. Since $Z(G)$ is finite, it follows that $N'$ and $N_H(K)$ are finite extensions of $K$. Now $T_E\in {\mathbb{W}}$ is a fixed point of ${{K}}$ and $N_H({{K}})$ is a finite extension of ${{K}}$, so the orbit $N_H({{K}}).
T_E$ is a finite set, hence it is closed. Notice that both $H$ and $K$ are reductive. We can therefore apply Luna’s criterion (Thm. \[luna\]) to the effect that the orbit $H.T_E$ is closed. To finish we claim that $H.T_E = {\operatorname{SL}}(V).T_E$. Since the action of ${\operatorname{SL}}(V)$ and the restriction of $\varpi$ to ${\operatorname{SL}}(V)\times \{1\}
\subset H$ agree, the inclusion $H.T_E \supset {\operatorname{SL}}(V).T_E$ is obvious. For the other let $h=(a,g)\in H$. Then $$\begin{gathered}
\varpi(h) T_E = \sigma(g) {\circ}T_E {\circ}{{\, \Lambda^ra}}{^{-1}}= \sigma(g)
{\circ}T_E {\circ}{{\, \Lambda^r}}\bigl (a \rho(g){^{-1}}\cdot \rho( g) \bigr){^{-1}}=\\
= \sigma(g) {\circ}T_E {\circ}{{\, \Lambda^r}}\rho(g){^{-1}}{\circ}{{\, \Lambda^r}}( \rho(g)
a{^{-1}}) = \\
=T_E {\circ}{{\, \Lambda^r}}( \rho(g) a{^{-1}}) = \varpi(a\rho(g){^{-1}}, 1)T_E\end{gathered}$$ and $a\rho(g{^{-1}}) \in {\operatorname{SL}}(V)$. Therefore $H.T_E \subset {\operatorname{SL}}(V).T_E$ so the two orbits coincide. This shows that the orbit of $T_E$ is closed. Let $S$ and ${\widetilde{S}}$ be the stabilisers defined as in . By Thm. \[ramajan\], ${\operatorname{Aut}}(E) = {\mathbb{C}}^* \cdot I_E$, therefore $\{f\in {\operatorname{Aut}}(E): \det f=1\}$ is finite, which implies, by Lemma \[lemma-aut\], that ${\widetilde{S}}$ and a fortiori $S$ are finite. [$\Box$\
]{}We remark that this proof works over any algebraically closed field of characteristic zero.
We come now to the second proof of this result. Recall that if $E$ is a globally generated bundle on $X$ and $\boldsymbol{s}=\{s_1,
\ldots{}, s_N\}$ is a basis of $H^0(X,E)$ there is an induced map $
{\varphi}_{\boldsymbol{s}} : X {\rightarrow}G(r,N)$. Consider on $G(r,N)$ the standard symmetric Kähler structure which coincides with the pull-back of the Fubini-Study metric via the Plücker embedding. Denote by $\mu: G(r,N) {\rightarrow}{\mathfrak{su}}(N)$ the moment map for the standard action of ${\operatorname{SU}}(N) $ on $G(r,N)$.
\[[[@wang-xiaowei-balance Thm. 3.1]]{}\] \[wang\] Let $(X^m,{\omega})$ be a compact [Kähler ]{}manifold and let $E$ be a globally generated bundle on $X$. Then $T_E$ is stable if and only if there is a basis $ {\boldsymbol{s}} $ of $H^0(X,E)$ such that $$\label{omega-bil}
\int_X \mu \bigl ( {\varphi}_ {\boldsymbol{s}} (x) \bigr ) {\omega}^m (x) =0.$$
For the reader’s convenience we briefly sketch the proof.\
[**Proof. **]{} Fix an arbitrary Hermitian metric $h$ on $E$ and consider on $V$ the corresponding $L^2$–scalar product. Let ${{{\boldsymbol{s}}}}$ be an orthonormal basis with respect to this product. On the line bundle $\det E$ consider the metric ${\varphi}_{{{\boldsymbol{s}}}}^* h_G$ where $h_G$ is the metric on $\mathcal{O}_{G(r,N)}(1)$. Consider on $V'$ the corresponding $L^2$–scalar product. Finally denote by ${\langle \cdot \, , \cdot \rangle}_{{\mathbb{W}}}$ the Hermitian inner product on ${\mathbb{W}}$, $|| \cdot ||_{\mathbb{W}}$ being the corresponding norm. Since we have fixed a basis we may identify ${\operatorname{SL}}(V)$ with ${\operatorname{SL}}(N,
{\mathbb{C}})$. For $g\in {\operatorname{SL}}(N, {\mathbb{C}})$ set $\nu(g) = \log || g{^{-1}}. T_E
||_{\mathbb{W}}$. We consider $\nu$ as a function on ${\operatorname{SL}}(N, {\mathbb{C}})/ {\operatorname{SU}}(N)$. On this space Wang introduces another functional $$L(g):= \int_M \biggl ( \sum_I || (g^{-1}T_E) (s_{I})(x)
||^2_{{\varphi}_{{{\boldsymbol{s}}}}^*h_G} \bigr )
\frac{{\omega}^n}{n!}(x),$$ which is strictly convex on ${\operatorname{SL}}(N,{\mathbb{C}})/SU(N)$ [@wang-xiaowei-balance Lemma 3.5]. (Here $s_I=s_{i_1} \wedge \cdots \wedge s_{i_r}\in \Lambda^r V$.) Critical points of $L$ correspond to $g\in {\operatorname{SL}}(N)$ such that the basis $\{gs_1, \ldots{}, gs_N\} $ satisfies . For some constants $C_2, C_4 \in \mathbb{R}$ and $C_1, C_3 >0$ the inequalities $$L
\geq C_1 \nu + C_2 \geq C_3 L + C_4
\label{wang-puppa}$$ hold on $ {\operatorname{SL}}(N,{\mathbb{C}})/SU(N)$. The first is proved by Wang [@wang-xiaowei-balance p.406]. The second is simply an application of Jensen inequality to the convex function $-\log$. If $T_E$ is stable, then $\nu$ is proper by the Kempf-Ness theorem [@kempf-ness]. Hence $L$ is proper too, so admits a minimum and there is a basis ${{{\boldsymbol{s'}}}}$ such that is satisfied. On the other hand if there is such a basis, $L$ has a minimum and being strictly convex this means it is proper. By , $\nu$ is proper as well and, again by Kempf-Ness theorem, this implies that $T_E$ is stable. It should be noted that the identification of the moment map for a projective action with the differential of a convex functional is standard in analytic Geometric Invariant Theory [@mumford-GIT Ch. 8], [@donaldson-kronheimer §6.5], [@heinzner-huckleberry-MSRI]. [$\Box$\
]{}
**Second proof of Theorem \[gieseker-intro\].** Let $K$ be a compact form of $G={\operatorname{Aut}}(X)$. By averaging on $K$ we can find $K$-invariant metrics ${\omega}$ and $h$ on $X$ and $E$ respectively. Let $\boldsymbol{s}$ be a basis of $H^0(X,E)$ that is orthonormal with respect to the $L^2$-scalar product obtained using $h$ and ${\omega}$. By Bott-Borel-Weil theorem (Thm. \[bbw\] below) $G$ and hence $K$ act irreducibly on $H^0(X,E) \cong {\mathbb{C}}^N$. Denote by $\sigma : K {\rightarrow}{\operatorname{SU}}(N)$ this representation (recall that $K$ is semisimple). Then $\mu {\circ}{\varphi}_{\boldsymbol{s}} $ is $K$-equivariant and $$B=\int_X \mu \bigl ( {\varphi}_ {\boldsymbol{s}} (x) \bigr ) {\omega}^m (x)$$ is a fixed point of ${{\operatorname{ad}}}(\sigma(K)) \subset {\operatorname{GL}}( {\mathfrak{su}}(N))$, that is $\sigma(k) B = B \sigma(k)$ for $k\in K$. By Schur lemma this implies that $B={\lambda}I$, so $B=0$ since $B\in {\mathfrak{su}}(N)$. By Thm. \[wang\] the Gieseker point $T_E$ is stable. [$\Box$\
]{}
In order to clarify the meaning of the above result it might be good to notice that together with the numerical criteria of [@gieseker-vector-surfaces] it allows an easy proof of the Gieseker stability (see e.g. [@kobayashi-vector p.189]) of irreducible homogeneous bundles. We sketch this argument, although a stronger result (Mumford-Takemoto stability) is well-known (see [@ottaviani-rat p.65] and references therein).
.
\[[[@gieseker-vector-surfaces Prop. 2.3]]{}\] \[gieseker123\] Let $T\in {\mathbb{W}}$ be a stable point. Let $V''\subset V$ be a subspace and let $d$ a number $1 \leq d < r $. Assume that for any $d+1$ vectors $v_1, \ldots{}, v_{d+1} \in V''$, $T(v_1, \ldots{}, v_d,
v_{d+1} , \cdots) \equiv 0$. Then $ \dim V'' < (d/r) \, \cdot \, \dim V.$ If $T$ is only semistable, then equality can hold.
In [@gieseker-vector-surfaces] there is a proof in the semistable case, which works as well in the stable case.
Let $E{\rightarrow}X$ be an irreducible homogeneous vector bundle of rank $r$ over a rational homogeneous space $X=G/P$. If $H^0(E) \neq 0$, and $F \subset E$ is a subsheaf of rank $d$, then $
h^0(F) < (d/r) \cdot h^0(E).
$
Fix now an irreducible homogeneous bundle $E$ of rank $r$ and let $F\subset E$ be a subsheaf of rank $d$, with $0<d<r$. Let $H$ be any polarisation on $X$. Since any line bundle is homogeneous, $E(k)=E
\otimes H^{\otimes k}$ is homogeneous. By Serre Theorem there is a $k_0$ such that for $k\geq k_0$ $$H^i(X, F(k) ) = H^i(X, E(k))=\{0\} \qquad i > 0$$ and both $E(k)$ and $F(k)$ are globally generated. By Thm. \[gieseker-intro\], $T_{E(k)}$ is stable, so by the above corollary $ \chi(X, F(k)) = h^0(X,F(k)) < (d/r) \cdot h^0(X,E(k)) =
(d/r) \cdot \chi(X,E(k)). $ This proves that any irreducible homogeneous bundle is Gieseker stable with respect to any polarisation.
The first eigenvalue of Hermitian symmetric\
spaces
============================================
Here we want to apply the previous stability result to a problem in spectral geometry. Let $X$ be a projective manifold and $L$ an ample line bundle on $X$. Let ${{\mathcal{K}(L)}}$ be the set of [Kähler ]{}metrics $g$ with [Kähler ]{}form ${\omega}_g$ lying in the class $2\pi{\operatorname{c}}_1(L)$. For $g$ in ${{\mathcal{K}(L)}}$ let $\Delta_g $ be the Laplacian on functions, $$\Delta_g f = -d^* d f =
2 \ g^{i\bar{j}}\frac{\partial^2 f}{\partial z^i \partial \bar{z}^j}.$$ It is well-known that $\Delta_g$ is a negative definite elliptic operator and has therefore discrete spectrum: denote its eigenvalues by $0 > -{\lambda}_1(g) > -{\lambda}_2(g) > \cdots$. The following result of Lichnerowicz relates ${\lambda}_1$ to [[Kähler-Einstein]{} ]{}geometry.
\[futaki\] If $X$ is a Fano manifold and $g_{KE}$ is a [[Kähler-Einstein]{} ]{}metric, i.e. $Ric(g_{KE})=g_{KE}$, then ${\lambda}_1(g_{KE}) =2$ if ${\operatorname{Aut}}(X)$ has positive dimension and ${\lambda}_1(g_{KE}) > 2$ otherwise.
We are interested in upper estimates for ${\lambda}_1(g)$ of general metrics in the class ${{\mathcal{K}(L)}}$. Bourguignon, Li and Yau [@bourguignon-li-yau] first studied this problem and showed that the supremum $$\label{eq:def-sup}
I(L)= \sup_{{{\mathcal{K}(L)}}} {\lambda}_1(g)$$ is always finite. (This heavily depends on the restriction to [Kähler ]{}metrics, see [@colbois-dodziuk].) They gave an explicit upper bound for $I(L)$ in terms of numerical invariants of a globally generated line bundle $E$. For $(X,L)=(\mathbb{P}^m,
\mathcal{O}_{\mathbb{P}^m} (1))$ they were able to show that $I(L)=2$. The following criterion, due to Arezzo, Loi and the second author, is an extension of Bourguignon, Li and Yau’s theorem. It allows to attack this problem using holomorphic vector bundles instead of just line bundles.
\[igi\] Let $(X,L)$ be a polarised manifold and $E $ a holomorphic vector bundle of rank $r$ over $X$. Assume that $E$ is globally generated and nontrivial and put $$\label{eq:def-J}
J(E,L):= \frac{2\, \dim_{\mathbb{C}}X \cdot h^0(E)\,
\langle c_1(E) \cup c_1(L)^{m-1} ,[X]\rangle }
{r\, (h^0(E)-r)\, \langle c_1(L)^m, [X]\rangle}.$$ If the Gieseker point $T_E$ is stable, then $$\label{mainest1}
I(L)\leq J(E,L).$$
The result of [@arezzo-ghigi-loi] is slightly more general since there is no projectivity assumption on $X$.
We want to apply this result to the case where $X=G/P$ is a rational homogeneous space and $E$ is homogeneous. In this case $J(E,L)$ can be computed, at least in principle, in terms of Lie algebra data. To proceed we fix the following (standard) notation. (See e.g. [@fels-huckleberry-wolf Ch. 1],[@ottaviani-rat], [@baston-eastwood].) $G$ is a simply connected complex semisimple Lie group, ${\mathfrak{g}}= {Lie}\, G$, ${\mathfrak{h}}\subset {\mathfrak{g}}$ is a Cartan subalgebra, $l=\dim {\mathfrak{h}}$ is the rank of $G$, $B$ is the Killing form of ${\mathfrak{g}}$, $\Delta$ is the root system of $({\mathfrak{g}}, {\mathfrak{h}})$, $\Delta_+$ is a system of positive roots, $\Delta_-=-\Delta_+$, $\Pi=\{{\alpha}_1, \ldots{}, {\alpha}_l\}$ is the set of simple roots, ${\varpi}_1, \ldots, {\varpi}_l$ denote the fundamental weights. $\Lambda=
{{\mathbb{Z}}}{\varpi}_1 \oplus \cdots \oplus {{\mathbb{Z}}}{\varpi}_l \subset {\mathfrak{h}}^*$ is the weight lattice of ${\mathfrak{g}}$ relative to the Cartan subalgebra ${\mathfrak{h}}$. For ${\alpha}\in \Delta$ let $H_{\alpha}\in {\mathfrak{h}}$ be such that ${\alpha}(X) = B(X,H_{\alpha})$. ${\mathfrak{b}}$ is the standard [negative]{} Borel subalgebra: $$\label{eq:borel}
{\mathfrak{b}}={\mathfrak{h}}\oplus \bigoplus _{{\alpha}\in \Delta_-} {\mathfrak{g}}_{\alpha}$$ Parabolic subalgebras containing ${\mathfrak{b}}$ are of the form $${\mathfrak{p}}(\Sigma) = {\mathfrak{b}}\oplus \bigoplus _{{\alpha}\in {\,\operatorname{span}}(\Pi - \Sigma) \cap
\Delta_+} {\mathfrak{g}}_{\alpha}$$ where $\Sigma $ is some subset of $ \Pi$. For example $\Sigma = \Pi$ corresponds to ${\mathfrak{b}}$, $\Sigma = \varnothing$ to ${\mathfrak{g}}$ and maximally parabolic subalgebras are of the form $ {\mathfrak{p}}({\alpha}_k)$. The algebra ${\mathfrak{p}}(\Sigma)$ admits a Levi decomposition $
{\mathfrak{p}}(\Sigma) = {\mathfrak{l}}(\Sigma) \oplus \mathfrak{u}(\Sigma) $, where $\mathfrak{u}(\Sigma)$ is the nilpotent radical and $${\mathfrak{l}}(\Sigma) = {\mathfrak{h}}\oplus \bigoplus_{{\alpha}\in {\,\operatorname{span}}(\Pi-\Sigma)
\cap \Delta} {\mathfrak{g}}_{\alpha}$$ is the reductive part. This latter admits a further decomposition $
{\mathfrak{l}}(\Sigma ) = {\mathfrak{z}}(\Sigma) \oplus {\mathfrak{s}}(\Sigma)$, ${\mathfrak{z}}(\Sigma) $ being the center and ${\mathfrak{s}}(\Sigma)$ being semisimple. Moreover $${\mathfrak{z}}(\Sigma) = \bigcap_{{\alpha}\in\Pi-\Sigma} \ker {\alpha}\subset {\mathfrak{h}}$$ and $${\mathfrak{s}}(\Sigma) = {\mathfrak{h}}' (\Sigma) \oplus \bigoplus_{{\alpha}\in {\,\operatorname{span}}(\Pi-\Sigma) \cap
\Delta} {\mathfrak{g}}_{\alpha}$$ where ${\mathfrak{h}}'(\Sigma)= {\,\operatorname{span}}\{ H_{\alpha}: {\alpha}\in \Pi- \Sigma \}
\subset {\mathfrak{h}}$ is a Cartan subalgebra for ${\mathfrak{s}}(\Sigma)$ and ${\mathfrak{h}}={\mathfrak{z}}(\Sigma) \oplus {\mathfrak{h}}'(\Sigma)$. We denote by $B,
P(\Sigma), L(\Sigma), U(\Sigma), Z(\sigma), S(\Sigma)$ the corresponding closed subgroups of $G$. Note that $S(\Sigma)$ is simply connected. One can describe ${\mathfrak{p}}(\Sigma), P(\Sigma)$ and the homogeneous space $G/P(\Sigma)$ by the Dynkin diagram of $G$ with the nodes corresponding to roots in $\Sigma$ crossed.
A weight ${\lambda}=\sum_i m_i {\varpi}_i \in {\Lambda}$ is [dominant for $G$]{} or simply [dominant]{} if $m_i \geq 0$ for any $i$. It is said to be [dominant with respect to ${\mathfrak{p}}(\Sigma)$]{} if $m_i\geq 0$ for any index $i$ such that ${\alpha}_i\not \in \Sigma$. By highest weight theory, the irreducible representations of $G$ are parametrised by dominant weights, while irreducible representations of a parabolic subgroup $P(\Sigma)$ are parametrised by weights that are dominant with respect to ${\mathfrak{p}}(\Sigma)$. If ${\lambda}$ is dominant we let $W_{\lambda}$ denote the irreducible representation of $G$ with highest weight ${\lambda}$. If ${\lambda}$ is dominant for ${\mathfrak{p}}(\Sigma)$ we let $V_{\lambda}$ denote the irreducible representation of $P(\Sigma)$ with highest weight ${\lambda}$. We let moreover $E_{\lambda}$ denote the homogeneous vector bundle on $X=G/P(\Sigma)$ defined by the representation $V_{\lambda}$, that is $ E_{\lambda}= G \times_{P(\Sigma)} V_{\lambda}. $
\[Bott-Borel-Weil\] \[bbw\] If ${\lambda}\in {\Lambda}$ is dominant for $G$, then $$H^0(X, E_{\lambda}) = W_{\lambda}.$$ Otherwise $H^0(X,E_{\lambda})=\{0\}$.
Bott’s version of the theorem is much more general, but this partial statement is enough for what follows. We also remark that if one chooses ${\mathfrak{b}}$ to be the Borel subalgebra with [positive]{} instead of negative roots, which is customary for example in the usual picture of $\mathbb{P}^n$ as the set of lines in ${\mathbb{C}}^{n+1}$, then one has to consider [lowest weights]{} instead of highest ones. This amounts to dualize both representations. With this choice the statement of the theorem becomes $ H^0(X, E^*_{\lambda}) = \bigl ( W_{\lambda}\bigr )^*.$ (The book [@baston-eastwood] follows this convention.)
Recall that the set of simple roots $\Pi=\{{\alpha}_1, \ldots, {\alpha}_l\}$ is a basis of ${\Lambda}\otimes \mathbb{Q}$. For a weight ${\lambda}\in {\Lambda}$, let ${\lambda}=\sum_i \xi_i ({\lambda}){\alpha}_i$ be its expression in this basis. We say that the (rational) number $\xi_i({\lambda})$ is the [coefficient of ${\alpha}_i$ in ${\lambda}$]{}. We denote by ${\lambda}_{{\operatorname{ad}}}$ the highest weight of the adjoint representation of $G$ (that is the largest root).
\[c1\] Let $X=G/P({\alpha}_k)$. The bundle $E_{{\varpi}_k}$ associated to the fundamental weight ${\varpi}_k$ is a very ample line bundle over $X$. Moreover $\operatorname{Pic}(X)\cong H^2(X,{{\mathbb{Z}}})={{\mathbb{Z}}}{\operatorname{c}}_1(E_{{\varpi}_k})$. For any weight ${\lambda}\in {\Lambda}$ that is dominant for $P({\alpha}_k)$ $${\operatorname{c}}_1(E_{\lambda}) = \dim V_{\lambda}\, \frac{\xi_k({\lambda})}{\xi_k({\varpi}_k)} \, {\operatorname{c}}_1(E_{{\varpi}_k}).$$
(For the proof see e.g. [@ramanan-homogeneous §5.2], [@ottaviani-rat p.56].)
In the following statement we summarise what we need of the structure theory of Hermitian symmetric spaces.
\[classifico\] An irreducible Hermitian (globally) symmetric space of the compact type is a rational homogeneous space. Moreover a rational homogeneous space $X=G/P$ is symmetric if and only if the representation of ${\mathfrak{p}}$ on ${\mathfrak{g}}/ {\mathfrak{p}}$ induced from the adjoint representation of ${\mathfrak{g}}$ is irreducible. The actual possibilities are explicitly listed in Table 1.
The characterisation in terms of irreducibility of ${\mathfrak{g}}/{\mathfrak{p}}$ is due to Kobayashi and Nagano [@kobayashi-nagano-filtered-II Thm. A] (see also [@baston-eastwood p.26]).
\[tab1\]
Klein form Type
------------------------- ---------------------------------------------------------- ------------ ------
Grassmannian $G_{k,n}= {\operatorname{SL}}(n)/P({\alpha}_k)$ $n\geq 2$ AIII
Odd quadrics $Q_{2n-1} = {\operatorname{Spin}}(2n+1) /P({\alpha}_1)$ $n\geq 2 $ BI
Even quadrics $Q_{2n-2} = {\operatorname{Spin}}(2n) /P({\alpha}_1)$ $n\geq 3 $ DI
Spinor variety $X={\operatorname{Spin}}(2n)/P({\alpha}_n)$ $n\geq 4$ DIII
Lagrangian Grassmannian $X= {\operatorname{Sp}}(n,{\mathbb{C}}) / P({\alpha}_n)$ $n \geq 2$ CI
$X= E_6 / P({\alpha}_1)$ EIII
$X= E_7 / P({\alpha}_7) $ EVII
: Irreducible Hermitian symmetric spaces of the compact type.
\[prop-J\] Let $X=G/P({\alpha}_k)$ be a compact irreducible Hermitian symmetric space and let ${\lambda}\in {\Lambda}$ be a nontrivial dominant weight. Then $$\label{eq:J-hom}
J(E_{\lambda}, -K_X) = \frac{ 2\, \dim W_{\lambda}} { \dim W_{\lambda}- \dim V_{\lambda}} \cdot
\frac{\xi_k({\lambda})}{\xi_k({\lambda}_{{\operatorname{ad}}})}.$$
[**Proof**. ]{}The tangent bundle to $X=G/P$ is the homogeneous bundle obtained from the representation of $P$ on ${\mathfrak{g}}/{\mathfrak{p}}$. For symmetric $X$ this is irreducible by Theorem \[classifico\], so Bott-Borel-Weil theorem and Lemma \[c1\] apply. Since $H^0(X,TX)= {\mathfrak{g}}=
W_{{\lambda}_{{\operatorname{ad}}}} $ (see [@akhiezer-libro p.75, p.131]), ${\mathfrak{g}}/{\mathfrak{p}}= V_{{{\lambda}_{{\operatorname{ad}}}}}$ and $TX=E_{{\lambda}_{{\operatorname{ad}}}}$. Set $m=\dim X =
\dim V_{{\lambda}_{{\operatorname{ad}}}} $. By Lemma \[c1\] $$\begin{gathered}
{\operatorname{c}}_1(-K_X) ={\operatorname{c}}_1(TX)= m \cdot
\frac{\xi_k({\lambda}_{{{\operatorname{ad}}}})}{\xi_k({\varpi}_k)} \, {\operatorname{c}}_1(E_{{\varpi}_k}) , \\
{\operatorname{c}}_1(E_{\lambda}) = \dim V_{\lambda}\cdot \frac{\xi_k({\lambda})}{\xi_k({\varpi}_k)}
\, {\operatorname{c}}_1(E_{{\varpi}_k}) , \\
\frac{ \langle c_1(E_{\lambda}) \cup c_1(-K_X)^{m-1} ,[X]\rangle } { \langle
c_1(-K_X)^m, [X]\rangle} = \frac{\dim V_{\lambda}}{m} \cdot
\frac{\xi_k({\lambda})}{\xi_k({\lambda}_{{\operatorname{ad}}})}.\end{gathered}$$ The rank of $E_{\lambda}$ is $\dim V_{\lambda}$, while $h^0(X,E_{\lambda})=\dim W_{\lambda}$ by Bott-Borel-Weil theorem. Therefore $$\begin{gathered}
J(E_{\lambda},-K_X)= \frac{2\, m\, h^0(E_{\lambda})} {r\, (h^0(E_{\lambda})-r)} \cdot
\frac{ \langle c_1(E_{\lambda}) \cup c_1(-K_X)^{m-1} ,[X]\rangle }
{ \langle c_1(-K_X)^m, [X]\rangle} = \\
= \frac{ 2\, m\, \dim W_{\lambda}} { \dim V_{\lambda}( \dim W_{\lambda}- \dim V_{\lambda})
} \cdot \frac{\dim V_{\lambda}}{m} \cdot
\frac{\xi_k({\lambda})}{\xi_k({\lambda}_{{\operatorname{ad}}})} = \\
= \frac{ 2\, \dim W_{\lambda}} { \dim W_{\lambda}- \dim V_{\lambda}} \cdot
\frac{\xi_k({\lambda})}{\xi_k({\lambda}_{{\operatorname{ad}}})}.\end{gathered}$$ [$\Box$\
]{}We are now ready for the proof of theorems 2 and 3.\
**Proof of Theorem \[classical-intro\].** Let $X$ be a compact irreducible Hermitian symmetric space. Denote by $g_{KE}$ the symmetric (Kähler-Einstein) metric with [Kähler ]{}form in $2\pi c_1(X)$. We need to show that $$I(-K_X) = 2 = {\lambda}_1(g_{KE}).$$ The second equality follows from Thm. \[futaki\]. So $ I(-K_X) \geq
2$ by definition . It is enough to prove that $I(-K_X) \leq 2$. For each space in the first five families in Table \[tab1\] we find a homogeneous bundle $E_{\lambda}{\rightarrow}X$ such that $J(E_{\lambda}, -K_X)=2$. The result then follows applying Thm. \[igi\]. The relevant information regarding weights and degrees can be found for example in [@humphreys-algebras p.66, p.69].\
1. The case of the Grassmannians (type $AIII$) is settled by hand in [@arezzo-ghigi-loi Thm. 1.3]. The vector bundle $E$ is the dual of the universal subbundle. If we choose the Borel group as in then $ E=E_{{\varpi}_1}$.\
2. For odd quadrics the Dynkin diagram is:
The largest root is ${\lambda}_{{\operatorname{ad}}}= {\varpi}_2$. Put ${\lambda}={\varpi}_n$. Then $W_{\lambda}$ is the spin representation, while $V_{\lambda}$ corresponds to the spin representation of the semisimple part $S({\alpha}_1) \cong
{\operatorname{Spin}}(2n-1)$ of $P({\alpha}_1)$ . The bundle $E_{\lambda}$ is the [spinor bundle]{} studied e.g. by Ottaviani [@ottaviani-spinor]. Of course $\dim W_{\lambda}= 2^{n}$, $\dim V_{\lambda}= 2^{n-1}$. Finally $\xi_1({\lambda}) =\xi_1({\varpi}_n) = 1/2$, $\xi_1({\lambda}_{{\operatorname{ad}}}) =\xi_1({\varpi}_2)
= 1$, so $$J(E_{\lambda}, -K_X) = \frac{ 2\, \dim W_{\lambda}} { \dim W_{\lambda}- \dim V_{\lambda}} \cdot
\frac{\xi_1({\lambda})}{\xi_1({\lambda}_{{\operatorname{ad}}})} =
2 \cdot \frac{2^{n}} {2^{n} -2^{n-1}} \cdot \frac{1/2}{1} = 2.$$\
3. The situation is very similar for even quadrics. The Dynkin diagram is:
The largest root is again ${\lambda}_{{\operatorname{ad}}}= {\varpi}_2$. We take $W_{\lambda}$ to be either one of the half-spin representation. ($E_{\lambda}$ is one of the two spinor bundles on $Q_{2n-2}$, [@ottaviani-spinor]). Say $W_{\lambda}=\mathcal{S}_+$. Then ${\lambda}= {\varpi}_n$ and $V_{\lambda}$ is the half-spin representation $\mathcal{S}_+$ of $S({\alpha}_1) \cong
{\operatorname{Spin}}(2n-2)$. Now $ \dim W_{\lambda}= 2^{n-1}$, $V_{\lambda}= 2^{n-2}$, $\xi_1({\varpi}_n) = 1/2$, $\xi_1({\varpi}_2) = 1$, so again $J(E_{\lambda}, -K_X)
=2$.\
4. For the Lagrangian Grassmannian the Dynkin diagram is:
The highest weight of the adjoint representation is ${\lambda}_{{\operatorname{ad}}}= 2
{\varpi}_1$. $W_{{\varpi}_1}$ is the standard representation of ${\operatorname{Sp}}(n,{\mathbb{C}})$ on ${\mathbb{C}}^{2n}$. The semisimple part of $P({\alpha}_n)$ is $S({\alpha}_n)={\operatorname{SL}}(n)$, so $V_{{\varpi}_1}$ is the standard representation of ${\operatorname{SL}}(n)$ on ${\mathbb{C}}^n$. So choosing $E=E_{{\varpi}_1}$ we get $$J(E, -K_X) = 2 \cdot \frac {2n}{2n -n} \cdot \frac{\xi_n({\varpi}_1) }
{\xi_n(2{\varpi}_1)} = 2.$$\
5. For the Spinor varieties the Dynkin diagram is:
Take $E=E_{{\varpi}_1}$. $W_{{\varpi}_1}$ is the standard representation of ${\operatorname{Spin}}(2n)$. The semisimple part of $P({\alpha}_n)$ is $S({\alpha}_n)={\operatorname{SL}}(n)$, so $V_{{\varpi}_1}$ is the standard representation of ${\operatorname{SL}}(n)$ on ${\mathbb{C}}^n$. The largest root is ${\lambda}_{{\operatorname{ad}}}= {\varpi}_2$, $\xi_n({\varpi}_1)=1/2$, $\xi_n({\varpi}_2) = 1$. So $$J(E, -K_X) =
2 \cdot \frac {2n}{2n -n} \cdot \frac{1/2 }{1} = 2.$$ [$\Box$\
]{}**Proof of Theorem \[exceptional-intro\].** 1. For $X=E_6/P({\alpha}_1)$ the Dynkin diagram (with Bourbaki numbering) is:
The largest root is ${\lambda}_{{\operatorname{ad}}}= {\varpi}_2$. An easy computation gives $J(E_{{\varpi}_6} , -K_X) $ $= 36/17$ and $J(E_{{\varpi}_2} , -K_X) =
78/31$. If ${\lambda}= \sum_i a_i {\varpi}_i$, then $$\begin{gathered}
J(E_{\lambda}, -K_X) \geq 2 \frac{\xi_1({\lambda})} {\xi_1({\varpi}_2)} =
\frac{8}{3} a_1 + 2 a_2 + \frac{10}{3} a_3 + 4 a_4 + \frac{8}{3} a_5
+ \frac{4}{3} a_6 .\end{gathered}$$ The right hand side is $< 36/17$ if and only if ${\lambda}={\varpi}_2$ or ${\lambda}={\varpi}_6$. Therefore the best estimate is gotten with ${\lambda}={\varpi}_6$.\
2. For $X=E_7/P({\alpha}_7)$ the Dynkin diagram (with Bourbaki numbering) is:
The largest root is ${\lambda}_{{\operatorname{ad}}}= {\varpi}_1$. We have $J(E_{{\varpi}_1} ,
-K_X) = 133/53$. If ${\lambda}= \sum_i a_i {\varpi}_i$, then $$\begin{gathered}
J(E_{\lambda}, -K_X) \geq 2 \frac{\xi_7({\lambda})} {\xi_7({\varpi}_1)} = 2
\xi_7({\lambda})
= \\
= 2 a_1 + 3 a_2 +4 a_3 + 6 a_4 + 5 a_5 + 4 a_6 + 3 a_7.\end{gathered}$$ The right hand side is $< 133/53$ if and only if ${\lambda}={\varpi}_1$. Therefore the best estimate is gotten with ${\lambda}={\varpi}_1$. [$\Box$\
]{}
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.3cm
Università degli Studi di Parma,\
*E-mail:* `leonardo.biliotti@unipr.it`
.3cm
Università di Milano Bicocca,\
*E-mail:* `alessandro.ghigi@unimib.it`
|
---
abstract: 'We prove that each lower-dimensional face of a quasi-arithmetic Coxeter polytope, which happens to be itself a Coxeter polytope, is also quasi-arithmetic. We also provide a sufficient condition for a codimension $1$ face to be actually arithmetic, as well as a few computed examples.'
address:
- 'Skolkovo Institute of Science and Technology, Skolkovo, Russia'
- 'Moscow Institute of Physics and Technology (State University), Dolgoprudny, Moscow region, Russia'
- 'Caucasus Mathematical Centre, Adyghe State University, Maikop, Russia'
- 'Institut de Mathématiques, Université de Neuchâtel, 2000 Neuchâtel, Suisse/Switzerland'
author:
- Nikolay Bogachev
- Alexander Kolpakov
title: 'On faces of quasi-arithmetic Coxeter polytopes'
---
Introduction {#section:intro}
============
Let $\mathbb{X}^n$ be one of the three spaces of constant curvature, i.e. either the Euclidean $n$-space ${\mathbb{E}}^n$, or the $n$-dimensional sphere ${\mathbb{S}}^n$, or the $n$-dimensional hyperbolic (Lobachevsky) space ${\mathbb{H}}^n$.
Let $P$ be a convex polytope in ${\mathbb{X}}^n$. The group $\Gamma$ generated by reflections in the supporting hyperplanes of the facets (i.e. codimension $1$ faces) of $P$ is a *discrete reflection group*, if the images of $P$ under the action of $\Gamma$ tessellate $\mathbb{X}^n$ (i.e. $\mathbb{X}^n$ is entirely covered by copies of $P$, such that their interiors do not overlap). The polytope $P$ in this case is *the fundamental polytope* for $\Gamma$. In particular, $\Gamma$ being discrete implies that any two hyperplanes $H_i$ and $H_j$ bounding $P$ either do not intersect or form a dihedral angle of $\pi/n_{ij}$, where $n_{ij} \in {\mathbb{Z}}$, $n_{ij} \geq 2$.
If $P$ is compact, then $\Gamma$ is called *a cocompact reflection group*, and if $P$ has finite volume (in which case $P$ may or may not be compact), then $\Gamma$ is called *cofinite* or a discrete group of *finite covolume*.
Discrete reflection groups of finite covolume acting on spheres and Euclidean spaces were classified by Coxeter in 1933 [@Cox34]. Therefore, their fundamental polytopes are called Coxeter polytopes. They belong to the class of so-called *acute-angled polytopes*, i.e. those whose dihedral angles are less than or equal to $\pi/2$.
Suppose that $\mathbb{F} \subset {\mathbb{R}}$ is a totally real number field, and let ${\mathcal{O} }_{\mathbb{F}}$ denote its ring of integers. Let $G = \mathbf{PO}(n,1)$ be the isometry group of ${\mathbb{H}}^n$, and $\widetilde{G}$ be an *admissible* simple algebraic ${\mathbb{F}}$-group, i.e. $\widetilde{G}({\mathbb{R}}) = G$ and ${\widetilde{G}}^\sigma({\mathbb{R}})$ is a compact group for any non-identity embedding $\sigma \colon \mathbb{F} \to \mathbb{R}$[^1].
It is known from the general theory, c.f. [@BHC62] and [@MT62], that if $\Gamma \subset G$ is commensurable[^2] with $\widetilde{G}({\mathcal{O} }_{\mathbb{F}})$ then $\Gamma$ is a *lattice* (i.e. a discrete isometry group with a finite volume fundamental polytope). Such groups are called *arithmetic*.
A lattice $\Gamma \subset G = \widetilde{G}({\mathbb{R}})$ is called *quasi-arithmetic* if $\Gamma \subset \widetilde{G}({\mathbb{F}})$ and *properly quasi-arithmetic* if it is quasi-arithmetic, but not actually arithmetic.
The history of discrete reflection groups acting on Lobachevsky spaces and, in particular, arithmetic reflection groups goes back to the 19th century, to the works of Poincare, Fricke, and Klein. A systematic study was started by Vinberg in 1967 [@Vin67]. Namely, he developed practically efficient methods that allow to determine compactness or volume finiteness of a given Coxeter polytope according to its Coxeter diagram, and provided a (quasi-)arithmeticity criterion for hyperbolic reflection groups. Later on, Vinberg created an algorithm (colloquially known as “Vinberg’s algorithm”) [@Vin72], that constructs a fundamental Coxeter polytope for any hyperbolic reflection group. Practically, it is most efficient for arithmetic reflection groups associated with Lorentzian lattices (see **§** \[sec:prel\], **§** \[section:t\], **§** \[section:l\]).
Due to the further results of Vinberg [@Vin84], Long, Maclachlan, Reid [@LMR05], Agol [@Ag06], Nikulin [@Nik81; @Nik07], Agol, Belolipetsky, Storm and Whyte [@ABSW2008], it became known that there are only finitely many maximal arithmetic hyperbolic reflection groups in all dimensions. Moreover, compact Coxeter polytopes in $\mathbb{H}^n$ can exist only for $n < 30$ [@Vin84] and finite volume Coxeter polytopes do not exist in ${\mathbb{H}}^{> 995}$ [@Hov86; @Pro86].
Let $P\subset {\mathbb{H}}^n$ be a finite volume hyperbolic Coxeter polytope. We say that $P$ is (quasi-) arithmetic with ground field ${\mathbb{F}}$ if it is a fundamental domain for a hyperbolic reflection group $\Gamma \subset \operatorname*{Isom}({\mathbb{H}}^n)$ with ground field ${\mathbb{F}}$ and $\Gamma$ is (quasi-)arithmetic. Here, $\Gamma = \Gamma(P)$ is the group generated by reflections in the bounding hyperplanes of $P$. By [@Vin84] arithmetic Coxeter polytopes with ${\mathbb{F}}\ne {\mathbb{Q}}$ in $\mathbb{H}^n$, can exist only for $n < 30$ (because of compactness), and for $14 \le n < 30$ only finitely many ground fields ${\mathbb{F}}$ are possible. Non-compact arithmetic Coxeter polytopes (i.e. for ${\mathbb{F}}= {\mathbb{Q}}$) can exist only for $n \le 21$, $n \ne 20$ [@Ess96].
Recently, Belolipetsky and Thomson provided infinitely many commensurability classes of properly quasi-arithmetic hyperbolic lattices in any dimension $n > 2$ (cf. [@BT11; @Thom16]), as well as Emery [@Em17] proved that the covolume of any quasi-arithmetic hyperbolic lattice is a rational multiple of the covolume of an arithmetic subgroup.
By considering faces of a higher-dimensional Coxeter polytope in $\mathbb{H}^n$, one can try to find more examples of lower-dimensional Coxeter polytopes in $\mathbb{H}^k$, $k<n$. A good indication that such lower-dimensional Coxeter faces should appear is the work by Allcock [@Allcock Theorems 2.1 & 2.2]. Even earlier, Borcherds [@Bor87] used a $25$-dimensional infinite volume polytope due to Conway in order to build a $21$-dimensional Coxeter polytope of finite volume as its face.
In the context of passing to lower-dimensional faces, our first theorem shows that quasi-arithmeticity is inherited from the initial polytope.
\[th-face\] Let $P$ be a quasi-arithmetic Coxeter polytope in ${\mathbb{H}}^n$ with ground field ${\mathbb{F}}$ and $P'$ be its $k$-dimensional face, for $2 \le k \le n-1$. If $P'$ is itself a Coxeter polytope, then $P'$ is also quasi-arithmetic with ground field ${\mathbb{F}}$.
In the sequel, we enhance the definition of quasi-arithmeticity to all acute-angled polytopes, c.f. Definition \[def-quasi-arithm\]. Then, we have the following easy consequence.
\[cor\] Let $P$ be a Coxeter polytope in ${\mathbb{H}}^n$ with ground field ${\mathbb{F}}= {\mathbb{F}}(P)$, and let $P'$ be a codimension $k \geq 1$ face of $P$ with ground field ${\mathbb{F}}' = {\mathbb{F}}(P')$. If $|{\mathbb{F}}:{\mathbb{F}}'| > 1$, then $P$ cannot be quasi-arithmetic.
Another result shows that arithmetic polytopes can often have arithmetic Coxeter facets.
\[th-facet-arithm\] Let $P$ be an arithmetic Coxeter polytope in ${\mathbb{H}}^n$ with ground field ${\mathbb{F}}$ and $P'$ be its facet (i.e. a codimension $1$ face). Moreover, assume that $P'$ meets all its adjacent facets at dihedral angles of the form $\frac{\pi}{2m}$, for some natural $m\geq 1$, not all necessarily equal to each other. Then $P'$ is itself an arithmetic Coxeter polytope with ground field ${\mathbb{F}}$.
The paper is organised as follows. In **§** \[sec:prel\] some preliminary facts are given. Then, **§** \[sec:th1\] is devoted to the proof of Theorem \[th-face\] and **§** \[sec:th2\] is devoted to the proof of Theorem \[th-facet-arithm\].
At the end of the paper, some computed examples are provided (c.f. **§** \[section:t\]–**§** \[section:l\]) in order to illustrate the above theorems, as well as to enrich the collection of known Coxeter polytopes. All computations were performed by using `SageMath` computer algebra system [@Sage].
The closing remarks (**§** \[sec:open\]) contain a number of questions that we find quite enticing, although likely hard to approach.
Acknowledgements. {#acknowledgements. .unnumbered}
-----------------
The authors are grateful to È. B. Vinberg and D. Allcock for their encouragement and helpful remarks. N.B. was supported by the Russian Science Foundation (project no. 18-71-00153). A.K. was supported by the Swiss National Science Foundation (project no. PP00P2-170560). N.B. is thankful to Institut des Hautes Études Scientifiques — IHES, and especially to Fanny Kassel, for their hospitality while this work was carried out.
Preliminaries {#sec:prel}
=============
Let $\mathbb{F} \subset {\mathbb{R}}$ be a totally real number field, and let ${\mathcal{O} }_{\mathbb{F}}$ denote its ring of integers. For convenience, we assume that ${\mathcal{O} }_{\mathbb{F}}$ is a principal ideal domain.
A free finitely generated ${\mathcal{O} }_{\mathbb{F}}$–module $L$ with an inner product of signature $(n,1)$ is called a $\textit{Lorentzian lattice}$ if, for any non-identity embedding $\sigma \colon \mathbb{F} \to \mathbb{R}$, the quadratic space $L \otimes_{\sigma({\mathcal{O} }_{\mathbb{F}})} \mathbb{R}$ is positive definite (we say that the inner product in $L$ is associated with some admissible Lorentzian quadratic form).
As a matter of terminology, Lorentzian lattices, as defined in the present paper, are also called hyperbolic lattices, cf. [@BP18; @Bog19; @Bog19a; @Vin84].
Let $L$ be a Lorentzian lattice. Then the vector space ${\mathbb{E}}^{n, 1} = L \otimes_{{\mathrm{Id} }({\mathcal{O} }_{\mathbb{F}})} {\mathbb{R}}$ is identified with the $(n + 1)$-dimensional real *Minkowski space*. The group $\Gamma = \mathcal{O}'(L)$ of integer (i.e. with coefficients in ${\mathcal{O} }_{\mathbb{F}}$) linear transformations preserving the lattice $L$ and mapping each connected component of the cone $$\mathfrak{C} = \{v \in {\mathbb{E}}^{n,1} \mid (v, v) < 0\} = \mathfrak{C}^+ \cup \mathfrak{C}^-$$ onto itself is a discrete group of motions of the Lobachevsky space ${\mathbb{H}}^n$. Here and below we use the *hyperboloid model* for ${\mathbb{H}}^n$, which is the set $\{v \in {\mathbb{E}}^{n,1} \cap \mathfrak{C}^+ \mid (v, v) = -1\}$. Its isometry group is $$\operatorname*{Isom}({\mathbb{H}}^n) = {\mathbf{PO}}(n,1) \simeq {\mathrm{O}}(L \otimes_{{\mathrm{Id} }({\mathcal{O} }_{\mathbb{F}})} {\mathbb{R}})/\{\pm {\mathrm{Id} }\},$$ which is the group of orthogonal transformations of the Minkowski space ${\mathbb{E}}^{n, 1}$ that leaves $\mathfrak{C}^+$ invariant.
It is known [@BHC62; @MT62] that if ${\mathbb{F}}= {\mathbb{Q}}$ and the lattice $L$ is isotropic (that is, the quadratic form associated with it represents zero), then the quotient space ${\mathbb{H}}^n/\Gamma$ (the fundamental domain of $\Gamma$) is not compact, but is of finite volume, and in all other cases it is compact. The case ${\mathbb{F}}= {\mathbb{Q}}$ was first studied by Venkov [@Ven37].
Any group $\Gamma$ obtained in the way described above, and any subgroup of $\operatorname*{Isom}({\mathbb{H}}^n)$ commensurable to such one, is called an arithmetic group (or lattice) of simplest type. The field ${\mathbb{F}}$ is called the ground field (or the field of definition) of $\Gamma$.
A primitive vector $e$ of a Lorentzian lattice $L$ is called a *root* or, more precisely, a *$k$-root*, where $k = (e, e) \in ({\mathcal{O} }_{\mathbb{F}})_{>0}$ if $2\cdot (e, x) \in k {\mathcal{O} }_{\mathbb{F}}$ for all $x \in L$. Every root $e$ defines an *orthogonal reflection* (called a *$k$-reflection* if $(e, e) = k$) in $L \otimes_{{\mathrm{Id} }({\mathcal{O} }_{\mathbb{F}})} \mathbb{R}$ given by $$\mathcal {R}_e: x \mapsto x - \frac{2 (e, x)} {(e, e)} e,$$ which preserves the lattice $L$. Geometrically speaking, this is a reflection of $\mathbb{H}^n$ with respect to the hyperplane $H_e = \{x \in \mathbb {H} ^ n \mid (x, e) = 0 \},$ called the *mirror* of $\mathcal{R}_e$.
Let ${\mathcal{O} }_r (L)$ denote the subgroup of ${\mathcal{O} }'(L)$ generated by all reflections contained in it.
A Lorentzian lattice $L$ is called reflective if the index $[{\mathcal{O} }'(L): {\mathcal{O} }_r(L)]$ is finite.
Clearly, $L$ is reflective if and only if the group ${\mathcal{O} }_r(L)$ has a finite volume fundamental Coxeter polytope.
The results described in **§** \[section:intro\] give hope that all reflective Lorentzian lattices, as well as maximal arithmetic hyperbolic reflection groups, can be classified. For more information about the recent progress on the classification problem c.f. [@Bel16; @Bog19; @Bog19a].
By a result of Vinberg [@Vin67 Lemma 7], any arithmetic hyperbolic reflection group is an arithmetic lattice of simplest type with ground field ${\mathbb{F}}$, and therefore is commensurable with ${\mathcal{O} }_r(L)$, where $L$ is some (necessarily reflective) Lorentzian lattice over a totally real number field ${\mathbb{F}}$. It also shows that every quasi-arithmetic hyperbolic reflection group is contained in a group ${\mathrm{O}}(f, {\mathbb{F}})$, where $f$ is some admissible Lorentzian form over ${\mathbb{F}}$.
As a consequence of Vinberg’s algorithm [@Vin72] (its recent software implementations are `AlVin` [@Guglielmetti2], for Lorentzian lattices with an orthogonal basis over several ground fields, and `VinAl` (cf. [@VinAlg2017; @BP18]), for Lorentzian lattices with an arbitrary basis over ${\mathbb{Q}}$), reflective lattices provide many explicit examples of arithmetic reflection groups and arithmetic Coxeter polytopes. In contrast, finding properly quasi-arithmetic reflection groups appears to be a harder task.
The record example of a compact Coxeter polytope was found by Bugaenko in dimension $8$ [@Bug92], although the maximal possible dimension is bounded by $30$.
The record example of a finite volume Coxeter polytope is due to Borcherds in dimension $21$ [@Bor87]. It is known that Coxeter polytopes of finite volume can exist only in dimensions smaller than $996$ [@Hov86; @Pro86].
It is worth mentioning that both these examples come from arithmetic reflection groups.
For every hyperplane $H_e$ with unit normal $e$, let us set $H_e^- = \{x \in {\mathbb{E}}^{n,1} \mid (x,e) \le 0\}$. If $$P = \bigcap_{j=0}^N H_{e_j}^-$$ is an acute-angled polytope of finite volume in ${\mathbb{H}}^n$, then $G(P) = G(e_0, \ldots, e_N)$ is its Gram matrix, ${\widetilde{\mathbb{F}}}(P) = {\mathbb{Q}}[\{g_{ij}\}^N_{i,j=0}]$, ${\mathbb{F}}(P)$ denotes the field generated by all possible cyclic products of the entries of $G(P)$ and is called the ground field of $P$. The set of all cyclic products of entries of a matrix $A$ (i.e. the set consisting of all possible products of the form $a_{i_1 i_2} a_{i_2 i_3} \ldots a_{i_k i_1}$) will be denoted by ${\mathrm{Cyc}}(A)$. Thus, ${\mathbb{F}}(P) = {\mathbb{Q}}[{\mathrm{Cyc}}(G(P))] \subset {\widetilde{\mathbb{F}}}(P)$.
Given a finite covolume hyperbolic reflection group $\Gamma$, the following criterion allows us to determine if $\Gamma$ is arithmetic, quasi-arithmetic, or neither.
\[V\] Let $\Gamma$ be a cofinite reflection group acting on ${\mathbb{H}}^n$ with fundamental Coxeter polytope $P$. Then $\Gamma$ is arithmetic if and only if the following conditions hold:
- ${\widetilde{\mathbb{F}}}(P)$ is a totally real algebraic number field;
- for any embedding $\sigma \colon \widetilde{\mathbb{F}}(P) \to {\mathbb{R}}$, such that $\sigma\!\mid_{{\mathbb{F}}(P)} \ne {\mathrm{Id} }$, $G^\sigma(P)$ is positive semi-definite;
- ${\mathrm{Cyc}}(2 \cdot G(P)) \subset {\mathcal{O} }_{{\mathbb{F}}(P)}$.
A cofinite reflection group $\Gamma$ acting on ${\mathbb{H}}^n$ with fundamental polytope $P$ is quasi-arithmetic if and only if it satisfies conditions **(V1)**–**(V2)**, but not necessarily **(V3)**.
Obviously, every arithmetic reflection group is quasi-arithmetic. For most of our arguments, we shall need a wider definition of quasi-arithmetic polytopes that will allow us to capture their main geometric and algebraic properties without paying attention to whether their dihedral angles are of Coxeter type. This will become important in order to transfer from higher-dimensional faces to lower-dimensional ones in a chain of inclusions, which does not necessarily consist entirely of Coxeter polytopes, c.f. Proposition \[quasi-facet:acute\].
\[def-quasi-arithm\] Let $P$ be a finite volume acute-angled polytope in ${\mathbb{H}}^n$, and let it have well-defined fields ${\widetilde{\mathbb{F}}}= {\widetilde{\mathbb{F}}}(P)$, ${\mathbb{F}}= {\mathbb{F}}(P) \subset {\widetilde{\mathbb{F}}}$, as described above. Then $P$ is called quasi-arithmetic if ${\widetilde{\mathbb{F}}}$ and ${\mathbb{F}}$ satisfy conditions **(V1)**–**(V2)**.
After presenting the proof of Theorem \[th-face\], we provide some computed examples of Coxeter polytopes and their faces. For each polytope $P \subset \mathbb{H}^n$, let its *facet tree* $\mathcal{T}(P)$ be a rooted tree with root $P$, where each vertex of level $0 \leq i \leq n-3$ represents the isometry type of a codimension $i$ face $P^{(i)}_j$ of $P$, while the level $i+1$ descendants of $P^{(i)}_j$ represent all distinct isometry types of codimension $1$ faces of $P^{(i)}_j$. Thus, $\mathcal{T}(P)$ shows all possible isometry types of faces of $P$ (in codimensions $1$ through $n-2$), as well as the set of mutually non-isometric facets for each lower-dimensional face. Since $\mathcal{T}(P)$ does not take face adjacency into account, it represents only some features of the geometric structure of $P$. Mostly, we shall be interested in determining its Coxeter facets, and classifying them into arithmetic and properly quasi-arithmetic ones.
Proof of Theorem \[th-face\] {#sec:th1}
============================
Auxiliary results
-----------------
Below we formulate a few auxiliary lemmas. Their main point is that the properties **(V1)**–**(V2)** are inherited by facets of an acute-angled quasi-arithmetic polytope.
In **§** 3.1 – **§** 3.2, $P$ always denotes a finite volume acute-angled polytope in ${\mathbb{H}}^n$.
\[lemma:projections\] Let $P' \subset H_{e_0}$ be a facet of $P$, bounded by the respective hyperplanes $H_{e_1}, \ldots, H_{e_k}$ of $P$ (i.e. $H_{e_0} \cap H_{e_j}$ are the supporting hyperplanes of $P'$ in $\mathbb{H}^{n-1}$). Let $G(P) = \{ g_{ij} \}$ be the Gram matrix of $P$. Then the Gram matrix $G(P') = \{ g'_{ij} \}$ of $P'$ has entries $$g'_{ij} = \frac{g_{ij} - g_{0i}\, g_{0j}}{\sqrt{(1-g^2_{0i})\,(1-g^2_{0j})}}.$$
Let $e^0_i$, $e^0_j$ be the projections of $e_i$, $e_j$ onto the hyperplane $H_{e_0}$. Then we obtain $e^0_j = e_j - (e_j,e_0)e_0$ and $(e^0_i, e^0_j) = (e_i, e_j) - (e_0, e_i)(e_0, e_j)$. Therefore, the lemma follows by setting $e'_i = \frac{e^0_i}{\| e^0_i \|}$, $e'_j = \frac{e^0_j}{\| e^0_j \|}$ to be the respective unit normals of $P'$.
\[lemma:Gram\] In the notation of Lemma \[lemma:projections\] and its proof, let $G'_s = G(e'_1, \ldots, e'_s)$ be a corner submatrix of $G(P')$, and let $G_s = G(e_0, e_1, \ldots, e_s)$ be the respective corner submatrix of $G(P)$. Then $\det G'_s \geq 0$ if and only if $\det G_s \geq 0$.
Let us consider the matrix $G_s$. One can perform the following transformation on its rows: the $j$-th row ($1 \leq j \leq s$) is replaced by the difference of itself and the row corresponding to $e_0$ multiplied by $g_{j 0}$.
After that, each $i$-th column ($1 \leq i \leq s$) of the resulting matrix is divided by $\| e^0_i \| = \sqrt{1 - g^2_{0i}}$, and each $j$-th row ($1 \leq j \leq s$) is divided by $\| e^0_j \| = \sqrt{1 - g^2_{0j}}$.
According to Lemma \[lemma:projections\], this transformation results in the matrix $$G''_s =
\begin{pmatrix}
1 & * \\
0 & G'_s(e'_1,\ldots, e'_s)
\end{pmatrix},$$ and thus, we have $\det G_s = \kappa^2 \cdot \det G''_s = \kappa^2 \cdot \det G'_s$, where $$\kappa^{-2} = \prod_{i,j=1}^s \|e^0_i\| \cdot \|e^0_j\| = \prod_{i=1}^s \|e^0_i\|^2.$$
\[lemma:field-inclusion\] Under the above assumptions, ${\mathbb{F}}(P') = {\mathbb{F}}(P)$.
Let ${\mathbb{F}}= {\mathbb{Q}}[{\mathrm{Cyc}}(G(P))]$ and ${\mathbb{F}}' = {\mathbb{Q}}[{\mathrm{Cyc}}(G(P'))]$. We shall consider a cyclic product $$\label{eq:cycl}
g'_{i_1 i_2} g'_{i_2 i_3} \ldots g'_{i_s i_1} =
\frac{g_{i_1 i_2} - g_{0 i_1}\, g_{0 i_2}}{\sqrt{(1-g^2_{0 i_1})\,(1-g^2_{0 i_2})}}
\frac{g_{i_2 i_3} - g_{0 i_2}\, g_{0 i_3}}{\sqrt{(1-g^2_{0 i_2})\,(1-g^2_{0 i_3})}}
\ldots
\frac{g_{i_s i_1} - g_{0 i_s}\, g_{0 i_1}}{\sqrt{(1-g^2_{0 i_s})\,(1-g^2_{0 i_1})}}.$$ The denominator of the above expression equals $$(1-g^2_{0 i_1})(1-g^2_{0 i_2}) \ldots (1-g^2_{0 i_s}) \in {\mathbb{F}}.$$ Thus, it remains to consider the numerator $$\label{eq:cycl3}
(g_{i_1 i_2} - g_{0 i_1}\, g_{0 i_2})
(g_{i_2 i_3} - g_{0 i_2}\, g_{0 i_3})\ldots
(g_{i_s i_1} - g_{0 i_s}\, g_{0 i_1}).$$ While expanding the above expression, we take from each pair of parentheses either $g_{i_m i_{m+1}}$ or $g_{0 i_m } g_{0 i_{m+1}}$, and obtain a sum of cyclic products where each term looks like $g_{i_1 i_2} g_{i_2 i_3} \ldots g_{i_s i_1}$ with some terms of the form $g_{i_m i_{m+1}}$ being replaced by the respective products $g_{0 i_m } g_{0 i_{m+1}}$.
This is equivalent to replacing some transpositions $(i_m, i_{m+1})$ in $(i_1, i_2)(i_2, i_3) \ldots (i_{s-1}, i_s) = (i_1, i_2, \ldots,$ $i_s)$, with $(0, i_m)(0, i_{m+1})$. Obviously, this operation creates another permutation with a different cycle structure, where either new non-trivial cycles of the elements in $I = \{0, i_1, i_2, \ldots, i_s\}$ will be formed, or some fixed points appear. The latter happens whenever a square of some $g_{ij}$, $i,j \in I$, is present. Thus, each term in after expansion is a cyclic product from ${\mathrm{Cyc}}(G(P))$.
Therefore, a cyclic product of the form (\[eq:cycl\]) is a linear combination of cyclic products of $G(P)$ divided by some elements of the field ${\mathbb{F}}$. This implies $g'_{i_1 i_2} g'_{i_2 i_3} \ldots g'_{i_s i_1} \in {\mathbb{F}}$, and hence ${\mathbb{F}}' \subset {\mathbb{F}}$.
Now, let us suppose that ${\mathbb{F}}' \neq {\mathbb{F}}$. This implies that there exists an embedding $\sigma: {\mathbb{F}}\to {\mathbb{R}}$ such that $\sigma|_{{\mathbb{F}}'} = {\mathrm{Id} }$. Then $G^\sigma(P)$ is positive semi-definite, and so is $G^\sigma(P')$, by Lemma \[lemma:Gram\]. However, the latter is impossible, since $G^\sigma(P') = G(P')$ is the Gram matrix of a hyperbolic polytope.
Quasi-arithmeticity of a facet of a quasi-arithmetic acute-angled polytope
--------------------------------------------------------------------------
\[quasi-facet:acute\] Let $P$ be a quasi-arithmetic acute-angled polytope in ${\mathbb{H}}^n$ with ground field ${\mathbb{F}}= {\mathbb{F}}(P)$, and let $P'$ be its facet. Then $P'$ is also a quasi-arithmetic acute-angled polytope with the same ground field ${\mathbb{F}}' = {\mathbb{F}}(P') = {\mathbb{F}}$.
It is well-known that a face of an acute-angled polytope is also acute-angled. Now the proof follows from verifying conditions **(V1)**–**(V2)** of Vinberg’s arithmeticity criterion (Theorem \[V\]).
*Verification of **(V1)**.* The field $\widetilde{{\mathbb{F}}'}$ equals $\widetilde{{\mathbb{F}}}[\{k^2_{ij}\}^k_{i,j=1}]$, where $k^{-1}_{ij} = \|e^0_i\|\cdot \|e^0_j\|$, and thus is a finite extension of $\widetilde{{\mathbb{F}}}$. It remains to show that $\widetilde{{\mathbb{F}}'}$ is totally real. Instead, we prove that the larger field $\widetilde{{\mathbb{F}}''} = \widetilde{{\mathbb{F}}}[\{\sqrt{1 - g^2_{0i}}\}^k_{i=1}]$ is totally real, so that $\widetilde{{\mathbb{F}}'} \subset \widetilde{{\mathbb{F}}''}$ is totally real, as well.
To this end, recall that $|g_{0i}| = \cos \angle(H_0, H_i) \le 1$, and thus $1 - g^2_{0i} \geq 0$. Also, $1 - g^2_{0i} = \det G(e_0, e_i)$ is a corner minor of $G(P)$ and thus remains positive for any non-identity embedding by Theorem \[V\]. Thus $\widetilde{{\mathbb{F}}}[\sqrt{1 - g^2_{0i}}]$ is totally real and, consequently, so is $\widetilde{{\mathbb{F}}''}$.
*Verification of **(V2)**.* By Lemma \[lemma:Gram\] we have that, up to squares in ${\mathbb{F}}$, all the corner minors of $G' = G(P')$ coincide with the corresponding corner minors of $G = G(P)$. Since ${\mathbb{F}}' = {\mathbb{F}}$ by Lemma \[lemma:field-inclusion\], we have that $\det G^\sigma_s \geq 0$ for every embedding $\sigma: \widetilde{{\mathbb{F}}} \to {\mathbb{R}}$, such that $\sigma|_{{\mathbb{F}}} \ne {\mathrm{Id} }$, by Theorem \[V\], and thus $\det (G')^\sigma_s \geq 0$.
Proof of Theorem \[th-face\] {#proof-of-theoremth-face}
----------------------------
Let $P$ be a quasi-arithmetic Coxeter polytope in ${\mathbb{H}}^n$ with ground field ${\mathbb{F}}= {\mathbb{F}}(P)$, and let $P'$ be its face of any dimension $\ge 2$ that is also a Coxeter polytope. Then we need to prove that $P'$ is also a quasi-arithmetic Coxeter polytope with the same ground field ${\mathbb{F}}$.
Clearly, the face $P'$ can be included in the following chain of polytopes, by inclusion: $$P' = P_1 \subset P_2 \subset \ldots \subset P_t = P,$$ where $P_j$ is a facet of $P_{j+1}$, for every $1 \leq j \leq t-1$.
Since $P = P_t$ is a quasi-arithmetic Coxeter polytope, then it is an acute-angled one and, by Proposition \[quasi-facet:acute\], $P_{t-1}$ is also a quasi-arithmetic acute-angled polytope with the same ground field. Thus, clearly, each $P_j$ is a quasi-arithmetic acute-angled polytope with ${\mathbb{F}}(P_{j}) = {\mathbb{F}}(P_{j+1})$ for $1 \le j \le t-1$.
A Coxeter prism and its ground field
------------------------------------
The $3$-dimensional compact prism $P \subset \mathbb{H}^3$ with Coxeter diagram depicted in Figure \[prism\] has one of its bases $P'$ (namely, facet 1) orthogonal to all neighbours (namely, facets 3, 4, and 5). The Coxeter diagram of $P'$ is the sub-diagram in the diagram of $P$ spanned by vertices 3, 4, and 5.
(5.5,0) circle \[radius=.1\] node \[above\] [$1$]{} node \[right\] [ $\cosh \ell = \frac{1}{2} \sqrt{5 + 3 \sqrt{2} + 2 \sqrt{5} + \sqrt{10}}$]{} (3,0) circle \[radius=.1\] node \[above\] [$2$]{} (0,-2) circle \[radius=.1\] node \[below\] [$3$]{} (-3,0) circle \[radius=.1\] node \[above\] [$4$]{} (0,2) circle \[radius=.1\] node \[above\] [$5$]{} (3,0) – (0,-2) (0,-1.9) – (-3,0.1) (0,-2.1) – (-3,-0.1) (3,0) – (0,2) (0,2.1) – (-3,0.1) (0,2) – (-3,0) (0,1.9) – (-3,-0.1) ; (5.5,0) – node\[above\] [$\ell$]{} (3,0);
From the Coxeter diagram in Figure \[prism\], one easily gets that ${\mathbb{F}}(P) = {\mathbb{Q}}[\sqrt{2}, \sqrt{5}]$, while ${\mathbb{F}}(P') = {\mathbb{Q}}[\sqrt{5}]$. Thus, $P$ cannot be quasi-arithmetic by Corollary \[cor\]. Another obstruction to quasi-arithmeticity is the fact that $\cosh^2 \ell$ is not totally positive. However, we do not need to make this computation in order to come up with our conclusion.
Proof of Theorem \[th-facet-arithm\] {#sec:th2}
====================================
We start with an auxiliary lemma, that will become useful in the computations below.
Even algebraic integers.
------------------------
Let $\rho_m = \sin^{-2} \frac{\pi}{2m}$, for $m\geq 2$. We claim that $\rho_m$ is *even*, meaning that $\frac{\rho_m}{2}$ is an algebraic integer.
\[lemma:even\_integer\] For all $m\geq 2$, we have that $\frac{\rho_m}{2}$ is an algebraic integer.
Let $p_m(z)$ be the following function: $$p_m(z) = \left\{ \begin{array}{cc}
T_m\left( \frac{z}{\sqrt{2}} \right) \cdot T_m\left( -\frac{z}{\sqrt{2}} \right), & \text{if $m$ is even}, \\
U_m\left( \frac{z}{\sqrt{2}} \right) \cdot U_m\left( -\frac{z}{\sqrt{2}} \right) + 1, & \text{if $m$ is odd}, \\
\end{array} \right.$$ where $T_m$, resp. $U_m$, is the $m$-th Chebyshev polynomial of the $1$st, resp. $2$nd, kind. Their basic properties and some relations that will be used below are collected in [@AbrSte Chapter 22]. From the relation $T_{m}(z) = T_{m/2}(2z^2 - 1)$, for even $m$, we see that $T_m\left(\frac{z}{\sqrt{2}}\right)$ is a polynomial. From the recurrence $U_{m+1}(z) = 2z U_m(z) - U_m(z)$, with $U_0(z)=1$ and $U_1(z)=2z$, we get that $U_m(z)$ is a polynomial in $2z$. This, together with the fact that the odd powers of $\frac{z}{\sqrt{2}}$ in the above expression for $p(z)$ cancel out, means that $p(z)$ is a polynomial with integer coefficients. Moreover, $T_m(z)$ has constant term $\pm 1$ for even $m$, and $U_m(z)$ has constant term $0$ for odd $m$. Thus, $p_m(z)$ has constant term $1$. By using the trigonometric definitions of $T_m(z)$ and $U_m(z)$, let us observe that $\tau_m = \sqrt{2} \cdot \sin \frac{\pi}{2m}$ is a root of $p_m(z)$.
Let $\widetilde{p}_m(z) = z^{\deg p_m} \cdot p_m(z^{-1})$ be the reciprocal of $p_m(z)$. Then $\widetilde{p}_m(z)$ is a unitary polynomial having $\tau^{-1}_m = \sqrt{\frac{\rho_m}{2}}$ among its roots. Hence, $\frac{\rho_m}{2}$ is an algebraic integer.
Proof of Theorem \[th-facet-arithm\].
-------------------------------------
Let $P'$ be a facet of an arithmetic Coxeter polytope with ground field ${\mathbb{F}}$. Let $H_{e_0}$ be the supporting hyperplane of $P'$ as a facet of $P$, and let $F_i$, $i=1, \ldots, s$, be the facets of $P'$. Let $H_{e_i}$, $i = 1, \ldots, s$ be the hyperplanes of $P$ such that $H_{e_i} \cap H_{e_0}$ is the supporting hyperplane of $F_i$ in $H_{e_0}$. Since the stabiliser of $H_{e_i} \cap H_{e_0}$ in the reflection group of $P$ is the dihedral group of order $4m$, and $H_{e_0}$ is one of its mirrors, there exists another mirror $H'_i$ (not necessarily a supporting hyperplane for $P$), such that $H'_i$ and $H_{e_0}$ are orthogonal, while $F_i \subset H'_i \cap H_{e_0}$. Thus, the facets of $P'$ come from orthogonal projections of some of the mirrors of the reflection group of $P$ onto the hyperplane $H_{e_0}$. Therefore, $P'$ is a Coxeter polytope.
By Theorem \[th-face\], $P'$ is quasi-arithmetic with ground field ${\mathbb{F}}$, and it remains to verify condition **(V3)** of Vinberg’s arithmeticity criterion.
Each cyclic product from ${\mathrm{Cyc}}(2 G(P'))$ has the form $2^s g'_{i_1 i_2} g'_{i_2 i_3} \ldots g'_{i_s i_1}$, which is similar to . Taking into account that the denominator in has each $g_{0i_k} = \cos \frac{\pi}{2m_{i_k}}$, it can be written as $$\label{eq:cycle4}
2^s \cdot \prod^s_{k=1} \sin^{-2}\left( \frac{\pi}{2m_{i_k}} \right) \cdot (g_{i_1 i_2} - g_{0 i_1}\, g_{0 i_2})
(g_{i_2 i_3} - g_{0 i_2}\, g_{0 i_3})\ldots
(g_{i_s i_1} - g_{0 i_s}\, g_{0 i_1}).$$
Same as in , $g_{ij}$’s form some cyclic products from ${\mathrm{Cyc}}(G(P))$. Let $t$ be the cardinality of the set $I = \{i_k\, |\, m_{i_k} = 1\}$. Since $\rho_{m_{i_k}} = \sin^{-2} \left( \frac{\pi}{2m_{i_k}} \right) \in 2 {\mathcal{O} }_{\mathbb{F}}$ by Lemma \[lemma:even\_integer\] for each $i_k \notin I$, we can rewrite as $$\label{eq:cycle5}
2^{2 s-t} \cdot \rho \cdot (g_{i_1 i_2} - g_{0 i_1}\, g_{0 i_2})
(g_{i_2 i_3} - g_{0 i_2}\, g_{0 i_3})\ldots
(g_{i_s i_1} - g_{0 i_s}\, g_{0 i_1}),$$ with $\rho \in {\mathcal{O} }_{\mathbb{F}}$. Let $\mu = 2s - t$. The longest cyclic product that appears in the above expression has length $\lambda = 2s$, if $t=0$, or $\lambda = 2s-t-1$, if $t\geq 1$ (which happens whenever we have $g_{0i_j} = \ldots = g_{0i_{j+t-1}} = 0$ for $t$ consecutive terms). Thus, each term in of length $\lambda$ is multiplied by $2^{\mu}$ with $\mu \geq \lambda$. Due to the arithmeticity of $P$, each such product belongs to ${\mathcal{O} }_{\mathbb{F}}$ by Vinberg’s criterion, c.f. Theorem \[V\].
A polytope by Bugaenko {#section:t}
======================
In this section we consider the polytope $P$ with Coxeter diagram depicted in Figure \[polytope-compact\]. This is a compact polytope in $\mathbb{H}^7$ first described by Bugaenko [@Bugaenko], and later on included into a larger census by Felikson and Tumarkin [@FT08]. This is an arithmetic polytope: $P$ is the fundamental polytope for $\mathcal{O}_r(L)$, where the lattice $L$ is associated with the quadratic form $q(x) = - \frac{1+\sqrt{5}}{2}\, x^2_0 + x^2_1 + \ldots + x^2_7$.
(5\*0.75, 5\*0.7) circle \[radius=.1\] (5\*0.54, 5\*1.0) circle \[radius=.1\] (5\*0.1, 5\*1.1) circle \[radius=.1\] (-0.35\*5, 5\*1.0) circle \[radius=.1\] (-0.6\*5, 5\*0.7) circle \[radius=.1\] (-0.3\*5, 5\*0.4) circle \[radius=.1\] (5\*0.1, 5\*0.5) circle \[radius=.1\] (5\*1.15, 5\*0.7) circle \[radius=.1\] (-1\*5, 5\*0.7) circle \[radius=.1\] (5\*0.1, 5\*0.8) circle \[radius=.1\] (5\*0.54, 5\*0.4) circle \[radius=.1\] (5\*0.54, 5\*0.4) – (5\*0.75, 5\*0.7) (5\*0.75, 5\*0.7) – (5\*0.54, 5\*1.0) (5\*0.54, 5\*1.0) – (5\*0.1, 5\*1.1) (5\*0.1, 5\*1.1) – (-0.35\*5, 5\*1.0) (-0.35\*5, 5\*1.0) – (-0.6\*5, 5\*0.7) (-0.6\*5, 5\*0.7) – (-0.3\*5, 5\*0.4) (5\*0.1, 5\*1.1) – (5\*0.1, 5\*0.8) (-0.3\*5, 5\*0.4+0.05) – (5\*0.1, 5\*0.5+0.05) (-0.3\*5, 5\*0.4-0.05) – (5\*0.1, 5\*0.5-0.05) (5\*0.1, 5\*0.5+0.05) – (5\*0.54, 5\*0.4+0.05) (5\*0.1, 5\*0.5-0.05) – (5\*0.54, 5\*0.4-0.05) (-0.6\*5, 5\*0.7-0.1) – (-1\*5, 5\*0.7-0.1) (-0.6\*5, 5\*0.7) – (-1\*5, 5\*0.7) (-0.6\*5, 5\*0.7+0.1) – (-1\*5, 5\*0.7+0.1) (5\*0.75, 5\*0.7-0.1) – (5\*1.15, 5\*0.7-0.1) (5\*0.75, 5\*0.7) – (5\*1.15, 5\*0.7) (5\*0.75, 5\*0.7+0.1) – (5\*1.15, 5\*0.7+0.1) ; (5\*0.1, 5\*0.5) – (5\*0.1, 5\*0.8); (-0.3\*5, 5\*0.4) – (5\*0.54, 5\*0.4);
The outer normals to the facets of $P$ can be easily computed by using `AlVin` [@Guglielmetti1; @Guglielmetti2] and are included in the `PLoF`[^3] worksheet [@plof].
The geometric characteristics of its lower-dimensional faces can be found by using our `PLoF` `SageMath` worksheet [@plof]. These include Coxeter diagrams and Gram matrices of all isometry types of faces from dimension $7$ down to $2$.
Furthermore, `PLoF` builds the facet tree $\mathcal{T}(P)$ for $P$. Here, we would like to stress the fact that on each level of the tree only *isometry types* of faces are given, and thus the adjacency structure of $P$ is not entirely revealed. Another feature is that each isometry type of non-Coxeter faces happens only once in $\mathcal{T}(P)$, since we prune the tree by removing duplicates. However, Coxeter faces are always preserved, so that their inclusion chains can be easily observed.
(-0.8\*0.6301687854462221,0.8\*1.5960492890907991)– (0.8\*1.0770546473922904,0.8\*2.821417146526391); (0.8\*1.0770546473922904,0.8\*2.821417146526391)– (0.8\*2.770009787007278,0.8\*1.5764106682619587); (-0.8\*0.6301687854462221,0.8\*1.5960492890907991)– (0.8\*2.770009787007278,0.8\*1.5764106682619587); (0.8\*1.0885979390908462,0.8\*4.819991966575433)– (0.8\*1.0770546473922904,0.8\*2.821417146526391); (-0.8\*1.6622043099335837,0.8\*4.835879944572051)– (-0.8\*0.6301687854462221,0.8\*1.5960492890907991); (-0.8\*1.6622043099335837,0.8\*4.835879944572051)– (0.8\*1.1001412307894027,0.8\*6.818566786624475) node \[midway, above\] [$10\,\,\,$]{}; (0.8\*1.1001412307894027,0.8\*6.818566786624475)– (0.8\*1.0885979390908462,0.8\*4.819991966575433); (0.8\*1.1001412307894027,0.8\*6.818566786624475) – (0.8\*3.839400188115276,0.8\*4.804103988578815) node \[midway, above\] [$\,\,\,10$]{}; (0.8\*3.839400188115276,0.8\*4.804103988578815)– (0.8\*2.770009787007278,0.8\*1.5764106682619587);
(-0.8\*0.6301687854462221,0.8\*1.5960492890907991) circle (.1) node \[left\] [$4$]{}; (0.8\*2.770009787007278,0.8\*1.5764106682619587) circle (.1) node \[right\] [$7$]{}; (0.8\*3.839400188115276,0.8\*4.804103988578815) circle (.1) node \[above right\] [$1$]{}; (0.8\*1.1001412307894027,0.8\*6.818566786624475) circle (.1) node \[above\] [$2$]{} ; (-0.8\*1.6622043099335837,0.8\*4.835879944572051) circle (.1) node \[above left\] [$3$]{}; (0.8\*1.0885979390908462,0.8\*4.819991966575433) circle (.1) node \[above right\] [$6$]{}; (0.8\*1.0770546473922904,0.8\*2.821417146526391) circle (.1) node \[above right\] [$5$]{};
Let us remark that $P$ has properly quasi-arithmetic faces only in dimension $2$. One of them, the complete list being computed by `PLoF` [@plof], belongs to a $3$-dimensional Coxeter face $P'$ with Coxeter diagram in Figure \[fig:3dim\]. The subdiagram of $P$ giving rise to $P'$ is generated by the two triple bonds in Figure \[polytope-compact\]. Then, the diagram of $P'$ can be easily computed by using the method of [@Allcock].
The face $P''_1$ in question has label $2$ in the Coxeter diagram of $P'$. This is a right-angled hexagon with Gram matrix $$\resizebox{0.95\linewidth}{!}{$\displaystyle{
G(P^{\prime\prime}_1) = \left( \begin{array}{cccccc} 1 & -2 \sqrt{5}-5 & 0 & 0 & -\sqrt{\frac{1}{3} \left(2 \sqrt{5}+5\right)} & -2 \sqrt{5}-4 \\
-2 \sqrt{5}-5 & 1 & -2 \sqrt{5}-4 & 0 & -\sqrt{\frac{1}{3} \left(2 \sqrt{5}+5\right)} & 0 \\
0 & -2 \sqrt{5}-4 & 1 & -\frac{1}{2} \sqrt{\sqrt{5}+5} & 0 & -\frac{1}{2} \left(3 \sqrt{5}+5\right) \\
0 & 0 & -\frac{1}{2} \sqrt{\sqrt{5}+5} & 1 & -\frac{\sqrt{5}+1}{\sqrt{6}} & -\frac{1}{2} \sqrt{\sqrt{5}+5} \\ -\sqrt{\frac{1}{3} \left(2 \sqrt{5}+5\right)} & -\sqrt{\frac{1}{3} \left(2 \sqrt{5}+5\right)} & 0 & -\frac{\sqrt{5}+1}{\sqrt{6}} & 1 & 0 \\
-2 \sqrt{5}-4 & 0 & -\frac{1}{2} \left(3 \sqrt{5}+5\right) & -\frac{1}{2} \sqrt{\sqrt{5}+5} & 0 & 1 \end{array} \right).}$}$$
One can easily verify that $P''_1$ is not arithmetic. Let us point out that $P''_1$ makes angles of $\frac{\pi}{2}$, $\frac{\pi}{3}$, and $\frac{\pi}{10}$ with its neighbours. Thus, the “evenness of the dihedral angles” condition in Theorem \[th-facet-arithm\] seems fairly reasonable.
Another face $P''_2$ of $P'$ is labelled $1$ in Figure \[fig:3dim\], and makes angles of $\frac{\pi}{2}$ and $\frac{\pi}{10}$ only with its neighbours. This face is therefore arithmetic by Theorem \[th-facet-arithm\]. This is a right-angled pentagon with Gram matrix $$\resizebox{0.95\linewidth}{!}{$\displaystyle{
G(P^{\prime\prime}_2) = \left( \begin{array}{ccccc} 1 & -\sqrt{2 \sqrt{5}+5} & 0 & 0 & -\frac{1}{2} \left(\sqrt{5}+1\right) \\ -\sqrt{2 \sqrt{5}+5} & 1 & -\frac{1}{2} \left(\sqrt{5}+3\right) & 0 & 0 \\ 0 & -\frac{1}{2} \left(\sqrt{5}+3\right) & 1 & -\frac{1}{2} \sqrt{\sqrt{5}+5} & 0 \\ 0 & 0 & -\frac{1}{2} \sqrt{\sqrt{5}+5} & 1 & -\frac{\sqrt{5}+1}{2 \sqrt{2}} \\ -\frac{1}{2} \left(\sqrt{5}+1\right) & 0 & 0 & -\frac{\sqrt{5}+1}{2 \sqrt{2}} & 1 \\ \end{array} \right).}$}$$
We also computed the faces of Bugaenko’s compact polytope in ${\mathbb{H}}^8$, which did not give very informative outcome. Indeed, it happens to have only few Coxeter faces, all of which are arithmetic. The complete computation can be found in [@plof].
A curious reflective lattice {#section:l}
============================
In this example, we consider the reflective lattice $L$ associated with the Lorentzian quadratic form $f(x) = -15\, x^2_0 + x^2_1 + \ldots + x^2_5$. Let $P$ be the fundamental polytope for $\mathcal{O}_r(L)$. An apparently interesting fact is that $P$ has a descending chain entirely of Coxeter faces starting from the polytope itself that ends up with two $2$-dimensional faces: one arithmetic, and the other properly quasi-arithmetic. None of the previous examples has such a long chain of Coxeter faces. This can be observed by using `PLoF` [@plof]. Namely, this chain ends in a $2$-dimensional face $P'_1$ with Gram matrix $$G(P'_1) = \left(\begin{array}{rrr}
1 & -\frac{1}{2} \, \sqrt{2} & -1 \\
-\frac{1}{2} \, \sqrt{2} & 1 & 0 \\
-1 & 0 & 1
\end{array}\right),$$ and another $2$-dimensional face $P'_2$ with Gram matrix $$G(P'_2) = \left(\begin{array}{rrrr}
1 & -\frac{1}{2} \, \sqrt{2} & -\frac{2}{3} \, \sqrt{3} & 0 \\
-\frac{1}{2} \, \sqrt{2} & 1 & 0 & -\frac{1}{2} \, \sqrt{10} \\
-\frac{2}{3} \, \sqrt{3} & 0 & 1 & 0 \\
0 & -\frac{1}{2} \, \sqrt{10} & 0 & 1
\end{array}\right).$$ It is easy to check that $P'_1$ is arithmetic, while $P'_2$ is properly quasi-arithmetic.
Moreover, since $15$ is not a sum of three rational squares, $P$ has a descending chain of faces corresponding to the restrictions of $f$ onto the subspaces $x_5 = \ldots = x_{5-i} = 0$, for $i=0,1,2,3$, that starts with two non-compact finite volume faces and ends with two compact ones. All of them are obviously arithmetic (cf. [@Bug92 Theorem 2.1]).
Some open questions {#sec:open}
===================
Below we list some, to the best of our knowledge, open problems, that seems interesting to us and are related to the above discussion of (quasi-)arithmeticity.
Do there exist compact Coxeter polytopes in ${\mathbb{H}}^{\ge 9}$?
Let us recall that the record example is due to Bugaenko in ${\mathbb{H}}^8$, and there have been no compact polytopes found in higher dimensions since almost 30 years to date.
Do there exist properly quasi-arithmetic Coxeter polytopes in ${\mathbb{H}}^{\ge 6}$, compact or non-compact?
Some related work was done by Vinberg in the non-compact case [@Vin14] for all dimensions $n \leq 12$ and $n = 14, 18$ by constructing analogues of the non-arithmetic hybrids due to Gromov — Pyatetski–Shapiro (later on, Thomson [@Thom16] proved that these are never quasi-arithmetic). There are also some examples among Coxeter prisms, including properly quasi-arithmetic ones [@Vin67]. If more Coxeter polytopes become available, one can use the same idea and try to fill in the gaps for $n=13, 15, 16, 17$, as well as to answer the above question.
Is it true that a quasi-arithmetic Coxeter polytope in ${\mathbb{H}}^{\ge 4}$ with ground field ${\mathbb{Q}}$ cannot be compact?
This question is motivated by the examples of cocompact quasi-arithmetic subgroups of ${\mathrm{SL}}_2({\mathbb{Q}})$ constructed by Vinberg in [@Vin-Bielefeld] that preserve an isotropic quadratic form.
If Question 3 has affirmative answer, then Theorem \[th-face\] limits us only to non-compact polytopes appearing as Coxeter faces of arithmetic polytopes in $\mathbb{H}^{\geq 4}$ with ground field ${\mathbb{Q}}$.
Does there exist an arithmetic Coxeter polytope in ${\mathbb{H}}^{\ge 4}$ that has a properly quasi-arithmetic Coxeter face of small codimension?
The fact that the only properly quasi-arithmetic faces we found in Coxeter polytopes from **§** \[section:t\] – **§** \[section:l\] have dimension $2$ is the main motivation for Question 4.
Are there only finitely many maximal quasi-arithmetic hyperbolic reflection groups in all dimensions? Is it true at least for the fixed dimension and degree $d = [{\mathbb{F}}: {\mathbb{Q}}]$ of the ground field ${\mathbb{F}}$?
This question is naturally motivated by the affirmative answer for arithmetic groups. It is also interesting, whether Nikulin’s methods (cf. [@Nik81; @Nik07]) or the spectral method (cf. [@ABSW2008; @Bel16]). can be applied.
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[^1]: Here $H^\sigma$ denotes the algebraic group defined over $\sigma({\mathbb{F}})$ and obtained from an abstract algebraic group $H$ by applying $\sigma$ to the coefficients of all polynomials that define $H$.
[^2]: Two subgroups $\Gamma_1 $ and $\Gamma_2$ of some group are called commensurable if the group $\Gamma_1 \cap \Gamma_2$ is a subgroup of finite index in each of them.
[^3]: **P**olytope’s **Lo**wer-dimensional **F**aces
|
---
abstract: 'We consider the bosonic fractional quantum Hall effect in the presence of a non-Abelian gauge field in addition to the usual Abelian magnetic field. The non-Abelian field breaks the twofold internal state degeneracy, but preserves the Landau level degeneracy. Using exact diagonalization, we find that for moderate non-Abelian field strengths the system’s behaviour resembles a single internal state quantum Hall system, while for stronger fields there is a phase transition to either two internal state behaviour or the complete absence of fractional quantum Hall plateaus. Usually the energy gap is reduced by the presence of a non-Abelian field, but some non-Abelian fields appear to slightly increase the gap of the $\nu=1$ and $\nu=3/2$ Read-Rezayi states.'
address: 'School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom'
author:
- 'Rebecca N. Palmer and Jiannis K. Pachos'
bibliography:
- 'references.bib'
title: 'Fractional quantum Hall effect in a U(1)$\times$SU(2) gauge field'
---
Introduction
============
The fractional quantum Hall (FQH) effect [@fqh-frac-gauge] occurs in two-dimensional interacting systems in a strong perpendicular Abelian gauge field at sufficiently low temperatures. It manifests as incompressible strongly correlated ground states, at several simple fractional values of the filling factor (ratio of particles to flux quanta), with anyonic excitations. The FQH effect is usually realized with electrons in a magnetic field, but theoretically exists for both bosons and fermions, and both long- and short-ranged interactions [@rotating-gas-review; @rotating-gas-review2]. Several methods have been proposed for implementing it with ultracold atoms [@artificial-gauge-review; @fqh-laserhop; @fqh-laserhop2; @fqh-bec-immersion; @fqh-quadrupole; @fqh-stark; @fqh-rotation; @fqh-oam], and some few-atom experiments have been done [@fqh-few-atom].
One motivation for studying the FQH effect is that it can support anyons [@topo-qc-review; @na-anyon-review]. These are particles whose exchange statistics is not the $\pm 1$ of bosons or fermions, but some other phase $\rme^{\rmi\theta}$ for Abelian anyons, or a unitary matrix for non-Abelian anyons. This statistical effect depends only on the topology of the anyons’ paths and not on the details of their motion. In the case of non-Abelian anyons, the states on which the statistical matrix acts are a global property of the pair, indistinguishable if each anyon is measured individually, but can be measured by bringing them together. Hence, a quantum computer using well separated non-Abelian anyons as qubits and statistical matrices as gates would be immune to error from either local decoherence or imperfect control of the anyon motion [@topo-qc-orig], as long as there were no uncontrolled stray anyons in the system. For some types of anyons, including those expected in some FQH states [@rr-qc-braid; @topo-qc-review], such a quantum computer would be universal.
However, errors will occur if uncontrolled stray anyons, such as thermally created anyons [@thermal-anyon1; @thermal-anyon2], move around the computational anyons. If the temperature is too high compared to the anyon pair creation gap, these stray anyons will make the system unusable for computation, and eventually destroy the FQH effect completely. In standard FQH systems, the states supporting non-Abelian anyons have small enough energy gaps that they are difficult to observe [@fqh-2ll-expt; @mr-interference-expt] and would be even more difficult to use for computation. This provides a motivation to look for modified FQH systems with more robust non-Abelian anyons.
The modification we consider here is to add a uniform SU(2) non-Abelian component to the magnetic field, coupled to a two-dimensional internal state space of the particles. Such a system can be realized using ultracold atoms in an optical lattice [@artificial-gauge-review; @u2f-const-iqh; @u2f-graphene-iqh; @u2f-rashba-zeeman-iqh; @u2f-laserhop; @u2f-valley; @u2f-u3f-darkstate], or using electrons in a material with spin-orbit splitting [@spin-orbit-equal; @spin-orbit-levels]. We study this system using lowest Landau level exact diagonalization, for bosonic particles and a two-parameter range of SU(2) field strengths. When both parameters are of the same order as the Abelian field strength, the system behaves similarly to a *one* internal state FQH system, i.e. incompressible states at filling factors $\nu=\frac{1}{2},1,\frac{3}{2},\ldots$ with large overlaps with the Read-Rezayi states [@rr-torus-numeric], despite actually having two internal states. For most parameter values, these have smaller energy gaps than in a true one internal state system, but in the cases $\nu=1,\frac{3}{2}$, it appears that some choices of non-Abelian field can slightly increase the gap. When both parameters are large, we find no incompressible states at all, while when only one is large, we find similar behaviour to when both are zero, i.e. a two internal state FQH system.
In section \[model\] we introduce our model, starting from a lattice Hamiltonian, then taking the continuum limit. In subsection \[sec1part\] we derive the single particle Landau levels in the presence of the SU(2) gauge field. Section \[numerics\] gives many-particle exact diagonalization results for the density profile, the energy gap, and the overlap with some trial states. In particular, we consider the $\nu=1/2$ Laughlin state and the $\nu=1$ Moore-Read state. We conclude in section \[conc\].
The Model {#model}
=========
We consider bosons in two dimensions, in either continuous space or a lattice that can be well approximated by continuous space. We apply a uniform classical U(2)=U(1)$\times$SU(2) gauge field, minimally coupled to two internal states of the particles. The U(1) part is the Abelian field of standard FQH, while the SU(2) non-Abelian part is our new ingredient. The FQH effect also requires an interaction between the particles, which we take to be a contact interaction. A possible physical implementation of this system is described in [@u2f-const-iqh; @artificial-gauge-review], using ultracold bosonic atoms in an optical lattice with laser induced hopping.
Lattice Hamiltonian
-------------------
We consider bosons with two internal states on a square lattice, described by the Bose-Hubbard Hamiltonian with a classical gauge field [@artificial-gauge-review], $$\begin{aligned}
\fl H=\sum\limits_{x,y}-J\left[a^\dagger_{x,y}\exp\left(\rmi\int\limits_{x-1}^x\bi{A}(x^\prime,y)dx^\prime\right)a_{x-1,y}\right.\nonumber\\
\left.+a^\dagger_{x,y}\exp\left(\rmi\int\limits_{y-1}^y\bi{A}(x,y^\prime)dy^\prime\right)a_{x,y-1}+\mathrm{H.c.}\right]\nonumber\\
+Ua^\dagger_{x,y}a^\dagger_{x,y}a_{x,y}a_{x,y}.\label{Hgauge}\end{aligned}$$ Here $\bi{A}(x,y)$ is the vector potential of the gauge field, $a^\dagger_{x,y}$ creates a particle in lattice site $(x,y)$ (we use the lattice spacing as the unit of length), $J$ is the hopping rate and $U$ is the interaction strength. For simplicity, we assume isotropic hopping, and state-independent hopping and interaction.
In the case of a non-Abelian gauge field, the particles have multiple internal (e.g. hyperfine) states, and $\bi{A}$ is a matrix in internal state space as well as a vector field in real space, i.e. hopping can change the particle’s internal state. We consider the case of a uniform U(2)=U(1)$\times$ SU(2) gauge field [@u2f-const-iqh], $$\begin{aligned}
\bi{A}(x,y)&=(\alpha\sigma_y,\beta\sigma_x+2\pi\Phi x)\nonumber\\
&=\left(\left(\begin{array}{cc}0&-\rmi\alpha\\\rmi\alpha&0\end{array}\right),\left(\begin{array}{cc}2\pi\Phi x&\beta\\\beta&2\pi\Phi x\end{array}\right)\right),
\label{gauge}\end{aligned}$$ where the $\sigma_i$ are the Pauli matrices. The Abelian part of this field is parameterized by $\Phi$, and the non-Abelian part by $\alpha,\beta$. As we see below, the Abelian part is necessary to have an FQH effect; a pure SU(2) field [@su2-bec] does not produce Landau levels.
For this choice of field, $A_x$ is independent of $x$ and $A_y$ of $y$, so the integrals in [(\[Hgauge\])]{} simplify to $\rme^{\rmi A_x}$, $\rme^{\rmi A_y}$, giving $$\begin{aligned}
\fl H=\sum\limits_{x,y}-J\left[a^\dagger_{x,y}(\cos\alpha+\rmi\sigma_y\sin\alpha)a_{x-1,y}\right.\nonumber\\
\left.+\rme^{2\pi \rmi\Phi x}a^\dagger_{x,y}(\cos\beta+\rmi\sigma_x\sin\beta)a_{x,y-1}+\mathrm{H.c.}\right]\nonumber\\
+Ua^\dagger_{x,y}a^\dagger_{x,y}a_{x,y}a_{x,y}.
\label{Hlattice}\end{aligned}$$
Continuum approximation
-----------------------
When the lattice spacing is small compared to the magnetic length (defined below) and the typical interparticle spacing, we can approximate the site operators $a_{x,y}$ by a continuous field $\psi(x,y)$. Taylor expanding $\psi(x-1,y)$ and similar terms in [(\[Hlattice\])]{} about $(x,y)$, and discarding the (dynamically irrelevant) constant $-4J\psi^\dagger\psi$, gives the approximate Hamiltonian $$\begin{aligned}
\fl H\approx H_c\equiv\int dxdy J\psi^\dagger\left[\left(-\rmi\frac{\partial}{\partial x}+\alpha\sigma_y\right)^2\right.\nonumber\\
+\left.\left(-\rmi\frac{\partial}{\partial y}+\beta\sigma_x+2\pi\Phi x\right)^2\right]\psi +\frac{1}{2}U\psi^\dagger\psi^\dagger\psi\psi,\label{Hcont}\end{aligned}$$ where all occurrences of $\psi$ are taken at position $(x,y)$. This Hamiltonian has a magnetic length $l=1/\sqrt{2\pi\Phi}$, which sets the length scale of the quantum Hall physics. Hence, the above continuum approximation is valid in the limit $l\gg 1$. For convenience we also define the dimensionless parameters $a\equiv\alpha/\sqrt{\Phi}$ and $b\equiv\beta/\sqrt{\Phi}$. Within this continuum limit, varying $\Phi$ at fixed $a,b$ changes only the overall length and energy scales. We can assume without loss of generality that $a,b,\Phi\geq 0$, as a spin space rotation by $\sigma_x$ ($\sigma_y$) changes the sign of $a$ ($b$), while a real space reflection in the $y$ axis changes the signs of $a$ and $\Phi$.
Expanding out the brackets and discarding the dynamically irrelevant constants $\sigma_x^2=\sigma_y^2=I$, we find that the non-interacting part of [(\[Hcont\])]{} is equal to the Hamiltonian of 2D electrons with spin-orbit coupling [@spin-orbit-levels], with the coordinate axes rotated by $45^\circ$ (in both real and spin space). The SU(2) terms correspond to the spin-orbit interaction, which is pure Rashba if $\alpha=-\beta$, pure Dresselhaus if $\alpha=\beta$, and a combination of the two for all other ratios. An optical lattice loaded with fermionic atoms can hence be used as a quantum simulator of spin-orbit coupled electrons [@artificial-gauge-review]. However, we consider only the bosonic case.
Single particle degeneracy: Landau levels {#sec1part}
-----------------------------------------
In this subsection we consider the behaviour of a single particle subject to the Hamiltonian [(\[Hcont\])]{}. Since $H_c$ is independent of $y$, its single particle eigenstates are plane waves in the $y$ direction, $\psi\propto\rme^{\rmi ky}$. Substituting this into [(\[Hcont\])]{}, we find that the eigenstates have the form $\psi=\rme^{\rmi ky}\Psi_n(x+k/2\pi\Phi)$, where $\Psi_n$ and the energy $E_n$ satisfy the equation $$\begin{aligned}
\fl E_n\Psi_n(X)=J[(-\rmi\partial/\partial X+\alpha\sigma_y)^2\nonumber\\
+(\beta\sigma_x+2\pi\Phi X)^2]\Psi_n(X).\label{lll-wavefn}\end{aligned}$$ Like the Abelian case, the energy is hence independent of $k$, so each single particle energy level $n$ (called a Landau level) has a macroscopic degeneracy of $\Phi$ states per unit area (one per Abelian flux quantum). In the many-particle case, this degeneracy is broken by the contact interaction, so a weak interaction can produce strongly correlated states. As in standard FQH, we define the filling factor $\nu$ as the number of particles divided by the number of states in a Landau level, $\nu\equiv\rho/\Phi$ where $\rho$ is the 2D density.
In the Abelian case ($\alpha=\beta=0$), $E_n=2,2,6,6,10,10,\ldots\times\pi J\Phi$, the additional 2-fold degeneracy coming from interchange of internal states. In contrast, in the non-Abelian case, the Landau level energies $E_n$ are in general all distinct and follow no simple pattern. The $E_n$ can be found analytically [@spin-orbit-levels], but this result is complicated, and gives the corresponding wavefunctions $\Psi_n$ only as infinite series; we instead solve [(\[lll-wavefn\])]{} numerically. Full analytic solutions exist in the special cases $\alpha=0$ or $\beta=0$ (2-fold degenerate [@spin-orbit-equal]), or $\alpha=\pm\beta$ (non-degenerate [@spin-orbit-Rashba; @u2f-naqh-symm]).
![Lowest few Landau level energies $E_n$, and symmetries $+-$ (blue/dark) or $-+$ (green/light), for $a=b$ (thin lines) and $a=b\pm 1$ (thick lines).[]{data-label="llevels"}](naqh_levels.pdf){width="8.6cm"}
Figure \[llevels\] plots the lowest few Landau level energies $E_n$ along two diagonal lines in $(a,b)$ space, while figure \[1part\]a plots the difference between the lowest two energies against $a,b$. Figure \[1part-wavefn\] plots the lowest Landau level wavefunction $\Psi_0$ at four $(a,b)$ settings. Because [(\[lll-wavefn\])]{} has the $Z_2$ symmetry $\Psi_n(x)\rightarrow\sigma_z \Psi_n(-x)$, in each Landau level either the first internal state component is symmetric about $x=-k/2\pi\Phi$ and the second antisymmetric ($+-$ symmetry, figure \[1part-wavefn\]a,b,c), or vice versa ($-+$, figure \[1part-wavefn\]d). Levels of the same symmetry avoided cross, touching only in conical intersections on the $a=\pm b$ diagonals, but levels of opposite symmetry cross freely, on lines in $(a,b)$ space. For small positive $a,b$, the lowest Landau level is $+-$ and the second $-+$, but at larger $a,b$ these cross infinitely many times [@u2f-naqh-symm], causing a discontinuous change in the wavefunction $\Psi_0$ each time.
![Single particle lowest Landau level wavefunctions $\Psi_0(x)$, for non-Abelian field strengths $(a,b)=$ (a) (1,1), (b) (1,5), (c) (5,1), (d) (5,5). Blue and green are the two internal states $\Psi_0(x,\uparrow)$, $\Psi_0(x,\downarrow)$, black their rms total amplitude $\sqrt{|\Psi_0(x,\uparrow)|^2+|\Psi_0(x,\downarrow)|^2}$, thin grey the Abelian-field ground state for comparison.[]{data-label="1part-wavefn"}](naqh_1part_wavefn.pdf){width="8.6cm"}
The exactly solvable cases $a=0$ and $b=0$ can be used as perturbative approximations for the cases when $a$ or $b$ is small but not zero [@spin-orbit-equal]. When $a=0$, the two degenerate ground states at a given $k$ are Gaussians with shifted centres, $\psi(x,y)=\exp\{\rmi k y-[x+(k\pm\beta)/2\pi\Phi]^2/2l^2\}$, in the $\sigma_x=\pm 1$ internal states (equal superpositions with relative phase $\pm 1$). When $a\ll b$, the ground state is close to the symmetric combination of these (figure \[1part-wavefn\]b), and hence has two peaks approximately $\beta/\pi\Phi=bl/\sqrt{\pi}$ apart (figure \[1part\]d). When $b=0$, the ground states have a Gaussian amplitude but an extra phase, $\psi(x,y)=\exp[\rmi (ky\pm\alpha x)-(x+k/2\pi\Phi)^2/2l^2]$ with internal states $\sigma_y=\pm 1$. The symmetric combination, and approximate $a\gg b$ ground state (figure \[1part-wavefn\]c), has a Gaussian total amplitude but a position dependent internal state.
However, this apparent asymmetry between $a$ and $b$ arises entirely from our choice to have states extended along $y$ and localized in $x$, and to use a gauge that simplifies such states; we could instead have made the Abelian part of $\bi{A}$ proportional to $y$ (by a gauge transformation) and had states $\psi\propto\rme^{\rmi kx}$, which would interchange the roles of $a$ and $b$.
![Single particle state properties: (a) Energy difference $(E_1-E_0)/4\pi J\Phi$ between the lowest two Landau levels, and symmetry ($+-$ or $-+$) of the lowest level. (b) Overlap $\langle\Psi_{0,\mathrm{Abelian}}|\Psi_0\rangle$ of the lowest Landau level wavefunction with the Abelian lowest Landau level. (c) RMS spread (standard deviation) of the lowest Landau level wavefunction, $({\left\langle \Psi_0 \right|}x^2{\left| \Psi_0 \right\rangle})^{1/2}/l$. The small dark features on the right hand edge are numerical artefacts, caused by the gap between the two lowest Landau levels being too small to resolve. (d) Distance between the two highest maxima of $\Psi_0$ (equal height by symmetry, see figure \[1part-wavefn\]b,d), or zero if there is a single central maximum.[]{data-label="1part"}](naqh_1part.pdf){width="8.6cm"}
Numerical results {#numerics}
=================
We now consider the multi-particle case. We assume weak interaction, by which we mean that the interaction energy per particle ($\sim U\Phi$ at $\nu\sim 1$) is small compared to the Landau level spacing ($\sim J\Phi$), i.e. $U\ll J$. We can hence assume that all the particles are in the lowest Landau level, and use a Fock basis over the single particle eigenstates ${\left| k \right\rangle}$. Since all momenta $k$ within a Landau level are degenerate, the only non-trivial term is the interaction term, with the matrix elements $$\begin{aligned}
\fl{\left\langle k_1^\mathrm{out},k_2^\mathrm{out} \right|}H_\mathrm{int}{\left| k_1^\mathrm{in},k_2^\mathrm{in} \right\rangle}=U\delta(k_1^\mathrm{in}+k_2^\mathrm{in}-k_1^\mathrm{out}-k_2^\mathrm{out})\nonumber\\
\times\int dx \Psi_0^*(x+k_1^\mathrm{out}/2\pi\Phi)\Psi_0^*(x+k_2^\mathrm{out}/2\pi\Phi)\nonumber\\
\times\Psi_0(x+k_1^\mathrm{in}/2\pi\Phi)\Psi_0(x+k_2^\mathrm{in}/2\pi\Phi).\end{aligned}$$ We adopt periodic boundary conditions (a torus geometry) [@qh-torus-llstates; @qh-torus-2dsymm], which makes the number of allowed $k$ finite, and use the full 2D Haldane symmetry [@qh-torus-2dsymm]. The interaction Hamiltonian then becomes a finite matrix, which we diagonalize to find the ground and lowest few excited states. This method is exact within the lowest Landau level, and in particular can describe the strong correlations typical of FQH states, but is limited to small particle numbers by the exponentially growing basis size.
Density profiles {#secdensity}
----------------
![Filling factor (zeros offset for clarity) against chemical potential, found by exact diagonalization on a square torus. (a) Abelian field (single internal state), (b-d) non-Abelian fields $(a,b)=$ (b) (2,2), (c) (5,5), (d) (5,1). System sizes 6 (red), 8 (yellow), 10 (green), 12 (cyan), 16 (magenta) flux quanta; due to computational limitations, the larger sizes can only be used at small filling factors. Horizontal lines mark the Read-Rezayi densities $\nu=1/2,1,3/2,\ldots$.[]{data-label="density"}](naqh_density.pdf){width="8.6cm"}
![Filling factor (lines mark plateau edges, Read-Rezayi series $\nu=p/2$ labelled in black text, composite fermion series $\nu=p/(p\pm 1)$ in blue, others in magenta), and projected overlap with the corresponding Abelian-field state (shading), against chemical potential and $a$. Overlap is with the lowest Abelian state of the same Haldane momentum $\bi{K}$, projected into the non-Abelian lowest Landau level; green crosses indicate that this is not the same $\bi{K}$ as the Abelian ground state. (a) $a=b$, (b) $b=1$. System sizes 12 ($\nu<1$, cyan lines), 8 ($1\leq\nu\leq 2$, yellow), 6 ($\nu>2$, red) flux quanta, on a square torus. Dark red triangles indicate densities skipped because the exact diagonalization failed to converge.[]{data-label="density2"}](naqh_density2.pdf){width="8.6cm"}
We perform exact numerical diagonalization for different numbers of particles (at a fixed number of Abelian flux quanta, i.e. number of single particle states per Landau level) to obtain the ground state energy as a function of density. We then convert this to density as a function of chemical potential [@fqh-density-steps]. This approach makes incompressible states visible as plateaus in the density. Furthermore, in the local density approximation (valid for weak traps) the local chemical potential equals the global chemical potential minus the trap potential [@fqh-density-steps], so these results can be directly compared with experiment if the in-trap density profile can be measured.
At moderate non-Abelian field strengths ($a,b\lesssim 4$), the density profile (figure \[density\]b, lower halves of figures \[density2\]a,b) resembles the [*single*]{} internal state Abelian-field case (figure \[density\]a and [@fqh-density-steps]). Furthermore, most of the ground states have large ($>90\%$) overlaps with the projections of their Abelian-field equivalents onto the new lowest Landau level. This makes sense, as though we have two internal states, the lowest Landau level is a single level, not a degenerate pair. The main plateaus are the Read-Rezayi series $\nu=p/2$ ($p$ integer), clearly present for $\nu=1/2,1,\ldots,3$ and possibly present all the way up to our calculation’s limit of $\nu=4$. (Because the number of particles must be an integer, finite system density profiles are stepped even in compressible phases, so narrow incompressible plateaus cannot be reliably distinguished from regions of low but non-zero compressibility.) Of the composite fermion series $\nu=p/(p\pm 1)$, we see hints of $\nu=2/3,3/4,4/5,7/6,6/5,5/4$ (but not $5/6$ or $4/3$). We also see $\nu=7/4$, $9/4$ and $11/4$, but do not have any theory as to what these states are (the previous Abelian-field study [@fqh-density-steps] used 6 flux quanta, at which these $\nu$ are not possible). They may be finite size artefacts, as the $5/4$ plateau narrows greatly on increasing the system size from 8 to 12 states. Due to computational limitations, we were unable to do this test for the higher densities. There are two main differences from the Abelian case at these values of $a,b$. First, all plateaus are shifted towards higher chemical potential. This would cause a trapped system (with fixed particle number) to slightly expand. Second, the Laughlin state ($\nu=1/2$) no longer has exactly zero interaction energy, and possible new plateaus appear below it at $\nu=1/4,1/3$, though again these are too narrow to be sure they are real.
If the non-Abelian field strength is further increased along the $a=b$ diagonal (figure \[density\]c, upper half of figure \[density2\]a), all the plateaus disappear suddenly, in a first order phase transition. As this happens at the same point as the single particle Landau levels cross (figure \[llevels\]), it is probably caused by this sudden change of single particle states.
If $a$ is increased while $b$ is kept constant (figure \[density\]d, upper half of figure \[density2\]b) or vice versa, most of the plateaus disappear gradually, though at some system sizes a few plateaus survive or even grow ($\nu=1/3,2/3,1$ for 12 flux quanta), then at even larger $a$ the normal plateaus reappear. This process becomes slower and occurs at a larger $a$ as the system size is increased, with the minimum plateau width occuring very close to the point where the single particle peak separation equals half the torus circumference ($a=4.1,4.6,5.5$ for 6,8,12 particles and $b=1$, figure \[1part\]d). We hence believe the revival, and possibly some of the initial decrease, to be a finite size artefact, caused by single particle states wrapping right round the torus. In the infinite size limit, it is not clear whether this will become a second order phase transition to a gapless phase at finite $a$ (as suggested by simple $1/N$ extrapolation, figures \[laughlin\_varyn\]Cb, \[read\_varyn\]Cd), or a decay that tends to zero only in the $a\rightarrow\infty$ limit.
However, as the gap between the lowest two Landau levels is exponentially decreasing in this region ($\sim 5\%$ of its peak value at the point of minimum plateau width, figure \[1part\]a), the weak interaction assumption must eventually fail, allowing the second Landau level to be populated. In the $a\gg b$ or $b\gg a$ limit, the two Landau levels become degenerate, and by taking superpositions of them we can recover two sets of single particle states with Gaussian amplitudes. The two states centred at $x=x_0$ are $\psi(x,y)=\exp(-(x-x_0)^2/2l^2-2\rmi x_0 y/l^2\mp\rmi\alpha x)$ ($a\gg b$, internal states $\sigma_y=\pm 1$) or $\psi(x,y)=\exp(-(x-x_0)^2/2l^2-2\rmi x_0 y/l^2\mp\rmi\beta y)$ ($a\ll b$, internal states $\sigma_x=\pm 1$). Because the interaction does not change the internal state, the extra phase factors cancel out in the interaction Hamiltonian $\psi^\dagger\psi^\dagger\psi\psi$, so in this limit the Hamiltonian matrix is identical to that at $a=b=0$, i.e. Abelian-field FQH with *two* internal states. This is known to have a $\nu=2/3$ 221 state [@rotating-gas-review], and is conjectured to have a $\nu=2p/3$ NASS series [@nass-def].
The weak interaction assumption would also fail close to Landau level crossings [@u2f-naqh-symm], but here the two degenerate single particle states are not close to Gaussian. We hence do not know whether these parameter values would be in a two internal state FQH phase, or a gapless phase similar to the one internal state gapless phase above the first such crossing.
$\nu=1/2$: Laughlin state
-------------------------
![$\nu=1/2$, (A) $N=6$ particles, (B) $N=8$ particles, (C) $1/N$ extrapolation to infinitely many particles, based on data for 4-8 particles. (a) Overlap of the numerically calculated lowest $\bi{K}=(0,0)$ state with the projection of the exact Laughlin state onto the lowest Landau level. (b) Energy gap between this state and the first excited state (of any $\bi{K}$). (c) Energy gap between the excitation branch and the continuum (as shown in figure \[energies\_k\]a), smallest gap of $\bi{K}=(N/2,0),(0,N/2),(N/2,N/2)$ ($\bi{K}$ scaled to take integer values $0,\ldots,N-1$). The small features near $(a,b)=(4,8),(8,4)$ are numerical artefacts.[]{data-label="laughlin_varyn"}](naqh_l_varyn.pdf){width="10cm"}
![Lowest few energies found by exact diagonalization, plotted against $|\bi{K}|$. (a) $\nu=1/2$ (8 particles, 16 flux quanta), $(a,b)=(2,2)$. (b,c) $\nu=1$ (12 particles, 12 flux quanta), $(a,b)=$ (b)$(2,2)$, (c)$(2,1)$. (d) $\nu=3/2$ (15 particles, 10 flux quanta), $(a,b)=(2,2)$. Colour denotes projected overlap with the corresponding Abelian-field state, from none (red) to full (black). The green circles mark the ground state(s) predicted by Read-Rezayi theory.[]{data-label="energies_k"}](naqhEDenergies.pdf){width="8.6cm"}
We now consider the $\nu=1/2,1$ states in more detail. The bosonic Laughlin state, $$\psi(z_1,\ldots,z_N)=\prod\limits_{i,j}(z_i-z_j)^2\rme^{-\sum_i|z_i|^2/4l^2}$$ where $z_i=x_i+\rmi y_i$ is the position of particle $i$ as a complex number, has exactly zero interaction energy as it is zero whenever two particles come together. In Abelian field, it is also a lowest Landau level state, so is the exact ground state at $\nu=1/2$. It is a non-degenerate state (except for the 2-fold centre of mass degeneracy of all states at half-integer filling [@qh-torus-2dsymm]) with zero Haldane momentum, with a gap to a well-defined branch of excitations [@rotating-gas-review].
In non-Abelian field, the Laughlin state is no longer a lowest Landau level state, but it can be projected onto the new lowest Landau level and used as a trial state, which has large overlap with the ground state for nearly all $a,b$ (figure \[laughlin\_varyn\]a). Its interaction energy is no longer zero after this projection, but for $a,b\lesssim 4$ the energy spectrum remains qualitatively similar (figure \[energies\_k\]a), retaining the gap (figure \[laughlin\_varyn\]b) and well defined excitation branch (figure \[laughlin\_varyn\]c).
At stronger fields, if $a\approx b$ the gap suddenly disappears, while for $a\gg b$ or $a\ll b$ it continuously decreases to near zero then increases again. As discussed in subsection \[secdensity\] for general $\nu$, the first is probably a first order phase transition driven by the sudden change of single particle state, while in the second case, it is not clear whether or not the gap actually reaches zero at finite field strength, while the revival is probably a finite size artefact caused by single particle states wrapping round the torus.
Our method does not reveal the excitation statistics. However, the excitations of any non-degenerate gapped state at non-integer filling have fractional particle number by Laughlin’s gauge argument [@fqh-frac-gauge], which is still valid in the presence of the non-Abelian field, and hence fractional statistics [@fqh-charge-stat]. This does not distinguish Abelian from non-Abelian statistics, but given the lack of a visible phase transition we expect the same statistics as the Laughlin state, i.e. Abelian $\theta=\pi/2$ (semions) [@laughlin-stat].
$\nu=1$: Moore-Read state
-------------------------
{width="15cm"}
The $p$th Read-Rezayi state for bosons is defined as the $\nu=p/2$ state which vanishes when any $p+1$ particles come together; it is unique on a disc but $p+1$-fold degenerate (including the twofold centre of mass degeneracy if $p$ is odd) on a torus [@rr-cluster]. It is equal to the symmetrized product of $p$ Laughlin states [@rr-lprod]. It is not an exact eigenstate for $p\neq 1$, but in Abelian field has been found to be a good trial state for small $p$ [@rr-torus-numeric].
The $p=2$ ($\nu=1$) case is also called the Moore-Read or Pfaffian state. Its excitations are Ising anyons, whose non-Abelian statistics have been tentatively detected in a related fermionic state [@mr-interference-expt]. These are not computationally universal on their own [@mr-qc], but can be made so with relatively noisy non-topological operations [@mr-qc; @mr-qc2]. On a torus the Moore-Read state is 3 states at Haldane momenta $\bi{K}=(N/2,0),(0,N/2),(N/2,N/2)$ [@rr-torus-numeric] (where $\bi{K}$ is scaled to take integer values $0,\ldots,N-1$); for a system to be in the Moore-Read phase, in the infinite size limit these 3 states must be topologically degenerate [@mr-torus-degen] and have an energy gap between them and all other states. However, the Moore-Read state has a smaller energy gap and a longer correlation length than the Laughlin state, and hence is more sensitive to finite size error, which can split the degeneracy. At the sizes we can access (up to 12 particles) it is not well converged, as shown by the strong system size dependence in figure \[read\_varyn\].
We find a positive energy gap (i.e. the three lowest energy states are at the right three $\bi{K}$ for a Moore-Read state, figure \[read\_varyn\]d) and large overlaps between the exact ground state and the projected (into the lowest Landau level) Moore-Read state at all three $\bi{K}$ (figure \[read\_varyn\]b,c) over roughly the same region as the Laughlin state, i.e. $a,b\lesssim 4$ plus probably-artefact revivals when $a$ or $b$ is large. However, the gap and degeneracy splitting (figure \[read\_varyn\]e) vary significantly within this region, with 2-3 “good" stripes (large gap, small splitting, energy spectrum shown in figure \[energies\_k\]b), separated by “bad" stripes (small gap, large splitting, figure \[energies\_k\]c). Simple $1/N$ extrapolation suggests that only the “good" stripes are in the Moore-Read phase (zero splitting in the infinite size limit, figure \[read\_varyn\]Ce), but the strong system size dependence makes this result unreliable, and it is a plausible alternative that the entire $a,b\lesssim 4$ region is in the Moore-Read phase. The stripes arise from crossings of the three Moore-Read state energies (marked by lines in figure \[read\_varyn\]d,e); near triple crossings, giving a greatly reduced splitting, seem to be more common than would be expected by chance.
At stronger fields, the Moore-Read state is replaced by smectic (stripe ordered), non-degenerate gapped, or gapless states (figure \[read\_varyn\]a). Which of these occur in what regions of $a,b$ parameter space is strongly system size dependent, and we hence do not know which, if any, of these survive in the infinite size limit. As discussed above for general $\nu$, at these fields the width of a single particle eigenstate is a large fraction ($\gtrsim\frac{1}{2}$) of the the torus circumference, and strong finite size artefacts are hence likely.
We did not perform a similarly detailed study of the $\nu=3/2$ state (which if Read-Rezayi supports universal topological quantum computation [@rr-qc-braid]), as at the sizes we can reach, the two $\bi{K}=0$ states that should be its degenerate ground states are mostly not even the lowest two states (figure \[energies\_k\]d); this is consistent with previous Abelian-field results [@3rr-numeric]. However, we note that it too can have its splitting reduced and its “gap" made less negative by some choices of non-Abelian field, and that its components sometimes avoided cross (one such crossing causes the rapid decrease of overlap at $a\approx 2$ in figure \[density2\]b), suggesting similar behaviour to $\nu=1$.
State-dependent atomic interactions
-----------------------------------
The above assumes equal inter- and intraspecies interaction, $U_{\uparrow\uparrow}=U_{\downarrow\downarrow}=U_{\uparrow\downarrow}$. For hyperfine states of real atoms, these are usually nearly but not exactly equal, and they can be far from equal near a Feshbach resonance. Also, an alternative method of generating the U(2) field [@u2f-valley] uses the two pseudo-internal states (momentum valleys) that occur at $\Phi\approx 1/2$, which have a fixed $1:1:2$ interaction ratio [@fqh-high-flux], instead of true internal states.
We tested the effect of asymmetric interactions by repeating some of the above calculations for $U_{\uparrow\uparrow}:U_{\downarrow\downarrow}:U_{\uparrow\downarrow}=1.024:0.973:1$ (the ${\left| F,m_F \right\rangle}={\left| 1,-1 \right\rangle},{\left| 2,1 \right\rangle}$ states of $^{87}$Rb [@rotating-gas-review]), $1:1:0$ (i.e. no interspecies interaction) and $1:1:2$. The small deviation from 1 had no visible effect; since there is only one lowest Landau level, not a degenerate pair, there is nothing to phase separate. The larger increase (decrease) of interspecies interaction increased (decreased) the parameter range over which FQH was seen and its energy gap, but did not alter its qualitative features.
Conclusions {#conc}
===========
We numerically studied the fractional quantum Hall effect in bosons in a U(2) gauge field. For moderate non-Abelian field strengths, we find that it behaves similarly to a *one* internal state Abelian-field quantum Hall system, despite actually having two internal states. Within this regime, the energy gap of the non-Abelian anyon states $\nu=1,3/2$ is strongly dependent on the non-Abelian field parameters, with some settings giving larger gaps than in the true one internal state system. However, as the gap is also strongly dependent on the system size, it is not clear whether this effect persists in the infinite size limit.
In strongly asymmetric non-Abelian fields, the system reverts to behaving as a *two* internal state quantum Hall system (i.e. like it would without any non-Abelian field), while in strong near-symmetric non-Abelian fields, it does not exhibit the fractional quantum Hall effect at all.
We thank Michele Burrello for useful discussions. R.N.P. acknowledges financial support from the European Commission of the European Union under the FP7 STREP Project HIP (Hybrid Information Processing).
References {#references .unnumbered}
==========
|
---
abstract: |
New results, namely the independent determination of the deuterium abundance in several quasar absorption systems, and the complementary determination of the cosmological baryon density by observations of anisotropies in the cosmic microwave background (CMB), allow for a reevaluation of the constraints on the relativistic particle content of the universe at primordial nucleosynthesis. Expressed in terms of the neutrino energy density, we find $1.7 < N_\nu < 3.5\
(95\%\rm~CL)$. In particular, we show that phenomenological four neutrino models including a sterile state (not participating in $SU(2)_L\times U(1)_Y$ interactions) unavoidably thermalize a fourth neutrino, and are highly disfavored in the standard minimal model of primordial nucleosynthesis, if the systematic uncertainty in the primordial helium abundance is small. We describe plausible extensions of the minimal model which evade this constraint.
address: 'NASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory, Batavia, Illinois 60510-0500, USA'
author:
- 'Kevork N. Abazajian'
date: 7 August 2002
title: |
Telling Three from Four Neutrinos\
with Cosmology
---
Introduction
============
With the simplifying conditions of isotropy, homogeneity, thermal equilibrium, and the particle content of the standard model of particle physics, big bang nucleosynthesis (BBN) is a one parameter model dependent only on the baryon-to-photon ratio, $\eta$, at that epoch. The appeal of this simple yet successful standard model [@bbnreview1] has motivated a predictive ability, for example, in the number of leptons in the standard model of particle physics [@Steigman:kc].
The light nuclides D, $^3$He, $^4$He and $^7$Li are produced in measurable quantities in the first three minutes of the standard cosmology. Considerable attention has been devoted to the analysis of uncertainties in the predicted abundances of the light elements and their consistency with the observed light element abundances [@bbnconsist; @Burles:2000zk; @Cyburt:2001pq; @rpp]. Though systematic uncertainties likely dominate the helium abundance measurements [@rpp; @sauerjedamzik; @os] and may be present in observations of the deuterium-to-hydrogen ratio D/H in high-redshift quasar systems [@omeara], and $^7$Li may be partially depleted by stellar processes [@lideplete], standard BBN is remarkably successful predictor for the abundances of these light elements with abundances that differ by nine orders of magnitude.
There exist a variety of ways of modifying the standard BBN paradigm, including altering the spatial distribution of baryon number, out-of-equilibrium decays of massive particles, or new neutrino physics (for a summary, see Ref. [@karsten]).
We focus our attention on a minimal extension to BBN by a modification of the neutrino sector, and specifically to models which attempt to simultaneously account for the indications of neutrino mixing and masses from the atmospheric neutrino results of Super-Kamiokande [@superk], the observations of the transformed solar neutrino flux [@solar; @sno], and the Liquid Scintillator Neutrino Detector (LSND) signal [@lsnd]. A class of models that can accommodate all of these results introduce a fourth mass eigenstate [@maltoni]. As is well known, the fourth flavor state must be sterile, [*i.e.*]{}, not participating in $SU(2)_L\times
U(1)_Y$ interactions, due to its being both light ($m \ll 1\rm\ GeV$) and not observed in the invisible width of the $Z^0$ boson [@rpp]. In current manifestations of four neutrino mixing models, the sterile neutrino is not necessarily closely associated with a single mass eigenstate, since the atmospheric and solar observations each disfavor large sterile components. However, recent global analyses [@maltoni] of the available neutrino oscillation data, including short baseline limits [@choozpaloverde], leave four-neutrino models viable. There are several ways that light sterile neutrinos can be accommodated in neutrino mass models. For a review see, [*e.g.*]{}, Ref. [@Volkas:2001zb].
The standard contribution to the energy density by the three active neutrinos can be augmented by the complete or partial equilibration of the sterile mode. The increased energy density in units of the energy density in one neutrino and its antiparticle ($\rho_{\bar\nu\nu} =
7\pi^2 T^4/120$) is then $N_\nu = 3 + \Delta N_\nu$, where $\Delta
N_\nu = \rho_s/\rho_{\bar\nu\nu}$ is the relative contribution of the sterile state. Here, we use $\Delta N_\nu$ strictly as a parameterization of extra (or missing) relativistic energy density. Sterile neutrinos in the early universe can also give rise to lepton asymmetry generation [@leptasym], which can alter or strongly suppress sterile neutrino thermalization, or, if the asymmetry is generated in the $\nu_e/\bar\nu_e$ sector, alter beta-equilibrium and thus light element abundance production, primarily in the production of $^4$He. As an alternate measure, the baryon-to-photon ratio $\eta$ is related to the cosmological baryon density $\Omega_b$ (as a fraction of the cosmological critical density) as $\eta \simeq
2.74\times 10^{-8}\, \ob$, where $h$ is the present Hubble parameter in units of $100\rm\ km\ s^{-1}\ Mpc^{-1}$.
Letting $N_\nu$ be a free parameter of BBN, its primary effect is altering the predicted $^4$He abundance $Y_p$. The remaining parameter, the baryon content $\eta$, is (over) constrained by D/H, $^3$He and $^7$Li. In one analysis, Lisi, Sarkar and Villante [@lisi] used four permutations of primordial light element abundance determinations to derive limits roughly in the range $2<N_\nu < 4$. In one combination of light element abundance determinations (their data set A), the 99.7% CL region allowed $N_\nu
\sim 4.5$. This value is widely cited as allowing for an additional neutrino (or relativistic degree of freedom) at BBN. Though this limit was correct, it relied on the possibility of a “high” primordial deuterium abundance. Since that work, deuterium has been observed or bounded to have a “low” value in six high-redshift quasar absorption systems (QAS) by three groups [@omeara; @sixdh], and the “high” deuterium QAS observation [@webb] has not been verified in other systems and is disputed [@highbye]. Using the low deuterium abundance and a small uncertainty in $Y_p$, Burles [*et al.*]{} [@Burles:1999zt] found the limit $N_\nu < 3.20$ with the prior $N_\nu > 3$. The work by Cyburt, Fields & Olive [@Cyburt:2001pq] took the possibility of two values of the baryon density, that given by D/H+BBN and CMB, $\Omega_b h^2 \simeq
0.02$, and a value of $\ob \simeq 0.01$ preferred by one inferred primordial value of $Y_p$ [@oss] and the $^7$Li abundance [@ryan] (if undepleted). Ref. [@Cyburt:2001pq] finds that these densities give 95% CL upper bounds of $N_\nu < 3.6$ and $N_\nu < 3.9$, respectively.
Motivated by the six quasar absorption system measurements of a “low” D/H and the analysis of the observations of the CMB anisotropy experiments DASI, BOOMERanG, and MAXIMA, we adopt that the inferred value of the cosmic baryon density from D/H plus standard BBN $\ob
(\rm D/H)$ and the shape of the acoustic peaks in the CMB angular power spectrum $\ob ({\rm CMB})$, are approaching the actual value of $\ob$, within statistical and systematic uncertainties. The forthcoming analysis of the Microwave Anisotropy Probe (MAP) satellite’s observations will potentially reduce the uncertainty in $\ob$ to approximately 10% [@map]. In Section \[bbn\], we analyze in detail the constraints arising from accurate calculations of the primordial helium abundance, the inferred primordial helium abundance, using either $\ob (\rm D/H)$ or $\ob ({\rm CMB})$, and show that a thermalized fourth neutrino is highly disfavored by standard BBN. In Section \[neutrino\], we analyze current four neutrino mixing schemes and their behavior in the early universe and show that thermalization of a fourth neutrino state is unavoidable. In Section \[caveat\], we describe various means and methods of extending the standard model to evade this constraint.
The BBN Prediction of the Number of Neutrinos {#bbn}
=============================================
The consistency of BBN as a predictor of the light element abundances already been explored in some detail [@bbnconsist; @Burles:2000zk; @Cyburt:2001pq; @rpp]. We instead focus on the current uncertainties in the cosmic baryon density $\ob$ and the observed primordial $^4$He abundance $Y_p$, the light nuclide whose abundance is most sensitive to the energy content of the universe at the BBN epoch.
The baryon content of the universe can be estimated in a variety of ways, of which the most precise measures currently are the deuterium abundance at high redshift and the shape of the acoustic peaks in the CMB [@newsarkar]. Deuterium has been observed in high-redshift metal-poor neutral hydrogen systems which are seen as absorbers in the spectrum of back-lighting quasars. The deuterium in these extremely metal-poor systems is inferred to be close to the primordial value due to minimal stellar processing which produces metals and only destroys deuterium. Due to the extreme sensitivity of D/H to the baryon content at BBN, the baryon density required to produce the observed deuterium abundance is rather precisely determined [@Burles:2000zk; @omeara]: $$\label{omegabdh}
\ob({\rm D/H}) = 0.020\pm 0.002\ ({\rm 95\% CL}),$$ for the standard energy density content $N_\nu=3$. The baryon density inferred from D/H has a small dependence on the relativistic energy density of the plasma, which is [@Cardall:1996ec] $$\label{omegabdhnnu}
\ob({\rm D/H}, N_\nu) = \ob({\rm D/H}, N_\nu=3)(1+0.125\,\Delta N_\nu).$$
The baryon content also alters the amplitude of acoustic oscillations in the primordial plasma at CMB decoupling and the relative height of the first three acoustic peaks (for a review of the physics of the CMB, see Ref. [@hudodelson]). The first three acoustic peaks in the angular power spectrum of the CMB have been detected in the analysis of CMB anisotropy measurements by the DASI [@dasi], BOOMERanG [@boom] and MAXIMA [@maxima] experiments. The results of these experiments’ analyses find $$\begin{aligned}
\ob\,({\rm D}) &= 0.022^{+0.004}_{-0.003}\ &({\rm 95\% CL})
\label{dasiomegab} \cr
\ob\,({\rm B}) &= 0.022^{+0.004}_{-0.003}\ &({\rm 95\% CL})\cr
\ob({\rm M}) &= 0.033\pm 0.013\ &({\rm 68\% CL})\end{aligned}$$ respectively. The BOOMERanG value above is that given by their Bayesian approach. For concreteness in our analysis, we quantify the uncertainty in the cosmic baryon density inferred from the CMB with the likelihood function given in Ref. [@dasi] by DASI+DMR.
If analyses of the CMB anisotropy measurements change and provide a value for $\ob$ that is higher than that inferred from standard BBN, then this could have been an indication for a model with large and disparate neutrino degeneracy parameters known as degenerate BBN [@dbbn]. However, if the favored large mixing angle (LMA) neutrino mixing parameters of the solution to the solar neutrino problem are verified ([*e.g.*]{}, by the KamLAND experiment [@kamland]), then synchronized neutrino flavor transformation in the early universe stringently limits neutrino degeneracies [@synch] and degenerate BBN is no longer a rescue.
In order to precisely predict the abundance of $^4$He from standard BBN for varying baryon density and neutrino number, Lopez and Turner [@lopezturner] included finite-temperature radiative, Coulomb and finite-nucleon-mass corrections to the weak rates; order-$\alpha$ quantum-electrodynamic correction to the plasma density, electron mass, and neutrino temperature; and incomplete neutrino decoupling. Ref. [@lopezturner] provides a fitting formula for their results of the predicted helium abundance as a function of $\eta$, $N_\nu$ and neutron lifetime $\tau$. We employ this fitting formula for the predicted $^4$He abundance, taking into account the typographical correction of signs noted in Ref. [@Burles:2000zk]. After all of the above corrections are applied, the uncertainty in the predicted helium abundance is dominated by the neutron lifetime uncertainty, which is now known better than 0.1% [@rpp]. Therefore, we can safely ignore the theoretical errors, as they are dwarfed by observational uncertainty, which we now address.
The primordial helium abundance $Y_p$ has been estimated in observations of hydrogen and helium emission lines from regions of hot, ionized metal-poor gas in dwarf galaxies (H[ii]{} regions). By extrapolating the helium abundance and metallicity relationship for these regions to zero metallicity, Olive, Steigman and Skillman [@oss], and Fields and Olive [@Fields:1998gv] find $$Y_p(\text{OSS-FO}) = 0.238 \pm 0.002\ \text{(stat.)}\pm 0.005\
\text{(sys.)}\,,$$ while Izotov and Thuan [@it] find $$Y_p({\rm IT}) = 0.244 \pm 0.002\ \text{(stat.)}\,.$$ Uncertainties regarding the ionization structure and temperature uniformity of the H[ii]{} regions as well as underlying stellar absorption are sources of significant systematic error. Refs. [@os; @oss] estimate systematic effects in the primordial helium abundance are 2%. Ref. [@sauerjedamzik] finds that systematic effects can lead to 2-4% uncertainties that tend to [*overestimate*]{} $Y_p$. In an attempt to avoid bias, in this work we adopt the central value of $$Y_p = 0.241 \pm 0.002\ \text{(stat.)}\pm \sigma_{\rm sys}\,,
\label{yp}$$ and characterize the systematic uncertainty as the disparity between competing claims $$\sigma_{\rm sys} = {|Y_p(\text{OSS-FO}) -Y_p({\rm IT})|}\,,$$ or approximately 3%. The shape of systematic uncertainty in likelihood space is certainly not well defined, therefore we make the simplifying ansatz of a Gaussian distribution, as done, [*e.g.*]{}, in Ref. [@Cyburt:2001pq], and combine the statistical and systematic errors in quadrature.
We produce probability distribution functions (p.d.f.’s) for $N_\nu$ versus $\ob$, using Gaussian distributions for $Y_p$ \[Eq. (\[yp\])\] and $\ob({\rm D/H},N_\nu)$ \[Eq. (\[omegabdhnnu\])\], and the likelihood function given in Ref. [@dasi] for $\ob({\rm DASI})$. We find, using either the information from deuterium or the CMB on $\ob$: $$\begin{aligned}
N_\nu{\rm(D/H)} &=& 2.60^{+0.90}_{-0.90} \\
N_\nu{\rm(DASI)} &=& 2.46^{+1.03}_{-1.01} \,,\end{aligned}$$ at 95% CL, which is consistent with the standard BBN prediction. We show the shapes of the likelihood contours in Fig. \[nnufull\] (a). To illustrate the difference between adopting the IT or OSS-FO helium values, we plot the 99% likelihood contours for the choices $Y_p(\text{OSS-FO})$, $Y_p({\rm IT})$ (using the systematic uncertainty of $\pm 0.005$) and our choice (\[yp\]). As seen in Fig. \[nnufull\] (b) the range of uncertainty does not depend on the choice of the central value but the size of systematic effects.
![\[nnufull\] Shown are the contours of 68%, 95% and 99% CL (inner to outer contours) when using the baryon density inferred from the deuterium abundance (D/H) at high-redshift (solid lines) [*or*]{} that from the DASI+DMR analysis of CMB anisotropies (dashed lines), in frame (a). In both cases, we use our our adopted determination of the primordial $^4$He abundance. In frame (b), we show the 99% CL contours using the Izotov & Thuan (IT), Olive, Steigman & Skillman (OSS) and Fields & Olive (FO) and our adopted value and uncertainty of the primordial $^4$He abundance along with the D/H determination of $\ob$. The uncertainty in $N_\nu$ does not depend on the choice between IT and OSS-FO as much as the size of systematic uncertainty. See text for details.](nnufull3.eps){width="9cm"}
For a sterile neutrino to be thermalized with the bath of the early universe, the active neutrinos must be thermalized initially. This constitutes prior information that may [*loosen*]{} the constraints shown in Fig. \[nnufull\]. Prior information can be included in a Bayesian approach [@rpp], integrating the p.d.f. only in the physically allowed region, $N_\nu > 3$, which we have done using a Monte Carlo integration. We show the confidence level intervals for this case in Fig. \[nnu\]. As seen there, a fully populated fourth neutrino is excluded at approximately the 99% CL.
![\[nnu\]Shown are the contours of 68%, 95% and 99% CL (inner to outer contours) given the prior condition that the active neutrinos are thermalized, our adopted value of primordial $^4$He abundance, and $\ob$ determined by D/H (solid lines) or the CMB observations by DASI+DMR (dashed lines). See text for details.](nnu3.eps){width="9cm"}
Measurements of CMB anisotropies by the MAP satellite may measure $\ob$ to 10%, giving a precise value independent of BBN. If consistent with $\ob$ inferred from D/H and BBN, then $\ob$ becomes a “nuisance parameter” that can be marginalized and the confidence level for $N_\nu$ is simply the integral of the p.d.f., $${\rm CL}(N_\nu) = \int_3^{N_\nu}{p(N_\nu^\prime|Y_p)\,dN_\nu^\prime}.$$
Four Neutrino Models in the Early Universe {#neutrino}
==========================================
There is now convincing evidence for neutrino flavor states to be composed of large amplitudes of more than one mass state from two experiments: Super-Kamiokande [@superk] and the Sudbury Neutrino Observatory (SNO) [@sno]. There is also an indication of a neutrino oscillation signal at short baselines from the Liquid Scintillator Neutrino Detector (LSND) experiment [@lsnd]. To accommodate all three of these results, a four neutrino model must be invoked (or $CPT$ is violated; see below). The mass and flavor state bases are related by a unitary transformation $$\nu_\alpha = \sum_i^4 U_{\alpha i} \nu_i \,,$$ where $\alpha=e,\mu,\tau,s$ denotes the flavor state and $i$ is the mass state. The matrix $U_{\alpha i}$ generally has 6 rotation (mixing) angles and 3 $CP$ violating phases. The transformation probability has the form: $$\begin{aligned}
P(\nu_\alpha\rightarrow\nu_\beta) =\ &\delta_{\alpha\beta} - 4
\sum_{i>j}{\rm Re}\left(U_{\alpha i}^\ast U_{\beta i} U_{\alpha j}
U_{\beta j}^\ast\right)\ \sin^2 \left(\delta
m^2_{ij}\frac{L}{4E}\right)\cr &+ 2 \sum_{i>j}{\rm
Im}\left(U_{\alpha i}^\ast U_{\beta i} U_{\alpha j} U_{\beta
j}^\ast\right)\ \sin \left(\delta m^2_{ij}\frac{L}{2E}\right),\end{aligned}$$ which provides rich neutrino oscillation phenomenology and experimental possibilities [@telling]. Here $\delta m^2_{ij}
\equiv m_i^2 - m_j^2$, $L$ is the distance from where the flavor $\nu_\alpha$ was created, and $E$ is its energy. For reviews of neutrino phenomenology see, [*e.g.*]{}, Refs. [@nureview].
For neutrino physics in the early universe, we are interested in the magnitude of mixing amplitudes of active neutrinos converting to sterile states. First derived by Langacker [@Langacker:1989sv], and Dolgov and Barbieri [@Barbieri:1989ti], it has been known for some time that a sterile neutrino coupling with a single active neutrino via the unitary neutrino mass matrix must not have [@enqvist; @shischrammfields] $$\delta m_{\alpha s}^2 \sin^4 2\theta_{\rm BBN} \lesssim
\begin{cases}
5\times 10^{-6},\qquad\text{for $\alpha=e$}
\\
3\times 10^{-6},\qquad\text{for $\alpha=\mu,\tau$}
\end{cases}
\,,
\label{oldconstraints}$$ in order to not fully thermalize the sterile neutrino prior to BBN via nonresonant collisional processes. Such constraints (\[oldconstraints\]) certainly do not directly apply to multiple neutrino mixing schemes including a sterile which nature may have given us. Multiple mixing angles and the phenomenon of lepton number generation via neutrino mixing complicate the BBN bound.
We adopt a rotation angle ordering for $U$ so that $(\nu_e,\nu_\mu,\nu_\tau,\nu_s) = (\nu_1,\nu_2,\nu_3,\nu_4)$ when all mixing angles are set to zero. Four neutrino mixing scheme constraints have been examined recently in detail by Di Bari [@dibari], which we summarize and expand on here in view of the recent global analyses of four-neutrino models by Maltoni, Schwetz and Valle [@maltoni]. The effective oscillation amplitude of active-sterile neutrino mixing between flavor $\alpha$ and the sterile can be written as $$A_{\alpha;s} = 4|U_{\alpha 4}|^2 |U_{s 4}|^2 \simeq \sin^2
2\theta_{\rm BBN}\,.$$ Therefore, BBN constraints for active-sterile mixing through a pair of neutrino mass eigenstates $\nu_4,\nu_i$ are $$\delta m_{4i}^2 A^2_{\alpha;s} \lesssim
\begin{cases}
5\times 10^{-6},\qquad\text{for $\alpha=e$}
\\
3\times 10^{-6},\qquad\text{for $\alpha=\mu,\tau$}
\end{cases}
\,,
\label{constraint}$$ for nonresonant sterile production, where $\delta m_{4i}^2 >0$. This constraint (\[constraint\]) applies primarily to the active-sterile mixing that leads the oscillation, [*i.e.*]{}, that which has the shortest oscillation length or largest $\delta m_{ij}^2$. Small vacuum mixing amplitudes in the leading oscillation mode may avoid thermalization, and so secondary oscillation modes with larger mixing amplitudes may thermalize the sterile.
3+1
---
Models referred to as (3+1) may satisfy all experimental indications of neutrino oscillations with a triplet of mass eigenstates that provide the atmospheric and solar mass-scales, and a sterile-dominated mass eigenstate with a large mass-scale splitting with the triplet providing the LSND result via indirect $\bar\nu_\mu \rightarrow
\bar\nu_e$ mixing through the sterile. In order to satisfy the mixing amplitude that would provide the LSND signal, the oscillation amplitude must be [@maltoni] $$\begin{aligned}
A_{\mu;e}\ &=&\ 4 |U_{e4}|^2 |U_{\mu 4}|^2\cr
&>&\ 3 \times 10^{-4}\qquad ({\rm 99\%\ CL})\,,
\label{lsnd31}\end{aligned}$$ at a $\delta m^2_{\rm LSND}\simeq 2\ {\rm eV}^2$ from Fig. 8 of Ref. [@maltoni]. Indirect mixing of this form has two mixing amplitudes that may thermalize the sterile. In this case, the mixing amplitudes $A_{\mu;s}$ and $A_{e;s}$ may participate in the thermalization. Consider the slightly less-constrained \[cf. (\[constraint\])\] $A_{\mu;s}= 4|U_{\mu 4}|^2 |U_{s 4}|^2$. The mixing matrix element $$|U_{s 4}|^2 > 0.54\quad ({\rm 99\%\ CL})\,,
\label{atm31}$$ is bounded from below from constraints on the fraction of sterile neutrinos participating in atmospheric oscillations (see Fig. 3 of Ref. [@maltoni]).
Evading constraints from BBN on $A_{\mu;s}$ and $A_{e;s}$ would require minimizing both $|U_{e4}|$ and $|U_{\mu 4}|$ while satisfying (\[lsnd31\]). This gives $|U_{e4}|^2 = |U_{\mu 4}|^2 \simeq
10^{-2}$. Combined with the limit (\[atm31\]), the BBN constrained combination has the minimum value $$\delta m^2_{\rm LSND} A^2_{\mu;s} \gtrsim 7\times 10^{-4}$$ which exceeds the limits (\[constraint\]) by at least two orders of magnitude and invariably thermalizes the sterile. This constraint comes from the conservative case where $\delta m_{4i}^2 > 0$. The inverted case $\delta m_{4i}^2 < 0$ is resonant and more stringently constrained. Therefore, (3+1) models are strongly disfavored by standard BBN.
2+2 {#twoplustwo}
---
Four neutrino models may also accommodate all indications for neutrino mixing with mass eigenstates in a pair of doublets that provide the solar and atmospheric mass scales and a large mass gap between the doublets providing the LSND mass scale. The global analysis by Maltoni [*et al.*]{} [@maltoni] finds that such (2+2) models are consistent within 99% CL either with the sterile neutrino completely participating in the atmospheric or solar solutions. Therefore, it is possible to choose a very small amplitude mixing between the doublet not participating in sterile oscillations and the sterile state such that the large LSND mass splitting does not populate the sterile neutrino. On the other hand, unitarity constrains the sterile state to be present among some linear combination of mass eigenstates. Whether (2+2) scenarios are compatible with the exclusion of large sterile components in both the atmospheric and solar neutrino observations is controversial [@strumiabargerbahcall].
Whether the sterile flavor is in the atmospheric doublet or solar doublet is not of concern for the early universe. In either case, the sterile flavor will be thermalized. For our notation, we employ the mass scheme used in Ref. [@maltoni], where the (2+2) model has the solar scale between $\nu_1$ and $\nu_4$ and the atmospheric scale between $\nu_2$ and $\nu_3$, with the hierarchy only being determined by the condition for resonance in the sun, $m_1<m_4$. One can avoid the effects of thermalization of the largest mass scale by setting the inter-doublet mixings to zero, $U_{s1}=U_{s3}=0$, but by unitarity $|U_{s2}|^2 + |U_{s4}|^2 = 1$, whereby the sterile neutrino participates in large part in the solar scale, atmospheric scale or both. Having a complete sterile solution for either scale has already been known to thermalize the sterile neutrino [@enqvist; @shischrammfields]. One could consider democratically separating the sterile into both the atmospheric and solar scales to minimize its presence in both, so that $|U_{s2}|^2 =
|U_{s4}|^2 =1/2$. However, the oscillation amplitudes $A_{\mu;s} = 4
|U_{\mu 2}|^2 |U_{s 2}|^2$ and $A_{e;s} = 4 |U_{e 4}|^2 |U_{s 4}|^2$ still grossly exceed the BBN bounds (\[constraint\]) since the magnitudes $|U_{e4}|^2 = |U_{e1}|^2 \tan^2\theta_{\rm LMA}$ and $|U_{\mu 2}|$ must be large to accommodate the large to maximal mixing angle solutions of the solar and atmospheric neutrino problems. Therefore, (2+2) models are also strongly disfavored by standard BBN.
Self-Suppression
----------------
The possibility that a four neutrino mass scheme could be arranged in such a way as to evade sterile-thermalization constraints were considered by Bell, Foot & Volkas [@bell] and Shi, Fuller & Abazajian [@shi]. One could potentially either self-generate a lepton number and suppress the large-mixing-amplitude thermalization or offset the effects of sterile thermalization by altering the electron neutrino-antineutrino asymmetry through alteration of beta-equilibrium, $$\begin{aligned}
{\rm n} + \nu_e\, &\leftrightarrow& {\rm p} + e^-\cr
{\rm n} + e^+&\leftrightarrow&{\rm p} + \bar\nu_e \,.
\label{beta}\end{aligned}$$ In the models considered in Refs. [@bell; @shi], the direct thermalization bounds (\[constraint\]) were avoided by placing the sterile neutrino in the small-mixing-angle solution to the solar neutrino problem, a region of parameter space still viable at the time and outside of the constraint region (\[constraint\]).
In addition, Refs. [@bell; @shi] explored methods of generating asymmetries between electron neutrinos and antineutrinos by resonant lepton number generation [@leptasym]. The resonance condition in the early universe requires $m_4 < m_i$, where $m_i$ is a mass eigenstate (more) closely associated with an active flavor. A positive electron neutrino number will suppress the $^4$He abundance by shifting the rates (\[beta\]), which would be necessary if standard BBN is inconsistent by having too high of a predicted $^4$He abundance for a given $\ob$. As shown above, standard BBN remains consistent within observational uncertainty. The sign of resonantly generated electron neutrino/antineutrino asymmetry can be chaotic [@shichaos], or at least not well determined [@chaosfinns], having an significant chance (50% if chaotically random) of being negative and actually [*increasing*]{} $Y_p$ by altering beta-equilibrium in the opposite direction. If the sign of the asymmetry [*is*]{} randomly chaotic, then causally disconnected regions will have different sign asymmetries, which leads to an enhancement of the transformation of active neutrinos into sterile neutrinos at the boundaries of regions of different sign [@shifuller] and potentially placing more stringent constraints on four neutrino mass schemes [@abazajianfullershi].
There is considerable evidence now that the solar solution lies in the LMA region of parameter space [@sno]. Therefore, as discussed in the previous sections, thermalization of the sterile is unavoidable in either the (3+1) or (2+2) scenarios. And, importantly, it was shown by Di Bari [@dibari] that thermalization of the sterile in these four neutrino models suppresses lepton number generation, and electron neutrino/antineutrino asymmetries are not effective in avoiding the BBN bounds (\[constraint\]).
Constraint Evasion and New Physics {#caveat}
==================================
The simplifying and appealing principle of Occam’s razor has proven to be a powerful tool as a predictor in science, yet nature does not always take the most simple form. The minimal model for four neutrino mixing or the standard BBN described above may certainly not be the entire framework of the early universe or particle physics. Importantly, if all experimental indications for neutrino oscillations remain, [*viz.*]{}, if the MiniBooNE detector [@boone] verifies the LSND signal, K2K [@k2k] and MINOS [@minos] verify the atmospheric oscillation solution and KamLAND detects the LMA signal [@kamland], then new physics must be at play beyond standard three-neutrino mixing and standard BBN. There exist a number of ways of accommodating such a scenario, several of which are described below. The aesthetic value of these scenarios are left to the judgment of the reader.
[*Pre-existing lepton asymmetry*]{} — A lepton number in the active neutrino flavors will suppress sterile neutrino population by magnifying the associated lepton potential and dwarfing the vacuum mixing amplitude [@leptasymsupp]. This lepton number would have to be produced by an unspecified mechanism earlier than the population of the sterile neutrino would take place.
[*A fifth mass eigenstate*]{} — Appropriate insertion of a mass eigenstate with a major sterile component with $m_5 < m_i\ (i=1...4)$ in degenerate neutrino mass models may resonantly generate lepton number sufficiently prior to sterile thermalization as to suppress it. This possibility was explored in Ref. [@dibari].
[*Majoron fields*]{} — One mechanism for generating neutrino mass involves a massless Nambu-Goldstone boson (a majoron) from models where either the total or partial lepton number is spontaneously broken [@Chikashige:1980ui]. In such models, a coherent majoron field creates potentials for the neutrinos proportional to the gradient of the field and suppresses sterile thermalization in a similar way as a pre-existing lepton asymmetry [@Bento:2001xi]. Interestingly, this mechanism arises from the neutrino mass model itself.
[*A low reheating temperature universe*]{} — There is no direct evidence that the neutrino background is thermalized. As explored in Refs. [@lowrh], the highest temperature of the universe could have been only $0.7\rm\ MeV$. The neutrino background may never have been thermalized, but the observed light element abundances could still be created. In this case, sterile neutrinos may modify the nucleosynthesis processes by partial population but are not directly excluded.
[*Baryon-antibaryon inhomogeneity*]{} — Detailed calculations of diffusion and nucleosynthesis in universes containing baryon number asymmetries [@Steigman:ev] have found that small-scale antibaryon domains are not excluded by BBN and the observed light element abundances [@antibbn], and may lift constraints on relativistic energy density present at BBN to $N_\nu \lesssim 7$, even with total baryon densities consistent with CMB observations [@Giovannini:2002qw].
[*Extended quintessence*]{} — Non-minimally coupled quintessence models (where the quintessence field is not only coupled to gravity) that provide a negative-pressure vacuum energy density to explain the acceleration phase that the universe may be entering can alter BBN [@Chen:2000xx]. In certain cases of such “extended quintessence” scenarios, the quintessence field may behave to decrease the expansion rate during the freeze-out of beta equilibrium (\[beta\]), and therefore [*decrease*]{} the predicted helium abundance. This reduction of the expansion rate could offset the increase in the expansion rate due to the presence of an extra neutrino degree of freedom and allow for four-neutrino models.
[*$CPT$ violating neutrinos*]{} — There exists a radical proposal that fits all indications for neutrino oscillation and invokes $CPT$ violation in the neutrino sector [@nocpt]. The success of this model lies in the fact that LSND’s indication for neutrino oscillation lies primarily in the antineutrino $\bar\nu_\mu \rightarrow\bar\nu_e$ channel [@lsnd], is motivated by braneworld scenarios with extra dimensions, and gives dramatic predictions for the MiniBooNE [@boone] and KamLAND [@kamland] experiments. This model has no effect on standard BBN since in the standard case the neutrinos and antineutrinos are equally thermally populated and therefore $CPT$ violating neutrino oscillations do not disturb the detailed balance of thermal equilibrium, leading to no direct conflicts between light element abundances and standard BBN.
Discussion and Conclusions
==========================
In minimal models of big bang nucleosynthesis with no new physics, we have shown that four neutrino models explaining current indications for neutrino oscillations are disfavored at the 99% CL. This conclusion depends on systematic effects not being larger than that expected ($\sim 3\%$) in determining the $^4$He abundance in ionized H[ii]{} regions and that the baryon density inferred by the D/H abundance in six high-redshift quasar absorption systems and the anisotropies in the cosmic microwave background are approaching the true cosmic value of $\ob$.
The MAP satellite will verify or disprove the value of $\ob$ inferred by the above methods to high precision in the near future [@map]. If consistency remains in cosmological determinations of the baryon density, then $\ob$ would then become a “nuisance” parameter in determining the cosmological energy density at standard BBN. The primordial helium abundance is currently the best probe of the energy density of the universe present at BBN, yet there remain no concrete proposals in the literature for the reducing systematic uncertainties present in determining the primordial $^4$He abundance, which is the dominant uncertainty in constraining the energy density present in the universe at the age of one second via standard BBN.
In addition, we have summarized several scenarios that evade the standard BBN model constraints presented here. Remarkably, if all experimental indications for neutrino oscillations are confirmed, new physics must be present not only in the particle content of the neutrino sector but also in the early universe.
Acknowledgments
===============
I would like to thank Kaladi Babu, Gabriela Barenboim, Nicole Bell, Scott Burles, Janet Conrad, Scott Dodelson, Josh Frieman, George Fuller, Manoj Kaplinghat, Jim Kneller, Rabi Mohapatra, Sandip Pakvasa, Subir Sarkar, Mike Turner and Jose Valle for fruitful discussions, and the Institute for Nuclear Theory at the University of Washington for hospitality and the DOE for support in hosting a Mini-Workshop on Neutrino Masses & Mixing which initiated this project. I would especially like to thank John Beacom for extremely valuable discussions regarding my statistical approach. This research was supported by the DOE and NASA grant NAG 5-10842 at Fermilab.
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|
---
author:
- |
Jonathan Baxter\
Department of Systems Engineering\
Australian National University\
Canberra 0200, Australia
bibliography:
- 'bib.bib'
date: 'August 16, 1999'
title: 'Some observations concerning Off Training Set (OTS) error.'
---
Introduction
============
A new measure of generalisation error called Off Training Set (OTS) error was introduced recently in [@wolly96a; @wolly96b]. Under quite weak assumptions it was shown that with respect to OTS error there are no [*a priori*]{} distinctions between learning algorithms, at least if it is assumed that the target functions are uniformly distributed. Thus, as far as OTS error is concerned, an algorithm that minimizes error on the training set will do no better than random guessing. If OTS error accurately models the concept of generalization then this is a depressing conclusion indeed.
However, in this paper it is argued that OTS error does not model what is normally meant by generalization error. In particular, it is shown that the assumptions underlying one of the main “no free lunch” (NFL) theorems (theorem 2) in [@wolly96a] imply that the distributions used to generate training data and testing data have disjoint supports. Thus, training a neural network to recognise faces by showing it images of handwrittten characters is the kind of learning problem covered by the NFL theorem. Not surprisingly, one cannot conclude anything about generalisation performance in such circumstances, but it would seem that such a scenario is of little interest in machine learning and statistics anyway.
OTS error {#OTS}
=========
For simplicity of exposition, the following restricted learning scenario is considered. In the notation of [@wolly96a], the learning algorithm is supplied with a training set $d =
\{d_X(i),d_Y(i)\}$, $i=1,\dots,m$, where each $d_X(i)$ is chosen from the (finite) input space $X$ according to a distribution $\pi$, and $d_Y(i) = f(d_X(i))$ is some fixed Boolean target function, $f\colon X\to
\{0,1\}$. The set of all $d_X(i) \in d$ is denoted by $d_X$. The learning algorithm is assumed to be deterministic, so that in response to the taining set $d$, the algorithm produces a hypothesis $h_d\colon X\to \{0,1\}$.
The generalization performance of the algorithm is measured by the [*off training set error*]{} (OTS error): $$E_\OTS(d,f) := \frac{1}{\sum_{x\in X-d_X} \pi(x)}
\sum_{x\in X - d_X} \pi(x) |h_d(x) - f(x)|.$$ Note that OTS error is just the expected error of the algorithm’s hypothesis on those inputs [*not*]{} appearing in the training set. Another way of expressing OTS error is as the expected loss of the learner with respect to the [*testing distribution*]{}: $\pibar_d(x) := 0$ if $x \in d_X$ and $\pibar_d(x) := \pi(x)/\sum_{x\in X-d_X} \pi(x)$ if $x\notin d_X$. Note that $\pibar_d$ depends on the training set $d$. The more general case where $\pibar_d(x)$ is any distribution on $X$ with the property that $\pibar_d(x) = 0$ if $x\in d_X$ is also considered in [@wolly96a] (see the remarks preceding theorem 2 in that paper). In either case we can write, $$\label{ots}
E_\OTS(d,f) = \sum_{x\in X} \pibar_d(x) |h_d(x) - f(x)|.$$ As the input space is finite and we are only considering Boolean target functions, there is no difficulty with the concept of choosing a target function $f$ uniformly at random. The uniform average over all target functions will be denoted by $\E_f$. The following theorem is essentially theorem 2 from [@wolly96a], applied to the particular scenario of the present paper.
\[bong\] Suppose that $P(d|f)$ is independent of $f(x)$ for all $x\in X - d_X$ (such a $P(d|f)$ is called a [*vertical likelihood*]{} in [@wolly96a]). Furthermore, suppose that for all training sets $d$ of size $m$, the testing distribution $\pibar_d(x) = 0$ if $x\in d_X$. Then, $$\label{bongeq}
\E_d \E_f E_\OTS(d,f) = \frac12,$$ where $\E_d$ is the expectation over all training sets $d$ of size $m$.
Discussion
==========
On face value, theorem \[bong\] looks rather negative. It says that [*any*]{} algorithm for choosing a hypothesis $h_d$ based on training data $d$ will have an expected OTS error of 1/2. As random guessing would give an expected error of 1/2, theorem \[bong\] would appear to show that no algorithm can do better than random guessing.
However, closer inspection reveals that while OTS error is a reasonable candidate for generalisation error when considered in the context of a [*single*]{} training set $d$, it is quite pathological when expectations are taken over [*all*]{} training sets, as is the case in theorem \[bong\]. Specifically, substituting into the left hand side of gives $$\label{boing}
\E_d \E_f \sum_{x\in X} \pibar_d(x) |h_d(x) - f(x)| = \frac12.$$ As the testing distribution $\pibar_d$ varies with $d$, this expression cannot be interpreted as the expected OTS error of the algorithm with respect to some [*fixed*]{} testing distribution. A test distribution that depends on the training data is too hard a target for machine learning because it encompasses the situation in which an adversary generates training sets according to some fixed distribution $\pi$, but then varies the distribution generating test sets depending on the particular training set produced.
For theorem \[bong\] to be more relevant to machine learning a [*fixed*]{} test distribution $\pibar(x)$ must be chosen. However, one of the crucial assumptions in the proof of theorem \[bong\] is that the testing distribution satisfies $\pibar(x) = 0$ if $x\in d_X$, for [*any*]{} training set $d$ of size $m$ of positive probability (see [@wolly96a], appendix C. The condition used there is actually that $\pibar(x) = 0$ for any training set $d$ of size $m$ (regardless of whether it has positive probability), but the theorem still holds under the weaker assumption above). This implies that $\pibar(x) = 0$ for any $x$ such that $\pi(x) >0$ (recall that $\pi(x)$ is the probability of input $x$ appearing in the training set). In other words, for a fixed test distribution $\pibar$, the assumptions behind theorem \[bong\] imply that the training distribution $\pi$ and the testing distribution $\pibar$ have disjoint supports. This means that no matter how large the training set is, there is always [*zero*]{} probability of seeing an example in testing that was already seen in training.
Clearly under such circumstances one cannot conclude anything about the generalisation behaviour of a learning algorithm, which is the content of theorem \[bong\]. However, disjoint training and testing distributions is unlikely to be interesting from a machine learning perspective. Some kind of relationship between training and test data is always assumed, otherwise there would be no point feeding the training data into the learning algorithm in the first place. In fact in practice, where possible, the assumption that the training and testing data are generated according to the same distribution is usually [ *engineered*]{}.
Put another way, no-one would train a neural network to recognize faces by feeding it a training set consisting of images of handwritten characters.
Very large input spaces {#very-large-input-spaces .unnumbered}
-----------------------
If the input space $X$ is very large then in practice the training set and testing set will almost always be disjoint, even if the training and testing distributions are identical. Under such circumstances one might expect the negative conclusion of theorem \[bong\] to apply. However, it does not, the reason being the subtle difference between “almost never seeing the same example in testing as seen in training” and ruling out a-priori any possibility of seeing the same example in testing as seen in training. The latter has to hold if the NFL theorems are to apply.
Conclusion
==========
We have seen that the negative conclusions of the “No Free Lunch” theorems can be avoided by assuming that the training and test distributions have some overlap. Note that although we have to assume something about the input-space distributions, we do not have to assume anything about the distribution over target functions.
|
---
abstract: 'We study theoretically the capturing of paramagnetic beads by a magnetic field gradient in a microfluidic channel treating the beads as a continuum. Bead motion is affected by both fluidic and magnetic forces. The transfer of momentum from beads to the fluid creates an effective bead-bead interaction that greatly aids capturing. We demonstrate that for a given inlet flow speed a critical density of beads exists above which complete capturing takes place.'
author:
- Christian Mikkelsen and Henrik Bruus
date: 6 May 2005
title: 'Microfluidic capturing-dynamics of paramagnetic bead suspensions'
---
Introduction {#sec:introduction}
============
Recently, there has been an increasing interest in using magnetic beads in separation of, say, biochemical species in microfluidic systems [@Choi:00a; @Pankhurst:03a]. The principle is to have biochemically functionalized polymer beads with inclusions of superparamagnetic nanometersize particles of, for example, magnetite or maghemite. They attach to particular biochemical species and can be separated out from solution by applying external magnetic fields. As most biological material is either diamagnetic or weakly paramagnetic, this separation is specific. Paramagnetic particles in fluids are also used to measure the susceptibility of, for example, magnetically labelled cells by measuring particle capture or motion in a known field [@Zborowski:95a; @McCloskey:03a].
In this paper we study microfluidic capturing of paramagnetic beads from suspension by modeling the beads as a continuous distribution [@Warnke:03a]. The separation of suspended paramagnetic beads from their host fluid is an important process as it decides operating characteristics for practical microfluidic devices. It involves an interplay between forces of several kinds governing the dynamics of the process: (a) Magnetic forces from the application of strong magnetic fields and field gradients. (b) Drag forces due to the motion of the beads with respect to the host fluid. (c) The trivial effect of gravity, which we ignore in the following. We emphasize the effects of bead motion on the fluid flow as this gives rise to a hydrodynamic interaction between the beads. As we have noted in a previous few-bead study, this interaction is more important than the magnetic bead-bead interactions [@Mikkelsen:05a]. It is created by drag forces in two steps: First, drag transfers momentum to the fluid from the beads moving under the influence of external forces. Second, the modified flow changes the drag on and thus motion of other beads.
![(a) \[fig:GeometryConc\] Sketch of the microfluidic system with $L=350~\mu$m and $h=50~\mu$m. A suspension of paramagnetic beads enters at $x=0$ with a parabolic Poiseuille flow profile, $\mathbf{u}_{\text{o}}$, and leaves at $x=L$. Beads are caught by the pair of wires placed $100~\mu$m from the outlet at the top and carrying currents $\pm I$. (b)–(d) Simulated stationary density of the beads ranging from zero (white) to $c_\text{o}$ (black) for increasing values of the current-distance product $Id$ as indicated. At $x=0$ the concentration is $c_\text{o} =
10^{13}$ m$^{-3}$ and the maximum flow speed is $300~\mu$m/s.](fig01.eps)
Model
=====
As sketched in Fig. \[fig:GeometryConc\](a) we consider a viscous fluid (water) flowing in the $x$ direction between a pair of parallel, infinite, planar walls. The walls are placed parallel to the $xy$ plane at $z=0$ and $z=h$, respectively. A steep magnetic field gradient is generated by a parallel pair of closely spaced, infinitely long, thin wires along the $y$ direction separated by $d$ and carrying opposite currents $\pm I$. The system is translation invariant in the $y$ direction thereby reducing the simulation to a tractable problem in 2D. The simulation domain is defined by $0<x<L$ and $0<z<h$ with $L =
350~\mu$m and $h = 50~\mu$m. The wires intersect the $xz$ plane near $(x,z)=(250~\mu\text{m},\ 55~\mu\text{m})$ just above the top plate. Paramagnetic beads in suspension are injected into the microfluidic channel by the fluid flow at $x=0$. They are either exiting the channel at $x=L$ or getting collected at the channel wall near the wires.
When a suspension of beads is viewed on a sufficiently large scale compared to the single bead radius $a$ but on a scale comparable to density variations, we can describe the distribution of beads in terms of a continuous, spatially varying bead number density $c$. We consider a suspension of beads with radius $a=1~\mu$m and denote the initial number density at $x=0$ by $c_{\text{o}}$. The four basic constituents of the model are described in the following.
*Magnetic force.* The beads are paramagnetic with a magnetic susceptibility $\chi=1$. In an external magnetic field ${\ensuremath{\textbf{H}_{\text{ext}}}}(\textbf{r})$ the force on such a bead is $$\begin{aligned}
\textbf{F}_{\text{ext}} & = \mu_\text{o}\!\int_{\text{bead}}
(\textbf{M}\cdot{\boldsymbol{\nabla}}){\ensuremath{\textbf{H}_{\text{ext}}}}\,dV\nonumber \\
\label{eq:magforce}
& = 4\pi\mu_\text{o} a^3 \frac{\chi}{\chi+3}({\ensuremath{\textbf{H}_{\text{ext}}}}\cdot{\boldsymbol{\nabla}}){\ensuremath{\textbf{H}_{\text{ext}}}}\end{aligned}$$ assuming that the bead is so small that we can take the external field ${\ensuremath{\textbf{H}_{\text{ext}}}}$ to be approximately constant across the bead, i.e., $a |{\boldsymbol{\nabla}}{\ensuremath{\textbf{H}_{\text{ext}}}}| \ll |{\ensuremath{\textbf{H}_{\text{ext}}}}|$ when determining the magnetization $\textbf{M}$.
As mentioned, ${\ensuremath{\textbf{H}_{\text{ext}}}}$ in this study arises from a pair of current carrying wires. It is determined in the following manner. From Ampère’s law, we readily find the magnetic field, $\textbf{H}$, around a straight circular wire, $\textbf{H}
(\textbf{r}) = \textbf{J}\times\textbf{r}/(2\pi r^2)$, where the electrical current vector $\textbf{J}$ is along the wire orthogonal to the position vector $\textbf{r}$ which is in the $xz$ plane. The magnetic field from the two closely spaced anti-parallel wires is found by decreasing the separation $d$ and increasing the current, $I=|\textbf{J}|$, while keeping the product $I d$ constant, $$\textbf{H}_{\text{ext}}
= \frac{1}{2\pi r^2}\left(\textbf{J}\times\textbf{d} -
\frac{2(\textbf{J}\times\textbf{r})(\textbf{d}\cdot\textbf{r})}{r^2}\right).$$ This together with [Eq. (\[eq:magforce\])]{} yields $$\textbf{F}_{\text{ext}} = -\frac{2}{\pi}
\frac{\chi}{\chi+3}\mu_\text{o}
a^3 (I\, d)^2 \,\frac{\textbf{r}}{r^6}\: ;$$ a manifestly attractive central force (from the mid-point of the wires), independent of the direction of $\textbf{d}$.
*Fluid flow.* \[Sec:flow\] The beads are suspended in a fluid of viscosity $\eta$ and density $\varrho$ that is launched at $x=0$ with a parabolic velocity profile, $\mathbf{u}_{\text{o}}$, and flows past the wires. In microfluidics inertial effects are unimportant compared to drag, so the small beads in suspension almost always move with constant velocity relative to the fluid. Except for acceleration phases shorter than microseconds the external forces are exactly balanced by drag [@Newton2]. The momentum transfer from beads to fluid is included by adding a bulk force term, $c\textbf{F}_{\text{ext}}(\textbf{r})$ to the Navier–Stokes equation. This bulk force term is proportional to the number density $c$ of beads and the magnetic external force $\textbf{F}_{\text{ext}}$ on an individual bead at position $\textbf{r}$. The velocity $\textbf{u}$ of the fluid is given by $$\varrho\partial^{{}}_t \textbf{u}+\varrho(\textbf{u}\cdot{\boldsymbol{\nabla}}){\textbf{u}}
= -{\boldsymbol{\nabla}}p +\eta\nabla^2 \textbf{u} + c\textbf{F}_{\text{ext}},
\label{eq:NS}$$ along with the incompressibility condition ${\boldsymbol{\nabla}}\cdot\textbf{u} =
0$.
*Bead motion.* \[Sec:beads\] To complete the set of equations, it is necessary to have an equation of motion for the bead number density $c$. As the beads neither appear nor disappear in the bulk, $c$ must obey a continuity equation $$\partial^{{}}_t c + {\boldsymbol{\nabla}}\cdot\textbf{j} = 0,
\label{eq:particlecontinuity}$$ where the bead current **j** is defined by the Nernst–Planck equation [@Probstein:94] $$\label{eq:NP}
\textbf{j} = -D{\boldsymbol{\nabla}}c + c
\textbf{u} + cb\textbf{F}_{ \text{ext}}$$ with diffusivity $D$ and bead mobility $b = 1/(6\pi\eta a)$.
For our spherical beads the diffusivity is given by the Einstein expression $D = kT/(6\pi\eta a)$ which for water at room temperature equals $2.2\times 10^{-13}~$m$^2$/s. In the simulations below, however, we artificially increase the magnitude of $D$ in order to stabilize the computations and to use a coarser mesh and thus save computation time.
![\[fig:BetaGamma\] The fraction $\beta$ of beads caught as function of the current-distance product $Id$ for twenty different flow speeds (50 – 1000 $\mu$m/s; indicated by the arrows). Larger current leads to higher $\beta$; faster flow to smaller $\beta$. In this simulation the initial concentration is low, $c_\text{o} = 10^{13}$ m$^{-3}$. *Inset:* Rate $\gamma_{\text{cap}}$ of bead capture as function of $Id$, for the flows above. The faster flow or the larger current, the higher $\gamma_{\text{cap}}$.](fig02.eps){width="\columnwidth"}
*Boundary conditions.* In addition to the bulk equations (\[eq:NS\]), (\[eq:particlecontinuity\]), and (\[eq:NP\]), we need appropriate boundary conditions. As the beads move out to the walls of the domain and settle there, merely demanding that the normal component of the bead current vector $\textbf{j}$ vanishes is not correct, rather, it must be free to take on any value as long as it is directed into the wall. As beads do not enter the bulk from the walls (by assumption once settled, beads stick) we demand that the normal current component is never directed into the liquid. For the fluid we demand the usual no-slip condition at the walls.
At the inlet $x=0$ of the microfluidic channel we assume that the fluid comes in with the constant initial number density $c_{\text{o}}$ and with a parabolic fluid velocity profile $\mathbf{u}_{\text{o}}$. At the outlet $x=L$ we let the bead current take on any value, while the fluid pressure is zero.
Results {#Sec:results}
=======
Having set up the equations for bead and fluid motion, they are solved with the finite element method on a mesh with $\sim 10^4$ elements refined in the vicinity of the wires. To this end we employ the finite element solver software package Femlab [@Femlab]. The parameter values for the fluid are those of water, $\eta = 1$ mPa$\:$s and $\varrho = 10^3$ kg/m$^3$, while for the beads $a=1~\mu$m and $c_{\text{o}} = 10^{13}$ to $10^{16}$ m$^{-3}$.
To study capturing we must keep track of which beads are captured and which are flushed through the channel with the flow. This is done by calculating the rates $\gamma_i$ by which the beads are either captured or transported in/out at each of the four boundary segments $i$ of the channel (inlet, outlet, upper wall, and lower wall). By integration of the normal components of the bead currents along each segment $i$, we find $$\gamma_i=\int_i\textbf{j}\!\cdot\!\textbf{n}\: d\ell_i.$$ The rate of capture is $\gamma_{\text{cap}} =
\gamma_{\text{lower}} + \gamma_{\text{upper}}$. In steady state the conservation of beads enforces $\gamma_{\text{inlet}} +
\gamma_{\text{cap}} + \gamma_{\text{outlet}} = 0$, which provides a useful check of the simulation results. The primary control parameters are the current-wire distance product $Id$, the maximum fluid in-flow speed $u_{\text{o}}$, and the bead number density, $c_{\text{o}}$. The product $Id$ decides the magnetic force which captures the beads against the fluid flow. As we are investigating effects of bead-bead interaction, our interest is properties that depend on the bead number density, in particular those that do so nonlinearly.
![\[fig:collapsed\] The fraction $\beta$ of beads caught versus $(Id)^2/u_{\text{o}}$, the ratio of the current-distance product squared and the fluid flow velocity. This demonstrates scaling in the competition between capturing and flushing: the twenty curves from Fig. \[fig:BetaGamma\] approximately collapse to one single.](fig03.eps){width="\columnwidth"}
*Electrical current and fluid flow.* The effects of having electrical wires near, and thus a magnetic field gradient in, the channel is illustrated in Fig. \[fig:GeometryConc\](b)–(d) for three values of the current-distance product $Id$. At small values of $Id$ only a narrow region is emptied but increasing the current the region expands until it covers the width of the channel.
A simple measure of the capturing is the ratio $\beta$ of the bead capture rate $\gamma_{\text{cap}}$ to the bead in-flow rate $\gamma_{\text{inlet}}$, $$\beta =
\frac{\text{``capture rate''}}{\text{``in-flow rate''}} =
\frac{\gamma_{\text{cap}}}{\gamma_{\text{inlet}}}.$$ If capturing dominates $\beta$ tends to unity, if flushing dominates $\beta$ tends to zero. Fig. \[fig:BetaGamma\] shows this in that slow flow and strong current leads to a high $\beta$ whereas fast flow and weak current leads to a small value. The rate $\gamma_{\text{cap}}$ of bead capture as function of wire current and flow velocity is illustrated in the inset of Fig. \[fig:BetaGamma\].
If there is a competition between magnetic capturing and flushing, then we expect that the data can be described essentially by the ratio of the magnetic forces to the inlet fluid flow speed $u_{\text{o}}$. The force is proportional to the square of the current-distance product $Id$. In Fig. \[fig:collapsed\], we plot the data from Fig. \[fig:BetaGamma\] as function of $(Id)^2/u_{\text{o}}$ and see that the data mostly collapses onto a single curve. The collapse is not perfect and is not expected to be as the underlying flow and bead distribution patterns (see Fig. \[fig:GeometryConc\]) are different for different flows and magnetic fields.
![\[fig:captureconc\] Fraction $\beta$ of beads caught as function of initial bead number density with and without the bulk force term $c\textbf{F}_\text{ext}$ in [Eq. (\[eq:NS\])]{}. The fixed values for the current-distance product $Id$ and the maximum in-flow speed $u_{\text{o}}$ are shown. At low densities less than 50%are caught; at high densities the collective motion of the beads leads to 100% capture.](fig04.eps){width="\columnwidth"}
*Interactions and concentration.* The second point we wish to make is that modification of the overall flow, and the effective bead-bead interaction this entails, is significant for bead capturing. We can study the effect by excluding momentum transfer to the fluid flow due to the bulk force term $c\textbf{F}_\text{ext}$ in the Navier–Stokes equation (\[eq:NS\]). At high bead number densities the force acting on the beads contributes a significant force on the fluid affecting fluid flow and spawning the effective interaction. The strength of this interaction must thus depend on the density of particles. This is illustrated in Fig. \[fig:captureconc\]; capturing was simulated at fixed in-flow speed, $u_{\text{o}}=300~\mu$m/s, and a fixed value of the current-wire distance product, $Id=8~\mu$Am, but for varying bead number densities $c_ \text{o}$ ranging from $10^{13}$ to $10^{16}$ m$^{-3}$. At low densities we find that capturing is roughly independent of density and the fraction $\beta$ of beads captured has some intermediate value, however, for high densities all beads are caught. In contrast, leaving out the bulk force term $c\textbf{F}_{\text{ext}}$ in the Navier–Stokes equation, i.e., the force acting on the fluid, gives concentration independence as shown in Fig. \[fig:captureconc\].
As can be seen in Fig. \[fig:DeltaBeta\], a complementary way of exhibiting the importance of the bulk force term is to plot the difference $\Delta \beta = \beta_\text{incl} - \beta_\text{excl}$ between including and excluding $c\textbf{F}_{\text{ext}}$ as a function of concentration and the current-distance product. This shows that interactions makes an appreciable difference at high concentrations and intermediate magnetic fields.
*Diffusion constant.* Even for the small beads of radius $a=1~\mu$m, the diffusion constant given by the Einstein relation is small compared to the dimensions entering the problem. The time-scale for a bead to diffuse across the channel is $\tau_{\text{diff}}\sim h^2/D$. If we are to see the influence of diffusion competing with bead advection, then the relevant quantity is the Péclet number $h u/D$ which is advection time-scale $\tau_{\text{adv}}\sim h/u$ over the diffusion time-scale. When this number is larger than unity, which it is except for artificially large diffusion constants, then convection dominates. In the simulations the diffusion constant is increased artificially up to $10^{-11}$ m$^2$/s in order to help numerical convergence. But we have verified that values smaller than $10^{-10}$ m$^2$/s have no influence on the results.
![\[fig:DeltaBeta\] The difference $\Delta\beta =
\beta_{\text{incl}} - \beta_{\text{excl}}$ in captured bead fractions between two situations: including and excluding the bulk force term $c\textbf{F}_{\text{ext}}$ in the Navier–Stokes equation (\[eq:NS\]). At high concentrations ($c_\text{o}
> 10^{15}$ m$^{-3}$) there is an appreciable difference between including and excluding the bulk force term, corresponding to hydrodynamic bead-bead interactions. ](fig05.eps){width="\columnwidth"}
Discussion and conclusion
=========================
We have studied microfluidic capture of paramagnetic beads in suspension. The three main findings of work are: the approximate scaling shown in Fig. \[fig:collapsed\], the existence of a critical bead density for capture shown in Fig. \[fig:captureconc\], and the qualitative difference for capturing between models with and without the hydrodynamic bead-bead interaction shown in Figs. \[fig:captureconc\] and \[fig:DeltaBeta\].
Clearly, it is very important for the capture process to include the action of the beads on the host fluid medium. Simply leaving it out can give qualitatively wrong results for high concentrations of beads. This casts some doubt on the measurement of cell susceptibility through capturing as it depends on cell concentration [@Zborowski:95a; @McCloskey:03a]. Deduction of susceptibilities from single bead or cell considerations together with measurements at high bead or cell concentration is suspect. Care must be taken to compare with standards of known and similar susceptibility, size, and concentration.
The effective bead-bead interaction greatly helps capturing. It should make detectable differences depending on whether there are a few or hundreds of particles in a channel at a time in actual experiments especially when the flow and magnetic field are such that the beads are barely caught one by one. This interaction should be considered when choosing operating conditions for microfluidic devices based on capturing of beads as higher bead number densities potentially eases requirements for external magnets and allows faster flushing. We hope that experimental studies will be initiated to verify this prediction of our work.
**Acknowledgements**. We thank Mikkel Fougt Hansen and Kristian Smistrup for valuable discussions on magnetophoresis in general and of their experiments in particular.
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By Newton’s second law and $\textbf{F}_{\text{drag}} = 6\pi\eta
a\textbf{v}$, a spherical bead with radius $a$ approaches terminal velocity exponentially with a time constant $\tau =
{\scriptstyle \frac{2}{9}}\varrho_{\text{bead}} a^2/\eta$, where $\eta$ is the fluid viscosity. In this work $\tau = 0.2~\mu$s.
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FEMLAB *version 3.1* (COMSOL AB, Stockholm, 2004).
|
---
abstract: 'We have developed a tunneling theory to describe the temperature dependence of tunneling magnetoresistance (TMR) of the magnetic tunnel junctions (MTJs) with periodic grating barrier. Through the Patterson function approach, the theory can handle easily the influence of the lattice distortion of the barrier on the tunneling process of the electrons. The lattice distortion of the barrier is sensible to the temperature and can be quite easily weakened by the thermal relaxation of the strain, and thus the tunneling process of the electrons gets changed highly with the variation of the temperature of the system. That is just the physical mechanism for the temperature dependence of the TMR. From it, we find that the decrease of TMR with rising temperature is mostly carried by a change in the antiparallel resistance ($R_{AP}$), and the parallel resistance ($R_{P}$) changes so little that it seems roughly constant, if compared to the $R_{AP}$, and that, for the annealed MTJ, the $R_{AP}$ is significantly more sensitive to the strain than the $R_{P}$, and for non-annealed MTJ, both the $R_{P}$ and $R_{AP}$ are not sensitive to the strain. They are both in agreement with the experiments of the MgO-based MTJs. Other relevant properties are also discussed.'
author:
- Henan Fang
- 'Mingwen Xiao[^1]'
- Wenbin Rui
- Jun Du
- Zhikuo Tao
title: Temperature effects of the magnetic tunnel junctions with periodic grating barrier
---
. INTRODUCTION
==============
Magnetic tunnel junctions (MTJs) have received considerable attention for many years. They can be applied to the promising spintronic devices such as high-density magnetic reading head [@rf1]. Early experimental studies were limited within the MTJs of amorphous aluminum oxide (Al-O) barriers. In 2001, W. H. Butler et al. [@rf2] predicted theoretically that, if MgO single crystal is used to prepare the MTJ barrier, the tunneling magnetoresistance (TMR) can acquire a very high value. The prediction was verified soon by S. S. P. Parkin et al. [@rf3] and S. Yuasa et al. [@rf4]. Since then, the MgO-based MTJs have been widely investigated over the last decade [@rf5; @rf6; @rf7; @rf8; @rf9; @rf10; @rf11; @rf12; @rf13]. One of the most important and distinguished properties of MgO-based MTJs is that the parallel resistance $\left( R_{P}\right) $, the antiparallel resistance $\left( R_{AP}\right) $, and the TMR all oscillate with the barrier thickness [@rf4; @rf10; @rf11; @rf12; @rf13], which is radically different from the case of Al-O-based MTJs where no such oscillation is found. Those oscillations have already been well interpreted by the spintronic theory developed previously by us [@rf14]. The theory is founded on the traditional optical scattering theory [@rf15]. Within it, the barrier is treated as a diffraction grating with intralayer periodicity. It is found that the periodic grating can bring strong coherence to the tunneling electrons, the oscillations being a natural result of this coherence. Besides the oscillations, the theory can also explain the puzzle why the TMR is still far away from infinity when the two electrodes are both half-metallic.
Experimentally, there is another important property for MgO-based MTJs, that is, the temperature dependences of the $R_{P}$, $R_{AP}$ and TMR. It is found that, as usual, the TMR will decrease when the temperature of the system increases. However, the decrease of TMR with rising temperature is mostly carried by a change in the $R_{AP}$. The $R_{P}$ changes so little that it seems roughly constant, if compared to the $R_{AP}$ [@rf3; @rf16; @rf17; @rf18; @rf19; @rf20; @rf21; @rf22; @rf23]. Theoretically, the modified version of the magnon excitation model [@rf24] is at hand for the mechanism of the above temperature dependence. However, this model can not explain the TMR oscillation on the thickness of MgO barrier. Physically, that is because it dose not include the effect of the periodicity of the single-crystal barrier which plays a key role in the scattering process when the electrons tunnel through the barrier. Based on this reason, we would like to extend our previous theory to interpret the temperature dependences of the $R_{P}$, $R_{AP}$ and TMR of MgO-based MTJs.
As well known, the MgO-based MTJs are fabricated through epitaxial growth. Hence there will be lattice mismatch and interfacial defects between the barrier and its neighbouring layers. Obviously, both of them can cause some lattice distortion to the barrier. The influences of this lattice distortion have been investigated widely by the experiments [@rf4; @rf25; @rf26]. In particular, Ref. \[25\] discovers that, if the MTJ is annealed, the $R_{AP}$ will increase with raising of strain, which is much more sensitive than the $R_{P}$, and if it is non-annealed, the $R_{AP}$ will unchange with the strain. In addition, Ref. \[26\] finds that the lattice distortion can modify the band gap of the MgO barrier. Based on those facts, we shall take into account the effect of the lattice distortion of the barrier upon the $R_{P}$, $R_{AP}$ and TMR within the framework of our previous work. Our aim is to interpret theoretically the temperature dependences of the $R_{P}$, $R_{AP}$ and TMR of MgO-based MTJs. As will be seen in the following, this effect can account for the temperature dependences of the $R_{P}$, $R_{AP}$ and TMR of MgO-based MTJs.
. METHOD
========
To begin with, let us consider a MTJ consisting of a perfect single-crystal barrier. As in Ref. \[14\], we suppose that the atomic potential of the barrier is $v(\mathbf{r})$, and that the total number of the layers of the barrier is $n$. Then, the periodic potential $U(\mathbf{r})$ of the barrier can be written as $$U(\mathbf{r})=\sum_{l_{3}=0}^{n-1}\sum_{\mathbf{R}_{h}}v\left( \mathbf{r}-\mathbf{R}_{h}-l_{3}\,\mathbf{a}_{3}\right) ,$$ where $\mathbf{R}_{h}$ is a two-dimensional lattice vector of the barrier: $\mathbf{R}_{h}=l_{1}\,\mathbf{a}_{1}+l_{2}\,\mathbf{a}_{2}$, with $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ being the primitive vectors of the atomic layers, and $l_{1}$ and $l_{2}$ the corresponding integers. The $\mathbf{a}_{3}$ is the third primitive vector of the barrier, with $l_{3}$ the corresponding integer. Letting $\mathbf{e}_{z}=\mathbf{a}_{1}\times \mathbf{a}_{2}/|\mathbf{a}_{1}\times \mathbf{a}_{2}|$, we shall set $\mathbf{e}_{z}$ point from the upper electrode to the lower one, which is antiparallel to the direction of the tunneling current.
Now, let us consider the effect of the lattice distortion of the barrier. Physically, the periodic potential $U(\mathbf{r})$ of the barrier will be modified by the lattice distortion, as shown by Ref. \[26\]. In order to elucidate the effect of the distortion on the potential $U(\mathbf{r})$, we would employ the Patterson function approach, which is a standard and very powerful method for studying the diffraction by imperfect crystals [@rf15]. Within the framework of two-beam approximation [@rf14; @rf15], this leads to that the Fourier transform $v(\mathbf{K}_{h})$ of the atomic potential undergoes a modification as follows, $$\label{vKh0}
v(\mathbf{K}_{h}) = \left( 1+2\frac{\sigma }{1-\sigma }\cos \left(
\mathbf{K}_{h}\cdot \mathbf{\alpha }\right) \right) \left( 1-\sigma \right)
v_{0}(\mathbf{K}_{h}),$$ where $\mathbf{K}_{h}$ is a planar vector reciprocal to the intralayer lattice vectors $\mathbf{R}_{h}$, $\sigma $ is the defect concentration, $\mathbf{\alpha }$ represents the effect of strain of the barrier [@rf15], and $v_{0}(\mathbf{K}_{h})$ is the Fourier transform of the atomic potential of ideal perfect barrier, $$v_{0}(\mathbf{K}_{h}) = \Omega ^{-1}\int d\mathbf{r}\,v(\mathbf{r})e^{-i\mathbf{K}_{h}\cdot \mathbf{r}}.$$ Here, $\Omega$ is the volume of the primitive cell of the barrier: $\Omega =(\mathbf{a}_{1}\times \mathbf{a}_{2})\cdot \mathbf{a}_{3}$.
With regard to the strain $\alpha$, Ref. \[27\] has studied it both experimentally and theoretically on some oxide heterostructures, it is found that, within the low temperature region, the strain decreases linearly with temperature $T$ as follows, $$\label{alpha}
\alpha = \alpha_{0}\left(1-\frac{T}{T_{c}}\right), \quad T < T_{c},$$ where $\mathbf{\alpha }_{0}$ is the strain of the oxide film at zero temperature, and $T_{c}$ is the recovery temperature above which the strain disappears. Generally, $T_{c}$ is around $800\; \mathrm{K}$. As pointed in Ref. \[27\], this result can be applied to other oxide heterostructures. Therefore, we would like to employee it to handle the strain of MgO barrier. As to the defect concentration $\sigma$, it should be independent on the temperature because the energy to excite defects within a lattice is too high.
Combining the Eqs. (\[vKh0\]) and (\[alpha\]) above, we obtain $$\label{vKh}
v(\mathbf{K}_{h}) = \left[ 1+2\frac{\sigma }{1-\sigma }\cos \left(\mathbf{K}_{h}\cdot \mathbf{\alpha }_{0}\left( 1-\frac{T}{T_{c}}\right)\right) \right] \left(1-\sigma \right) v_{0}(\mathbf{K}_{h}).$$ This equation builds the relationship between the Fourier transform $v(\mathbf{K}_{h})$ of the atomic potential of realistic imperfect barrier and the temperature $T$.
Now, according to Ref. \[14\], the transmission coefficient for the channel of the spin-up to spin-up tunneling reads as follows, $$\begin{aligned}
T_{\uparrow \uparrow }(\mathbf{k}) &= \frac{1}{8k_{z}} \Big\{ p_{+}^{z}\mathrm{e}^{i[p_{+}^{z}-(p_{+}^{z})^{\ast }]d}+p_{-}^{z} \mathrm{e}^{i[p_{-}^{z}-(p_{-}^{z})^{\ast }]d}+q_{+}^{z} \mathrm{e}^{i[q_{+}^{z}-(q_{+}^{z})^{\ast }]d}+q_{-}^{z}\mathrm{e}^{i[q_{-}^{z}-(q_{-}^{z})^{\ast }]d} \notag \\
&\quad +\Big[p_{+}^{z}\mathrm{e}^{i[p_{+}^{z}-(p_{-}^{z})^{\ast }]d}+p_{-}^{z} \mathrm{e}^{i[p_{-}^{z}-(p_{+}^{z})^{\ast }]d}-q_{+}^{z} \mathrm{e}^{i[q_{+}^{z}-(q_{-}^{z})^{\ast }]d}-q_{-}^{z}\mathrm{e}^{i[q_{-}^{z}-(q_{+}^{z})^{\ast }]d}\Big] + \mathrm{c}.\mathrm{c}. \Big\}\end{aligned}$$ where $\mathbf{k}$ is the incident wave vector of tunneling electrons, and $k_{z}$ its $z$-component, $d$ is the thickness of MgO barrier, and
$$\begin{aligned}
p^{z}_{\pm } &= \left[ \mathbf{k}^{2}-\mathbf{k}_{h}^{2}\pm 2m\hbar ^{-2}\,v(\mathbf{K}_{h})\right] ^{1/2}, \\
q^{z}_{\pm } &= \left[ \mathbf{k}^{2}-(\mathbf{k}_{h}+\mathbf{K}_{h})^{2}\pm 2m\hbar ^{-2}\,v(\mathbf{K}_{h})\right]^{1/2}.\end{aligned}$$
Here, $\mathbf{k}_{h}$ is the planar component of $\mathbf{k}$. Since $v(\mathbf{K}_{h})$ is a function of $T$ now, the transmission coefficient $T_{\uparrow \uparrow }(\mathbf{k})$ will also be a function of $T$. That is to say, the tunneling process will vary with temperature.
From $T_{\uparrow \uparrow }$, the conductance $G_{\uparrow \uparrow }$ for the channel of the spin-up to spin-up tunneling can be obtained as follows, $$G_{\uparrow \uparrow } = \frac{e^{2}}{16\pi ^{3}\hbar }\int_{0}^{\pi /2}\text{d}\theta \int_{0}^{2\pi }\text{d}\varphi \,k_{F\uparrow }^{2}\,\sin (2\theta)\,T_{\uparrow \uparrow }\left( k_{F\uparrow },\theta ,\varphi \right),$$ where $e$ denotes the electron charge, $\theta$ the angle between $\mathbf{k}$ and $\mathbf{e}_{z}$, $\varphi$ the angle between $\mathbf{k}_{h}$ and $\mathbf{a}_{1}$, and $k_{F\uparrow}$ the Fermi wave vector of the spin-up electrons. Here, we have ignored the effect of temperature on the Fermi-Dirac distribution of the electrons of ferromagnetic electrodes, which is fairly weak in the present case because $T \leq 400\, \mathrm{K} \ll T_{F}$ where $T_{F} > 10^{4}\; \mathrm{K}$ is the Fermi temperature for either of the electrodes. Since $T_{\uparrow \uparrow }$ is a function of $T$, the above equation shows that $G_{\uparrow \uparrow }$ will depend on the temperature, too.
The other three conductances, $G_{\uparrow \downarrow}$, $G_{\downarrow\uparrow}$, and $G_{\downarrow\downarrow}$, can be obtained similarly. With them, one can get $G_{P} = G_{\uparrow\uparrow} + G_{\downarrow\downarrow}$, $G_{AP} = G_{\uparrow\downarrow} + G_{\downarrow\uparrow}$, $R_{P} = G_{P}^{-1}$, $R_{AP} = G_{AP}^{-1}$, and $\mathrm{TMR} = G_{P}/G_{AP} - 1 = R_{AP}/R_{P} - 1$.
With the same reason as for $G_{\uparrow \uparrow }$, $G_{\uparrow\downarrow}$, $G_{\downarrow\uparrow}$, and $G_{\downarrow\downarrow}$ will also depend on the temperature of the system. Physically, that arises from the fact $v(\mathbf{K}_{h})$ varies with temperature, as shown in Eq. (\[vKh\]). In a word, the four conductances, $G_{\uparrow\uparrow}$, $G_{\uparrow\downarrow}$, $G_{\downarrow\uparrow}$, and $G_{\downarrow\downarrow}$, as well as the $\mathrm{TMR}$ will all changes with the variation of temperature $T$.
The rest calculations are analogous to the Ref. \[14\]. The parameters of the ferromagnetic electrodes are chosen as follows: the chemical potential $\mu $ is $11\, \mathrm{eV}$, the half of the exchange splitting $\Delta $ for the ferromagnetic electrodes is $10\, \mathrm{eV}$, and the Fourier transform of the atomic potential of the ideal perfect barrier $v_{0}(\mathbf{K}_{h})$ is set to be $15.3\, \mathrm{eV}$.
. RESULTS AND DISCUSSIONS
=========================
As a preparatory step to the temperature effects of the MgO-based MTJs, we shall first study the dependences of $R_{P}$ and $R_{AP}$ on $v(\mathbf{K}_{h})$. The results are shown in Fig. 1 where the thickness of the barrier varies from $1.5\, \mathrm{nm}$ to $3\, \mathrm{nm}$. Obviously, both the $R_{P}$ and $R_{AP}$ oscillate with $v(\mathbf{K}_{h})$. As pointed out in Ref. \[14\], that originates from the interference among the diffracted waves. In addition, Fig. 1 shows that the amplitude of $R_{AP}$ is much larger than that of $R_{P}$. It can be understood as follows: As stated in Ref. \[14\], there exist two kinds of integral regions for the transmission coefficients, for the one of them, the transmission coefficients contain oscillating term, for the other, they do not. Only when both $p_{+}^{z}$ and $p_{-}^{z}$ are real or both $q_{+}^{z}$ and $q_{-}^{z}$ are real there can arise oscillating term. For the channel $T_{\downarrow \uparrow }$, the integral regions where the transmission coefficient contains oscillating term is more extensive than the other three channels. It leads to that $R_{AP}$ oscillates more strongly than $R_{P}$. At last, it can be seen from Fig. 1 that the thicker the width of barrier, the shorter the period of the oscillation. Equation (6) indicates that when the width $d$ of the barrier gets thicker, the frequency of $T_{\uparrow \uparrow }$ with respect to $p_{+}^{z}-p_{-}^{z}$ and $q_{+}^{z}-q_{-}^{z}$ will become larger. At the same time, it is easy to know from Eq. (7) that both the $p_{+}^{z}-p_{-}^{z}$ and $q_{+}^{z}-q_{-}^{z}$ are monotonically increasing functions of $v(\mathbf{K}_{h})$. Therefore, the thicker the width $d$ of the barrier, the larger the frequencies of $R_{P}$ and $R_{AP}$ with respect to the variable $v(\mathbf{K}_{h})$.
With those results, the temperature effects of the MgO-based MTJs can be explained as follows. From Eq. (5), it can be found that $v(\mathbf{K}_{h})$ oscillates with temperature $T$. Since both $
R_{P}$ and $R_{AP}$ oscillate with $v(\mathbf{K}_{h})$, as stated above, they will also oscillate with temperature $T$. In addition, the amplitude of $R_{AP}$ will be much stronger than that of $R_{P}$, that is because $R_{AP}$ shows stronger oscillations with regard to $v(\mathbf{K}_{h})$ than $R_{P}$. This accounts for the physical mechanism of the temperature effects of the MTJ.
In the following, we shall try to use this mechanism to explain in detail the experimental results of the MTJ.
First, we would like to investigate the effect of the strain on $R_{P}$ and $R_{AP}$. The theoretical results are depicted in Fig. 2 where $\mathbf{K}_{h}\cdot\mathbf{\alpha }_{0}$ varies from $\pi/6$ to $\pi/2$, $\sigma = 0.08$, $T_{c} = 800\, \mathrm{K}$, and $d = 1.5\, \mathrm{nm}$. Figure 2(a) shows the dependence of $R_{P}$ and $R_{AP}$ on the strain when $T = 10\, \mathrm{K}$. Clearly, both $R_{P}$ and $R_{AP}$ oscillate with $\mathbf{K}_{h}\cdot\mathbf{\alpha }_{0}$ but the amplitude of $R_{AP}$ is much larger than that of $R_{P}$. In order to interpret this result, we draw up Fig. 3 to demonstrate the dependence of $v(\mathbf{K}_{h})$ on $\mathbf{K}_{h}\cdot\mathbf{\alpha }_{0}$. It can be seen that $v(\mathbf{K}_{h})$ decreases monotonously from $16.2\, \mathrm{eV}$ to $14.1\, \mathrm{eV}$ when $\mathbf{K}_{h}\cdot\mathbf{\alpha }_{0}$ increases from $\pi/6$ to $\pi/2$. Combining the Figs. 3 and 1, one can easily deduce that the amplitude of $R_{AP}$ is much larger than that of $R_{P}$. As to the annealed MTJ of Ref. \[25\], it shows that the $R_{AP}$ increases with raising of strain, that can be explained if the strain lies within the range from $\pi/6$ to 1.14 in Fig. 2(a). Of course, if the strain can overcome the region, the $R_{AP}$ would be experimentally expected to decrease or even oscillate with variation of strain. On the other hand, if the MTJ is non-annealed, the barrier of non-annealed MTJ is not well crystallized, the interference arising from the diffraction by the barrier will disappear. Therefore, the $R_{AP}$ will unchange with the strain. In other words, the $R_{AP}$ can not oscillate with the strain. As such, the theoretical results explain the experiments of Ref. \[25\]: For the annealed MTJ, the $R_{AP}$ is significantly more sensitive to the strain than the $R_{P}$; for non-annealed MTJ, both the $R_{P}$ and $R_{AP}$ are not sensitive to the strain. Figure 2(b) displays the temperature dependence of $R_{P}$ and $R_{AP}$ under different strains. Evidently, both $R_{P}$ and $R_{AP}$ become more sensitive to the temperature when the strain goes larger. That can be easily understood from Eq. (5): The larger the strain is, the more sensitive to the temperature the $v(\mathbf{K}_{h})$ will be.
Secondly, we shall study the effect of $\sigma $ on $R_{P}$ and $R_{AP}$. The results are shown in Fig. 4 where $\sigma $ varies from 0.01 to 0.16, $\mathbf{K}_{h}\cdot
\mathbf{\alpha }_{0} = \pi/3$, $T_{c} = 800\, \mathrm{K}$, and $d = 1.5\, \mathrm{nm}$. Figure 4(a) shows that both $R_{P}$ and $R_{AP}$ are nearly independent on $\sigma $ at $10\,\mathrm{K}$. Physically, that is because $v(\mathbf{K}_{h})$ changes little when $\sigma $ increases from 0.01 to 0.16, as shown in Fig. 5. This can be understood as following: From Eq. (5), we can obtain $$\label{vKh1}
v(\mathbf{K}_{h}) = v_{0}(\mathbf{K}_{h})+v_{0}(\mathbf{K}_{h})\left[ 2\cos \left(\mathbf{K}_{h}\cdot \mathbf{\alpha }_{0}\left( 1-\frac{T}{T_{c}}\right)\right)-1\right]\sigma.$$ Equation (9) shows that there is a linear relationship between $v(\mathbf{K}_{h})$ and $\sigma $. With the present parameters, the slope is very small, therefore, the $v(\mathbf{K}_{h})$ will change little with $\sigma $. Figure 4(b) displays the temperature dependences of $R_{P}$ and $R_{AP}$ with different $\sigma $. Evidently, the larger the $\sigma $, the more sensitive to temperature the $R_{P}$ and $R_{AP}$. It comes from the fact that the larger the $\sigma $, the more sensitive to the temperature the $v(\mathbf{K}_{h})$, as can be easily seen from Eq. (9).
Thirdly, we will discuss the effect of $T_{c}$ on $R_{P}$ and $R_{AP}$. The theoretical results are shown in Fig. 6 where $T_{c}$ varies from $600\, \mathrm{K}$ to $1000\, \mathrm{K}$, $\mathbf{K}_{h}\cdot
\mathbf{\alpha }_{0} = \pi/3$, $\sigma = 0.08$, and $d = 1.5\, \mathrm{nm}$. Figure 6(a) shows the dependence of $R_{P}$ and $R_{AP}$ on $T_{c}$ when temperature is at $10\, \mathrm{K}$. It can be seen that both $R_{P}$ and $R_{AP}$ are nearly independent on $T_{c}$. This can be interpreted from Fig. 7 which shows that $v(\mathbf{K}_{h})$ changes little when $T_{c}$ increases from $600\, \mathrm{K}$ to $1000\, \mathrm{K}$. That is because $T/T_{c}$ is much smaller than 1 when $600\, \mathrm{K} \leq T_{c} \leq 1000\, \mathrm{K}$. From Eq. (5), it means that $v(\mathbf{K}_{h})$ will change little. Figure 6(b) displays the temperature dependence of $R_{P}$ and $R_{AP}$ for different $T_{c}$: The larger the $T_{c}$, the less sensitive to temperature the $R_{P}$ and $R_{AP}$. The result can be easily understood from Eq. (5): The larger the $T_{c}$, the less sensitive to the temperature the $v(\mathbf{K}_{h})$.
Finally, we will compare our theory with experiments. As stated above, the most fundamental feature discovered by the experiments is that the decrease of TMR with rising temperature is mostly carried by a change in the $R_{AP}$, and the $R_{P}$ changes so little that it seems roughly constant, if compared to the $R_{AP}$. In order to reproduce this feature, we draw up Fig. 8 to show the temperature dependences of the $R_{P}$, $R_{AP}$ and TMR where $\mathbf{K}_{h}\cdot
\mathbf{\alpha }_{0} = \pi/3$, $\sigma = 0.08$, $T_{c} = 800\, \mathrm{K}$, and $d = 1.5\, \mathrm{nm}$. With those parameters, the $R_{AP}$ just lies in the dropping region of the oscillation. And because the amplitude of $R_{AP}$ is much larger than that of $R_{P}$, the $R_{P}$ changes so little that it seems roughly constant. It can be seen from Fig. 8 that the theoretical results agree qualitatively well with the experiments [@rf3; @rf16; @rf17; @rf18; @rf19; @rf20; @rf21; @rf22; @rf23]. Here, it should be pointed out that, the experimental results are only within part range of the parameters in the present model. If the whole range is taken into consideration, $R_{P}$ and $R_{AP}$ may decrease, or increase, or even oscillate with increasing temperature, which case occurs depends on the varing range of $v(\mathbf{K}_{h})$ when the temperature changes, as can be easily seen from Fig. 1. This suggests that, if the MgO barrier is replaced by another kind of material, the $R_{P}$, $R_{AP}$ and TMR may decrease, or increase, or even oscillate with temperature, that is to say, the situation can be quite different from MgO-based MTJs discussed here.
On the other hand, we also calculate the influence of temperature on the TMR oscillations. The result are shown in Fig. 9 where $\mathbf{K}_{h}\cdot
\mathbf{\alpha }_{0} = \pi/3$, $\sigma = 0.08$, and $T_{c} = 800\, \mathrm{K}$. Figure 9 indicates that both the amplitude and period decrease weakly with temperature. This can be understood as follows. When temperature varies from $10\, \mathrm{K}$ to $300\, \mathrm{K}$, $v(\mathbf{K}_{h})$ will vary correspondingly from $15.3\, \mathrm{eV}$ to $16\, \mathrm{eV}$. As pointed out in Ref. \[14\], the amplitude and period of TMR will both decrease as $v(\mathbf{K}_{h})$ increases. This means that the weak decrease of the amplitude and period roots from the small variation of $v(\mathbf{K}_{h})$, from $15.3\, \mathrm{eV}$ to $16\, \mathrm{eV}$. This theoretical result is in agreement with the experiments [@rf4; @rf12].
. CONCLUSION
============
So far, we have developed a tunneling theory to study the temperature effects of the MTJ with periodic grating barrier. The theory is an extension of our previous work where the barrier is treated as a diffraction grating with intralayer periodicity. Physically, the extension is done mainly through the so-called Patterson function approach. Within the framework of this extension, one can easily take into account the influence of the lattice distortion of the barrier on the tunneling process of the electrons. We find that the distortion can account for the temperature effects of the MTJ with periodic grating barrier.
Theoretically, the distortion of the lattice of the barrier can be described by the defect concentration and the strain, they can both modify highly the scattering potential of the barrier. Although the defect concentration is nearly independent on the temperature, the strain depends strongly upon the temperature of the system. As a result, with the thermal activation of the scattering potential of the barrier, the tunneling process of the electrons will be highly changed by the temperature of the system, that is just the origination of the temperature effects of the MTJ with periodic grating barrier. With this mechanism, the $R_{P}$, $R_{AP}$ and TMR can all oscillate with the variation of temperature. For a certain concrete range of temperature, the three can occur as increasing, decreasing, or oscillating with temperature. As such, the theory can explain the experiments on the MgO-based MTJs: First, it reproduces the most fundamental feature of the temperature effects: The decrease of TMR with rising temperature is mostly carried by a change in the $R_{AP}$, and the $R_{P}$ changes so little that it seems roughly constant, if compared to the $R_{AP}$. Second, it shows that both the amplitude and period of oscillation of the TMR with regard to the barrier thickness decrease weakly with temperature. And third, it demonstrates that, for the annealed MTJ, the $R_{AP}$ is significantly more sensitive to the strain than the $R_{P}$, and for non-annealed MTJ, both the $R_{P}$ and $R_{AP}$ are not sensitive to the strain.
Recently, Hu and co-workers \[28\] find an interesting result of the MgO-based MTJs with Co$_{2}$MnSi electrodes. One can easily see from the Fig. (4) of Ref. \[28\] that the $R_{P}$ oscillates with the temperature, but the $R_{AP}$ dose not. Here, it is worth noting that the situation of Co$_{2}$MnSi electrodes is much distinct from the case considered in this paper because Co$_{2}$MnSi is half-metallic but the present electrodes are conventional. In order to discuss this intriguing property, one needs further to take into account the half-metallic characteristics of the electrodes. We believe that it can be interpreted within the framework of our model. The work is in progress and will be published elsewhere.
ACKNOWLEDGMENTS
===============
This work is supported by the National Natural Science Foundation of China (11704197, 61106009, 51471085, 51331004), the State Key Program for Basic Research of China (2014CB921101), the Nature Science of Foundation of Jiangsu province (BK20130866), the University Nature Science Research Project of Jiangsu province (14KJB510020), the Scientific Research Foundation of Nanjing University of Posts and Communications (NY213025, NY215083, NY217046).
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[^1]: Email: xmw@nju.edu.cn
|
---
address:
- '$^1$ Omega Mathematical Institute/University of Toronto, Toronto, Ontario, Canada'
- '$^2$ University of Bucharest, Bucharest, Romania'
- '$^3$ Department of Computer Science and Technology, Nanjing University, Nanjing, China'
author:
- 'Akira Kanda$^1$, Mihai Prunescu$^2$ and Renata Wong$^3$'
bibliography:
- 'references-LAoRT.bib'
nocite: '[@*]'
title: |
Presented at\
Physics Beyond Relativity 2019 conference\
Prague, Czech Republic, October 20, 2019\
(invited presentation)\
Logical Analysis of Relativity Theory
---
Prelude
=======
Newton v.s. Galileo: Reclining Tower Experiment
-----------------------------------------------
$m$$m\prime $$m>m\prime $$r$$m$$m\prime $$M$$m$$F=GMm/r^{2}.$$$a_{m}=GM/r^{2},\quad a_{M}=Gm/r^{2}$$$a_{m}$$m$$M$$M$$-a_{M}$$M$$m$$a_{M}+a_{m}=GM/r^{2}+Gm/r^{2}.$ $m$$m\prime $$M$$m=m\prime $$m>m\prime $$$t_{m\prime }>t_{m}.$$
Newton’s absolutism v.s. Galileo’s relativism: God v.s. human
-------------------------------------------------------------
$v$ $-v$
*.*
$m$$M$$M$$M$
$M$$M$$a_{m}$$m $$M$$GM/r^{2}.$$a_{m^{\prime }}$$m^{\prime }$$M$$GM/r^{2}.$$m$$m\prime $$m$$M$
Classical Electromagnetism
--------------------------
$$\mathbf{\nabla \times E}=-\frac{1}{c}\frac{\partial \mathbf{H}}{\partial t},\quad \mathbf{\nabla \cdot E}=4\pi \rho ,\quad \mathbf{\nabla \cdot H}=0,\quad \mathbf{J}=\rho \mathbf{v,\quad \nabla \times H}=\frac{4\pi }{c}\mathbf{J}+\frac{1}{c}\frac{\partial }{\partial t}\mathbf{E}$$$\mathbf{J}$$\rho $
$E_{em}$ $m=4E_{em}/3c^{2}$
$E_{em}/c^{2}$
.
$k^{3}\epsilon $$k\varepsilon $$k=\sqrt{1-v^{2}/c^{2}}$$\epsilon $$\epsilon =1,$
$$m_{L}=m_{0}/\sqrt{1-v^{2}/c^{2}}^{3},\quad m_{T}=m_{0}/\sqrt{1-v^{2}/c^{2}}\quad \mathrm{where\ }m_{0}\mathrm{\ }=(4/3)(E_{em}/c^{2}).$$$c$
$\mathbf{F}=q(\mathbf{E+v\times }\ \mathbf{B}).$
Michelson-Morley experiment
----------------------------
*,* $c$$c+v=c$
All of these are rather philosophical and conceptual questions. One of the major deficiencies of the current practice of physics is the failure to consider these conceptually important questions. The Michelson-Morley experiment is one of the main examples of this deficiency. Despite its overvalued importance, the more we think the more we get lost.
Fitzgerald contraction and electromagnetic Lorentz transformation
-----------------------------------------------------------------
The first group of physicists who took the result of the Michelson-Morley experiment very seriously were researchers in the electromagnetic field theory, which was expected as the theory of light was a part of the EM field theory of Maxwell. Fitzgerald prematurely concluded that a moving body shrinks in the direction of the motion. Theoretically, there is no such thing as physical bodies as Newton reduced all of them to point masses in order to make the mathematics work to form a theory of dynamics. This is to say that a body is not the subject of theoretical study in dynamics. (In modern day term what Fitzgerald called body is a massively complex system of particles, each of which obeys the laws of quantum mechanics.) It was Lorentz who embraced this idea of Fitzgerald and developed it into the concept of what we now call Lorentz transformation. For Lorentz, who was not a relativist, this transformation however was limited only to between the absolute electromagnetic field and a frame moving inside it.
Special theory of relativity: kinematics
========================================
Special theory of relativity: kinematics
----------------------------------------
Time dilation and length contraction
------------------------------------
$c$$c$
This is a contradiction. and it is consistent with Aristotle’s warning that a point on a line may not be a part of the line. the same is held by contemporary topologists but in a more modern way, namely that real numbers (even rational numbers on the real line) are defined through a limit. Therefore there is no finite access to any real number on a real line. If we cannot access it, how can we move it? If we cannot move even a single point in our 3D space, how can we move the entire 3D space inside another 3D space? In short, topology says that a point does not exist on a topological space.
We tend to take mathematics we use in physics lightly just as a language. This is a perfect example of the price we pay for our ignorance and arrogance. Mathematical results at the level of topology, etc., are obtained with utmost care and precision. So often, unless we pay due attention and effort to understand, we take the results wrongly and end up with this kind of devastating mistakes.
Relativistic Lorentz transformation
-----------------------------------
$$t^{\prime }=t/\sqrt{1-(v/c)^{2}}.$$$$v^{\prime }=\sqrt{1-(v/c)^{2}}x$$$$x^{\prime }=(x-vt)/\sqrt{1-(v/c)^{2}},\quad y^{\prime }=y,\quad z^{\prime
}=z^{\prime },\quad t^{\prime }=(t-vx/c^{2})/\sqrt{1-(v/c)^{2}}.$$
$x^{\prime }=(x-vt)/\sqrt{1-(v/c)^{2}}.$$$x=(x^{\prime }+vt^{\prime })/\sqrt{1-(v/c)^{2}}.$$$t^{\prime }$$$t^{\prime }=(t-vx/c^{2})/\sqrt{1-(v/c)^{2}}.$$
$$t^{\prime }=\left( t-vx/c^{2}\right) /\sqrt{1-(v/c)^{2}}$$ $x=0$$$t^{\prime }=t/\sqrt{1-(v/c)^{2}}.$$$x=0$$x$-
Lorentz transformation v.s. principle of relativity
---------------------------------------------------
The Lorentz transformation plays yet other questionable roles. We can shown that this transformation fails to respect Newton’s law of gravitation, Coulombs’ law, Newton’s second law and wave equations. For example, despite the claimed advantage of conserving wave equations, Lorentz transformation astoundingly fails to conserve the more fundamental second law and the law of gravitation.
Is wave equation invariant under the Lorentz transformation?
------------------------------------------------------------
$$\begin{aligned}
\frac{\partial \psi (x^{\prime },t^{\prime })}{\partial x} &=&\frac{\partial
\psi (x^{\prime },t^{\prime })}{\partial x^{\prime }}\frac{\partial
x^{\prime }}{\partial x}+\frac{\partial \psi (x^{\prime },t^{\prime })}{\partial x^{\prime }}\frac{\partial t^{\prime }}{\partial x} \\
&=&\frac{\partial \psi (x^{\prime },t^{\prime })}{\partial x^{\prime }}\frac{\partial \gamma (x^{{}}-vt)}{\partial x}+\frac{\partial \psi (x^{\prime
},t^{\prime })}{\partial x^{\prime }}\frac{\partial \gamma (t-\frac{vx}{c^{2}})}{\partial x} \\
&=&\gamma \frac{\partial \psi (x^{\prime },t^{\prime })}{\partial x^{\prime }}-\frac{\gamma v}{c^{2}}\frac{\partial \psi (x^{\prime },t^{\prime })}{\partial t^{\prime }}\end{aligned}$$
$$\frac{\partial \psi (x^{\prime },t^{\prime })}{\partial t}=-\gamma v \frac{\partial \psi (x^{\prime },t^{\prime })}{\partial x^{\prime }}+\gamma \frac{\partial \psi (x^{\prime },t^{\prime })}{\partial t^{\prime }}$$$$\frac{\partial \psi ^{2}(x^{\prime },t^{\prime })}{\partial x^{2}}=\left(
\gamma \frac{\partial }{\partial x^{\prime }}-\frac{\gamma v}{c^{2}}\frac{\partial }{\partial t^{\prime }}\right) \left( \gamma \frac{\partial }{\partial x^{\prime }}-\frac{\gamma v}{c^{2}}\frac{\partial }{\partial
t^{\prime }}\right) =\gamma ^{2}\frac{\partial ^{2}}{\partial x^{\prime 2}}-2\frac{\gamma ^{2}v}{c^{2}}\frac{\partial ^{2}}{\partial x^{\prime }\partial
t^{\prime }}+\frac{\gamma ^{2}v^{2}}{c^{4}}\frac{\partial ^{2}}{\partial
t^{\prime 2}}$$$$\frac{\partial \psi ^{2}(x^{\prime },t^{\prime })}{\partial t^{2}}=\gamma
^{2}v^{2}\frac{\partial ^{2}}{\partial x^{\prime 2}}-2\gamma ^{2}v^{{}}\frac{\partial ^{2}}{\partial x^{\prime }\partial t^{\prime }}+\gamma ^{2}\frac{\partial ^{2}}{\partial t^{\prime 2}}$$$v=c=\omega .$$\omega $$c$$v=\omega $
$c$
Inconsistency of the special theory of relativity
-------------------------------------------------
### The power pole - power line paradox
### Deductive paradox
$c+v=c+v.$$c+v=c $
### Speed paradox
$v=d/t$$d$$t$$v\prime =d\prime /t\prime $$$d\prime =d/\sqrt{1-(v/c)^{2}},\qquad t^{\prime }=t\sqrt{1-(v/c)^{2}}.$$$v\neq v^{\prime }.$ $$\lbrack speed]=[length]/[time].$$
$d$$t$$0$$t$$d^{\prime }$$d$$t^{\prime }$$t$$d^{\prime }/t^{\prime }$$d/t$$v$$v\prime $$v\neq v^{\prime }.$
### Dingle’s paradox
Michelson-Morley experiment revisited I
---------------------------------------
$c$
Special theory of relativity: dynamics
======================================
Einstein’s ambition and its fallout
-----------------------------------
Galilean theory of relativity did not consider reference frames that are under acceleration relative to each other. This was because acceleration, through the second law, violates the principle of relativity. This only restriction imposed by kinematics on relativity theory was too limiting for Einstein. Considering that way before this setback, already at the most basic level of Galilean theory of relativity the concept of relativity is insurrectionist, Einstein should have abandoned the idea of relativity. It is unfortunately not what happened.
### Relativistic collision, relativistic mass, relativistic momentum and relativistic energy
$$m=m_{0}/\sqrt{1-(v/c)^{2}}$$$m_{0}$$v$ $$\mathbf{F}=d\mathbf{p}/dt$$$\mathbf{p}=m\mathbf{v}$ $$dE=\mathbf{F}\cdot d\mathbf{r=}\frac{d(m\mathbf{v})}{dt}\cdot d\mathbf{r}=d(m\mathbf{v})\cdot \mathbf{v=}dm\mathbf{(v\cdot v)+}m(d\mathbf{v}\cdot \mathbf{v}).$$ $$E=mc^{2}$$$\mathbf{v}
$$$E=0$$
$$E^{2}-c^{2}p^{2}=m_{0}^{2}c^{4}.$$
### Impact on quantum field theory
$e=mc^{2}$
$$E=h\nu =pc\qquad h=h/\lambda$$$E$$p$
### More contradictions coming from $e=mc^{2}$
$$e=mc^{2}=m_{0}c^{2}/\sqrt{1-v^{2}/c^{2}}$$$v=c$$m_{0}=0.$$e=0/0$$0x=0$$0x=0$$0$$e=0/0$$0$$$E=\sqrt{(cp)^{2}+(m_{0})^{2}c^{4}}=cp=m_{0}vc/\sqrt{1-vc^{2}}\frac{{}}{{}}=(0/0)cv=c^{2}h\nu =h\nu .$$$$E=\sqrt{(pv)^{2}}=\sqrt{c^{2}m_{0}/\sqrt{1-(v/c)^{2}}=\sqrt{0/0}=\sqrt{h\nu }}=h\nu =1.$$$0$$c$$0$
Michelson-Morley experiment revisited II
----------------------------------------
### Light-as-photon interpretation of Michelson-Morley experiment
$v$$c+v$$v+c$$v$$v$$c-v$$v$$v$$v$$c+v=c+v$ $v$.
### Quantum mechanical interpretation of Michelson-Morley experiment
* *
From Einstein, through Dirac to material science: the particle-wave duality in full swing
-----------------------------------------------------------------------------------------
1. 2.
Minkowski’s relativity theory
=============================
$v$
$v$
$v$
$v$
$d\tau $ $$(d\tau ){{}^2}=(dt){{}^2}-(1/c)((dx){{}^2}+(dy){{}^2}+(dz){{}^2})$$$v$
1. 2. 3.
$$(d\tau ){{}^2}=(dt){{}^2}-(1/c)((dx){{}^2}+(dy){{}^2}+(dz){{}^2})$$$$dt^{\prime }=\sqrt{1-v^{2}/c^{2}}dt\quad dx^{\prime }=dx/\sqrt{1-v^{2}/c^{2}}\quad dy^{\prime }=dy\quad dz^{\prime }=dz$$$$\begin{aligned}
&&(dt^{\prime })^{2}-(1/c)((dx^{\prime })^{2}+(dy^{\prime })^{2}+(dz^{\prime
})) \\
&=&(1-v^{2}/c^{2})(dt)^{2}-(1/c)((dx)^{2}/(1-v^{2}/c^{2})+(dy){{}^2}+(dz){{}^2})) \\
&\neq &(dt){{}^2}-(1/c)((dx){{}^2}+(dy){{}^2}+(dz){{}^2}).\end{aligned}$$This reconfirms that Einstein’s STR and Minkowski’s STR are two different theories. There is no such thing as Minkowski distance in Einstein’s STR. There is no light cone either. This is a good news in a sense as the inconsistency of Einstein’s STR will not be deleterious to Minkowski’d STR. However, as we have stressed many times, nobody knows what Minkowski’s STR is and what it is for. There is no ontology associated with it. Furthermore we now have to cleanly detach Minkowski’s STR from Einstein’s STR. It is a lot of work, especially because most of popular results in STR came from Einstein’s version. This is however expected, because it is not clear what Minkowski was talking about.
General theory of relativity
============================
$e=mc^{2}. $
Principle of equivalence
------------------------
, **.**
$\mathbf{\alpha }.$$m$$\mathbf{f}$$m$$\mathbf{a}$$\mathbf{f}$$\mathbf{f}=m\mathbf{a}.$$m$$\mathbf{\alpha }+\mathbf{a}.$ $m$$\mathbf{f}=m(\mathbf{\alpha }+\mathbf{a}).$$$\mathbf{f}-m\mathbf{\alpha }=m\mathbf{a}\quad \quad \quad \quad \quad
\quad \quad \textrm{(IF)}$$$\mathbf{\alpha }$$-m\mathbf{\alpha }$$m,$$m$$m $$\mathbf{f}$$\mathbf{f}-m\mathbf{\alpha }$
1. $v\oplus v^{\prime }.$
2. 3. $\mathbf{f}$$m$$\mathbf{a}=\mathbf{f}/m.$ $m$$\mathbf{a}$$\mathbf{f}$
4. 5.
Violating the point mass assumption
-----------------------------------
$m$
The usual response is that we experience such force even if we are firmly attached to the inside wall.
However, our body is not just a solid. *Our body is beyond the category of physical objects.* Our body has incredibly complex internal system for perception. This is why we feel such a pressure.
Acceleration-induced gravitational field
----------------------------------------
Red shift and energy issue
--------------------------
$v+v^{\prime }$$v\oplus v^{\prime }.$
Centre of masses in general relativity theory
---------------------------------------------
Light bend
----------
$0$$c$$x$$a$$y$
$$x\prime =ct\qquad y\prime =-(at^{2})/2 \quad \quad(1)$$$\theta $$x$
$$tan(\theta )=-ax^{\prime }/c^{2},$$$\theta $
$$\theta \doteqdot -ax^{\prime }/c^{2} \quad \quad(2)$$
$$\theta =-(3a/2c^{2})x^{\prime 2} \quad \quad(3)$$$(3)$
$0$$0$$0$$0$$0$$0$
$0 $$c$
“Induced gravitational field" revisited
---------------------------------------
$g$
We must stop identifying entirely different things in approximation as it was done in quantum field theory.
Moreover, the force field that Einstein introduced to the frame of the spaceship is not gravitational at all. Gravitational fields are to be the field representation of the effect of gravitational force created by Newton’s law of gravity. So, it is not a uniform field.
General theory of relativity (II)
=================================
General coordinate system
-------------------------
$(x_{1},x_{2},x_{3})$
1. 2. 3. 4. $x_{0}.$
<!-- -->
1. 2. 3.
<!-- -->
1. $(x_{0},x_{1},x_{2},x_{3})$$(x_{0},x_{1},x_{2},x_{3})$$(x_{0},x_{1},x_{2},x_{3})$* *
2. It is a common understanding among researchers in “dynamical system theory" that time has a special status and is different from all other coordinates of the system. This is in agreement with the idea of Newton in his classical dynamics. Newton said that time, unlike other coordinates, has a natural flow that “moves" forward only. This makes it impossible to consider time as reading of clocks. Time is an entity that transcends empiricism and operationalism.
Once we violate the most fundamental assumption on time, anything can happen and relativity apparently made it happen.
3. 4.
Minkowskian local frame
-----------------------
$\emph{P}$$(\emph{Px,Py,Pz)}$$\emph{P}$$P$
$(Px,Py,Pz)$$P$
$\emph{P}$$(t,x,y,z)$
$(t,x,y,z)$$(t+dt,x+dx,y+dy,z+dz).$$d(\tau )$
$$(d(\tau )){{}^2}=(dt){{}^2}-(1/c)((dx){{}^2}+(dy){{}^2}+(dz){{}^2})$$
: * *
$x_{i}$$x_{i}\prime =\pi (x_{0},x_{1},x_{2},x_{3})$$(x_{0},x_{1},x_{2},x_{3})$ $(t,x,y,z)$
$$t=\theta (x_{0},x_{1},x_{2},x_{3}),\;x=\pi
(x_{0},x_{1},x_{2},x_{3}),\;y=\psi (x_{0},x_{1},x_{2},x_{3}),\;z=\gamma
(x_{0},x_{1},x_{2},x_{3}).$$
$x_{i}$$dx_{i}$$t,x,y,z$
$$dx=(\partial (\theta )/\partial (x_{0}))dx_{0}+(\partial (\pi
)/\partial (x_{1}))dx_1 +(\partial (\psi )/\partial (x_{2}))dx_2 +(\partial (\gamma )/\partial (x_{3}))dx_3$$$$(d(\tau )){{}^2}=(dt){{}^2}-(1/c)((dx){{}^2}+(dy){{}^2}+(dz){{}^2})$$ $d\tau
{{}^2}$$dx$$dx_{i}$$dx_{i}$
$$d\tau
{{}^2}=\sum_{i=0}^{3}\sum_{j=0}^{3}(g_{ij})dx_{i}dx_{j}\qquad \qquad (R)$$$(g_{ij})$$x_{i}$
$(R)$$(R)$
Here are some issues to be discussed:
1. $(R)$$$dx=(\partial (\theta )/\partial (x_{0}))dx_{0}+(\partial (\pi
)/\partial (x_{1}))dx_1 +(\partial (\psi )/\partial (x_{2}))dx_2 +(\partial (\gamma )/\partial (x_{3}))dx_3$$ $$(d(\tau )){{}^2}=(dt){{}^2}-(1/c)((dx){{}^2}+(dy){{}^2}+(dz){{}^2}).$$
2. More fundamentally, Einstein was clearly not aware of the difference between a countably infinite and a continuum. Cantor’s diagonal argument clearly shows that there are more points in the geometric continuum than the discrete collection of points. The Lebesgue integral of the Weierstrass function over $[0,1]$ shows that the geometric continuum has unimaginably more points than the “space" of countably many points has. For example, on the real number line almost all points are irrational numbers. So, one cannot cover the entire global space with clocks as there are only finitely many clocks. This makes the most fundamental assumptions of Einstein’s general theory of relativity untenable. There is no such thing as the “global spacetime" prescribed by Einstein. define neither differentiation nor integration.
3. $0$
4. In short, contrary to what Einstein proposed, the universe cannot be a sea of clocks.
In addition to this topological problem the general theory of relativity suffers, there is an even more fundamental issue of logical deficiency in the idea of the general reference frame which is the sea of clocks. Clocks are physical entities and it requires physics to make them. One cannot use clocks to define clocks at the pain of vicious circle. So, there is no such thing as metaphysical clocks though time is certainly a metaphysical entity, as Newton thought. It was relativity theory, the special and the general, which tried to use empirical clocks that lead the world of physics to the current confusion about time.
Logically speaking, modern physics started with the wrong idea of what is time. Contrary to the special theory of relativity, time cannot be defined in terms of speed as speed is defined in terms of time. And as we have discussed here, the universe is not a sea of clocks contrary to the general theory of relativity. From the combination of these wrong assumptions, it is expected that we ended up with questioning what time is.
Our understanding of time as in relativity theory is completely wrong.
We have shown that the general theory of relativity came from the special theory of relativity, which is false. Therefore, the general theory of relativity is also false. Any theory which contains an inconsistent theory is inconsistent.
Geodesics
---------
$P_{1}(t_{1},x_{1},y_{1},z_{1})$ $P_{2}(t_{2},x_{2},y_{2},z_{2})$ $$\overline{P_{1}P_{2}}=\sqrt{(t_{1}-t_{2})^{2}+(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(z_{1}-z_{2})^{2}}.$$$P_{1}P_{2}$ $$\left( (x_{1}-x_{2})/(t_{1}-t_{2}),\quad (y_{1}-y_{2})/(t_{1}-t_{2}),\quad (z_{1}-z_{2})/(t_{1}-t_{2})\right) .$$
$P_{1}$$P_{2}$ $$\widetilde{P_{1}P_{2}}=\sqrt{(t_{1}-t_{2})^{2}-(1/c^{2})\left\{
(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(z_{1}-z_{2})^{2}\right\} }.$$$P_{1}$$P_{2}$$$\widetilde{P_{1}P_{2}}<\overline{P_{1}P_{2}}.$$$l_{P_{1},P_{2}}$$P_{1}$$P_{2}$$$\overline{P_{1}P_{2}}\leq \overline{l_{P_{1},P_{2}}}$$$\overline{l_{P_{1},P_{2}}}$$l_{P_{1},P_{2}}.$
1. 2. $\widetilde{P_{1}P_{2}}$$P_{1}$$P_{2}$
$$d\tau
{{}^2}=\sum_{i=0}^{3}\sum_{j=0}^{3}(g_{ij})dx_{i}dx_{j}=(dt){{}^2}-(1/c)((dx){{}^2}+(dy){{}^2}+(dz){{}^2}).\qquad$$
$$d\tau
{{}^2}=\sum_{i=0}^{3}\sum_{j=0}^{3}(g_{ij})dx_{i}dx_{j}$$$$\frac{d}{d\tau }\left( \sum_{j=0}^{3}g_{ij}\frac{dx_{j}}{d\tau }\right) =\frac{1}{2}\sum_{j=0}^{3}\sum_{k=0}^{3}\frac{\partial g_{ik}}{\partial
x_{i}}\frac{dx_{j}}{d\tau }\frac{dx_{k}}{d\tau }\qquad (i=0,1,2,3).$$ $$(d(\tau )){{}^2}=(dt){{}^2}-(1/c)((dx){{}^2}+(dy){{}^2}+(dz){{}^2}),$$$$\frac{d^{2}x}{d\tau ^{2}}=\frac{d^{2}y}{d\tau ^{2}}=\frac{d^{2}z}{d\tau ^{2}}=\frac{d^{2}t}{d\tau ^{2}}=0.$$ $$x=\frac{dx}{d\tau }t+a,\quad y=\frac{dy}{d\tau }t+b,\quad z=\frac{dz}{d\tau }t+c\quad$$ $a,b,c$ .
Einstein’s equation of gravitation
----------------------------------
$g_{ij}$$T_{ij}$$g_{ij}$
1. 2. $m0$$mv$$(1/2)mv^{2},$ $m0$ $mv.$
3. 4. $e=mc^{2}$ $v$$m$
5.
References {#references .unnumbered}
==========
|
---
abstract: 'The scalar sector of the Standard Model is extended to include an arbitrary assortment of scalars. In the case where this assignment does not preserve $\rho=1$ at the tree-level, the departure from unity itself puts the most stringent constraint on the scalar sector, and where $\rho_{tree}=1$ is maintained, useful bounds on the parameter space of the charged Higgs mass and the doublet-nondoublet mixing angle can arise from data on $\bbbar$, $\kkbar$ mixing and the $\e$ parameter. These constraints turn out to be comparable (and in some cases, better) to those obtained from $\zbb$ data.'
---
=16 true pt
[**Constraints on a General Higgs Sector\
from $\kkbar$, $\bbbar$ Mixing\
and the $\e$ Parameter**]{}
1 true cm
[Debrupa Chakraverty]{} [^1] and [Anirban Kundu]{} [^2]\
Theory Group, Saha Institute of Nuclear Physics,\
1/AF Bidhannagar, Calcutta - 700 064, India
1 true cm
The electroweak symmetry breaking sector of the Standard Model (SM) is still as cloudy as it was in the time of its formulation; and the main factor responsible for this is the absence of any direct evidence of the Higgs boson. The minimal version of the SM requires one complex scalar doublet to break the electroweak symmetry; however, there is no a priori reason why more scalars cannot exist. Models with two or more doublets have been explored in this spirit [@hunter].
It is also pertinent to investigate the consequences of scalars belonging to non-doublet representations of $SU(2)$. This will enlarge the particle content of the SM, and change the gauge-scalar as well as the fermion-scalar interactions, without affecting the $SU(2)_L\times U(1)_Y$ gauge structure of the model. That these non-doublet scalar representations can induce Majorana masses for left-handed neutrinos has been shown [@gelmini]. Collider signatures of scalars belonging to a triplet representation have also been investigated [@collider].
However, there is one serious constraint on these higher dimensional ($>2$) scalar representations: they in general do not maintain $\rho=1$ at tree-level. Singlet and doublet representations do not suffer from this malady and that is why much work have been done on their phenomenological implications [@grossman; @gunion]. For an arbitrary assortment of scalars, one has three possibilities:
1. The higher dimensional multiplet does not [*incidentally*]{} contribute to $\rho$. This will happen, e.g., for a multiplet with weak isospin $T=3$ and weak hypercharge $Y=4$. However, being quite artificial, such representations will not be discussed anymore in this paper.
2. The vacuum expectation values (VEV) of the higher representations are much smaller than the doublet VEV so that $\rho-1$ is within experimental bound.
3. There is a remaining custodial $SU(2)$ symmetry among the higher representations. In this case, the effects of the ‘bad’ representations on $\rho-1$ cancel out. For such a cancellation to remain valid even at one-loop level, one requires a fine-tuning; however, it has been shown [@gunion] that the fine-tuning required is of the same order as one encounters in the SM. Following this prescription, some serious model-building has been done in recent times [@gunion; @georgi].
Recently, a general formulation to treat arbitrary representations of scalars was proposed [@km]. Only the constraint coming from the tree-level absence of flavour-changing neutral currents (FCNC) was assumed there. In simple terms, this constraint means that either a single weak doublet $\Phi_1$ couples with both $T_3=+1/2$ and $T_3=-1/2$ fermions, or one weak doublet $\Phi_1$ couples with $T_3=+1/2$ and another, $\Phi_2$, couples with $T_3=-1/2$ type fermions. For simplicity, we have assumed that the same doublet couples with quarks and leptons.
It was shown in Ref. [@km] that if the arbitrary assortment of multiplets do not keep $\rho=1$ at tree-level, then the constraint coming from $\rho-1$ is by far the strictest to limit the doublet-nondoublet mixing. (This mixing occurs because, in general, the weak and the mass basis of scalars are not identical and states in these two basis are related by some unitary matrices.) However, for those models which keep $\rho_{tree}
=1$ (either entirely consisting of doublets and singlets, or having compensating ‘bad’ representations — possibility (3) as listed above), a significant constraint on the parameter space of the singly-charged Higgs mass $m_{H^+}$ and doublet-nondoublet mixing angle $\theta_H$ can be obtained from $\zbb$ data.
In this paper we investigate what constraints on the abovementioned parameter space can be obtained from processes like $\bbbar$ and $\kkbar$ mixing, and from the experimental value of the $CP$-violating $\e$ parameter. Such a study was performed earlier for two-Higgs doublet models [@thdm]. We intend to show that the constraints sometimes turn out to be better than those obtained with the $\zbb$ data. As already pointed out by Grossman [@grossman], other processes do not play a significant role in constraining the parameter space. We will show that for those models where the scalar sector contains nondoublet representations, conclusions can differ significantly from the ones drawn in the case of multi-Higgs doublet model. It may again be stressed that we give a general treatment which yields the well-known results for multi-Higgs doublet model at proper limit.
Before proceeding further, let us set our notations, which will mostly follow Ref. [@km]. In the weak basis, the Higgs multiplets are denoted by $\Phi$ and the fields by $\phi$. In the mass basis, we use $H$ to denote the fields. $H$ and $\phi$ are related via unitary matrices; for our purpose it is sufficient to show a pair of such relations: H\_i\^+&=&\_[ij]{}\_j\^+; \_i\^+= \_[ji]{}\^H\_j\^+;\
H\_i\^-&=&\_[ij]{}\^\_j\^-; \_i\^-=\_[ji]{}H\_j\^-. We also set $H_1^+\equiv G^+$, which means G\^+=\_k\_[1k]{}\_k\^+.
Keeping the quarks in the weak basis, the Yukawa couplings are given by |u d H\_i\^+ : [igm\_W]{}[\_[i1]{}\_[11]{}]{} (m\_uP\_L-m\_dP\_R) for the case where only $\Phi_1$ gives mass to both $u$- and $d$-type quarks, and |u d H\_i\^+ : [igm\_W]{}([\_[i1]{}\_[11]{}]{} m\_uP\_L-[\_[i2]{}\_[12]{}]{}m\_dP\_R) for the case where $\Phi_1$ ($\Phi_2$) gives masss to $u$($d$)-type quarks. The projection operators are P\_L=(1-\_5)/2, P\_R=(1+\_5)/2. These two models will henceforth be called Model 1 and Model 2 respectively. Consideration of quarks in the mass basis will introduce the relevant elements of the quark mixing matrix.
In the SM, the short-distance part of $\Delta m_K$, the $K_L-K_S$ mass difference, is given by m\_K=[G\_F\^26\^2]{}\_1m\_KB\_Kf\_K\^2|(V\_[cd]{}\^ V\_[cs]{})|\^2m\_c\^2I\_1(x\_c), where $\eta_1$ takes care of the relevant short-distance QCD correction, and $f_K$ is the kaon decay constant. $B_K$ parametrizes the error in using vacuum insertion approximation to evaluate the matrix element $<\bar K|\bar d\gamma^{\mu}(1-\gamma_5)s\bar d\gamma_{\mu}(1-\gamma_5)s|
K>$, and lies between 0 and 1. Using chiral perturbation theory as well as hadronic sum rules, one obtains $B_K=1/3$ [@chipt], whereas lattice QCD studies give $B_K=0.85$ as the central value [@lattice]. Other values like $B_K=0.70$ is obtained from $1/N$ expansion technique [@onebyn].
The function $I_1(z)$ has the expression I\_1(z)=1-[3z(1+z)4(1-z)\^2]{}-[3z\^2(z)2(1-z)\^3]{}, and for any quark $q$, we use $x_q=m_q^2/m_W^2$.
Parametrizing the quark mixing matrix in an approximate form [@chau] V\_[CKM]{}=, where the cosines of the mixing angles have been approximated by unity and $s_{13}$ is assumed to be one order of magnitude smaller than $s_{12}$ and $s_{23}$, one obtains |(V\_[cd]{}\^V\_[cs]{})|\^2=s\_[12]{}\^2+s\_[23]{}\^4q\^2 +2s\_[12]{}s\_[23]{}\^2qwith $q=s_{13}/s_{23}$.
Expression for the $\bbbar$ mixing parameter, $x_d$, in the SM, is [@kdd] x\_d=[m]{}\_[B\_d]{}=\_b[G\_F\^26\^2]{} \_BB\_Bf\_B\^2m\_Bm\_t\^2I\_1(x\_t)|V\_[td]{}\^V\_[tb]{}|\^2 where $\eta_B$ is the corresponding short-distance QCD correction. $\sqrt{B_B}f_B$ is estimated from the lattice studies to be $0.14\pm
0.04$ GeV. $\tau_b|V_{td}^{\star}V_{tb}|^2$ can be written as \_b|V\_[td]{}\^V\_[tb]{}|\^2=\_b|V\_[cb]{}|\^2(s\_[12]{}\^2+q\^2-2s\_[12]{} q).
Lastly, the $CP$-violating parameter of the neutral kaon system, $\e$, has the following expression in the SM: ||&=&[G\_F\^212\^2]{}[m\_Km\_K]{} m\_W\^2B\_K f\_K\^2 , where, apart from the symbols previously explained, I\_2(z\_1,z\_2)=(z\_2/z\_1)-[34]{}[z\_21-z\_2]{}. $\eta_1$, $\eta_2$ and $\eta_3$ are three QCD correction factors, $\eta_1$ being the same as in eq. (6).
Now let us concentrate on the contributions to the abovementioned parameters coming from an extended Higgs sector. Our discussion will be limited within those assortment of scalar multiplets which keep $\rho_{tree}=1$; however, a generalization is straightforward but of little physical importance.
One has to consider two new box diagrams: one with two charged Higgses and two up-type quarks, and one with one charged Higgs, one $W^+$ and two up-type quarks. Note that as per eq. (2), the new physics contribution should exclude the diagrams containing only $H_1^+$ and no other charged Higgses.
To avoid cumbersome formulae which do not shed much light to new physics issues, we assume all charged Higgses to be degenerate in mass [@km]. This is not a too drastic approximation if one considers the fact that it is the mass of the charged Higgs, $m_{H^+}$, which we want to constrain. In case the charged Higgses do not have the same mass, $m_{H^+}$ corresponds to the lightest physical one. To do meaningful numerology, one has either to assume that all $H^+$s are degenerate, or that one of them is light enough to conribute and the others are so heavy that they effectively decouple. However, physically interesting models [@georgi] do have all scalar masses of the same order of magnitude, and so we stick to the first approximation. It may be mentioned that if the masses of the charged scalars are not exactly the same but similar in magnitude, bounds that we obtain change very little. We will also state what happens if one considers the second limit, i.e., existence of only one ‘light’ charged scalar.
Another reasonable approximation is to take all other quarks except the top to be massless while considering their couplings to the scalar fields. This makes eqs. (3) and (4) identical, and the results thus obtained will be more general. Note that as the scalar coupling to fermion-antifermion pair is proportional to the fermion mass, the GIM mechanism is not operative.
We give expressions for the contributions of the scalar-mediated diagrams to $\Delta m_K$, $x_d$ and $\e$. For any general quark $q$, we use $y_q=m_q^2/m_{H^+}^2$. As all $m_{H^+}$s are assumed to be same, $y_q$ is unique.
The contribution to $\Delta m_K$ is m\_K\^H=[G\_F\^224\^2]{}\_1m\_KB\_Kf\_K\^2|V\_[td]{}\^V\_[ts]{}| \^2m\_t\^2(J\_[HH]{}+J\_[HW]{}) where J\_[HH]{}=\_[i,j]{} y\_t[\_[i1]{}\^2\_[j1]{}\^2\_[11]{}\^4]{} , and J\_[HW]{}= \_[i=2]{}\^n x\_t([\_[i1]{}\^2\_[11]{}\^2]{}) . Here $\sum '$ means that the sum over both the mass indices runs from 1 to $n$, the number of charged scalars, but $i=1$, $j=1$ term corresponding to the Goldstone contribution is to be subtracted, as that is already considered in the SM amplitude. The same logic applies for the sum in eq. (16). The expression for the two functions, $I_3$ and $I_4$, are given by I\_3(x\_H,x\_t)&=&[x\_t(1-x\_t)(x\_H-x\_t)]{}-[x\_H\^2x\_H(1-x\_H) (x\_t-x\_H)\^2]{}\
&[ ]{}&+[x\_t(2x\_H-x\_t-x\_tx\_H)x\_t(x\_H-x\_t)\^2(1-x\_t)\^2]{},\
I\_4(x\_H,x\_t)&=&-[1(1-x\_t)(x\_H-x\_t)]{}+[x\_Hx\_H(1-x\_H) (x\_t-x\_H)\^2]{}\
&[ ]{}&-[(x\_H-x\_t\^2)x\_t(x\_H-x\_t)\^2(1-x\_t)\^2]{}, with $x_H=m_{H^+}^2/m_W^2$. Note that $I_3$ differs in sign from that given in eq. (B.3) of Ref. [@grossman].
$x_d$ is enhanced by x\_d\^H=\_b[G\_F\^224\^2]{}\_BB\_Bf\_B\^2m\_Bm\_t\^2|V\_[td]{}\^ V\_[tb]{}|\^2(J\_[HH]{}+J\_[HW]{}), and the contribution to $\e$ is ||\^H=[G\_F\^248\^2]{}[m\_Km\_K]{}m\_W\^2B\_Kf\_K\^2 m\_K\_2x\_t[Im]{}(V\_[td]{}\^V\_[ts]{})\^2(J\_[HH]{}+J\_[HW]{}).
None of the above charged scalar mediated processes are possible if $\alpha_{i1}=0$ for $i\not= 1$. In other words, the charged scalar of the weak doublet that gives mass to the top quark must mix with charged scalars of other multiplets to produce such contributions. This mixing is parametrised by $\theta_H$, i.e., $\alpha_{i1}=\cos\theta_H$. From the unitarity of the $\alpha$ matrix, $\sum_{i=2}^n |\alpha_{i1}|^2
=\sin^2\theta_H$.
Thus, if all $H^+$s are degenerate, $J_{HH}$ is proportional to $\sec^4\theta_H-1$ and $J_{HW}$ is proportional to $\sec^2\theta_H
-1$. However, if only the $k$-th charged scalar effectively contributes, the element $|\alpha_{k1}|^2$, and not the sum, gets paramount importance. It may happen that $|\alpha_{k1}|^2$ is very small or actually zero. Such a thing happens if $H_5^+$ is the lightest charged scalar in the triplet model of Ref. [@georgi]. In this case, all our discussions are invalidated, and we arrive at the well-known result of possible existence of a light charged scalar which does not couple to fermions.
Assuming the degeneracy of $m_{H^+}$, we try to put constraints on $m_{H^+}-\tan\theta_H$ plane. A major obstacle in that direction is the fact that a lot of quantities like $B_K$, $B_Bf_B^2$, $s_{23}$, $\delta$, and even $m_t$, are poorly known or estimated. To be consistent with the present experimental data, we take [@pdg; @cdf] G\_F&=&1.1663910\^[-5]{} [GeV]{}\^[-2]{}, m\_K=3.5210\^[-15]{} [GeV]{},\
m\_[B\_d]{}&=&5.28 [GeV]{}, m\_t=176 [GeV]{}, m\_W=80.41 [GeV]{},\
x\_d&=&0.77, ||=2.2610\^[-3]{}, f\_K=0.165 [GeV]{}, m\_K=0.498 [GeV]{},\
s\_[12]{}&=&0.2205, s\_[23]{}=0.040, \_b|V\_[cb]{}|\^2=3.510\^[-9]{} [GeV]{}\^[-1]{}. The numerical values of the QCD correction factors that we use are [@datta; @quinn] \_1=0.78, \_2=0.60, \_3=0.37, \_B=0.85.
First, let us concentrate on $\Delta m_K$. Assuming no long-distance contribution, $\Delta m_K$ does not limit $\tan\theta_H$ significantly. For $B_K=1/3$, the maximum value of $\tan\theta_H$ is $7.4$, $6.2$ and $7.6$ for $m_{H^+}=100$, 200 and 500 GeV respectively. This bound is one order of magnitude poorer than that derived from $\zbb$ data. Though formally eqs. (6) and (10) contain $\delta$, the result is insensitive to its specific value; the reason is the small coefficient of $\cos\delta$ in eq. (9). For $B_K=0.85$, the bounds are a shed better: $\thm = 3.5$, $2.9$ and $3.6$ for $m_{H^+}=100$, 200 and 500 GeV. We note that the bound is ‘strongest’ at around $m_{H^+}=m_t$.
The situation is different if one has, say, a 50% long-distance contribution. In that case, $B_K=1/3$ gives $\thm = 4.5$, $3.7$ and $4.6$ for $m_{H^+}=100$, 200 and 500 GeV. However, $B_K=0.85$ oversaturates the SM value and no room for new physics is left. $B_K=0.70$ yields a fairly strong constraint: $\thm = 1.2$, $1.0$ and $1.2$ for the three values of $m_{H^+}$ we have chosen to mention. This is comparable to those bounds obtained from partial width of $Z$ into $b\bar b$ pairs.
With $B_K=1/3$, $s_{23}=0.040$ and $q=0.10$, the strongest bound on $\thm$ is 1.1, which is for $\delta=7\pi/12$ and $m_{H^+}=300$ GeV. For $q=0.06$, the bound is somewhat less stringent; the results are shown in Figs. (1a) and (1b). Also, $q=0.14$ constrains the parameter space more tightly. Furthermore, one observes that $B_K=0.85$ does not allow $\delta > \pi/4$, and $B_K=0.70$ does not allow $\pi/4 < \delta < 3\pi/4$ — the SM value saturates the experimental number. Even for those values of $\delta$ which allows for a new physics contribution, $\thm$ is generally less than 1.0, which is a better constraint than that obtained from $\zbb$ data.
Currently favoured values of $B_Bf_B^2$ ($\approx 0.02$ GeV$^2$) also does not allow $\delta > \pi/2$ from measurements on $x_d$. For $\delta
=\pi/2$, one gets $\thm = 0.36$ for $m_{H^+}=100$ GeV. Even for $\delta$ as low as $\pi/6$, $\thm = 1.4$ for a charged scalar mass of 100 GeV. Lowering $B_Bf_B^2$ to $0.01$ GeV$^2$ results in a larger allowed value of $\thm$. Figs. (2a) and (2b) show the detailed result.
We conclude from this analysis that even in a model with arbitrary assortment of scalars, one can obtain fairly strong constraints on the parameter space of the scalar sector, with a very few reasonable assumptions, from $\bbbar$ mixing data and the $\e$ parameter, and maybe even from $\kkbar$ mixing data. These constraints are shown to be comparable, and sometimes better, to those obtained from $\Gamma(\zbb )$, which was calculated in Ref. [@km]. We want to remind our readers that such an analysis is only meaningful if the lightest physical charged scalar(s) couple with fermions, and if $\rho_{tree}=1$ is maintained (otherwise, $\rho$ parameter puts a better constraint). The error bar in $m_t$ turns out to be insignificant; however, quantities like $B_K$, $B_Bf_B^2$ and $\delta$, which are either poorly known or completely unknown, play a significant role. With a more accurate experimental determination of these quantities, one hopes to make these constraints more meaningful.
The authors thank T. De, B. Dutta-Roy and B. Mukhopadhyaya for useful discussions. AK thanks International Centre for Theoretical Physics, Trieste, Italy, for its hospitality, where a large part of the work was done.
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**Figure Captions**
[**1(a)**]{}. Upper limits on tan$\theta_H$ for different values of $m_{H^+}$, as obtained from the analysis of the $\e$ parameter. We take $q=0.10$. The uppermost curve is for $\delta=\pi/6$, and the successive ones are for $\delta=\pi/4$, $\pi/3$, $5\pi/12$, $3\pi/4$ and $7\pi/12$ respectively.
[**1(b)**]{}. Same as in 1(a), with $q=0.06$. The curves are for $\delta=\pi/6$, $\pi/4$, $\pi/3$, $3\pi/4$, $5\pi/12$ and $7\pi/12$ respectively.
[**2(a)**]{}. Upper limits on tan$\theta_H$ for different values of $m_{H^+}$, as obtained from the analysis of $B_d-\bar B_d$ mixing. We take $B_Bf_B^2=0.02$. The uppermost curve is for $\delta=\pi/6$, and the successive ones are for $\delta=\pi/4$, $\pi/3$, $5\pi/12$ and $\pi/2$ respectively. For $\delta>\pi/2$, the SM value saturates the experimental bound.
[**2(b)**]{}. Same as in 2(a), with $B_Bf_B^2=0.01$. The curves are for $\delta=\pi/6$, $\pi/4$, $\pi/3$, $5\pi/12$, $\pi/2$, $7\pi/12$, $2\pi/3$, $3\pi/4$ and $5\pi/6$ respectively.
[^1]: E-mail: rupa@tnp.saha.ernet.in
[^2]: E-mail: akundu@saha.ernet.in
|
---
author:
- Hauyu Baobab Liu
- 'Melvyn C. H. Wright'
- 'Jun-Hui Zhao'
- 'Elisabeth A. C. Mills'
- 'Miguel A. Requena-Torres'
- Satoki Matsushita
- Sergio Martín
- Jürgen Ott
- 'Mark R. Morris'
- 'Steven N. Longmore'
- 'Christiaan D. Brinkerink'
- Heino Falcke
date: 'Received February 15, 2016; accepted March XX, 2016'
title: 'The 492 GHz emission of Sgr A\* constrained by ALMA'
---
[Our aim is to characterize the polarized continuum emission properties including intensity, polarization position angle, and polarization percentage of Sgr A\* at $\sim$492 GHz. This frequency being well into the submillimeter-hump where the emission is supposed to become optically thin, allows us to see down to the event horizon. Hence the reported observations contain potentially vital information on black hole properties. We have compared our measurements with previous, lower frequency observations, which provides information in the time domain.]{} [We report continuum emission properties of Sgr A\* at $\sim$492 GHz, based on the Atacama Large Millimeter Array (ALMA) observations. We measured fluxes of Sgr A\* from the central fields of our ALMA mosaic observations. We used the observations of the likely unpolarized continuum emission of Titan, and the observations of C<span style="font-variant:small-caps;">i</span> line emission, to gauge the degree of spurious polarization.]{} [The flux of 3.6$\pm$0.72 Jy during our run is consistent with extrapolations from the previous, lower frequency observations. We found that the continuum emission of Sgr A\* at $\sim$492 GHz shows large amplitude differences between the XX and the YY correlations. The observed intensity ratio between the XX and YY correlations as a function of parallactic angle may be explained by a constant polarization position angle of $\sim$158$^{\circ}$$\pm$3$^{\circ}$. The fitted polarization percentage of Sgr A\* during our observational period is 14%$\pm$1.2%. The calibrator quasar J1744-3116 we observed at the same night can be fitted to Stokes I = 252 mJy, with 7.9%$\pm$0.9% polarization in position angle P.A. = 14.1$^{\circ}$$\pm$4.2$^{\circ}$.]{} [The observed polarization percentage and polarization position angle in the present work appear consistent with those expected from longer wavelength observations in the period of 1999-2005. In particular, the polarization position angle at 492 GHz, expected from the previously fitted 167$^{\circ}$$\pm$7$^{\circ}$ intrinsic polarization position angle and (-5.6$\pm$0.7)$\times$10$^{5}$ rotation measure, is 155$^{+9}_{-8}$, which is consistent with our new measurement of polarization position angle within 1$\sigma$. The polarization percentage and the polarization position angle may be varying over the period of our ALMA 12m Array observations, which demands further investigation with future polarization observations.]{}
Introduction
============
The sub-Eddington accretion of the nearest supermassive black hole, Sgr A\* ($\sim$4$\times$10$^{6}$ $M_{\odot}$, e.g. Sch[ö]{}del et al. 2002; Ghez et al. 2005; Gillessen et al. 2009), has inspired a tremendous amount of observational and theoretical activity (see Yuan & Narayan 2014 for a complete review of existing theories). This [**includes**]{} monitoring observations at multiple wavelengths to probe synchrotron emission, which may be from the innermost part of an accretion flow, or the footpoint of a jet (Falcke et al. 2000; Liu et al. 2007; Falcke et al. 2009; Huang et al. 2009; more below), and has motivated very long baseline millimeter interferometric observations (e.g. Johnson et al. 2015, and references therein).
---------- ------------- ------------------- ------------------------------- -------------------
Field ID Correlation Average amplitude Amplitude Standard Deviations Parallactic angle
(Jy) (Jy) ($^{\circ}$)
18 XX 2.23 0.54 -42.4
YY 1.85 0.52
25 XX 2.05 0.54 -40.3
YY 1.69 0.51
0 XX 4.18 0.54 -23.0
YY 3.14 0.53
94 XX 2.39 0.69 5.5
YY 2.03 0.64
101 XX 1.99 0.66 9.0
YY 1.76 0.63
133 XX 1.83 0.62 28.6
YY 2.00 0.64
134 XX 1.75 0.62 29.0
YY 1.89 0.63
---------- ------------- ------------------- ------------------------------- -------------------
Observations of the polarization position angle and the polarization percentage of the synchrotron emission over a broad range of frequency, may provide information about the geometry and the magnetic field configuration of the accretion flow (Bromley et al. 2001; Liu et al. 2007; Huang et al. 2009), and can diagnose the black hole accretion rate on small [**scales**]{} via deriving Faraday rotation (more below). Previous strong constraints on the linear polarization percentage in the 4.8-112 GHz bands (Bower et al. 1999a, 1999c, 2001), and the detected linear polarization at the 83-400 GHz bands (Aitken et al. 2000; Bower et al. 2003, 2005; Macquart et al. 2006; Marrone et al. 2006a, 2007), have given rise to a model in which linearly polarized radiation is emitted from within a few gravitational radii around Sgr A\*, and is further Faraday depolarized by the ionized accretion flow foreground to Sgr A\*. This model is supported by the detection of circularly polarized emission in the 1.4-15 GHz bands (Bower et al. 1999b; Bower et al. 2002; Sault & Macquart 1999; see also the measurements at 230 and 345 GHz by Mu[ñ]{}oz et al. 2012). These observations have constrained the accretion rate of Sgr A\* to be between 2$\times$10$^{-9}$ and 2$\times$10$^{-7}$ $M_{\odot}$yr$^{-1}$. On the other hand, the observed variation of Sgr A\*, including large millimeter flares (Zhao et al. 2003, 2004; Marrone et al. 2006), indicates that the accretion may not be stationary. In this work, we report new constraints on the polarized emission of Sgr A\* at 492 GHz, based on Atacama Large Millimeter Array (ALMA) 12m-Array and Compact Array (ACA) mosaic observations towards the Galactic center. Our new high-frequency observations provide important, long lever arms in the frequency and time domains for comparison with submillimeter, millimeter, and radio bands observations carried out between 1999 and 2005. In particular, our observing frequency should be above the turnover frequency at which the emission becomes optically thin (Marrone et al. 2006b). Moreover, we are able to reliably diagnose polarizion, which provides the highest frequency interferometric polarization observations so far, and hence tells of the intrinsic polarization. Our works are pioneering future observations to probe variability, which are crucial to understand the physics of Sgr A\*.
Details of our observations and data reduction are provided in Section \[chap\_obs\]. Our results are given in Section \[chap\_result\]. In Section \[chap\_discussion\] we address potential systematic biases, and present the comparison of our results with previous observations. A brief conclusion is provided in Section \[chap\_conclusion\].
Observations and Data Reduction {#chap_obs}
===============================
The ALMA 12m-Array (consisting of 12 m dishes) mosaic observations of 149 fields were carried out on 2015 April 30 (UTC 06:48:32.4–08:04:38.4), with 39 antennas. The array consisted of 19 Alcatel antennas (DA), 18 Vertex antennas (DV), and 2 Mitsubishi antennas (PM). These observations approximately covered a 55$''$$\times$80$''$ rectangular region. The pointing and phase referencing center of the central field was R.A. (J2000) =17$^{\mbox{\scriptsize{h}}}$45$^{\mbox{\scriptsize{m}}}$40$^{\mbox{\scriptsize{s}}}$.036, and decl. (J2000) =-29$^{\circ}$00$'$28$''$.17, which is approximately centered upon Sgr A\*. We configured the correlator to provide four 1.875 GHz wide spectral windows (spws), covering the frequency ranges of 491.3-493.2 GHz (spw 0), 489.3-491.2 GHz (spw 1), 479.2-481.1 GHz (spw 2), and 481.0-482.9 GHz (spw 3), respectively. The observations were designed to cover the C<span style="font-variant:small-caps;">i</span> line and the CS 10-9 line, with rest frequencies are 492.16065 GHz and 489.75093 GHz, respectively. The frequency channel spacing was 1953.125 kHz ($\sim$1.2 kms$^{-1}$). The receivers are aligned in a parallel-linear configuration, which yielded the XX and YY linear correlations. The X polarization of the receivers is aligned radially in the receiver cryostat, with Y being aligned perpendicular to X (private communications with Ted Huang and Shin’ichiro Asayama). According to ALMA specifications, the accuracy of this alignment is within 2 degrees. The absolute feed alignment was obtained from the raw data, using the CASA software package (McMullin et al. 2007) command [tb.getcol(’RECEPTOR\_ANGLE’)]{}, and can be referenced from the ALMA Cycle 3 [**and Cycle 4**]{} Technical Handbook[^1].
The range of [*uv*]{} spatial frequencies sampled by the 12m-Array observations is 25-570 k$\lambda$. The system temperature ($T_{sys}$) ranged from $\sim$500-1000 K. The mosaic field was Nyquist sampled in hexagonal packing, with an on-source integration time of 12.08 seconds for each of the 149 mosaic fields. We observed J1744-3116 approximately every 10 minutes for gain calibrations. We observed Titan and J1833-2103 for absolute flux and passband calibrations, respectively.
The Atacama Compact Array (ACA; consisting of ten 7 m dishes) observations were carried out on 2015 April 30 (UTC 05:35:00.0–07:30:00.1) with 10 available antennas. All 10 antennas shared an identical (Mitsubishi, 7m) design. The ACA observations approximately covered the same field of view as the 12m-Array mosaic. The pointing and phase referencing center of the central field was also on Sgr A\*. The correlator setup of the ACA observations was identical to that of the 12m-Array mosaic. The ACA observations sampled a [*uv*]{} spacing range of 14-80 k$\lambda$. The mosaic field was Nyquist sampled in hexagonal packing. Due to unspecified technical issues, the ACA observations were terminated at the middle of the track. Therefore, the southeastern half of the observed region had a 60.6 seconds on-source integration time for each mosaic field, while the northwestern half had a 30.3 seconds on-source integration time for each mosaic field. This led to different sensitivity and $uv$ coverages for the southeastern and the northwestern fields. Like the 12 m observations, $T_{sys}$ values ranged from $\sim$500-1000 K. We again observed J1744-3116 approximately every 10 minutes for gain calibrations, and observed Titan and J1517-2422 for absolute flux and passband calibrations, respectively.
There are currently no available single-dish data to provide information on the zero-spacing fluxes for these observations.
A priori calibrations including the application of $T_{sys}$ data, the water vapor radiometer (wvr) solution (which is only provided for the 12m-Array observations), antenna based passband calibrations, gain amplitude and phase calibrations, and absolute flux scaling, were carried out using the CASA software package (McMullin et al. 2007) version 4.3.1. To enhance the signal to noise ratio, we first solved for and applied phase offsets between the four spectral windows, based on scans on the passband calibrator. We then derived gain calibration solutions. The gain phase solutions were derived separately for the XX and YY correlations, while the gain amplitude solutions were derived from the average of XX and YY correlations. We derived gain phase solutions for both individual spectral windows and averaging all spectral windows together. We ultimately chose to use the latter, as the wvr solutions for the 12m-Array data in spw 1 and 3 have poorer qualities, which led to massive data flagging when deriving gain phase solutions for individual spectral windows independently. We also tested whether applying or not applying the wvr solutions changed the quality of our final images; ultimately although the difference was minimal, we chose to apply the wvr solutions to the 12m data. We confirmed that the qualities of continuum images generated from all spectral windows are consistent (e.g., any differences are a result of the available bandwidths in spectral line-free channels). There was also significant interference due to atmospheric lines in spw 3, which degraded its continuum sensitivity.
The absolute flux scaling was derived incrementally from the gain amplitude solutions, combining all scans. The scans on Titan were largely flagged due to interference from spectral lines. Therefore, absolute flux referencing for both the 12m-Array and ACA observations is subject to a large uncertainty (e.g. $\sim$20 %, empirically). This can lead to the mismatched flux levels between the 12m-Array and the ACA observations, and errors in the observed spectral indices.
We fitted the continuum baselines from line-free channels, using the CASA task [uvcontsub]{}. After executing [uvcontsub]{}, we generated a continuum data set for each spectral window, by averaging the line free channels. We then exported the calibrated continuum data and the continuum-subtracted line data in standard fits format files, using the CASA task [exportfits]{}. Finally, we used the Miriad 4.3.8 (Sault et al. 1995) task [fits]{} to convert the fits format data into the Miriad data format, for further analyses including imaging.
We then used Miriad to make synthesized images (i.e. dirty images) of the continuum using naturally weighed data for the 12m-Array and ACA with beam widths (FWHM) $\theta_{\mbox{\scriptsize{maj}}}$$\times$$\theta_{\mbox{\scriptsize{min}}}$ = $0\farcs70$$\times$$0\farcs42$ (P.A.=-88$^{\circ}$) and $\theta_{\mbox{\scriptsize{maj}}}$$\times$$\theta_{\mbox{\scriptsize{min}}}$ = $3\farcs4$$\times$$2\farcs2$ (P.A.=78$^{\circ}$) , respectively. For the C<span style="font-variant:small-caps;">i</span> line, we tapered the 12m-Array data using a Gaussian weighting function of FWHM = $1\farcs5$ to enhance the signal-to-noise ratio of the line, and then generated the synthesized images. For all of these images we do not make deconvolved C<span style="font-variant:small-caps;">i</span> line (i.e. [clean]{}ed) maps, to avoid any possibility of uncertainties caused by the [clean]{} process.
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![[]{data-label="fig:fields"}](gc_fields.eps "fig:"){width="10.5cm"}
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Results {#chap_result}
=======
Throughout this manuscript, the X and Y polarization, and Stokes Q, are defined in the receiver coordinate frame if not specifically mentioned. In the nearly ideal observational and instrumental condition, the polarization percentage and the polarization position angle of a non-variable source are related to these quantities based on the following formula: $$\frac{Q}{I} - \delta \equiv \frac{XX- YY}{2I} - \delta = P\cdot\cos(2(\Psi - \eta - \phi) ),$$ where $Q$ denotes the observed Stokes Q flux, $\delta$ (Q offset, hereafter) is an assumed constant normalized offset of observed Stokes Q due to amplitude calibration errors or polarization leakage; $P$ is the polarization percentage; $\Psi$, $\eta$, and $\phi$ are the polarization position angle in the sky (e.g. right ascension/declination) frame, the parallactic angle, and the angular separations between the X polarization and the local vertical (which is known as [*Evector*]{}). Evector of ALMA is 0$^{\circ}$ for the frequency band we observed. A wide coverage of $\eta$ during the observations, will allow unambiguously fitting $\delta$, $P$ and $\Psi$.
Continuum data
--------------
After a priori calibrations, we found that the continuum emission from Sgr A\* was significantly detected in the central 19 mosaic fields [**of the 12m-Array observations**]{}. To inspect the residual phase errors, we used the CASA task [fixvis]{} to shift the phase referencing centers of these fields to the position of Sgr A\*. We observed up to $\sim\pm$50$^{\circ}$ of residual phase offsets, and a phase RMS of $\sim$16.5$^{\circ}$. The phase offsets and phase RMS of the XX and YY correlations are consistent.
We attribute the phase errors to phase variations that are faster than our gain calibration cycle time, as well as phase offsets between the gain calibrator and the target source fields. To correct for these phase errors we used the Miriad task [demos]{}, assuming the nominal ALMA primary beam shape, to generate models of Sgr A\* for the central 7 mosaic fields (Figure \[fig:fields\]). We removed the phase errors of the central 7 fields using the Miriad task [selfcal options=mosaic]{}, with a 0.01 minute solution interval. Then, we used the Miriad task [uvflux]{} to fit the observed amplitudes from the visibility data. Our [**12m-Array**]{} measurements for Sgr A\* are summarized in Table \[tab:obs\] and Figure \[fig:xxyytime\]. After self-calibration, the averaged flux of Sgr A\* at 492 GHz is 3.6$\pm$0.72 Jy. The application of phase self-calibration solutions does not significantly change the observed amplitude (or flux) ratios between the XX and the YY correlations. We do not present flux measurements of Sgr A\* from outside of the central 7 mosaic fields due to the potential for large amplitude uncertainties induced by antenna pointing errors (e.g. up to $\sim$1$''$, according to private communication among members in the ALMA Regional Centers), and the poorly understood primary beam phase responses.
The Stokes I intensity of the Sgr A\* may be varying with time, however, cannot be clearly distinguished given our present flux calibration accuracy (Figure \[fig:xxyytime\]). In addition, we find that Sgr A\* and the gain calibrator J1744-3116 have several times higher fractional amplitude differences between the XX and the YY correlations, than that of the continuum emission of Titan. From the $<$100 meter baselines, the XX/YY flux ratios of Titan measured from spw 0, 1, 2, and 3, are 0.99, 0.98, 1.0, and 1.0, respectively. The relative amplitude differences of Sgr A\* and J1744-3116 cannot be attributed to decoherence due to phase errors. The observed XX and YY amplitudes of the gain calibrator J1744-3116 can be fitted to Stokes I = 252 mJy, with 7.9%$\pm$0.9% polarization in position angle P.A. = 14.1$^{\circ}$$\pm$4.2$^{\circ}$, and a constant normalized Stokes Q offset $\delta$=$-$0.02$\pm$0.02, which may be caused by amplitude calibration errors or polarization leakage (Figure \[fig:j1744\]). However, the XX and YY amplitudes of Sgr A\*, obtained from the inner 7 mosaic fields, do not vary smoothly with parallactic angle. To first order, taking the intensity ratio of these two correlations removes the total intensity variations. Plotting the XX to YY intensity ratio versus parallactic angle [**from the 12m-Array observations**]{} shows a peak at a parallactic angle of $-$22$^{\circ}$, with an intensity ratio close to 1 around parallactic angle +20$^{\circ}$, From a least square fit to constant polarization position angle, the measured XX to YY intensity ratios for Sgr A\* are consistent with the polarization percentage of $\sim$14%$\pm$1.2% and a position angle of $\sim$158$^{\circ}$$\pm$3$^{\circ}$ (Figure \[fig:stokesfit\]). For comparison, previously measured polarization position angles at 340 GHz were $\sim$136$^{\circ}$-163$^{\circ}$, and showed variations on daily timescales (Marrone et al. 2006a). The imperfect fits of Figure \[fig:stokesfit\], if not due to calibration issues (more discussion in Section \[chap\_discussion\]), may be attributed to time variation in the polarization percentage and position angles during the period of our ALMA observations. However we cannot easily verify this without observing and calibrating the XY and YX cross correlations. We refer to Bower et al. (2003) and Marrone et al. (2006a) for the observational evidence and discussion of polarization percentage variability at the 230 and the 340 GHz bands. We refer to Eckart et al. (2006), Fish et al. (2009), Zamaninasab et al. (2010) and references therein, for modeling frameworks of the polarized emission.
To determine whether there might be a spurious polarization signal due to the heterogeneity of dishes in the 12m-Array, we split the 12m-Array visibility data into subsets containing only correlation products between the DA antennas, only correlation products between the DV antennas, and a subset containing all correlation products between the DA and the DV antennas. We obtained identical measurements from these three subsets. Therefore, we are convinced that there is no detectable spurious polarizations due to different DA and DV antenna designs. There were only two PM antennas in our 12m-Array observations, so we could not reliably check the cross products independently. Nevertheless, we found that including or not including the PM antennas does not significantly change our measurements. The XX and YY intensity differences of Sgr A\* observed from the four spectral windows are also consistent [**(Figure \[fig:sgrAspws\])**]{}.
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![The normalized intensity difference of the XX and YY correlations of Sgr A\*, observed by the ALMA 12m-Array (symbols), and a black curve representing our best fit to these data. The constant polarization percentage and polarization position angle obtained from our best fit model are 14%$\pm$1.2% and 158$^{\circ}$$\pm$3$^{\circ}$, respectively. Gray curves show 50 independent random realizations of models with constant polarization percentage and polarization position angle, which characterize the error bars we give. We caution that these quantities are not fully constrained without the measurements of the XY and YX correlations.[]{data-label="fig:stokesfit"}](stokesfit_sgrAb8.eps){width="10cm"}
Spectral line data {#subsec:line}
------------------
We are not aware of any mechanism which can uniformly polarize C<span style="font-variant:small-caps;">i</span> line emission to a high percentage over our mosaic field of view. Thermal continuum emission of Titan is also not known to be polarized. Therefore, we use these observations to gauge the magnitude of spurious polarization caused by the offset of antenna response in XX and YY, and polarization leakage.
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We used the Miriad task [imdiff]{} to systematically estimate the multiplicative factor that minimizes the difference between the XX and YY synthesized images of C<span style="font-variant:small-caps;">i</span> in a maximum likelihood sense. We note that this multiplicative factor ($F_{XX}^{CI}(\nu, t)/F_{YY}^{CI}(\nu, t)$, hereafter) can depend on $v_{lsr}$ and time. To avoid the high noise at the edge of the 12m-Array mosaic field, we limited the derivation of $F_{XX}^{CI}(\nu, t)/F_{YY}^{CI}(\nu, t)$ to a box-shaped region containing the most significant C<span style="font-variant:small-caps;">i</span> emission. The coordinates of the bottom left and top right corners of this region are R.A. (J2000) =17$^{\mbox{\scriptsize{h}}}$45$^{\mbox{\scriptsize{m}}}$41$^{\mbox{\scriptsize{s}}}$.332, and decl. (J2000) =$-$29$^{\circ}$00$'$56$''$.77 and R.A. (J2000) =17$^{\mbox{\scriptsize{h}}}$45$^{\mbox{\scriptsize{m}}}$38$^{\mbox{\scriptsize{s}}}$.885, and decl. (J2000) =$-$28$^{\circ}$59$'$56$''$.77, respectively. We verify that using the full images for estimating $F_{XX}^{CI}(\nu, t)/F_{YY}^{CI}(\nu, t)$ does not change the results, although it can change the noise behavior. We also measured the XX to YY amplitude ratio of the 12m-Array continuum observations of Titan, using the same method. The continuum emission from Titan shows a $\sim$3% intensity difference between the XX and the YY correlations. The XX to YY continuum intensity ratios of both Sgr A\* and Titan are shown in Figure \[fig:xxyytime\].
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![[]{data-label="fig:xxyyfreq"}](CI_Imdiff_final.eps "fig:"){width="9.5cm"}
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We define $\int F_{XX}^{CI}(\nu, t) d\nu$ / $\int d\nu$ = $F_{XX}^{CI}(t)$, and $\int F_{XX}^{CI}(\nu, t) dt$ / $\int dt$ = $F_{XX}^{CI}(\nu)$. In practice, we measured $F_{XX}^{CI}(t)$ / $F_{YY}^{CI}(t)$ of the 12m-Array observations from spectral channels which are dominated by C<span style="font-variant:small-caps;">i</span> [*emission*]{} (the case in which it is dominated by absorption is described further below), for each of the target source scans (i.e. every time period bracketed by two adjacent gain calibration scans). $F_{XX}^{CI}(t)$ / $F_{YY}^{CI}(t)$ for the ACA observations were measured in the same way, but over the entire ACA observing period. We also measured $F_{XX}^{CI}(\nu)$ / $F_{YY}^{CI}(\nu)$ for every 2.5 kms$^{-1}$ wide velocity channels, by averaging over all 12m-Array integrations. However, we were not able to obtain a meaningful constraint of $F_{XX}^{CI}(\nu)$ / $F_{YY}^{CI}(\nu)$ from the ACA observations, due to their limited sensitivity. Figure \[fig:xxyyfreq\] shows the measured $F_{XX}^{CI}(t)$ / $F_{YY}^{CI}(t)$ and $F_{XX}^{CI}(\nu)$ / $F_{YY}^{CI}(\nu)$ from our observations.
Extended emission from the C<span style="font-variant:small-caps;">i</span> line is detected in channels over a range of velocities, following a similar velocity field to that of the molecular circumnuclear disk (Guesten et al. 1987; Wright et al. 2001; Liu et al. 2012, 2013, and references therein). Examples of the C<span style="font-variant:small-caps;">i</span> line velocity channel [*synthesized*]{} images from the 12m-Array observations, are given in Figure \[fig:images\]. The CI emission will be discussed in more detail in a separate paper (Liu et al. in prep.) However, we found that for several velocity channels around $v_{lsr}$$\sim$20 kms$^{-1}$, the extended CI line emission from the Galactic center is nearly completely absorbed by foreground gas. In these channels, the dominant feature is absorption against the continuum emission of Sgr A\*, which is not spatially resolved by our observations.
At the same velocity as the absorption we detect a local maximum of $F_{XX}^{CI}(\nu)$ / $F_{YY}^{CI}(\nu)$ (Figure \[fig:xxyyfreq\]). The local peak value of $F_{XX}^{CI}(\nu)$ / $F_{YY}^{CI}(\nu)$ is $\sim$1.3 (or 0.11 in logarithm). This peak value of $F_{XX}^{CI}(\nu)$ / $F_{YY}^{CI}(\nu)$ is consistent within 1$\sigma$ with the XX and YY continuum intensity ratio of Sgr A\*, measured from the inner 7 fields mosaic of the 12m-Array observations. In fact, the three most prominent absorption line features of C<span style="font-variant:small-caps;">i</span> against the continuum emission of the Sgr A\*, consistently present a deeper absorption in XX correlation than in YY (Figure \[fig:spectrum\]). In the ACA observations, the difference of the absorption line intensities between the XX and the YY correlations, are lower than the 1$\sigma$ noise level of the ACA observations.
$F_{XX}^{CI}(\nu)$ / $F_{YY}^{CI}(\nu)$ is close to 1 in the remaining velocity channels with significant emission. The standard deviation of $F_{XX}^{CI}(\nu)$ / $F_{YY}^{CI}(\nu)$, $\sigma_{\nu}^{CI}$, measured from velocity channels away from $v_{lsr}$ = 20 kms$^{-1}$ (Figure \[fig:xxyyfreq\]), is 0.043. For the velocity range in which we significantly detected C<span style="font-variant:small-caps;">i</span>, the value of \[Max($F_{XX}^{CI}(\nu)$ / $F_{YY}^{CI}(\nu)$ ) - Mean($F_{XX}^{CI}(\nu)$ / $F_{YY}^{CI}(\nu)$) \] / $\sigma_{\nu}^{CI}$ is 7.8 (Figure \[fig:xxyyfreq\]). We have visually inspected the XX and YY intensity maps ($I^{XX}(\nu, t)$, $I^{YY}(\nu, t)$)[^2], and the residual $R(\nu, t)$ $\equiv$ $I^{XX}(\nu, t)$ - ($F_{XX}^{CI}(\nu, t)$ / $F_{YY}^{CI}(\nu, t)$)$\times$$I^{YY}(\nu, t)$. Based on the statistics of pixel values and the visual inspection of images, we found that $R(\nu, t)$, and its time integration, are consistent with thermal noise. On the other hand, we found that for spectral channels away from $v_{lsr}$$\sim$20 kms$^{-1}$, $I^{XX}(\nu, t)$ $-$1.3$\times$$I^{YY}(\nu, t)$ presents significant (i.e. $>$3$\sigma$) features of over subtraction.
Figure \[fig:xxyyfreq\] and \[fig:spectrum\] may be understood considering the radiative transfer equation $
T_{b} = (T_{ex} - T_{bg})( 1 - e^{-\tau}),
$ where $T_{b}$ is the observed C<span style="font-variant:small-caps;">i</span> brightness temperature, $T_{ex}$ is the gas excitation temperature, $T_{bg}$ is the background brightness temperature, and $\tau$ is the optical depth of gas. For the foreground C<span style="font-variant:small-caps;">i</span> absorption against the continuum emission of Sgr A\*, it is safe to assume that $T_{ex}$ is negligible, and the gas optical depth $\tau$ is identical for the orthogonal linear polarizations X and Y. The assumption of the identical gas optical depth $\tau$ for the X and Y polarizations can be supported by the observed XX/YY$\sim$1 from emission line (Figure \[fig:xxyyfreq\]). Therefore, the C<span style="font-variant:small-caps;">i</span> absorption line ratio of the XX and YY correlations, is expected to be nearly identical to the XX/YY flux ratio of the continuum emission of Sgr A\*.
{width="18cm"}
Discussion {#chap_discussion}
==========
The significant difference between the XX and YY correlations can be used to make a reliable determination of Stokes Q at 492 GHz. However, lacking the cross-correlations XY and YX which were not sampled in these observations, we are not able to determine Stokes U. Nevertheless, the ALMA observations give a meaningful lower limit on the linear polarization continuum emission from Sgr A\* at this highest frequency that has been studied in polarization there with any submillimeter bands available on interferometer arrays to date. The maximum of the intensity differences between the XX and the YY correlations observed from the $\sim$492 GHz continuum emission of Sgr A\* (diff($I_{XX,YY}^{cont.}$), hereafter), implies a $\sim$14%$\pm$1.2% lower limit on its polarization percentage.
Potential causes of the observed diff($I_{XX,YY}^{cont.}$) are synchrotron emission from ionized gas close to Sgr A\* (Falcke et al. 1998; Aitken et al. 2000; Bower et al. 2001, 2003; Marrone et al. 2006a; Bromley et al. 2001; Liu et al. 2007; Huang et al. 2009), or instrumental effects including (1) beam squint, (2) relative drifts of instrumental gain amplitude between the XX and YY correlations, (3) phase decoherence for a certain polarization, and (4) primary beam polarization. As addressed in Section \[chap\_obs\], we find no evidence that the decoherence due to phase errors can lead to the differences of intensities measured by the XX and YY correlations. In addition, our analysis of the C<span style="font-variant:small-caps;">i</span> line emission has ruled out the possibilities that the relative drifts of instrumental gain amplitude as well as the effects of phase decoherence can lead to the observed diff($I_{XX,YY}^{cont.}$) in continuum emission (Section \[subsec:line\]). Beam squint does not apply to the observations on the central field (field 0, see Figure \[fig:fields\]). The observed diff($I_{XX,YY}^{cont.}$) from the other mosaic fields also appears too big to be explained by beam squint, unless the actual primary beam response functions of the ALMA antennas seriously deviate from the present understanding. Nevertheless, the comparisons of the diff($I_{XX,YY}^{cont.}$) taken from the pairs of fields (18, 25), (94, 101), and (133, 134), which were observed closely in time, empirically provide a limit on the scale of beam squint effects (Figure \[fig:xxyytime\]). On the other hand, each two exposures of these three pairs of fields show rather consistent diff($I_{XX,YY}^{cont.}$), which may indicate that there is no significant variation of polarization on the very short timescales probed by their time separations. Primary beam polarization cannot explain the observed highest diff($I_{XX,YY}^{cont.}$) from the central field (i.e. Field 0, see Figure \[fig:xxyytime\]), and cannot explain the frequency dependence of $F_{XX}^{CI}(\nu)$ / $F_{YY}^{CI}(\nu)$ (Figure \[fig:xxyyfreq\]). We are not aware of other instrumental defects which can cause similar effects, and consider polarized synchrotron emission as the most probable explanation for the diff($I_{XX,YY}^{cont.})$ we measured from Sgr A\*.
{width="9.5cm"}
\[fig:spectrum\]
The $\sim$14%$\pm$1.2% polarization percentage and 158$^{\circ}$$\pm$3$^{\circ}$ polarization position angle of the continuum emission of Sgr A\* appear realistic when compared with previous (sub)millimeter observations at other frequency bands (Aitken 2000, Bower et al. 2003, 2005, Marrone et al. 2006a, 2007), despite the large time separations of these observations (Figure \[fig:chiplot\], \[fig:fracplot\]). In particular, Marrone et al. (2007) reported the fitted intrinsic polarization position angle $\chi_{0}$=167$^{\circ}$$\pm$7$^{\circ}$ and the rotation measure RM=($-$5.6$\pm$0.7)$\times $10$^{5}$ radm$^{-2}$, which inplies a 155$^{+9}_{-8}$$^{\circ}$ polarization position angle at 492 GHz. This is consistent with our new measurement of polarization position angle within 1$\sigma$. Least square fitting of our measured polarization position angle at 492 GHz, together with the records provided by Bower et al. (2003, 2005), Macquart et al. (2006), and Marrone (2006a, 2007), yield $\chi_{0}$=167$^{\circ}$$\pm$7$^{\circ}$, and RM of ($-$4.9$\pm$1.2)$\times $10$^{5}$ radm$^{-2}$, which essentially cannot be distinguished from the aforementioned fitting results of Marrone et al. (2007), and the results of $\chi_{0}$=168$^{\circ}$$\pm$8$^{\circ}$ and RM=($-$4.4$\pm$0.3)$\times $10$^{5}$ radm$^{-2}$ given by Macquart et al. (2006). We note that there is a discrepancy between the intrinsic polarization position angle determined with millimeter and submillimeter band observations, and that determined with near infrared observations (Eckart et al. 2006; Shahzamanian et al. 2015). Assuming a thin Keplerian rotating disk geometry of the accretion flow, and the toroidal magnetic field perpendicular to the rotating axis, this nearly 90$^{\circ}$ flip of polarization position angle may be interpreted by the spatially (projected) shifted dominant polarization emission area, when the observations move gradually from the optically thicker (low frequency) to the optically thinner (high frequency) regime (e.g. Bromley et al. 2001; Liu et al. 2007; Huang et al. 2009). Therefore, at which exact frequency the 90$^{\circ}$ polarization position angle flip occurs, will provide a particular important constraint on the property of the accretion flow model. By comparing the Stokes I flux we detected at 492 GHz with the previous observations at lower frequencies (Marrone et al. 2006b), we found that the 492 GHz emission is very likely to be in the transition from the optically thick to the optically thin regime of the spectrum. Our 492 GHz measurement does not yet present the suggested 90$^{\circ}$ flip of polarization position angle, which may suggest that the blueshifted side of the accretion flow does not yet fully dominate the polarized emission at this observing frequency (Huang et al. 2009). Nevertheless, our observing frequency may not be high enough to research the turning point of polarization position angle, which is expected to be $>$1 THz in some recent radiative transfer modelings (Liu et al. 2007; Huang et al. 2009). In addition, the comparison of the Stokes I fluxes is subject to the large time separations of those measurements. Therefore, whether our 492 GHz observations were indeed probing the optically thin regime is uncertain. Resolving the nature of this discrepancy will require future coordinated monitoring observations.
We point out that the polarization position angle observed in the 230 GHz band is reported to present a larger time variability than that observed in the 340 GHz band (see also Figure \[fig:chiplot\]). Bower et al. (2005) favored an interpretation in which the variation is attributed to variations in the medium through which the polarization propagates (i.e. the variation of rotation measure), and thereby proposed a scenario of a hot and turbulent accretion flow. On the contrary, Marrone et al. (2007) argued that the observed time variation of the polarization position angle is more likely due to the variation of the emission source. Since we only have a single epoch of observations at 492 GHz, it is probable that the consistency of our observed polarization position angle with the extrapolation of the previous observations is merely a coincidence. We cannot yet distinguish between these two proposed scenarios, which require future multi-epoch observations. We note, however, that these two scenarios are not mutually exclusive.
As indicated in recent studies, Sgr A\* is believed to be the source for the major events episodically along with large flares emitting luminosity up to 10$^{41-42}$ ergs$^{-1}$. The time-interval between these events is about 100 year suggested by the front of fluorescent X-ray propagating away from Sgr A\* (Ponti et al. 2010; Clavel et al. 2013). Such extraordinary X-ray flares are also expected from the statistical analysis of the flux-density fluctuations observed in the past decades in the near IR band (Witzel et al 2012). In comparison to the measurements made about 10 years ago, our new measurements of the rotation measure with the ALMA may imply that both the accretion rate to and the magnetic configuration around Sgr A\* have not been significantly changed in the past decade. No extraordinary flares have been found from the monitoring programs in multiple wavelengths from radio, submillimeter, IR and X-ray launched in the past decade. Our current results may be expected given this inactivity. We note that the total flux may have varied by $\sim$50% during our observations (see Figure \[fig:xxyytime\]). This might also result from pointing errors or other calibration problems in these observations. More frequent calibrations in future observation will be more robust for addressing this point.
Finally, our simple analysis technique may also be applied to ALMA observations of quasars used for gain calibrations, in order to generate a large database of rotation measures. A similar technique has been used to estimate rotation measure of PKS 1830-211. For details see Marti-Vidal et al. (2016).
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![[]{data-label="fig:chiplot"}](chiplot_random.eps "fig:"){width="10cm"}
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![[]{data-label="fig:fracplot"}](frac.eps "fig:"){width="10cm"}
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Conclusions {#chap_conclusion}
===========
We have performed Band 8 (479-482 GHz; 489-493 GHz) mosaic observations towards the Galactic center, using the ALMA 12m-Array and the ACA. The observed Stokes I flux of Sgr A\* at 492 GHz is 3.6$\pm$0.72 Jy. We [**hypothesize**]{} that the continuum emission of Sgr A\*, and the C<span style="font-variant:small-caps;">i</span> absorption line against Sgr A\*, exhibit substantial intensity differences between the XX and the YY correlations. However, the XX and YY intensities of the C<span style="font-variant:small-caps;">i</span> line emission are essentially identical, at all velocity channels for which there is significant emission and over the entire time period of the 12m-Array observations. The maximum value of the observed intensity differences from Sgr A\* implies a $\sim$14%$\pm$1.2% lower limit on the polarization percentage. A comparable or higher polarization percentage of the continuum emission of Sgr A\* is expected from prior observations at other frequencies (Bower et al. 2003, 2005). The intrinsic polarization position angle we derived from the observed XX to YY intensity ratios is $\sim$167$^{\circ}$, which is surprisingly, in good agreement with the polarization position angles reported by the SMA observations at 230-340 GHz about one decade ago (Marrone et al. 2006a, 2007). Therefore, we attribute the observed intensity differences to linearly polarized synchrotron emission from hot ionized gas immediately surrounding Sgr A\*. We found that the polarization percentage at our observing frequency may be varying over the time period of our 12m-Array observations. Improved constraints on polarization will require new measurements that include the XY and YX correlations. We also detected 7.9%$\pm$0.9% polarization in position angle P.A. = 14.1$^{\circ}$$\pm$4.2$^{\circ}$ from the gain calibration quasar J1744-3116, which was observed at the same night with Sgr A\*.
We thank our referee for the very precise and useful opinions. HBL thanks ASIAA for support. HBL thanks Yu-Nung Su for the help when organizing the observational proposal; and thanks Lei Huang for some basic discussion made in 2004-2006. This paper makes use of the following ALMA data: ADS/JAO.ALMA 2013.1.00071.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. We thank Aaron Evans and Todd Hunter for providing information about ALMA primary beams. We thank Shin’ichiro Asayama, Ted Huang, Hiroshi Nagai, Dirk Petry, George Moellenbrock and Charles Hull for providing clarification on the ALMA feed orientation.
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[^1]: https://almascience.eso.org/proposing/call-for-proposals/technical-handbook
[^2]: Here $I^{XX, YY}(\nu, t)$ refers to the intensity maps taken at a specific time $t$, rather than time variation of intensity at any specific position.
|
---
abstract: |
Free energy of a layered superconductor with $\xi_{\perp} < d$ is calculated in a parallel magnetic field by means of the Gor’kov equations, where $\xi_{\perp}$ is a coherence length perpendicular to the layers and $d$ is an inter-layer distance. The free energy is shown to differ from that in the textbook Lawrence-Doniach model at high fields, where the Meissner currents are found to create an unexpected positive magnetic moment due to shrinking of the Cooper pairs “sizes” by a magnetic field. This paramagnetic intrinsic Meissner effect in a bulk is suggested to detect by measuring in-plane torque, the upper critical field, and magnetization in layered organic and high-T$_c$ superconductors as well as in superconducting superlattices.\
\
PACS numbers: 74.70.Kn, 74.25.Op, 74.20.Rp
author:
- 'A.G. Lebed$^*$'
title: Paramagnetic Intrinsic Meissner Effect in Layered Superconductors
---
The Meissner diamagnetic effect is known to be the most important property of a superconducting phase and is responsible for destruction of superconductivity both in type-I and type-II superconductors \[1\]. Meanwhile, as shown by us in Refs. \[2-4\] and independently by Tesanovic, Rasolt, and Xing in Ref. \[5\], quantum effects of an electron motion in a magnetic field result in the appearance of a qualitatively different phenomenon - superconductivity surviving in high magnetic fields in layered \[2-4\] and isotropic three-dimensional \[5\] type-II superconductors. In particular, it was shown \[2-4,6\] that in a layered conductor in a parallel magnetic field, where the Landau levels quantization is impossible, some other quantum effects - the Bragg reflections - play an important role. These quantum effects result in a “two-dimensionalization” (i.e., $3D \rightarrow 2D$ dimensional crossover) of an open electron spectrum in an arbitrary weak parallel magnetic field. This is known \[6,7\] to cause the field-induced spin-density-wave (FISDW) and field-induced charge-density-wave (FICDW) instabilities in layered quasi-one-dimensional (Q1D) conductors. More complicated $3D \rightarrow 1D \rightarrow 2D$ dimensional crossovers are shown \[8,9\] to be responsible for the experimentally observed non-trivial angular magnetic oscillations in a metallic phase of different layered organic conductors, including Interference Commensurate \[8\] and Lebed Magic Angle \[9\] oscillations.
As shown in Refs.\[2-4,7\], the similar $3D \rightarrow 2D$ dimensional crossovers have to be responsible for a stabilization of a superconducting phase in layered Q1D \[2,4\] and quasi-two-dimensional (Q2D) \[3\] conductors since 2D superconductivity is not destroyed in a parallel magnetic field. More precisely, it is shown \[2-4\] that: (i) the quantum effects make the upper critical field to be divergent, $H^{\parallel}_{c2}(T) \rightarrow \infty$ as $T \rightarrow 0$, and (ii) there is some critical filed, $H^*$, above which superconducting temperature grows in an increasing magnetic field. Such superconducting phase with $d T_c / dH >0$ is called the Reentrant Superconductivity (RS) \[2-5\]. The original predictions \[2-4\] have been theoretically confirmed by a number of studies \[10-14\], including a study \[14\], which takes into account a possibility of a pure 2D phase transition. Despite of a great success of $3D \rightarrow 1D \rightarrow 2D$ and $3D \rightarrow 2D$ dimensional crossovers concepts in the explanations of magnetic properties in a metallic \[7-9\], the FISDW \[6,7\], and the FICDW \[7\] phases of organic conductors, so far there has been no evidence that superconducting temperature can grow in high magnetic fields due to the the quantum $3D \rightarrow 2D$ dimensional crossovers \[2-5,10-14\]. A possibility of the RS phase to exist was experimentally studied in Q1D layered organic superconductors (TMTSF)$_2$X (X = PF$_6$ and X = ClO$_4$) by Naughton’s and Chakin’s groups \[15-17\]. Their experiments gave hints on a possibility for superconductivity to exceed significantly the quasi-classical upper critical field $H^{\parallel}_{c2}(0)$ - the effect predicted in Refs.\[2-5,10-14\] - but they were not able to confirm the appearance of the RS phase with $d T_c / d H > 0$. Analogous experiments, performed on a Q2D superconductor Sr$_2$RuO$_4$ \[18\], did not detect any stabilization of superconductivity at $H > H^{\parallel}_{c2}(0)$.
The main obvious difficulty in the above mentioned efforts to discover the RS phase is the Pauli spin-splitting destructive mechanism against superconductivity and the related Clogston paramagnetic limiting field, $H_p$ \[1\]. It is absent only for some triplet superconducting phases which are believed to exist in (TMTSF)$_2$X \[2,15,4\] and Sr$_2$RuO$_4$ \[19\] superconductors. On the other hand, recently there have appeared the NMR measurements \[20\] in favor of a singlet nature of superconductivity in (TMTSF)$_2$ClO$_4$ material as well as some doubts \[21\] in a triplet nature of superconductivity in Sr$_2$RuO$_4$ one.
The goal of our Letter is a three-fold one. Firstly, we show that, although in layered paramagnetically limited (singlet) superconductors the RS phase may not be characterized by $d T_c / d H > 0$ feature \[2-4, 10-14\], nevertheless the RS phase reveals itself as another unique phenomenon - paramagnetic intrinsic Meissner effect (PIME). Secondly, we extend microscopical theory \[3\] to describe the most important from experimental point of view $d$- and $s$-wave Q2D superconductors with $\xi_{\perp} < d$, where $\xi_{\perp}$ is a coherence length perpendicular to the conducting layers and $d$ is an inter-layer distance. And thirdly, we suggest simple experimental methods to detect the PIME phenomenon in Q2D organic and high-T$_c$ superconductors by using in-plane torque, the upper critical field, and magnetization measurements. In particular, we demonstrate that in-plane anisotropy due to anisotropic Ginzburg-Landau coherence lengths, which disappears in an intermediate region of magnetic fields (where the Lawrence-Doniach model is applicable), appears again in high magnetic fields as a consequence of the PIME phenomenon (see Figs. 1,2). We suggest to measure in-plane anisotropy of the upper critical field and magnetization as well as in-plane torque in high magnetic fields to discover the PIME and RS phenomena. For these purposes, we derive a free energy of a Q2D superconductor with $\xi_{\perp} < d$ in a parallel magnetic field from the Gor’kov formulation \[22-24\] of the microscopic superconductivity theory. Our results coincide with that of the Lawrence-Doniach model \[25,26\] only at low enough magnetic fields, $H \ll H^*$, where the Meissner effect is diamagnetic. We show that, at high magnetic fields, $H \sim H^* \leq H_p$, the field starts to shrink the Cooper pair “sizes” in perpendicular to conducting layers direction due to $3D \rightarrow 2D$ crossovers in a parallel magnetic field. The above mentioned $3D \rightarrow 2D$ crossovers of the Cooper pairs are not taken into account in the Lawrence-Doniach model and, as shown below, are responsible for the unique PIME phenomenon.
Let us consider a layered superconductor with a Q2D electron spectrum, $$\epsilon({\bf p})= \epsilon_{\parallel} (p_x, p_y) + 2 t_{\perp} \cos(p_z d) \ ,
\ \ \ t_{\perp} \ll \epsilon_F \ ,$$ in a parallel magnetic field, $${\bf H} = (0,H,0) \ , \ \ \ \ {\bf A} = (0,0,-Hx) \ ,$$ where $\epsilon_{\parallel} (p_x,p_y) \sim \epsilon_F$ is in-plane electron energy, $t_{\perp}$ is an overlapping integral of electron wave functions in a perpendicular to the conducting planes direction, $\epsilon_F$ is the Fermi energy. Electron spectrum (1) can be linearized near 2D Fermi surface (FS), $\epsilon_{\parallel} (p_x,p_y)=\epsilon_F$, in the following way, $$\epsilon({\bf p}) - \epsilon_F = v_x(p_y) [p_x - p_x(p_y)] + 2 t_{\perp} \cos(p_z d) \ ,$$ where $v_x(p_y) = \partial \epsilon_{\parallel} (p_x,p_y) / \partial p_y$ is a velocity component and $p_x(p_y)$ is the Fermi momentum along $x$-axis.
In the gauge (2), electron Hamiltonian in a magnetic field can be obtained from Eq.(3) by means of the Peierls substitution method, $p_x \rightarrow - i (d/dx)$, $p_z \rightarrow p_z + (e/c) H x$ \[6\]. Therefore, electron Green functions in a magnetic field satisfy the following differential equation, $$\begin{aligned}
\biggl\{ i \omega_n - v_x (p_y) \biggl[ - i \frac{d}{dx} - p_x(p_y) \biggl]
+2 t_{\perp} \cos \biggl( p_z d + \frac{eHdx}{c} \biggl) + 2 \mu_B H s \biggl\}
\nonumber\\
\times G_{i \omega_n} (x, x_1;p_y,p_z;s) = \delta (x-x_1) \ ,\end{aligned}$$ where $ \omega_n$ is the Matsubara frequency \[22\], $\mu_B$ is the Bohr magneton, $s = \pm \frac{1}{2}$ is an electron spin projection along quantization $y$-axis; $\hbar \equiv 1$. It is important that Eq.(4) can be solved analytically. As a result, we obtain, $$\begin{aligned}
G_{i \omega_n} (x, x_1;p_y,p_z;s) = - i \frac{ sgn \ \omega_n}{v_x(p_y)}
\exp \biggl[ - \frac{\omega_n (x-x_1)}{v_x(p_y)} \biggl] \exp[i p_x(p_y)(x-x_1)]
\nonumber\\
\times \exp \biggl[ \frac{2 i \mu_B s H (x-x_1)}{v_x(p_y)} \biggl]
\exp \biggl\{ \frac{ i \lambda(p_y)}{2} \biggl[ \sin \biggl( p_zd + \frac{eHdx}{c} \biggl)
- \sin \biggl( p_zd + \frac{eHdx_1}{c} \biggl)
\biggl] \biggl\} \ ,\end{aligned}$$ where $\lambda (p_y) = 4 t_{\perp} c / e v_x (p_y) H d$.
Linearized gap equation determining superconducting transition temperature, $T_c(H),$ can be derived using Gor’kov equations for non-uniform superconductivity \[3,23,24\]. As a result, we obtain, $$\begin{aligned}
\Delta(x) = V \oint \frac{d l}{v_{\perp}(l)} \
&&\int^{\infty}_{ |x-x_1| > |v_x(l)|/ \Omega} dx_1 \ \frac{2 \pi T }{v_x(l)
\sinh \biggl[ \frac{ 2 \pi T |x-x_1|}{ v_x (l)} \biggl] }
\cos \biggl[ \frac{2 \mu_B H (x-x_1)}{v_x (l)} \biggl]
\nonumber \end{aligned}$$ $$\begin{aligned}
&&\times J_0 \biggl\{ 2 \lambda (l) \sin \biggl[ \frac{ e H d (x-x_1)}{2 c} \biggl]
\sin \biggl[ \frac{ e H d (x+x_1)}{2 c} \biggl] \biggl\} \ \ \Delta(x_1) \ ,\end{aligned}$$ where integration in Eq.(6) is made along 2D contour, $\epsilon_{\parallel} (p_x,p_y)=\epsilon_F$, $v_{\perp}(l)$ is a velocity component perpendicular to the 2D FS, $V$ is an effective electron-electron interactions constant, $\Omega$ is a cut-off energy. \[Note that, although Eq.(6) is derived for singlet $s$-wave superconductors, it is also valid for $d$-wave superconductors \[27\] if we redefine properly anisotropic coherence lengths and the effective interactions constant $V$.\]
We point out that Eq.(6) is the most general one among the existing equations to determine the parallel upper critical field in a layered superconductor. In particular, it takes into account the Bragg reflections and related $3D \rightarrow 2D$ dimensional crossovers of electrons, which move in the extended Brillouin zone in a parallel magnetic field. As shown in Ref. \[7\], the above mentioned quantum effects result in a momentum quantization law for an electron momentum component along $x$-axis. This a reason why the kernel of the integral Eq.(6) is periodic \[2,4\] with respect to variables $x$ and $x_1$. In the case, where the destructive Pauli spin-splitting effects against superconductivity are absent \[i.e., at $\mu_B=0$ in Eq.(6)\], Eq.(6) possesses a periodic solution for $\Delta(x)$ at any magnetic field. In this case, superconductivity is stable in an arbitrary strong magnetic field and exists at high fields in a form of the RS phase with $d T_c / dH >0$. In the case of a singlet superconductivity, which is considered in the Letter, the Pauli spin-splitting effects may eliminate the superconductivity with $d T_c / dH >0$. Nevertheles, in the latter case, the RS phase reveals itself as unusual anisotropy of the upper critical field and magnetization in high magnetic fields, $H > H^* \sim (t_{\perp}/T_c)^{1/2} \phi_0 / \xi_x d
\ll H_p$ (see Figs. 1,2).
As the most general equation, Eq.(6) contains Ginzburg-Landau and Lawrence-Doniach descriptions as its limiting cases at low enough magnetic fields, $H \ll H^*$. For the so-called Josephson coupled layered superconductors with $\xi_{\perp} < d$ \[25,26\], Eq.(6) may be simplified and rewritten in the following differential form,
$$\biggl[ \frac{T_c-T}{T_c} - 2.1 \biggl( \frac{\mu_B H}{\pi T_c} \biggl)^2
+ \xi^2_x \frac{d^2}{dx^2} - A(H)
+ B(H) \cos \biggl( \frac{2 x \omega_c}{v_F} \biggl) \biggl]
\Delta(x) = 0 \$$
with $$A(H) = \frac{8 t^2_{\perp}}{\omega^2_c} \biggl < \biggl[ \frac{v_F}{v_x(l)} \biggl]^2
\int^{\infty}_0 \frac{d z}{\sinh(z)} \sin^2 \biggl[ \frac{\omega_c}{4 \pi T_c}
\frac{v_x(l)}{v_F} z \biggl]
\biggl>
\$$ and $$B(H) = \frac{8 t^2_{\perp}}{\omega^2_c} \biggl < \biggl[ \frac{v_F}{v_x(l)} \biggl]^2
\int^{\infty}_0 \frac{d z}{\sinh(z)} \sin^2 \biggl[ \frac{\omega_c}{4 \pi T_c}
\frac{v_x(l)}{v_F} z \biggl]
\cos \biggl[ \frac{\omega_c}{4 \pi T_c} \frac{v_x(l)}{v_F} z \biggl]
\biggl> \ ,$$ where $$\biggl < ... \biggl> = \oint \frac{dl}{v_{\perp} (l)} \biggl(...\biggl) \biggl/ \oint \frac{dl}{v_{\perp}(l)} \ .$$ \[Here, $\omega_c = eHv_Fd/c$ is a characteristic frequency of an electron motion along open FS (1) \[3\], $\xi_x = \sqrt{7 \zeta (3)} \bigl< v^2_x(l) \bigl>^{1/2} / 4 \pi T_c$ is in-plane Ginzburg-Landau coherence length, $\mu_BH \simeq \omega_c(H) \ll \pi T_c$.\]
Note that Eqs.(7)-(10) extend the Lawrence-Doniach model \[25,26\] to the case of strong magnetic fields and can be called extended Lawrence-Doniach equations. In contrast to the traditional Lawrence-Doniach equations, the coefficients $A(H)$ and $B(H)$ in Eqs.(7)-(10) depend on a magnetic field, which means that a probability for the Cooper pair to jump from one conducting layer to another depends on the field. This important feature of Eqs.(7)-(10) is a consequence of shrinking of the Cooper pairs “sizes” due to $3D \rightarrow 2D$ dimensional crossover in a parallel magnetic field \[2,3,7\].
Below, we are interested in descriptions of the RS and PIME phenomena, therefore, we consider Eqs.(7)-(10) at high magnetic fields. It is possible to show that at $H \geq H^*$ the solution of Eq.(7) can be represented as $\Delta(x) = \Delta = const $, which corresponds to the RS phase \[2,3\]. In this case, the corresponding second order term of a free energy with respect to the order parameter $\Delta$ can be written in the following simple form, $$F ^2(T,H) = - N(\epsilon_F) \biggl[ \frac{T_c(H)-T}{T_c} \biggl] \Delta^2 \ ,$$ where $$T_c(H) = T_c - 2.1 \frac{(\mu_B H)^2}{\pi^2 T_c} - 2.1 \frac{t^2_{\perp}}{\pi^2T_c}
+ 0.95 \frac{t^2_{\perp}}{\pi^2 T_c} \biggl( \frac{eHd}{c} \biggl)^2 \xi^2_x \ ,$$ where $N(\epsilon_F)$ is a density of states per one electron spin projection at $\epsilon = \epsilon_F$.
Note that the the first term in Eq.(12) describes destruction of a singlet superconductivity by the Pauli spin-splitting effects, whereas the last term in Eq.(12) is responsible for the restoration of superconductivity at high magnetic fields and for the appearance of the RS phase and PIME phenomenon. If we take into account that the fourth order term of a free energy with respect to the order parameter $\Delta$ can be calculated at $H \geq H^*$ in a standard manner \[23\], $F^4 = 7 \zeta (3) N(\epsilon_F) \Delta^4 / 16 \pi^2 T^2_c$, then we can minimize the total free energy and find that $$F (T,H) = - \frac{4 \pi^2}{7 \zeta(3)} N(\epsilon_F) [ T_c(H)-T]^2 \ .$$
Magnetization can be found by a differentiation of the free energy (13) with respect to a magnetic field, $$M(T, H) = - \frac{\partial F(T,H)}{\partial H} =
\frac{8}{7 \zeta (3)} N(\epsilon_F) \biggl( \frac{T_c-T}{T_c} \biggl)
\biggl[ -4.2 \ \mu^2_B + 1.9 \biggl( \frac{e t_{\perp} d \ \xi_x}{c} \biggl)^2
\biggl] H\ ,$$ Eqs.(12)-(14) are the main results of the Letter. Note that, in Eq.(14), the first term corresponds to a destruction of superconductivity due to the Pauli spin-splitting effects, whereas the second term represents unusual paramagnetic orbital contribution to a magnetic moment (i.e., the PIME phenomenon). It is important that $\xi_x$ in Eqs.(12)-(14) is anisotropic and depends on a direction of a magnetic field, since it is in-plane component of a coherence length perpendicular to the field. Therefore, the RS and PIME effects in Eqs.(12)-(14) can be detected by measuring a torque provided that spin-splitting effects are isotropic (see Eqs.1,2).
In conclusion, we discuss possible experiments to discover the PIME and RS phenomena. The most direct method is to create such layered superconducting super-lattice, where $\omega_c(H) \gg \mu_B H$ \[28\]. The latter condition means that the orbital effects are more important than the Pauli spin-splitting ones. Therefore, in this case, the increase of transition temperature (12) and the paramagnetic Meissner effect (14) can be directly observed. Nevertheless, in most real physical compounds with $\xi_{\perp} < d$, $\omega_c(H) \simeq \mu_B H$ and, thus, the PIME (14) and RS (12) phenomena can be observed only indirectly - by measurements of anisotropies of the in-plane upper critical field (12) and magnetization (14) as well as by measurements of in-plane torque. In our opinion, the most perspective superconductors for indirect observations of the PIME phenomenon in steady magnetic fields are organic compounds $\alpha$-(ET)$_2$NH$_4$Hg(SCN)$_4$, $\kappa$-(ET)$_2$Cu(NCS)$_2$, $\kappa$-(ET)$_2$Cu\[N(CN)$_2$\]X, $\alpha$-(ET)$_2$KHg(SCN)$_4$, and $\lambda$-(BETS)$_2$FeCl$_4$ \[29\]. The above mentioned studies of the in-plane anisotropies can be performed also in high-temperature superconductor Y$_1$Ba$_2$C$_3$O$_7$ but it will require ultra-high pulsed magnetic fields.
The author is thankful to N.N. Bagmet, P.M. Chaikin, and M.V. Kartsovnik for the useful discussions.
$^*$Also, Landau Institute for Theoretical Physics, 2 Kosygina Street, Moscow, Russia.
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In this Letter, we calculate the paramagnetic Meissner moment and its two-fold anisotropy. We disregard a weak four-fold anisotropy due to a $d$-wave gap since it is $(t_{\perp} / T_c)^2 \ll 1$ times smaller than the above calculated two-fold one.
This idea belongs to P.M. Chaikin (private communication).
The author is thankful to M.V. Kartsovnik for his help to select the above mentioned layered superconductors, where $\xi_{\perp} \leq d$.
![Superconducting transition temperature in a parallel magnetic field for a paramagnetically limited Q2D superconductor is sketched. GL - area of applicability of the Ginzburg-Landau theory \[1\], LD - area of applicability of the Lawrence-Doniach model \[25,26\], PIME - area, where both the GL and LD descriptions are broken. In the latter case, which corresponds to shrinking of the Cooper pairs “sizes” by a magnetic field, our Eqs. (6)-(14) are still valid and the Reentrant Superconducting (RS) phase appears. The RS phase may reveal itself as an increase of the transition temperature in a magnetic field, if the orbital effects of an electron motion are stronger than the Pauli spin-splitting effects (dashed line). The RS phase always reveals itself as a paramagnetic intrinsic Meissner effect (PIME), which results in unexpected in-plane anisotropy of the upper critical field and magnetization even in the case, where the Pauli spin-splitting effects are strong and, thus, the area with $d T_c/dH > 0$ is absent (solid line). We suggest to measure in-plane torque, the upper critical field, and magnetization to discover the RS phase.[]{data-label="fig1"}](PIME-1.eps){width="6.6in"}
![Solid line: in-plane magnetization, $4 \pi {\bf M}$, is sketched as a function of a magnetic field in the absence of the Pauli spin-splitting effects. At high magnetic fields, $H \geq H^*$, the Reentrant Superconducting (RS) phase reveals itself as a paramagnetic intrinsic Meissner effect (PIME). Dashed line: an absolute value of in-plane torque, $|{\bf \tau}|$, is sketched. It is important that the torque is independent on the Pauli spin-splitting effects since they are isotropic. Therefore, even in the case, where the destructive Pauli spin-splitting effects eliminate a positive sign of the Meissner effect in high magnetic field, the PIME phenomenon and the RS phase can still be detected by the in-plane torque measurements.[]{data-label="fig2"}](PIME-2.eps){width="6in"}
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abstract: 'We propose a new internal linear combination (ILC) method in the pixel space, applicable on large angular scales of the sky, to estimate a foreground minimized Cosmic Microwave Background (CMB) temperature anisotropy map by incorporating prior knowledge about the theoretical CMB covariance matrix. Usual ILC method in pixel space, on the contrary, does not use any information about the underlying CMB covariance matrix. The new approach complements the usual pixel space ILC technique specifically at low multipole region, using global information available from theoretical CMB covariance matrix as well as from the data. Since we apply our method over the large scale on the sky containing low multipoles we perform foreground minimization globally. We apply our methods on low resolution Planck and WMAP foreground contaminated CMB maps and validate the methodology by performing detailed Monte-Carlo simulations. Our cleaned CMB map and its power spectrum have significantly less error than those obtained following usual ILC technique at low resolution that does not use CMB covariance information. Another very important advantage of our method is that the cleaned power spectrum does not have any negative bias at the low multipoles because of effective suppression of CMB-foreground chance correlations on large angular scales of the sky. Our cleaned CMB map and its power spectrum match well with those estimated by other research groups.'
author:
- 'Vipin Sudevan, Rajib Saha'
title: A Global ILC Approach in Pixel Space over Large Angular Scales of the Sky using CMB Covariance Matrix
---
Introduction
============
For reconstruction of Cosmic Microwave Background (CMB) signal from multi-frequency observations an important method is Internal-Linear-Combination (ILC) [@Tegmark96; @Tegmark2003; @Bennett2003; @Eriksen2004; @Saha2006; @Hinshaw_07]. To obtain a foreground minimized CMB map the ILC method requires neither to explicitly model the frequency spectra of individual foreground components, nor does it require to model the foreground amplitudes (at some reference frequency) in terms of so called foreground templates. The only assumption one makes related to foregrounds is that each of them has a frequency spectrum that is different from the frequency spectrum of CMB component, which is assumed to be that of black-body in nature [@Mather1994; @Fixen1996]. The basic idea behind the ILC method is to linearly superpose the available foreground contaminated CMB maps using certain amplitude terms, a set of weights, to estimate a foreground minimized CMB map. The weights are obtained by minimizing the variance of the cleaned map and can be computed analytically by using a simple formula. In spite of being simple to design and yet a powerful technique to reconstruct a cleaned CMB map we see that it is necessary – for a few important reasons – to investigate performance of the usual ILC method in some hitherto unexplored cases. First, while estimating the weights the usual ILC method in pixel space does not take into account the covariance structure of the CMB maps. In other words, it does not use the fact that the final cleaned map, if perfectly cleaned of all foregrounds and detector noise is negligible, should have a covariance structure consistent with the underlying theoretical model. Secondly, some of the maximum likelihood methods [@Eriksen2007; @Eriksen2008; @Eriksen2008a; @Gold2011; @PlanckFg2016; @PlanckCMB2016] for component separation however use CMB and detector noise covariance matrices to reconstruct CMB and all foreground components. It is therefore natural to ask a question [*can we generalize usual ILC method in pixel space to incorporate CMB covariance information also?*]{}
In the present work we seek to find a solution to the above problem and generalize the pixel space ILC method taking into account prior information of the theoretical covariance matrix of the CMB maps. Therefore, instead of minimizing simple variance of the cleaned map we propose to estimate the weights by minimizing the reduced variance of the cleaned map, the reduced variance being defined by the CMB covariance weighted variance of the cleaned map, which is explained in Section \[formalism\]. Since storage space into the computer disks of such full pixel-space covariance matrix increases rapidly with the HEALPix pixel resolution parameter $N_{side}$ ($\sim N^4_{side})$, in the current work we use low pixel resolution maps. Further to focus largely on the low multipoles we smooth the input $N_{side} =16$ maps by a Gaussian window function of FWHM $9^\circ$. At this smoothing the input maps contain approximately $2.5$ pixels per beam width, which implies these maps are properly band-width limited. The larger beam smoothing also reduces detector noise contributions at different pixels.
Our method at low resolution bears an interesting complementarity in its approach when compared with the usual pixel space ILC method at high resolution, that do not use the CMB covariance matrix. Since the level of foreground contamination, and their spectral properties vary with the sky positions, in a high resolution analysis of usual ILC method one performs foreground cleaning individually over several smaller regions of the sky, in such a way that the foreground spectral properties and level of foreground contaminations in each region remains approximately constant. Because of low pixel resolution (and large smoothing on the low pixel resolution maps) of this work we chose either to perform foreground removal over the entire sky or by dividing the sky into small number of regions. In the second approach we divide the sky into two regions and clean them individually in a total of two iterations. Our aim is to use as much large sky fraction as possible during foreground removal and information about CMB theoretical covariance matrix from the corresponding large fraction, so that our method becomes a global method of foreground minimization. Thus our method may be seen as dual to usual high resolution ILC method, wherein the former uses global information from the covariance matrix and the data to estimate the foreground minimized CMB map and the later relies upon the local information of foregrounds properties.
By performing detailed Monte-Carlo simulations we find that the new ILC method of this work has significantly less reconstruction errors in cleaned maps and power spectrum than the usual ILC method in pixel space over large angular scales of the sky. The cleaned power spectrum of our method does not have a negative bias at the low multipole region that is present in usual ILC method and is caused by a chance correlations between CMB and foreground components on a particular realization of CMB sky.
The subject of component separation in the context of CMB is very rich. [@Bunn1994; @Bouchet1999] propose a Wiener filter approach. [@Saha2008] discuss in detail bias issues in CMB angular power spectrum for harmonic space ILC approach. [@Saha2016] apply an ILC technique to jointly estimate CMB and foreground components for Stokes Q polarization in presence of varying spectral index of synchrotron component. Iterative harmonic space ILC algorithm was applied on high resolution Planck and WMAP data, and one of its limitations arising due to foreground leakage was first discovered and remedied by [@Sudevan2017]. [@Delabrouille2012] and [@Delabrouille2013] implement a needlet space ILC algorithm to incorporate localization of foreground emissions both in pixel space and its ‘Fourier’ space. A variant of ILC technique by minimizing a measure of non-Gaussianity was implemented on WMAP temperature and Polarization data by [@Saha2011] and [@Purkayastha2017] respectively. [@Eriksen2007; @Eriksen2008; @Eriksen2008a] propose Gibbs sampling for component separation. [@Gold2011] use Markov Chain Monte Carlo method to jointly estimate CMB and foregrounds from WMAP data.
We organize our paper as follows. In Section \[formalism\] we discuss the formalism of the new method. We describe how to compute the theoretical CMB covariance matrix in Section \[comp\_C\] and comment on its singular nature in Section \[singC\]. In Section \[method\] we describe in detail our foreground minimization approaches on Planck and WMAP low resolution maps. We discuss the cleaned maps and CMB angular power spectra obtained from data on Section \[Result\]. We validate our foreground minimization methods by performing Monte Carlo simulations in Section \[validity\]. In Section \[advantage\] we show the advantage of the new ILC approach in pixel space over the usual ILC approach for analysis over large angular scales on the sky. We investigate the role of CMB-foreground chance correlation in not-so-efficient foreground removal by the usual ILC methods at low resolution in Section \[chancecorr\] and comment that using the CMB covariance matrix in our new method, we effectively suppress such chance correlations which leads to improved foreground minimization. Finally we conclude in Section \[Conclusion\].
Formalism
=========
Let we have $n$ full sky foreground contaminated CMB maps, ${\bf X}_{i}$ at a frequency $\nu_i$, with $i = 1, 2, ...., n$ at some beam and pixel resolution in thermodynamic temperature unit. We assume mean temperature corresponding to each frequency $\nu_i$ has already been subtracted from each ${\bf X}_{i}$. ${\bf y}$ represents the cleaned CMB map obtained by linear combination of $n$ input maps ${\bf X}_{i}$, with weight factor $w_{i}$, i.e., $$\begin{aligned}
{\bf y} = \sum_{i=1}^nw_{i} {\bf X}_{i}\, .
\label{cmap}\end{aligned}$$ Here each ${\bf X}_i$ and ${\bf y}$ are $N\times 1$ vectors describing full sky HEALPix[^1] map with $N$ pixels for a pixel resolution parameter $N_{side}$ ($N = 12N^2_{side}$), smoothed by Gaussian beam of certain FWHM. Instead of minimizing cleaned map variance ${\bf y}^T{\bf y}$ like the usual pixel space ILC method we propose a more general approach by incorporating the prior information about the theoretical CMB covariance matrix. We minimize, $$\begin{aligned}
\sigma^2 = {\bf y}^T{\bf C}^{\dagger}{\bf y}\, ,
\label{dispersion0}\end{aligned}$$ where ${\bf C}$ represents the CMB theoretical covariance matrix which as discussed in Section \[singC\] may not be always invertible. ${\bf C}^{\dagger}$ represents Moore-Penrose generalized inverse [@Moore1920; @Penrose1955] of matrix ${\bf C}$. Using Eqn. \[cmap\] we can write Eqn. \[dispersion0\] as $$\begin{aligned}
\sigma^2 = {\bf W} {\bf A W}^T \, , %\sum_{i',j'=1}^n w_{i'} A_{i'j'}w_{j'}\, ,
\label{dispersion1} \end{aligned}$$ where ${\bf W} = \left(w_1, w_2, w_3, ...., w_n\right)$ is a $1\times n$ row vector of weight factors of different frequency maps and ${\bf A}$ is an $n\times n$ matrix with its elements $A_{ij}$ satisfying $$\begin{aligned}
A_{ij} = {\bf X}^T_{i}{\bf C}^{\dagger} {\bf X}_{j}\, .
\label{AMatrix} \end{aligned}$$ Since spectral distribution of CMB photons is that of a blackbody to a very good approximation, CMB anisotropy in thermodynamic temperature unit is independent on frequency bands. To reconstruct CMB anisotropies without introducing any multiplicative bias in its amplitude we constrain the weights for all frequency bands to sum to unity, i.e., $\sum_{i=1}^n w_{i} = 1$. The choice of weights that minimize the variance given by Eqn. \[dispersion0\] is obtained following a Lagrange’s multiplier approach (e.g., see [@Saha2008] and also [@Tegmark96; @Tegmark2003; @Saha2006]) $$\begin{aligned}
{\bf W} = \frac{ {\bf e} {\bf A}^{\dagger}}{ {\bf e} {\bf A}^{\dagger} {\bf e}^T}\, ,
\label{weights}\end{aligned}$$ where ${\bf A}^{\dagger}$ represents Moore-Penrose generalized inverse of matrix ${\bf A}$ and ${\bf e} = \left(1, 1, ..., 1\right)$ is a $1\times n$ row-vector representing shape vector of CMB in thermodynamic temperature unit.
Computing CMB Covariance Matrix {#comp_C}
===============================
To compute elements of the CMB covariance matrix, ${\bf C}$ we assume principle of statistical isotropy of CMB anisotropy. Under this assumption the elements $C_{ij}$ of matrix ${\bf C}$, at the chosen beam and pixel resolution are given by $$\begin{aligned}
C_{ij} = \sum_{\ell =2}^{\ell=\ell_{max}}\frac{2\ell+1}{4\pi} C_{\ell}B^2_{\ell}
\mathcal P_{\ell}(\cos(\gamma_{ij}))P^2_{\ell} \, ,
\label{theory_cov}\end{aligned}$$ where $C_{\ell}$ is the fiducial CMB angular power spectrum [@PlanckCosmoParam2016], $B_{\ell}$ represents the beam transfer function, $\mathcal P_{\ell}$ denote Legendre polynomials and $P_{\ell}$ is pixel window function for the given $N_{side}$ parameter. The cosine of the angle $\gamma_{ij}$ is obtained following $$\begin{aligned}
\cos(\gamma_{ij}) = \cos(\theta_i)\cos(\theta_j) +
\sin(\theta_i)\sin(\theta_j)\cos(\phi_i - \phi_j) \, ,
\label{gammaij} \end{aligned}$$ where $(\theta_i, \phi_i)$ and $(\theta_j, \phi_j)$ are spherical polar angles respectively of $i$ and $j$th pixels of the map. Under the assumption of statistical isotropy ${\bf C}$ is independent on any particular choice of coordinate system (e.g., Galactic, Ecliptic, or any Euler rotated version of these coordinate systems) in which the input maps are provided. We note, however, the assumption of statistical isotropy is not a necessity in our method. If needed, we can also use a covariance matrix compatible to statistically anisotropic model which may be caused due to non-trivial primordial power spectrum [@Ghosh2016; @Contreras2017].
Is [**C**]{} singular? {#singC}
=======================
As is the case for this work, rank, $r$, of ${\bf C}$ is less than its dimension $N$. The rank of ${\bf C}$ is simply equal to effective number of independent $a_{\ell m}$ modes (real and imaginary) that are used in Eqn. \[theory\_cov\] to generate each element of the theoretical covariance matrix. A quick calculations shows that, $r = (\ell_{max}+1)(\ell_{max}+2)
- (\ell_{max}+1) - 4$, when the summation over multipoles in Eqn \[theory\_cov\] extends upto $\ell = \ell_{max}$. Since we use, $N_{side} = 16$ HEALPix maps in our analysis, $\ell_{max} = 2\times N_{side} = 32$ for us, implying $r = 1085$ which is less than dimension of ${\bf C}$, which is $N = 3072$. Since ${\bf C}$ is singular we use its generalized inverse in Eqn. \[dispersion0\].
Methodology {#method}
===========
Input maps and Data Processing
------------------------------
We use Planck 2015 released LFI 30, 44 and 70 GHz, HFI 100, 143, 217 and 353 GHz frequency maps along with the WMAP 9 year difference assembly (DA) maps in our analysis. For each of these maps we convert them to spherical harmonic space upto $\ell_{max} = 32$ and smooth the resulting $a_{\ell m}$ coefficients by the ratio $B^0_{\ell} P^0_{\ell}/B^i_{\ell} P^i_{\ell}$ where $B^i_{\ell}$ and $ P^i_{\ell}$ represent the beam and pixel window functions of the original maps whereas $B^0_{\ell}$ and $ P^0_{\ell}$ represent the corresponding window functions for the $N_{side} = 16$ maps. We take $B^0_{\ell}$ corresponding to a Gaussian beam of [FWHM = $9^\circ$]{}. We convert the smoothed spherical harmonic coefficients to $N_{side}=16$ maps using HEALPix supplied facility [synfast]{}. For each of WMAP Q, V and W bands we average all the DA maps for any given frequency band. We convert all these maps in $\mu K$ (thermodynamic) temperature unit and subtract the corresponding mean temperature from each frequency map. This results in a total of $12$ input maps for foreground removal at $N_{side} = 16$.
![The blue region shows sky portions dominated by the strong thermal dust emission and is removed by the [ThDust5000]{} mask[]{data-label="mask"}](f2)
Method-1 {#method1}
--------
Since we are interested in a global method of foreground minimization our aim is to use as much sky region as possible to estimate the weights. In the first method we therefore estimate the weights using information obtained from the entire sky. We first estimate full sky CMB theoretical covariance matrix using Eqn. \[theory\_cov\]. We obtain ${\bf C}^{\dagger}$ using singular value decomposition of ${\bf C}^{\dagger}$ and applying a cutoff of $1.0 \times 10^{-7}$ on the singular values. We show different square blocks across the diagonal of ${\bf C}$ matrix estimated for the entire sky in Fig. \[covmat\]. Non-diagonal elements of this matrix show significant coupling between different pixel pairs for a pure CMB map and justifies using Eqn. \[dispersion0\] for minimization instead of ignoring such correlations as is done in usual pixel space ILC approach. Using ${\bf C}^{\dagger}$ we obtain weights for foreground removal using Eqns. \[AMatrix\] and \[weights\]. The cleaned map obtained using these weights is discussed in Section \[Result\].
Method-2 {#method2}
--------
Since when compared with the expected level of CMB temperature anisotropy, the region near galactic plane is strongly contaminated by the foregrounds than the outside region, it is desirable to perform foreground removal separately on the sky region away from the plane and inside the plane. Moreover the spectral properties of the foregrounds vary with sky positions, specifically near the galactic plane. WMAP science team produce the internal linear combination map at $N_{side} = 512$ by dividing the galactic plane into 12 different regions. The sky region outside the plane was cleaned in a single iteration. The work of this paper, however, intends to use global information from the theoretical CMB covariance matrix and data. Keeping in mind such dual requirements we divide the sky into two regions and clean each as described below. The reason why we divide the sky into smaller number of regions than an usual ILC approach in pixel space at high resolution, is that we are interested in low resolution maps focusing on the low multipoles. The lack of structures on small scales in the input maps ensures that the sky regions need not be too small.
### Sky Division
To identify the region near the galactic plane that contains strong foreground emissions we take Planck 353 GHz and 70 GHz frequency maps at $N_{side} = 2048$. We downgrade these maps to $N_{side} = 256$ and smooth them by the ratio of window functions of a Gaussian beam of FWHM = $6^{\circ}$ and the original beam functions of the $N_{side} =2048$ maps at the their native resolutions. We subtract resulting reduced resolution 70 GHz map from 353 GHz map at $N_{side} = 256$. The difference map contains strong emissions from thermal dust at $353$ GHz. We identify pixels of the difference map with values $\ge 5000 $ $\mu K$ and assign a value of unity to them and zero to rest. We downgrade this binary map at $N_{side} = 16$. Finally we reassign all non-zero pixels of the downgraded map a value of zero and the rest to a value of unity. This sky region defined by the zero pixel values contains strong thermal dust emissions. The region complementary to this strong thermal dust emission is survived after application of the [ThDust5000]{} mask. The sky region removed by this mask is shown in deep blue color in Fig. \[mask\].
### Foreground cleaning {#fg_cleaning}
Based upon the discussions of the previous sections we perform the foreground cleaning following the second method in following three steps.
1. [ We estimate the covariance matrix $\tilde {\bf C}$ applicable for the sky region defined by the [ThDust5000]{} mask. This is done by using Eqn. \[theory\_cov\] for all the pixel pairs $(i,j)$ that survive after application of the mask. We estimate $\tilde {\bf C}^{\dagger}$ following the same procedure as described in Section \[method1\].]{}
2. [We use this generalized inverse of the partial sky CMB covariance matrix in Eqn. \[AMatrix\] to obtain elements of the partial sky $\tilde{\bf A}$ matrix. Using this partial sky matrix in Eqn. \[weights\] we obtain the weights corresponding to the [ThDust5000]{} sky region. Using these weights we obtain the cleaned [ThDust5000]{} sky region. ]{}
3. [Now we replace the [ThDust5000]{} sky region of all foreground contaminated input maps by the cleaned region obtained above. The resulting 12 maps have their galactic regions yet to be cleaned and strongly contaminated by the foregrounds. To clean the galactic region we repeat steps 1 and 2 above over the full sky. The cleaned map obtained at this point is the full-sky cleaned map obtained by Method-2.]{}
![Difference of full sky cleaned CMB maps ([CMap2-CMap1]{}) obtained from Method 1 and Method 2. [CMap2]{} appears to have lesser foreground contamination along the both sides of the galactic plane. Temperature scale is in $\mu K$ thermodynamic unit.[]{data-label="diffcmap"}](f4)
Results {#Result}
=======
Cleaned Maps
------------
Using the first method the weights for different WMAP and Planck channels become $-0.093$, $0.226$, $0.424$, $-0.392$, $-0.859$, $-0.105$, $0.195$, $0.390$, $0.890$, $0.906$, $-0.607$, $0.0245$ in the increasing order of frequency of the 12 input maps from 23 to 353 GHz. We use these weights to linearly combine the 12 input maps to estimate the cleaned CMB map at $N_{side} =16$ and at Gaussian beam resolution of FWHM = $9^\circ$ (henceforth we call this cleaned map [CMap1]{}). We show the [CMap1]{} in the top panel of Fig. \[cmap\_fig\]. Visually the [CMap1]{} does not contain any foreground residuals. We compare this map with other foreground minimized CMB maps each of which is obtained by employing a different algorithm at higher beam and pixel resolutions, as reported in the literature. [COMMANDER]{} CMB map was obtained following joint estimation of CMB and all foreground components, [NILC]{} CMB maps was obtained by employing an internal linear combination algorithm in the needlet space and [SMICA]{} CMB map was obtained by using spectral matching technique (e.g., see [@Planck2016_CMB] for detailed discussion about these maps). WMAP science team produced a CMB map by using usual ILC approach in pixel space [@Hinshaw_07; @Gold2011]. We downgrade these high resolution maps at $N_{side} = 16$ and bring them to a common beam resolution of $9^{\circ}$. We show the difference of [CMap1]{} from resulting [COMMANDER]{} and [NILC]{} maps respectively in the middle left and right panel of Fig. \[cmap\_fig\]. The lower left and right panel show differences of [CMap1]{} from [SMICA]{} and WMAP [ILC]{} maps. Since monopole and dipoles are not of any cosmological interests we have removed any residual dipole and monopole from all the four difference maps shown in this figure. Clearly our cleaned CMB map matches well with these cleaned CMB maps in the higher galactic plane. Along the galactic plane we find some differences. However, as one can easily make out such difference along the galactic plane exists for any pair of all five low resolution CMB maps discussed in this section.
![Top panel shows estimates of CMB angular power spectrum obtained from full sky region of [CMap1]{} (Method-1), [CMap2]{} (Method-2) along with the Planck 2015 theoretical LCDM power spectrum. Both these observed spectra of Method-1 and Method-2 match well with each other. The error-bars are compatible to Method-2. The bottom panel shows difference of spectra obtained from these two methods. The dashed line shows the zero level of the power spectrum. []{data-label="pow_spec_fig"}](f5)
Following the second method we recover a cleaned map ([CMap2]{}) similar to [CMap1]{}. The weights for the sky region survived after application of [ThDust5000]{} mask are $-0.066$, $0.083$, $0.500$, $-0.306$, $-0.562$, $ -0.757$, $0.021$, $0.917$, $0.876$, $0.948$, $-0.684$ and $0.031$ respectively for different frequencies increasing from 23 to 353GHz (e.g., see step 2 of Section \[fg\_cleaning\]). The corresponding weights for the full sky (step 3 of Method-2) linear combination are $-0.084$, $0.240$, $0.414$, $-0.399$, $-0.994$, $0.100$, $0.217$, $0.277$, $0.913$, $0.896$, $-0.604$ and $0.024$ respectively. A common feature of the weights for both these regions is that strongly contaminated frequency maps (e.g., K1 band or 353 GHz) get low (negative or positive) weights to cancel out foregrounds from all frequencies. The [CMap2]{} matches closely with the [CMap1]{}. We show the difference [CMap2 - CMap1]{} in Fig. \[diffcmap\]. Clearly the Method-2 has slightly less foreground residuals along the both sides of the galactic plane at the expense of some additional detector noise residuals along the ecliptic plane. We compare the full sky power spectra of [CMap1]{} and [CMap2]{} along with other CMB spectra in Section \[pow\_spec\].
Power Spectrum {#pow_spec}
--------------
We show the CMB angular power spectra after corrections of beam and pixel effects obtained from full sky of [CMap1]{} and [CMap2]{} in the top panel of Fig. \[pow\_spec\_fig\]. The theoretical CMB angular power spectrum is shown in red line to guide the eye. The error-bars show the reconstruction error in power spectrum obtained from Method-2 and agree well with the cosmic variance induced errors (e.g., see Section \[validity\]). The bottom panel of this figure show difference of the spectra of these two maps. As we see from this figure both spectra match very well with each other. Such close match is also expected from the very small difference between the two cleaned maps as shown in Fig. \[diffcmap\]. This results suggest that our new ILC approach is very weakly dependent on the sky divisions. This justifies following a global approach of foreground cleaning on large angular scales on the sky, as is done in this work. However, since method 2 simultaneously follows a global approach and performs foreground removal in an iterative fashion, we treat CMB angular power spectrum of [CMap2]{} as the main power spectrum of this work estimated using low resolution Planck and WMAP maps.
We compare full sky CMB angular power spectrum obtained from [CMap2]{} with the corresponding spectra obtained from [COMMANDER, NILC, SMICA]{} and WMAP [ILC]{} maps. We show these spectra in top left and top right panels of Fig. \[pow\_spec\_fig1\]. Also shown in these two panels is CMB theoretical angular power spectrum obtained from Planck 2015 results. The bottom panels of this figure show the difference of angular power spectra of this work with the other spectra of the corresponding top panels. As we see from this figure the CMB angular power spectrum from [CMap2]{} match closely with the angular spectra of these cleaned maps. A similar result was obtained considering CMB angular power spectrum from [CMap1]{} also. It is noteworthy that power spectra of [CMap2]{} and [NILC]{} map agree excellently for the entire multipole range $2 \le \ell \le 32$.
Monte Carlo Simulations {#validity}
=======================
Input CMB, Foreground and Noise Maps {#SimInput}
------------------------------------
We validate the methodology for the first and second methods by performing detailed Monte Carlo simulations of the entire foreground removal and power spectrum estimation procedures. For this purpose we first generate foreground maps at different WMAP and Planck frequency bands of this work. The free free, synchrotron and thermal dust emissions at different frequencies are first obtained at $N_{side} = 256$ and beam resolution $1^\circ$ following the procedure as described in [@Sudevan2017][^2]. We then downgrade the pixel resolution of each component map to $N_{side} = 16$ and smooth each one by Gaussian beam function of FWHM = $\sqrt{540^2 - 60^2} = 536.66^{\prime}$ so as to bring all component maps for all frequency maps to the common resolution of $9^\circ$. We generate CMB temperature anisotropy maps at $N_{side} = 16$ and FWHM = $9^\circ$ by using the theoretical CMB power spectrum consistent with cosmological parameters obtained by [@PlanckCosmoParam2016]. The procedure to generate the detector noise maps remains similar to [@Sudevan2017]. Following the same procedure as described by these authors, we first generate noise maps at $N_{side} = 512$ (for WMAP DA maps) or $1024$ and $2048$ (for Planck frequency maps). We then convert these maps to spherical harmonic space upto $\ell_{max} = 32$, and multiply the resulting spherical harmonic coefficients by the ratio of the window function corresponding to FWHM = $9^\circ$ and the native beam window function of each WMAP DA (or Planck frequency bands). For WMAP Q, V and W band each, we average the DA noise maps to generate a single noise map corresponding to the given frequency band. We generate a set of $200$ noise maps for each of $12$ frequency maps of our analysis. Each of these noise maps have uncorrelated noise properties. We add the CMB, foreground and noise maps generated above to obtain a set of frequency maps that represent realistic observations of WMAP and Planck missions at $N_{side} = 16$ and FWHM = $9^\circ$. We generate a total of $200$ such sets of input frequency maps for Monte Carlo simulations.
![Top panel shows standard deviation map obtained from the difference of foreground minimized CMB map and corresponding randomly generated input CMB map using $200$ Monte Carlo simulations of foreground minimization following Method-1 as described in Section \[method1\]. The bottom panel shows the standard deviation map obtained for the $200$ Monte Carlo simulations of Method-2 (e.g., see Section \[method2\]). All units are in $\mu K$ thermodynamic temperature. The reduction in reconstruction error for these two methods is discussed in Section \[sim\_results\].[]{data-label="err_map"}](f7)
Results {#sim_results}
-------
### Reconstruction Error in Cleaned Maps {#CMBErr}
If the input CMB map for the $i$th Monte Carlo simulations is denoted by $T_i(p)$, where $p$ denotes the pixel index, and the corresponding foreground minimized CMB map is $T^{\prime}_i(p)$, the map representing reconstruction error for the particular simulation is then given by $\Delta T_i(p) = T^{\prime}_i(p) - T_i(p)$. We estimate the standard deviation map using all $200$ error maps for each of our two methods of this work. The error-maps for method 1 and 2 are shown respectively is top and bottom panels of Fig. \[err\_map\]. As seen from this figure, using the iterative method reduces the reconstruction error in the north and southern hemisphere towards the galactic center region. Also seen from this figure is lower reconstruction error near the north polar spur region. The average variance per pixel over full sky for method 1 (estimated from the top panel Fig. \[err\_map\]) is $6.41 \mu K^2$ compared to a value of $5.25 \mu K^2$ for the method 2 (bottom panel). Corresponding average variances for [ThDust5000]{} mask region are $1.75$ and $0.97 \mu K ^2$ respectively. For galactic region not covered by the thermal dust mask the average variances become $22.84$ and $20.36 \mu K^2$ respectively for the Method-1 and Method-2. We conclude both methods work with comparable efficiencies, however, the second method performs better than the first method in terms of foreground removal.
### CMB Angular Power Spectrum
![Top panel shows the mean (in green) of $200$ full sky CMB angular power spectra obtained from Monte Carlo simulations of Method-2 of this work along with the theoretical CMB power spectrum (red line). The error bar computed from cosmic variance estimated from the theoretical power spectrum shown by the filled region. The reconstruction error on cleaned CMB power spectrum obtained from any one of the simulations is shown in green. The middle panel shows a close comparison of mean CMB angular power spectrum following Method-2 and theoretical CMB power spectrum. The error-bars of this plot represents error on the foreground cleaned mean CMB spectrum. The bottom panel shows the difference between the mean spectra of Method-2 and Method-1 along with the error bars applicable for mean spectrum of Method-1. []{data-label="sim_pow_spec"}](f8)
Using $200$ foreground cleaned maps obtained from Monte Carlo simulations of foreground removal and subsequent CMB angular power spectrum estimation over the complete sky region we assess reconstruction error in cleaned CMB power spectra obtain using Method-1 and Method-2. In top panel of Fig. \[sim\_pow\_spec\] we plot mean CMB angular power spectrum (green points) obtained following Method-2 along with the standard deviation of the cleaned power spectrum for any one of the simulations. The mean foreground cleaned power spectrum agrees well with the theoretical CMB power spectrum (red line) which is used to generate random (and isotropic) CMB realizations. The cosmic variance error limit is shown by the colored band around the theoretical CMB power spectrum. The close match of cosmic variance and the reconstruction error on the cleaned power spectrum at each multipole implies that the recovered angular power spectrum is only cosmic variance limited and reconstruction error due to foreground residuals (plus any error induced by detector noise) is a sub-dominant source of contamination on the angular scale chosen in this work. In the middle panel of Fig. \[sim\_pow\_spec\] we closely investigate any reconstruction biases that may exist in the foreground cleaned power spectrum of Method-2 by plotting the difference between foreground minimized mean CMB power spectrum and the CMB theoretical power spectrum. The error bar at each multipole plotted in this panel is applicable for the mean CMB angular power spectrum and therefore they are obtained by scaling the corresponding reconstruction error of top panel by $1/\sqrt{N_{sim}}$ where the number of simulations, $N_{sim} = 200$. For all the multipoles except ($\ell = 29$) the significance of any difference between the mean cleaned spectrum and the theoretical CMB spectrum is less than $2\sigma$. For $\ell=29$ the significance of deviation is $2.8\sigma$. This shows that power spectrum obtained from Method-2 has no significant bias that may arise due to imperfect foreground residuals. The bottom panel of Fig. \[sim\_pow\_spec\] shows the difference between mean CMB power spectra obtained from Method-1 and Method-2. The error-bars of this plot is computed from foreground cleaned maps of Method-1 and they are applicable for the mean power spectrum. Clearly mean spectra obtained by the two methods of this work agree very well with each other. Both methods produce comparable error-bars as well.
Advantage of global ILC method at low resolution {#advantage}
================================================
The global ILC method has two very important advantages over the usual ILC method in pixel space that does not take into account prior information about CMB theoretical covariance matrix. First, the globally cleaned CMB map has less reconstruction error at each pixel. Second, the usual ILC approach (without using the covariance information) at low resolution leads to a bias in the power spectrum which remains absent in the proposed methods of this work. The cause of these limitations in usual ILC approach at low resolution analysis is a chance-correlation between the CMB and foreground (and detector noise) components which can not be ignored over large scales of the sky. In this section we discuss about the advantages of our approach.
Using the simulated frequency maps at $N_{side}= 16$ and $9^{\circ}$ resolution (e.g., Section \[SimInput\]) we perform $200$ Monte Carlo simulations over the complete sky region using usual ILC approach, wherein no CMB covariance matrix is used. The error map in CMB reconstruction is then computed in the same fashion as discussed in Section \[CMBErr\]. The standard deviation map is plotted in Fig. \[ErrOldILC\] which indicates a strong residual, not only on the the galactic plane, but also in higher galactic latitudes. Unlike the small variance per pixel reported in Section \[CMBErr\] the average variance per pixel for Fig. \[ErrOldILC\] is large ($89.17 \mu K^2$). This clearly demonstrates the first advantage, i.e., sharp decrease in reconstruction error of cleaned CMB map, when we incorporate prior information about theoretical covariance of CMB component.
![The standard deviation map indicating the reconstruction error for usual pixel space ILC approach over the entire sky at $N_{side} = 16$ and $9^{\circ}$ resolution. Large reconstruction error compared to the methods (e.g., see Fig. \[err\_map\]) of this work is seen. Unit is in $\mu K$ thermodynamic.[]{data-label="ErrOldILC"}](f9)
![Top panel shows the mean CMB power spectrum (in green) obtained from $200$ Monte Carlo simulations of usual ILC approach over the entire sky on low resolution maps as discussed in Section \[advantage\] along with the theoretical CMB angular power spectrum (red line) consistent with Planck 2015 results. The filled color band shows the cosmic variance excursion limit of the observed CMB angular power spectrum. The green error-bars show reconstruction error in the cleaned power spectrum at different multipoles. The bottom panel closely compares the reconstruction error-bars with the cosmic variance induced errors. Residuals in the cleaned maps cause larger than cosmic variance error starting from $\ell \sim 10$. For low multipoles, $\ell \le 4$ reconstruction error becomes less than cosmic variance induced error since the cleaned power spectrum at low multipoles is biased low due to a chance correlation of CMB with foregrounds (and detector noise). []{data-label="ErrOldILC1"}](f10)
The larger reconstruction error in cleaned maps in usual ILC approach, causes a significant bias in the power spectrum which is a quadratic function of the data. We show the mean power spectrum computed from $200$ Monte Carlo simulations of usual ILC approach over the entire sky in green in top panel of Fig. \[ErrOldILC1\] along with the Planck 2015 theoretical power spectrum which is used to generate the input CMB maps. Clearly a positive bias exist due to imperfect foreground residuals in the cleaned spectrum starting from multipole $\ell = 8$. Another interesting feature of the top panel is existence of a negative bias for $\ell \le 5$. Such negative bias is expected and was first reported by [@Saha2006] and is discussed extensively in [@Saha2008] (see also [@Sudevan2017] for such bias in high resolution analysis) for multipole space ILC methods. In fact, observing the error pattern of Fig. \[ErrOldILC\] it is likely that a positive bias due to residual foregrounds exists even at low multipoles, $\ell \le 8$ on the top of the additional negative bias in this multipole range. The bottom panel of Fig. \[ErrOldILC1\] compares the reconstruction error in the cleaned power spectrum with the error due to cosmic variance alone. Starting from multipole $\ell \sim 10$ we see that the error in usual ILC power spectrum becomes larger than the cosmic variance induced error. Interestingly, due to existence of negative bias at the low multipoles the error in cleaned spectrum become biased low for $\ell \le 4$. The bias existing in the cleaned power spectrum of the usual ILC approach at low resolution along with larger error in reconstructed power spectrum from this approach justifies our second point of advantage (discussed at the beginning of the current section) of the new approach described in this article.
![The standard deviation map computed from the difference of cleaned CMB maps and corresponding input CMB maps using ILC method, when the weights are obtained from Eqn. \[weights2\]. The reconstruction error follows a noise pattern and is much smaller compared to Fig. \[ErrOldILC\] when CMB-foreground chance correlation affects the weight estimation. Unit is in $\mu K$ thermodynamic.[]{data-label="ErrCf"}](f11)
Role of CMB-Foreground (or CMB-Noise) Chance Correlation {#chancecorr}
========================================================
Having discussed in the previous section the advantages of the global ILC method of this work we now focus on the cause of excess residuals in the usual ILC method when applied to low resolution maps. If we apply usual ILC method on the input maps described in Section \[formalism\] variance of the cleaned map becomes, $$\begin{aligned}
\hat \sigma^2 = {\bf W }\hat{\bf C} {\bf W}^T\,, \end{aligned}$$ where $\hat {\bf C}$ is an $n \times n$ matrix representing the covariance between different input frequency maps (from which mean temperature anisotropies corresponding to each frequency is already subtracted). Similar to Eqn. \[weights\] the set of weights that minimizes variance of the cleaned map subject to the constraint CMB is preserved is given by [@Saha2008; @Tegmark2003; @Tegmark96], $$\begin{aligned}
{\bf W} = \frac{ {\bf e} \hat {\bf C}^{\dagger}}{ {\bf e} \hat {\bf C}^{\dagger} {\bf e}^T}\, . \end{aligned}$$
The data covariance matrix $\hat {\bf C}$ follows, $\hat {\bf C} = \hat \sigma^{2}_c {\bf e}^T{\bf e} + \hat {\bf C}_{fc} + {\bf C}_f$ where $\hat \sigma^{2}_c$ represents the variance of the CMB component which is independent on the frequency, $\hat {\bf C}_{fc}$ is an $n \times n$ matrix denoting the chance correlation between the CMB and all foreground components for a given realization of CMB (e.g., pure CMB signal in our Universe) and finally ${\bf C}_f$ is the $n \times n$ foreground covariance matrix[^3]. Following [@Saha2008] we note that, ${\bf e}^T \in \mathcal C(\hat {\bf C}_{fc} + {\bf C}_f)$ so that the generalized Sherman-Morrison formula for Moore-Penrose generalized inverse of rank one update becomes, $$\begin{aligned}
\hat {\bf C}^{\dagger} = \hat {\bf A}^{\dagger} - \frac{1}{\lambda} {\bf fg}^{T} \, ,
\label{GSM}\end{aligned}$$ where $\hat {\bf A} = \hat {\bf C}_{fc} + {\bf C}_f$, $\lambda = 1 + {\bf e}\hat {\bf A}^{\dagger}{\bf e}^T$, ${\bf f} = \hat {\bf A}^{\dagger}{\bf e}^T$ and ${\bf g} = \hat {\bf A}^{\dagger}{\bf e}^T$. Using Eqn. \[GSM\] we obtain, $$\begin{aligned}
{\bf W} = %\frac{ {\bf e} \hat {\bf C}^{\dagger}}{ {\bf e} \hat {\bf C}^{\dagger} {\bf e}^T} =
\frac{ {\bf e} \hat {\bf A^{\dagger}}}{ {\bf e} \hat {\bf A}^{\dagger} {\bf e}^T}\, .
\label{weights1}\end{aligned}$$ Using Eqn. \[weights1\] we conclude that the weights are independent on the exact level of CMB variance $\hat \sigma^2_c$ for the particular random realization. This is expected since the weights in the usual ILC method, in principle, should only be determined by the foregrounds as long as CMB follows blackbody distribution. One may interpret Eqn. \[weights1\] as the usual ILC weights minimizing the part of the variance in the cleaned map that arise due to CMB-foreground chance correlation and foreground components. Since $\hat {\bf A} = \hat {\bf C}_{fc} + {\bf C}_f $, we see from Eqn. \[weights1\] that in practice the ILC weights not only depend on foreground covariance matrix ${\bf C}_f$ but also they depend upon the CMB-foreground chance correlation matrix $\hat {\bf C}_{fc}$. What will happen if in Eqn. \[weights1\] we could replace $\hat {\bf A}$ by ${\bf C}_f$? We note that such a choice is not possible for analysis of the real data since the covariance matrix for the foregrounds is not known exactly a priori. However, in Monte Carlo simulations we can always assume that ${\bf C}_f$ is known. This will be the situation when weights are not affected by the chance-correlation matrix. If we know the true foreground covariance matrix accurately, in usual ILC procedure one will just minimize the part of the variance in the cleaned map that arise due to foreground components. Clearly this is $\sigma^2_f = {\bf W }{\bf C}_f {\bf W}^T$. Minimizing $\sigma^2_f $ subject to the constraint CMB is preserved gives, $$\begin{aligned}
{\bf W} = \frac{ {\bf e} {\bf C}^{\dagger}_f}{ {\bf e} {\bf C}^{\dagger}_f {\bf e}^T}\, .
\label{weights2}\end{aligned}$$ We perform detailed Monte Carlo simulations of foreground minimization at low resolution following usual ILC method, with simulated WMAP and Planck observations to investigate the difference in the cleaned maps obtained by two different ways. First, the weights are determined following Eqn. \[weights1\] and second, they are determined by Eqn. \[weights2\]. In the first case we recover results that are similar to those shown in Figs. \[ErrOldILC\] and \[ErrOldILC1\]. This implies in presence of CMB-foregrounds chance correlations usual ILC method perform a poor foreground subtraction on large scales on the sky. In the second case, when the chance correlation matrix is absent the method performs foreground removal very well. The standard deviation map computed from the difference of cleaned CMB maps and the corresponding input CMB maps, for this case, is shown in Fig. \[ErrCf\]. The standard deviation map is consistent with a detector noise pattern without any visible signature of residual foregrounds. The mean pixel variance of this map is only $0.14 \mu K^2$ indicating greatly improved foreground subtraction compared to the case when CMB-foreground chance correlation is present. We reemphasize that, although, we can use Eqn. \[weights2\] for the case of Monte Carlo simulations where the input foreground models are known, in practice, we can not use this equation to estimate ILC weights since the foreground covariance matrix ${\bf C}_f$ is unknown for the observed sky. We use Eqns. \[weights2\] and \[weights1\] in Monte Carlo simulations to establish that the CMB-foreground chance correlations cause significant residuals in usual ILC method. The global ILC method that propose to use CMB covariance information, thus, becomes greatly beneficial method, improving performance of usual ILC method without any need to know ${\bf C}_f$.
Apart from comparing the pixel reconstruction error maps (e.g., Figs. \[ErrOldILC\] and \[ErrCf\]) or the power spectra of cleaned maps there is another way in which we see that using the theoretical CMB covariance matrix helps to greatly improve usual ILC results. In Fig. \[W\] ($x,y$) coordinates of any blue point are given respectively by value of weight for a particular frequency band obtained using the usual ILC method and the corresponding value of the weight using Eqn. \[weights2\] while cleaning a given set of input frequency maps. The $y$-coordinate of the yellow points are same as the blue point for the same set of input frequency maps, however the $x$-coordinate of yellow points represents weights for the global ILC method using information about the theoretical CMB covariance matrix. The blue points show significantly larger dispersion along the horizontal axes for all frequency bands compared to the corresponding dispersion of yellow points. The new method of this work efficiently reduces the larger dispersion of weights of usual ILC method and produces better foreground minimized CMB maps at low resolution. The $y$-coordinates of all points of this figure show some level of fluctuations, even if we use Eqn. \[weights2\] to estimate the weights that represent the $y$-coordinates. This is because apart from the foregrounds ${\bf C}_f$ contain a small level of detector noise. The $x$-coordinate of vertical axis of each plot show the value of the weight when no detector noise is present in ${\bf C}_f$. Each of these values remains same for different Monte Carlo simulations and represent weights that will be necessary to remove foregrounds in an ideal noise-less experiment. We finally note that using CMB theoretical covariance matrix in Eqn. \[dispersion0\] we efficiently suppress CMB large angle covariances which leads to significantly smaller dispersion of weights because of smaller CMB-foreground chance correlation. The small dispersion of our weights results in a greatly improved foreground minimization than the usual ILC method on large scales of the sky.
Discussions & Conclusion {#Conclusion}
========================
We have developed a new ILC method for foreground minimization in pixel space for application on large angular scales on the sky using prior information about theoretical CMB covariance matrix. We apply the methodology on low resolution WMAP and Planck frequency maps and show that the cleaned CMB temperature anisotropy map obtained by us match very well with those obtained by other science groups of Planck and WMAP. [*This shows that results of CMB maps and its power spectrum are robust with respective to a variety of analysis pipeline.*]{} We validate the methodology of our foreground removal by detailed Monte Carlo simulations. Usage of this new approach has several benefits over naive application of usual ILC approach in pixel space over large scales of the sky.
1. First, the new approach generates cleaned CMB map that has significantly lower reconstruction error due to foreground residuals. The power spectrum from the cleaned map also has the lower reconstruction error for our case, the standard deviations of CMB angular power spectrum estimated from the Monte-Carlo simulations agree with those estimated from the cosmic variance alone.
2. Second, the CMB angular power spectrum obtained from our cleaned maps does not have any visible signature of negative bias at the low multipole region, which is seen to be present for pixel space application of usual ILC method over large scales on the sky. Such negative bias is also reported in harmonic space ILC method by [@Saha2006] and its property and origin were investigated in detail by [@Saha2008]. The negative bias arise due a chance correlation between CMB and foreground components on a particular realizations of the sky. Using inverse weight of CMB theoretical covariance matrix in Eqn. \[dispersion0\] we effectively get rid of such chance correlations and the as well as the resulting negative biases in the cleaned CMB angular power spectrum at low multipoles.
The new method complements the usual ILC approach in pixel space which so far has been applied on high resolution maps by incorporating local information available from input frequency maps to better remove foregrounds, the spectral property of which vary with the sky positions. On the very large scales the spectral properties of foregrounds are expected to vary by small amount over the entire sky. We show that, on the large scale it is sufficient to perform ILC foreground removal by dividing the sky merely into two regions, provided we use the prior information available from CMB covariance matrix globally on the sky. Although we have assumed a theoretical CMB covariance matrix consistent with assumption of statistical isotropy of CMB in Eqn. \[theory\_cov\], in principle, one can also use a covariance matrix in our method which is not statistically isotropic. This brings about a possibility to open up a new avenue to incorporate such additional information in our method which may be a signature of non trivial primordial power spectrum [@Ghosh2016; @Contreras2017]. Taking into account the global nature of our low resolution analysis and local nature of high resolution analysis of usual ILC method, we now consider pixel space ILC method in a general perspective that incorporates a very comprehensive duality in its nature. We hope that our method will be useful to analyze low resolution polarization maps from Planck or future generation CMB missions.
We use publicly available HEALPix [@Gorski2005] package available from http://healpix.sourceforge.net for some of the analysis of this work. We acknowledge the use of Planck Legacy Archive (PLA) and the Legacy Archive for Microwave Background Data Analysis (LAMBDA). LAMBDA is a part of the High Energy Astrophysics Science Archive Center (HEASARC). HEASARC/LAMBDA is supported by the Astrophysics Science Division at the NASA Goddard Space Flight Center.
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[^1]: Hierarchical Equal Area Isolatitude Pixellization of sphere, e.g., see [@Gorski2005]
[^2]: Unlike the work of [@Sudevan2017] in the current work we use a spatially constant spectral index ($\beta_s = -3.00$) for synchrotron component for all WMAP and Planck frequencies.
[^3]: In this discussions we have assumed that the detector noise contribution is small compared to the foreground or CMB signal. This is the case for WMAP and Planck temperature observation over the large scales of the sky. We emphasize that we do not require detector noise to be completely absent, we only assume that the data signal dominated. Accordingly, we interpret ${\bf C}_f$ to contain a small amount of detector noise also
|
---
abstract: 'A [*signed magic rectangle*]{} $SMR(m,n;r, s)$ is an $m \times n$ array with entries from $X$, where $X=\{0,\pm1,\pm2,\ldots, $ $\pm (ms-1)/2\}$ if $mr$ is odd and $X = \{\pm1,\pm2,\ldots,\pm mr/2\}$ if $mr$ is even, such that precisely $r$ cells in every row and $s$ cells in every column are filled, every integer from set $X$ appears exactly once in the array and the sum of each row and of each column is zero. In this paper we prove that a signed magic rectangle $SMR(m,n;r, 2)$ exists if and only if either $m=2$ and $n=r\equiv 0,3 \pmod 4$ or $m,r\geq 3$ and $mr=2n$.'
author:
- |
Abdollah Khodkar and Brandi Ellis\
Department of Mathematics\
University of West Georgia\
Carrollton, GA 30118\
[akhodkar@westga.edu]{}, [bellis5@my.westga.edu]{}
title: Signed magic rectangles with two filled cells in each column
---
Introduction {#SEC1}
============
A [*magic rectangle*]{} of order $m\times n$, $MR(m,n)$, is an arrangement of the numbers from 0 to $mn-1$ in an $m\times n$ rectangle such that each number occurs exactly once in the rectangle and the sum of the entries of each row is the same and the sum of entries of each column is also the same. The following theorem, whose proof can be found in [@TH1; @TH2] and [@sun], settles the existence of an $MR(m,n)$.
\[TH:sun\] An $m \times n$ magic rectangle exists if and only if $m \equiv n \pmod 2$, $m + n > 5$, and $m, n > 1$.
A [*$k$-magic square*]{} of order $n$ is an arrangement of the numbers from 0 to $kn-1$ in an $n\times n$ array such that each row and each column has exactly $k$ filled cells, each number occurs exactly once in the array, and the sum of the entries of any row or any column is the same. The study of magic squares with empty cells was initiated in [@KL1]. A magic square is called $k$-[*diagonal*]{} if its entries all belong to $k$ consecutive diagonals (this includes broken diagonals as well).
\[TH:KL\] [@KL1] There exists a $k$-diagonal magic square of order $n$ if and only if $n=k=1$ or $3\leq k\leq n$ and either $n$ is odd or $k$ is even.
A [*signed magic rectangle*]{} $SMR(m,n;r, s)$ is an $m \times n$ array with entries from $X$, where $X=\{0,\pm1,\pm2,\ldots,\pm (ms-1)/2\}$ if $mr$ is odd and $X = \{\pm1,\pm2,\ldots,\pm mr/2\}$ if $mr$ is even, such that precisely $r$ cells in every row and $s$ cells in every column are filled, every integer from set $X$ appears exactly once in the array and the sum of each row and of each column is zero. By the definition, $mr=ns$, $r\leq n$ and $s \leq m$. If $r=n$ or $s=m$, then the rectangle has no empty cell. In the case where $m = n$, we call the array a [*signed magic square*]{}. Signed magic squares represent a type of magic square where each number from the set $X$ is used once.
The following two theorems can be found in [@KSW].
\[TH:KSW1\] An $SMR(m,n)$ exists precisely when $m = n = 1$, or when $m = 2$ and $n \equiv 0, 3 \pmod4$, or when $n = 2$ and $m \equiv 0, 3 \pmod4$, or when $m, n > 2$.
In [@KSW] the notation $SMS(n;k)$ is used for a signed magic square with $k$ filled cells in each row and $k$ filled cells in each column.
\[TH:KSW2\] There exists an $SMS(n;k)$ precisely when $n=k = 1$ or $3\leq k\leq n$.
In this paper we prove that a signed magic rectangle $SMR(m,$ $n;r, 2)$ exists if and only if either $m=2$ and $n=r\equiv 0,3 \pmod 4$ or $m,r\geq 3$ and $mr=2n$.
Main constructions {#SEC2}
==================
A rectangular array is [*shiftable*]{} if it contains the same number of positive entries as negative entries in every column and in every row. Figure \[2,4;4,2\] displays a shiftable $SMR(2,4;4,2)$. These arrays are called *shiftable* because they may be shifted to use different absolute values. By increasing the absolute value of each entry by $k$, we add $k$ to each positive entry and $-k$ to each negative entry. If the number of entries in a row is $2\ell$, this means that we add $\ell k + \ell(-k) = 0$ to each row, and the same argument applies to the columns. Thus, when shifted, the array retains the same row and column sums.
$$\begin{array}{|c|c|c|c|}\hline
1&-2&-3&4\\\hline
-1&2&3&-4\\\hline
\end{array}$$
\[TH:kn;kr-km,kn\] Let there exist a shiftable $SMR(m,n;r,s)$. Then for every $k\geq 1$
1. there exists a shiftable $SMR(m,kn;kr,s)$ and
2. there exists a shiftable $SMR(km,kn;r,s)$ .
Let $A$ be a shiftable $SMR(m,n;r,s)$. Note that since $A$ is shiftable, it follows that $r$ and $s$ are both even. Partition an empty $m\times kn$ rectangle, say $B$, into $k$ empty rectangles of size $m\times n$, say $P_{\ell}$, where $0\leq\ell\leq k-1$. For each $(i,j;e)\in A$ we fill the cell $(i,j)$ of $P_{\ell}$ with $e+\ell(mr/2)$ if $e$ is positive or with $e-\ell(mr/2)$ if $e$ is negative. The resulting rectangle is a shiftable $SMR(m,kn;kr,s)$. See Figure \[2,12;12,2\].
We now prove that there exists a shiftable $SMR(km,kn;r,s)$ for $k\geq 1$. Partition an empty $km\times kn$ rectangle, say $C$, into $k^2$ empty rectangles of size $m\times n$, say $P_{a,b}$, where $0\leq a, b\leq k-1$. For each $(i,j;e)\in A$ we fill the cell $(i,j)$ of $P_{a,a}$ with $e+a(mr/2)$ if $e$ is positive or with $e-a(mr/2)$ if $e$ is negative for $0\leq a\leq k-1$. The resulting rectangle is a shiftable $SMR(km,kn;r,s)$. See Figure \[6,12;4,2\].
$$\begin{array}{|c|c|c|c||c|c|c|c||c|c|c|c|}\hline
1&-2&-3&4&5&-6&-7&8&9&-10&-11&12\\\hline
-1&2&3&-4&-5&6&7&-8&-9&10&11&-12\\\hline
\end{array}$$
$$\begin{array}{|c|c|c|c||c|c|c|c||c|c|c|c|}\hline
1&-2&-3&4&&&&&&&&\\\hline
-1&2&3&-4&&&&&&&&\\\hline\hline
&&&&5&-6&-7&8&&&&\\\hline
&&&&-5&6&7&-8&&&&\\\hline\hline
&&&&&&&&9&-10&-11&12\\\hline
&&&&&&&&-9&10&11&-1
2\\\hline
\end{array}$$
\[TH:kn+n’;kr+r’\] Let there exist a shiftable $SMR(m,n;r,s)$ and a (shiftable) $SMR(m,n';r',s)$ with $mr'$ even. Then there exists a (shiftable) $SMR(m,kn+n';kr+r',s)$ for $k\geq 1$.
Apply Part 1 of Theorem \[TH:kn;kr-km,kn\] with a shiftable $SMR(m,n;r,s)$ to obtain a shiftable $SMR(m,kn;kr,s)$, say $A$, for $k\geq 1$. Let $B$ be a (shiftable) $SMR(m,n';r',s)$ and let $C$ be the $m\times kn$ rectangle obtained from $A$ by adding $mr'/2$ to each positive entry of $A$ and subtracting $mr'/2$ from each negative entry of $A$. Finally, let $D$ be the $m\times (kn+n')$ rectangle obtained from $B$ and $C$ as follows: if $(i,j;e)\in B$, then $(i,j;e)\in D$ and if $(i,j;e)\in C$, then $(i,j+n';e)\in D$. It is easy to see that $D$ is a (shiftable) $SMR(m,kn+n';kr+r',s)$.
Figure \[2,11;11,2\] displays an $SMR(2,11;11,2)$ obtained by the construction given in the proof of Theorem \[TH:kn+n’;kr+r’\] using the shiftable $SMR(2,4;4,2)$ given in Figure \[2,4;4,2\], an $SMR(2,3;3,2)$ and $k=2$.
$$\begin{array}{|c|c|c||c|c|c|c||c|c|c|c|}\hline
-1&-2&3&-4&5&6&-7&-8&9&10&-11 \\ \hline
1&2&-3&4&-5&-6&7&8&-9&-10&11 \\ \hline
\end{array}$$
\[TH:km+m’,kn+n’\] Let there exist a shiftable $SMR(m,n;r,s)$ and a (shiftable) $SMR(m',n';r,s)$ with $m'r$ even., then there exists a (shiftable) $SMR(km+m',kn+n';r,s)$ for $k\geq 1$.
Apply Part 2 of Theorem \[TH:kn;kr-km,kn\] with a shiftable $SMR(m,n;$ $r,s)$ to obtain a shiftable $SMR(km,kn;r,s)$, say $A$, for $k\geq 1$. Let $B$ be a (shiftable) $SMR(m',n';r,s)$ and let $C$ be the $m\times kn$ rectangle obtained from $A$ by adding $m'r/2$ to each positive entry of $A$ and subtracting $m'r/2$ from each negative entry of $A$. Finally, let $D$ be the $(km+m')\times (kn+n')$ rectangle obtained from $B$ and $C$ as follows: if $(i,j;e)\in B$, then $(i,j;e)\in D$ and if $(i,j;e)\in C$, then $(i+m',j+n';e)\in D$. It is easy to see that $D$ is a (shiftable) $SMR(km+m',kn+n';r,s)$.
Figure \[7,14;4,2\] displays a shiftable $SMR(7,14;4,2)$ obtained by the construction given in the proof of Theorem \[TH:km+m’,kn+n’\] using the shiftable $SMR(2,4;4,2)$ given in Figure \[2,4;4,2\], the shiftable $SMR(3,6;4,2)$ given in Figure \[3,6;4,2\], and $k=2$.
$$\begin{array}{|c@{\hspace{0.3mm}}|c@{\hspace{0.3mm}}|c@{\hspace{0.3mm}}
|c@{\hspace{0.3mm}}|c@{\hspace{0.3mm}}|c@{\hspace{0.3mm}}
|c@{\hspace{0.3mm}}|c@{\hspace{0.3mm}}|c@{\hspace{0.3mm}}
|c@{\hspace{0.3mm}}|c@{\hspace{0.3mm}}|c@{\hspace{0.3mm}}
|c@{\hspace{0.3mm}}|c@{\hspace{0.3mm}}|} \hline
1&&-3&-4&&6&&&&&&&& \\ \hline
-1&2&&4&-5&&&&&&&&& \\ \hline
&-2&3&&5&-6&&&&&&&& \\ \hline\hline
&&&&&&-7&8&9&-10&&&& \\ \hline
&&&&&&7&-8&-9&10&&&& \\ \hline\hline
&&&&&&&&&&-11&12&13&-14 \\ \hline
&&&&&&&&&&11&-12&-13&14 \\ \hline
\end{array}$$
The existence of an $SMR(m,3m/2;3,$ $2)$ and an $SMR(m,5m/2;5,2)$ {#SEC3}
=================================================================
In this section we present direct constructions for the existence of an $SMR(m,3m/2;3,2)$, where $m\geq 2$ and even, and an $SMR(m,5m/2;5,2)$, where $m\geq 4$ and even. We will make use of these results in Section \[SEC4\]. Note that if $m$ is odd there is no $SMR(m,3m/2;3,$ $2)$ because $3m$ is odd and there is no $SMR(m,5m/2;5,2)$ because $5m$ is odd.
\[m,3m/2;3,2\] There exists an $SMR(m,3m/2;3,2)$ for $m$ even and $m\geq 2$.
Define an $m\times 3$ rectangle $A$ as follows.
Column 1: $ \left\{\begin{array}{l}
(i,1;i)\in A \mbox { for } 1\leq i\leq m/2,\\
(i,1;(m/2)-i)\in A \mbox { for } (m/2)+1\leq i\leq m.\\
\end{array}\right.$
Column 2: $ \left\{\begin{array}{lll}
(i,2;(3m/2)-2i+1)\in A \mbox { for } 1\leq i\leq m/2,\\
(i,2;-i)\in A \mbox { for } (m/2)+1\leq i\leq m.\\
\end{array}\right.$
Column 3: $ \left\{\begin{array}{l}
(i,3;(-3m/2)+i-1)\in A \mbox { for } 1\leq i\leq m/2,\\
(i,3;(-m/2)+2i)\in A \mbox { for } (m/2)+1\leq i\leq m.\\
\end{array}\right.$
By construction, it is easy to see that the entries in $A$ consist of $\{\pm1, \pm 2, \ldots, \pm 3m/2\}$, which are the numbers in an $SMR(m,$ $3m/2;3,2)$. Figure \[A.m=8.3.2\] displays the rectangle $A$ when $m=8, 10$. We now prove that the sum of each row of $A$ is zero. The row sum for row $i$ of $A$, where $1\leq i\leq m/2$, is $$i+ ((3m/2)-2i+1)+ ((-3m/2)+i-1)=0.$$ Similarly, the row sum for row $i$ of $A$, where $(m/2)+1\leq i\leq m$, is $$((m/2)-i)+(-i)+((-m/2)+2i) =0.$$
Let $a,k$ and $-k$ be the numbers in a row of $A$. Then $a+k+(-k)=0$, which implies that $a=0$. Since zero does not appear in $A$, it follows that the numbers $k$ and $-k$ do not appear in the same row of $A$.
Now let $B$ be an empty $m\times 3m/2$ rectangle. For each $(i,j;k)\in A$ let $(i,|k|;k)\in B$. By construction, the numbers in row $i$ of $B$ are precisely the numbers in row $i$ of $A$. Therefore the row sum for each row of $B$ is also zero. Since $\pm k$ are entries of $A$ for each $1\leq k\leq 3m/2$, it follows that column $k$ of $B$ contains only $k$ and $-k$. Hence, $B$ is an $SMR(m,3m/2;3,2)$ for $m$ even and $m\geq 2$.
Figure \[8,12;3,2\] displays an $SMR(8,12;3,2)$ obtained by the construction given in Proposition \[m,3m/2;3,2\].
$$\begin{array}{ccc}
\begin{array}{|c|c|c|}\hline
1&11&-12 \\ \hline
2&9&-11 \\ \hline
3&7&-10 \\ \hline
4&5&-9 \\ \hline
-1&-5&6 \\ \hline
-2&-6&8 \\ \hline
-3&-7&10 \\ \hline
-4&-8&12 \\ \hline
\end{array}&&
\begin{array}{|c|c|c|}\hline
1&14&-15 \\ \hline
2&12&-14 \\ \hline
3&10&-13 \\ \hline
4&8&-12 \\ \hline
5&6&-11 \\ \hline
-1&-6&7 \\ \hline
-2&-7&9 \\ \hline
-3&-8&11 \\ \hline
-4&-9&13 \\ \hline
-5&-10&15 \\ \hline
\end{array}\\
\mbox{ Array } A \mbox { when } m=8&&\mbox{ Array } A \mbox { when } m=10\\
\end{array}$$
$$\begin{array}{|c@{\hspace{1.0mm}}|c@{\hspace{1.0mm}}|c@{\hspace{1.0mm}}
|c@{\hspace{1.0mm}}||c@{\hspace{1.0mm}}|c@{\hspace{1.0mm}}
|c@{\hspace{1.0mm}}|c@{\hspace{1.0mm}}||c@{\hspace{1.0mm}}
|c@{\hspace{1.0mm}}|c@{\hspace{1.0mm}}|c@{\hspace{1.0mm}}|} \hline
1&&&&&&&&&&11&-12 \\ \hline
&2&&&&&&&9&&-11& \\ \hline
&&3&&&&7&&&-10&& \\ \hline
&&&4&5&&&&-9&&& \\ \hline
-1&&&&-5&6&&&&&& \\ \hline
&-2&&&&-6&&8&&&& \\ \hline
&&-3&&&&-7&&&10&& \\ \hline
&&&-4&&&&-8&&&&12 \\ \hline
\end{array}$$
It is an easy exercise to see that there is no $SMR(2,5;5,2)$. The following proposition shows how to build an $SMR(m,5m/2;5,2)$ for $m$ even and $m\geq 4$.
\[m,5m/2;5,2\] There exists an $SMR(m,5m/2;5,2)$ for $m$ even and $m\geq 4$.
Define an $m\times 5$ rectangle $C$ as follows.
Column 1: $ \left\{\begin{array}{l}
(i,1;i)\in C \mbox { for } 1\leq i\leq m/2,\\
(i,1;(m/2)-i)\in C \mbox { for } \frac{m+2}{2}\leq i\leq m.\\
\end{array}\right.$
Column 2: $ \left\{\begin{array}{l}
(i,2;(m/2)+2i-1)\in C \mbox { for } 1\leq i\leq m/2,\\
(i,2;(-3m/2)+i-1)\in C \mbox { for } \frac{m+2}{2}\leq i\leq m.\\
\end{array}\right.$
Column 3: $ \left\{\begin{array}{l}
(i,3;(-m)-i)\in C \mbox { for } 1\leq i\leq m/2,\\
(i,3;(5m/2)-2i+2)\in C \mbox { for } \frac{m+2}{2}\leq i\leq m.\\
\end{array}\right.$
Column 4: $ \left\{\begin{array}{l}
(i,4;(-3m/2)-i)\in C \mbox { for } 1\leq i\leq (m/2),\\
(i,4;(3m/2)+i)\in C \mbox { for } \frac{m+2}{2}\leq i\leq m.\\
\end{array}\right.$
Column 5: $ \left\{\begin{array}{l}
(i,5;2m-i+1)\in C \mbox { for } 1\leq i\leq m/2,\\
(i,5;-3m+i-1)\in C \mbox { for } \frac{m+2}{2}\leq i\leq m.\\
\end{array}\right.$
By construction, the entries in $C$ consist of $\{\pm1, \ldots, \pm 5m/2\}$, which are the numbers in an $SMR(m,5m/2;5,2)$. Figure \[C.m=8.5.2\] displays the rectangle $C$ when $m=8$. We now prove that the sum of each row of $C$ is zero. The row sum for row $i$ of $C$, where $1\leq i\leq m/2$, is $$i+ ((m/2)+2i-1)+ (-m-i) + ((-3m/2)-i) + (2m-i+1) =0.$$ Similarly, the row sum for row $i$ of $C$, where $(m/2)+1\leq i\leq m$, is $$\begin{array}{r}
((m/2)-i) + ((-3m/2)+i-1) + (5m/2)-2i+2)\\
+ ((3m/2)+i) + (-3m+i-1) =0.\end{array}$$
Let $a,b,c,d,e$ be the numbers in row $i$ and columns $1,2,3,4,5$ of $C$, respectively. It is straightforward to see that if $x,y\in \{a,b,c\}$ and $z\in \{d,e\}$, then $x+y\neq 0$ and $x+z\neq 0$. Now let $d+e=0$. If $1\leq i\leq m/2$, then $$d+e= ((-3m/2)-i)+ (2m-i+1)= (m/2)-2i+1=0.$$ This implies that $i=(m+2)/4$.
If $(m/2)+1\leq i\leq m$, then $$d+e=((3m/2)+i)+(-3m+i-1)=(-3m/2)+2i-1=0.$$ This implies that $i=(3m+2)/4.$
Therefore if $m\equiv 0\pmod 4$, then the numbers $k$ and $-k$ do not appear in the same row of $C$. If $m\equiv 2 \pmod 4$ and $i\neq (m+2)/2, (3m+2)/4$, then the numbers $k$ and $-k$ do not appear in row $i$ of $C$. When $m\equiv 2 \pmod 4$ we construct an $m\times 5$ array $C'$ by rearranging the eight entries of $C$ which are in the intersection of columns 1 and 2 with rows $(m-2)/2, (m+2)/2, (3m-2)/4$ and $(3m+2)/4$ as follows. Switch $$\begin{array}{l}
((m-2)/4,1;(m-2)/4) \mbox { and } (m+2)/4,1;(m+2)/4),\\
((m-2)/4,5;(7m+6)/4) \mbox{ and } ((m+2)/4,5;(7m+2)/4),\\
((3m-2)/4,1;(-m+2)/4) \mbox{ and } (3m+2)/4,1; (-m-2)/4), \\
\mbox{and }((3m-2)/4,5;(-9m-6)/4) \mbox{ and }((3m+2)/4,5;\\
(-9m-2)/4).\\
\end{array}$$ Figure \[C.m=8.5.2\] displays the rectangle $C'$ when $m=10$. It is easy to see that the sum of each row of $C'$ is zero and $k$ and $-k$ do not appear in any row of $C'$.
Now let $m\equiv 0 \pmod 4$, $m\geq 4$, and let $D$ be an empty $m\times 5m/2$ rectangle. For each $(i,j;k)\in C$ let $(i,|k|;k)\in D$. By construction, the numbers in row $i$ of $D$ are precisely the numbers in row $i$ of $C$. Therefore the row sum for each row of $D$ is also zero. Since $\pm k$ are entries of $C$ for each $1\leq k\leq 5m/2$, it follows that column $k$ of $D$ contains only $k$ and $-k$. Hence, $D$ is an $SMR(m,5m/2;5,2)$.
Similarly, if $m\equiv 2 \pmod 4$ and $m\geq 6$, we use the array $C'$ to build an $SMR(m,5m/2;5,2)$.
$$\begin{array}{ccc}
\begin{array}{|c|c|c|c|c|}\hline
1&5&-9&-13&16 \\ \hline
2&7&-10&-14&15 \\ \hline
3&9&-11&-15&14 \\ \hline
4&11&-12&-16&13 \\ \hline
-1&-8&12&17&-20 \\ \hline
-2&-7&10&18&-19 \\ \hline
-3&-6&8&19&-18 \\ \hline
-4&-5&6&20&-17 \\ \hline
\end{array}&&
\begin{array}{|c|c|c|c|c|}\hline
1&6&-11&-16&20 \\ \hline
3&8&-12&-17&18 \\ \hline
2&10&-13&-18&19 \\ \hline
4&12&-14&-19&17 \\ \hline
5&14&-15&-20&16 \\ \hline
-1&-10&15&21&-25 \\ \hline
-3&-9&13&22&-23 \\ \hline
-2&-8&11&23&-24 \\ \hline
-4&-7&9&24&-22 \\ \hline
-5&-6&7&25&-21 \\ \hline
\end{array}\\
\mbox {Array } C \mbox{ when } m=8&& \mbox{Array } C' \mbox{ when } m=10\\
\end{array}$$
The existence of an $SMR(m,n;r,2)$ with $m$ even {#SEC4}
================================================
Let there exist an $SMR(m,n;r,2)$. If $m=4b$ or $m=4b+2$, then $n=2br$ or $n=(2b+1)r$, respectively. We study the existence of an $SMR(4b,2br;r,2)$ and an $SMR(4b+2,(2b+1)r;r,2)$ in the following two subsections, respectively.
The existence of an $SMR(4b,2br;r,2)$
--------------------------------------
In this subsection we construct signed magic rectangles with parameters $(4b,8ab;4a,2)$, $(4b,2b(4a+2);4a+2,2)$, $(4b,2b(4a+1);4a+1,2)$, and $(4b,2b(4a+3);4a+3,2)$, where $a,b\geq 1$.
\[L2,4;4,2\] There exists a shiftable $SMR(2q,4pq;4p,2)$ for positive integers $p,q\geq 1$.
Figure \[2,4;4,2\] displays a shiftable $SMR(2,4;4,2)$. So by Part 1 of Theorem \[TH:kn;kr-km,kn\], there exists a shiftable $SMR(2,4p;4p,2)$ for $p\geq 1$. Now by Part 2 of Theorem \[TH:kn;kr-km,kn\] there exists a shiftable $SMR(2q,4pq;4p,2)$ for $p,q\geq 1$.
\[4b,8ab;4a,2\] There exists a shiftable $SMR(4b,8ab;4a,2)$ for $a,$ $b\geq 1$.
Apply Lemma \[L2,4;4,2\] with $p=a$ and $q=2b$ to obtain a shiftable $SMR(4b,8ab;4a,2)$ for all $a,b\geq 1$.
\[4b,8ab+4b;4a+2,2\] There exists a shiftable $SMR(4b,2b(4a+2);4a+2,2)$ for $a,b\geq 1$.
Figure \[4,12;6,2\] displays a shiftable $SMR(4,12;6,2)$. So by Part 2 of Theorem \[TH:kn;kr-km,kn\], there exists a shiftable $SMR(4b,12b;6,2)$, say $A$, for $b\geq 1$. On the other hand, by Lemma \[4b,8ab;4a,2\], there exists a shiftable $SMR(4b,8(a-1)b;4(a-1),2)$, say $B$, for $a\geq 2$ and $b\geq 1$. Now apply Theorem \[TH:kn+n’;kr+r’\] with $A$ and $B$ to obtain a shiftable $SMR(4b,2b(4a+2);4a+2,2)$ for $a,b\geq 1$.
$$\begin{array}{|c@{\hspace{0.5mm}}|c@{\hspace{0.5mm}}|c@{\hspace{0.5mm}}
|c@{\hspace{0.5mm}}||c@{\hspace{0.5mm}}|c@{\hspace{0.5mm}}
|c@{\hspace{0.5mm}}|c@{\hspace{0.5mm}}||c@{\hspace{0.5mm}}
|c@{\hspace{0.5mm}}|c@{\hspace{0.5mm}}|c@{\hspace{0.5mm}}|} \hline
-1&2&&&-5&6&&&9&-11&& \\ \hline
1&-2&&&5&-6&&&-9&11&& \\ \hline
&&-3&4&&&-7&8&&&10&-12 \\ \hline
&&3&-4&&&7&-8&&&-10&12 \\ \hline
\end{array}$$
\[4b,8ab+2b;4a+1,2\] There exists an $SMR(4b,2b(4a+1);4a+1,2)$ for $a,b\geq 1$.
By Proposition \[m,5m/2;5,2\], there exists an $SMR(4b,10b;5,2)$, say $A$, for $b\geq 1$. On the other hand, by Lemma \[4b,8ab;4a,2\], there exists a shiftable $SMR(4b,8(a-1)b;4(a-1),2)$, say $B$, for $a\geq 2$ and $b\geq 1$. Now apply Theorem \[TH:kn+n’;kr+r’\] with $A$ and $B$ to obtain an $SMR(4b,2b(4a+1);4a+1,2)$ for $a\geq 2$ and $b\geq 1$. When $a=1$ we apply Proposition \[m,5m/2;5,2\].
\[4b,8ab+2b;4a+3,2\] There exists an $SMR(4b,2b(4a+3);4a+3,2)$ for $a,b\geq 1$.
By Proposition \[m,3m/2;3,2\], there exists an $SMR(4b,6b;3,2)$, say $A$, for $b\geq 1$. On the other hand, by Lemma \[4b,8ab;4a,2\], there exists a shiftable $SMR(4b,8ab;4a,2)$, say $B$, for $a, b\geq 1$. Now apply Theorem \[TH:kn+n’;kr+r’\] with $A$ and $B$ to obtain an $SMR(4b,2b(4a+3);4a+3,2)$ for $a,b\geq 1$.
The existence of an $SMR(4b+2,(2b+1)r;r,2)$
-------------------------------------------
In this subsection we construct signed magic rectangles with parameters $(4b+2,2a(4b+2);4a,2)$, $(4b+2,(2a+1)(4b+2);4a+2, 2)$, $(4b+2,(4a+1)(2b+1);4a+1,2)$, and $(4b+2,(4a+3)(2b+1);4a+3,2)$ for all $a,b\geq 1$.
\[2,n,n,2)\] Let $n\equiv 3 \pmod 4$. Then there exists an $SMR(2,n;$ $n,2)$.
By Lemma \[L2,4;4,2\], there exists a shiftable $SMR(2,4k;4k,2)$, say $A$, for $k\geq 1$. Let $B$ be a $2\times 3$ array with first row $1,2,-3$ and second row $-1,-2,3$. Then $B$ is an $SMR(2,3;3,2)$. Now apply Theorem \[TH:kn+n’;kr+r’\] with $A$ and $B$ to obtain an $SMR(2,4k+3;4k+3,2)$. See Figure \[2,11;11,2\].
\[4b+2,8ab+4a;4a,2\] There exists a shiftable $SMR(4b+2,2a(4b+2);4a,2)$ for $a,b\geq 1$.
Apply Lemma \[L2,4;4,2\] with $p=a$ and q=$2b+1$ to obtain a shiftable $SMR(4b+2,2a(4b+2);4a,2)$ for $a,b\geq 1$.
\[(4b+2,3(4b+2);6,2)\] There exists a shiftable $SMR(4b+2,3(4b+2);6,2)$ for $b\geq 1$
Apply Part 2 of Theorem \[TH:kn;kr-km,kn\] with the shiftable $SMR(4, 12;$ $ 6,2)$ displayed in Figure \[4,12;6,2\] to obtain a shiftable $SMR(4(b-1), 12(b-1); 6, 2)$, say $A$. Then apply Theorem \[TH:km+m’,kn+n’\] with $A$ and the shiftable $SMR(6,18;6,2)$ displayed in Figure \[6,18;6,2\] to obtain a shiftable $SMR(4b+2,3(4b+2);6,2)$.
[$$\begin{array}{|c@{\hspace{0.2mm}}|c@{\hspace{0.2mm}}|c@{\hspace{0.2mm}}
|c@{\hspace{0.2mm}}|c@{\hspace{0.2mm}}|c@{\hspace{0.2mm}}
|c@{\hspace{0.2mm}}|c@{\hspace{0.2mm}}|c@{\hspace{0.2mm}}
|c@{\hspace{0.2mm}}|c@{\hspace{0.2mm}}|c@{\hspace{0.2mm}}
|c@{\hspace{0.2mm}}|c@{\hspace{0.2mm}}|c@{\hspace{0.2mm}}
|c@{\hspace{0.2mm}}|c@{\hspace{0.2mm}}|c@{\hspace{0.2mm}}|} \hline
-1&&3&&&&7&-8&&&&&13&-14&&&& \\ \hline
&-2&&4&&&&8&-9&&&&&14&-15&&& \\ \hline
&&-3&&5&&&&9&-10&&&&&15&-16&& \\ \hline
&&&-4&&6&&&&10&-11&&&&&16&-17& \\ \hline
1&&&&-5&&&&&&11&-12&-13&&&&&18 \\ \hline
&2&&&&-6&-7&&&&&12&&&&&17&-18 \\ \hline
\end{array}$$ ]{}
\[4b+2,(2a+1)(4b+2);4a+2, 2\] There exists a shiftable $SMR(4b+2,(2a+1)(4b+2);4a+2, 2)$ for $a,b\geq 1$.
By Lemma \[(4b+2,3(4b+2);6,2)\], there is a shiftable $SMR(4b+2,3(4b+2);6,2)$ for $b\geq 1$, say $A$. Apply Lemma \[L2,4;4,2\] with $p=a-1$ and $q=2b+1$ to obtain a shiftable $SMR(2(2b+1),4(a-1)(2b+1);4(a-1),2)$, say $B$, for $a\geq 2$ and $b\geq 1$. Finally, apply Theorem \[TH:kn+n’;kr+r’\] with arrays $A$ and $B$ to obtain a shiftable $SMR(4b+2,(2a+1)(4b+2);4a+2, 2)$ for $a\geq 2$ and $b\geq 1$. When $a=1$ apply Lemma \[(4b+2,3(4b+2);6,2)\].
\[4b+2,(4a+1)(2b+1;4a+1,2\] There exists an $SMR(4b+2,(4a+1)(2b+1);4a+1,2)$ for $a,b\geq 1$.
Apply Lemma \[L2,4;4,2\] with $p=a-1$ and $q=2b+1$ to obtain a shiftable $SMR(2(2b+1), 4(a-1)(2b+1); 4(a-1), 2)$, say $A$, for $a\geq 2$. By Proposition \[m,5m/2;5,2\] there is an $SMR(4b+2,5(2b+1);5,2)$, say $B$, for $b\geq 1$. Finally, apply Theorem \[TH:kn+n’;kr+r’\] with arrays $A$ and $B$ to obtain an $SMR(4b+2,(4a+1)(2b+1);4a+1,2)$ for $a,b\geq 1$.
\[4b+2,(4a+3)(2b+1)(4a+3);4a+3,2\] There exists an $SMR(4b+2,(4a+3)(2b+1);4a+3,2)$ for $a, b\geq 0$.
Apply Lemma \[L2,4;4,2\] with $p=a$ and $q=2b+1$ to obtain a shiftable $SMR(2(2b+1), 4a(2b+1); 4a, 2)$, say $A$. By Proposition \[m,3m/2;3,2\] there is an $SMR(4b+2,3(2b+1);3,2)$, say $B$, for $b\geq 1$. Finally, apply Theorem \[TH:kn+n’;kr+r’\] with arrays $A$ and $B$ to obtain an $SMR(4b+2,(4a+3)(2b+1);4a+3,2)$ for $a,b\geq 1$.
We conclude this section with the following theorem.
\[mainTHSEC4\] Let $m$ be even. There exists an $SMR(m,n;r,2)$ if and only if either $m=2$ and $n=r\equiv 0,3 \pmod 4$ or $m\geq 4$, $r\geq 3$ and $mr=2n$.
The existence of an $SMR(m,n;r,2)$ with $m$ odd and $r$ even {#SEC5}
============================================================
In this section we investigate the existence of a signed magic rectangle $(m,n;r,2)$ with $m$ odd and $r$ even. Note that if $m$ and $r$ are both odd, then there is no $SMR(m,n;r,2)$.
The existence of an $SMR(m,n;4a,2)$ with $m$ odd
------------------------------------------------
We consider two cases: $m=4b+1$ and $m=4b+3$.
\[4b+1,2a(4b+1);4a,2\] There exists a shiftable $SMR(4b+1,2a(4b+1);4a,2)$ for all $a,b\geq 1$.
Apply Lemma \[L2,4;4,2\] with $p=a=1$ and $q=2(b-1)$ to obtain a shiftable $SMR(4(b-1),8(b-1);4,2)$ for $b\geq 2$.
Figure \[5,10;4,2\] displays a shiftable $SMR(5,10; 4,2)$. Therefore there is a shiftable $SMR(4b+1,2(4b+1);4,2)$ by Theorem \[TH:km+m’,kn+n’\]. Now apply Part 1 of Theorem \[TH:kn;kr-km,kn\] to obtain a shiftable $SMR(4b+1,2a(4b+1);4a,2)$ for all $a,b\geq 1$.
$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline
1&&&&-5&-6&&&&10 \\ \hline
-1&2&&&&6&-7&&& \\ \hline
&-2&3&&&&7&-8&& \\ \hline
&&-3&4&&&&8&-9& \\ \hline
&&&-4&5&&&&9&10 \\ \hline
\end{array}$$
\[4b+3,2a(4b+3);4a,2\] There exists a shiftable $SMR(4b+3,2a(4b+3);4a,2)$ for all $a,b\geq 1$.
Apply Lemma \[L2,4;4,2\] with $p=1$ and $q=2b$ to obtain a shiftable $SMR(4b,8b;4,2)$ for $b\geq 1$. Figure \[3,6;4,2\] displays a shiftable $SMR(3,6; 4,2)$. Therefore, by Theorem \[TH:km+m’,kn+n’\], there is a shiftable $SMR(4b+3,2(4b+3);4,2)$. We now apply Part 1 of Theorem \[TH:kn;kr-km,kn\] to obtain a shiftable $SMR(4b+3,2a(4b+3);4a,2)$ for all $a,b\geq 1$.
$$\begin{array}{|c|c|c|c|c|c|}\hline
1&&-3&-4&&6 \\ \hline
-1&2&&4&-5& \\ \hline
&-2&3&&5&-6 \\ \hline
\end{array}$$
The existence of an $SMR(m,n;4a+2,2)$ with $m$ odd
--------------------------------------------------
We consider two cases: $m=4b+1$ and $m=4b+3$.
\[4b+1,3(4b+1);6,2\] There exists a shiftable $SMR(4b+1,3(4b+1);6,2)$ for all $b\geq 1$.
Apply Part 2 of Theorem \[TH:kn;kr-km,kn\] with the shiftable $SMR(4,12;$ $6,2)$ given in Figure \[4,12;6,2\] to obtain a shiftable $SMR(4(b-1),12(b-1);6,2)$ for $b\geq 1$. Figure \[5,15;6,2\] displays a shiftable $SMR(5,15; 6,2)$. Therefore there is a shiftable $SMR(4b+1,3(4b+1);6,2)$ for $b\geq 1$ by Theorem \[TH:km+m’,kn+n’\].
[$$\begin{array}{|c@{\hspace{0.1mm}}|c@{\hspace{0.1mm}}|c@{\hspace{0.1mm}}
|c@{\hspace{0.1mm}}|c@{\hspace{0.1mm}}|c@{\hspace{0.1mm}}
|c@{\hspace{0.1mm}}|c@{\hspace{0.1mm}}|c@{\hspace{0.1mm}}
|c@{\hspace{0.1mm}}|c@{\hspace{0.1mm}}|c@{\hspace{0.1mm}}
|c@{\hspace{0.1mm}}|c@{\hspace{0.1mm}}|c@{\hspace{0.1mm}}|} \hline
1&-2&&&&-6&&&&10&&12&&&-15 \\ \hline
&2&-3&&&6&-7&&&&&&-13&&15 \\ \hline
&&3&-4&&&7&-8&&&-11&&13&& \\ \hline
&&&4&-5&&&8&-9&&&-12&&14& \\ \hline
-1&&&&5&&&&9&-10&11&&&-14& \\ \hline
\end{array}$$ ]{}
\[4b+1,(2a+1)(4b+1);4a+2,2\] There exists a shiftable $SMR(4b+1,(2a+1)(4b+1);4a+2,2)$ for all $a,b\geq 1$.
Apply Lemma \[L2,4;4,2\] with $p=1$ and $q=2b-2$ to obtain a shiftable $SMR(2(2b-2),4(2b-2);4,2)$ for $b\geq 2$. Figure \[5,10;4,2\] displays a shiftable $SMR(5,10; 4,2)$. Therefore there is a shiftable $SMR(4b+1,2(4b+1);4,2)$ for $b\geq 1$ by Theorem \[TH:km+m’,kn+n’\]. Now apply Part 1 of Theorem \[TH:kn;kr-km,kn\] to obtain a shiftable $SMR(4b+1,2(a-1)(4b+1);4(a-1),2)$, say $A_1$, for all $a\geq 2$ and $b\geq 1$. By Lemma \[4b+1,3(4b+1);6,2\] there exists a shiftable $SMR(4b+1,3(4b+1);6,2)$ for $b\geq 1$, say $A_2$. Now apply Theorem \[TH:kn+n’;kr+r’\] with $A_1$ and $A_2$ to obtain a shiftable $SMR(4b+1,(2a+1)(4b+1);4a+2,2)$ for $a\geq 2$ and $b\geq 1$. When $a=1$, we apply Lemma \[4b+1,3(4b+1);6,2\].
\[4b+3,3(4b+3);6,2\] There exists a shiftable $SMR(4b+3,3(4b+3);6,2)$ for all $b\geq 1$.
Apply Part 2 of Theorem \[TH:kn;kr-km,kn\] with the shiftable $SMR(4,12;$ $6,2)$ given in Figure \[4,12;6,2\] to obtain a shiftable $SMR(4b,12b;6,2)$ for $b\geq 1$. Figure \[3,9;6,2\] displays a shiftable $SMR(3,9; 6,2)$. Therefore there is a shiftable $SMR(4b+3,3(4b+3);6,2)$ by Theorem \[TH:km+m’,kn+n’\].
$$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline
1&-2&&-4&&6&7&-8& \\ \hline
&2&-3&4&-5&&-7&&9 \\ \hline
-1&&3&&5&-6&&8&-9 \\ \hline
\end{array}$$
\[4b+3,(2a+1)(4b+3);4a+2,2\] There exists a shiftable $SMR(4b+3,(2a+1)(4b+3);4a+2,2)$ for all $a,b\geq 1$.
Apply Lemma \[L2,4;4,2\] with $p=1$ and $q=2b$ to obtain a shiftable $SMR(2(2b),4(2b);4,2)$ for $b\geq 1$. Figure \[3,6;4,2\] displays a shiftable $SMR(3,6; 4,2)$. Therefore there is a shiftable $SMR(4b+3,2(4b+3);4,2)$ by Theorem \[TH:km+m’,kn+n’\]. Now apply Part 1 of Theorem \[TH:kn;kr-km,kn\] to obtain a shiftable $SMR(4b+3,2(a-1)(4b+3);4(a-1),2)$, say $A_1$, for all $a\geq 2$ and $b\geq 1$. By Lemma \[4b+3,3(4b+3);6,2\] there exists a shiftable $SMR(4b+3,3(4b+3);6,2)$, say $A_2$, for $b\geq 1$. Now apply Theorem \[TH:kn+n’;kr+r’\] with $A_1$ and $A_2$ to obtain a shiftable $SMR(4b+3,(2a+1)(4b+3);4a+2,2)$ for $a\geq 2$ and $b\geq 1$. When $a=1$ we apply Lemma \[4b+3,3(4b+3);6,2\].
We summarise the results obtained in Lemmas \[4b+1,3(4b+1);6,2\]-\[4b+3,(2a+1)(4b+3);4a+2,2\] in the next theorem..
\[mainTHSEC5\] Let $m$ be odd and $r$ be even. Then there exists a shiftable $SMR(m,n;r,2)$ if and only if $m\geq 3$, $r\geq 4$ and $mr=2n$.
We are now ready to state the main theorem of this paper.
[**Main Theorem.**]{} [*There exists an $SMR(m,n;r, 2)$ if and only if either $m=2$ and $n=r\equiv 0,3 \pmod 4$ or $m,r\geq 3$ and $mr=2n$.*]{}
[99]{} T. Harmuth, [*Ueber magische Quadrate und ahnliche Zahlenfiguren*]{}, Arch. Math. Phys. [**66**]{} (1881), 286–313.
T. Harmuth, Ueber magische Rechtecke mit ungeraden Seitenzahlen, Arch. Math. Phys. [**66**]{} (1881), 413–447.
A. Khodkar and D. Leach [*Magic squares with empty cells*]{}, Ars Combinatoria (to appear).
A. Khodkar, C. Schulz and N. Wagner, [*Existence of Some Signed Magic Arrays*]{}, Discrete Mathematics [**340**]{} (2017), 906–926.
R. G. Sun, [*Existence of magic rectangles*]{}, Nei Mongol Daxue Xuebao Ziran Kexue, [**21**]{} (1990), 10–16.
|
---
abstract: 'We review the concept of the number wall as an alternative to the traditional linear complexity profile (LCP), and sketch the relationship to other topics such as linear feedback shift-register (LFSR) and context-free Lindenmayer (D0L) sequences. A remarkable ternary analogue of the Thue-Morse sequence is introduced having deficiency 2 modulo 3, and this property verified via the re-interpretation of the number wall as an aperiodic plane tiling.'
author:
- Fred Lunnon
title: 'The Pagoda Sequence: a Ramble through Linear Complexity, Number Walls, D0L Sequences, Finite State Automata, and Aperiodic Tilings'
---
Introduction
============
In the early 1970’s the availability of the Berlekamp-Massey algorithm led to the emergence of the Linear Complexity Profile (LCP), as a measure of how well a sequence of (say) binary digits could be approximated by a Linear Feedback Shift-Register (LFSR) — a topic of some practical importance in the design of cryptographic key-stream sequences.
A less established alternative, previously known to rational approximation specialists by the somewhat unimaginative term [*C-table*]{}, is the [*number wall*]{} — an array of Hankel determinants formed from consecutive intervals of the sequence — which lends itself better to geometrical interpretation than the traditional LCP.
An algorithm for number-wall computation, generalising the classical Jacobi recurrence to the previously intractable case of zero determinants, was later discovered by the author, who typically then failed to get around to actually publishing it for another 25 years. It is applicable to sequences over any integral domain, and with care can be implemented to cost constant time per entry computed.
A particular area of interest involves sequences whose complexity according to this model is in some way extreme, such as that proposed by Rueppel with a so-called ‘perfect’ LCP. Sequences with ‘perfect’ number-walls are harder to find, in fact over a finite domain they appear not to be possible: a probabilistic argument gives approximate bounds on the depth of such tables, confirmed by computer searches modulo 2 and 5.
Despite this in 1997 was discovered a remarkable sequence with modulo 3 deficiency 2, that is its ternary number-wall contains only isolated zeros — or in plainer language, no linear recurrence or LFSR of order $m$ spans any $2m+2$ consecutive terms, for any order at any point. More remarkably still, computational evidence suggests that the same sequence has deficiency 2 modulo other primes of the form $p = 4k-1$.
The construction of this [*Pagoda*]{} sequence resembles that of the classical square-free Thue-Morse ternary sequence: an auxiliary sequence is generated via a D0L system, then mapped to the target sequence via a final extension morphism. Such D0LEC (D0L with extension and constant width) or ‘automatic’ sequences have some claim to form a natural complexity class immediately above the LFSR class, combining greater flexibility with accessible distribution properties.
The proof of the deficiency modulo 3 was finally accomplished two years later, involving the recasting of the number wall as a tiling of the plane — essentially a two-dimensional D0LEC — by a tesselation using 107 different varieties of tile. Proof for other primes remains elusive.
Linear Complexity
=================
A sequence $[S_n]$ is a [*linear recurring*]{} or [*linear feedback shift register*]{} (LFSR) sequence of order $r$, when there exists a nonzero vector $[J_i]$ (the [*relation*]{}) of length $r+1$ such that $$\sum_{i=0}^rJ_iS_{n+i}\ =\ 0 \quad\hbox{for all integers $n$.}$$ If the relation has been established only for $a\le n\le b-r$ we say that the relation [*spans*]{} $S_a,\ldots,S_b$, with $a=-\infty$ and $b=+\infty$ permitted.
Sequences may have as elements members of any integral domain: in applications the domain will usually be the integers or some prime (often binary) finite field. LFSR sequences over finite fields are discussed comprehensively in [@Lid97] §6.1–6.4. It must be emphasised that the same sequence may have very different linear complexity behaviour, according to the domain considered: this caveat will apply in particular to profiles and walls of integer sequences modulo a prime, often 2 or 3.
Of practical importance in the design of secure cryptographic key-stream sequences is the question of how well a binary sequence is approximated by (one or more) LFSR’s. Developed in the early 1970’s, the [*(Shifted) Linear Complexity Profile*]{} (LCP/SLCP) represented an attempt to establish a relevant quantitive formalism: given $[S_n]$, its LCP is an auxiliary sequence with $m$-th term the order of the minimal LFSR spanning segment $S_0,\ldots,S_{m-1}$; the SLCP generalises this reluctantly into two dimensions by considering the order of $S_n,\ldots,S_{n+m-1}$, where both $m$ and $n$ vary.
In recent years linear complexity has made little progress; and it is my contention that the major culprit is the accidental manner in which LCP’s were contrived. The Berlekamp-Massey algorithm had recently been developed, providing a means of computing the minimal relation spanning $n$ terms of a sequence in time quadratic in $n$. This seems then to have been seized upon by both coding and complexity communities — the latter simply discarding the components $[J_i]$ of the relation, retaining only the order $r$.
To introduce a personal note at this point, I have to confess to having never felt comfortable with Berlekamp-Massey: its application is tricky — for instance, the intermediate vectors it generates cannot be relied upon to represent relations spanning a prefix of the segment — and its proof (see [@Lid97]) strikes me as both complicated and lacking obvious direction.
A more natural and elementary alternative considers instead the simultaneous linear equations for the relation components $[J_i]$ in terms of the sequence elements $[S_n]$. Easily, these have a solution just when the [*Toeplitz*]{} determinant \[or with an extra reflection, [*Hankel*]{} or [*persymmetric*]{}\] $$S_{mn}\ =\ \left|\matrix{S_n&S_{n+1}&\ldots&S_{n+m}\cr
S_{n-1}&S_{n}&\ldots&S_{n+m-1}\cr
\vdots&\vdots&\ddots&\vdots\cr
S_{n-m}&S_{n-m+1}&\ldots&S_n\cr}\right|$$ vanishes.
A zero entry $S_{mn}$ indicates a relation of order $r\le m$ spanning the segment $[S_{n-m},\ldots,S_{n+m}]$. If the sequence is in fact generated by a single LFSR of order $r$, the table will be zero from row $r$ onwards: therefore this [*number-wall*]{} bears the same relation to an LFSR sequence as does the difference table to a polynomial sequence (where $S_n$ is a polynomial function of $n$); in fact, one generalises the other, to the extent that every polynomial sequence of degree $r-1$ is de facto LFSR of order $r$, with relation given by the vanishing of its $r$-th difference.
These determinants can also be computed in quadratic time, via an algorithm not only progressive \[so the time becomes effectively linear for a table of many values\], but beguilingly simple and symmetrical, and classical — being a special case of a well-known pivotal condensation rule or extensional identity credited variously to Sylvester, Jacobi, Desnanot, Dodgson, Frobenius: $$S_{m,n}^2\ =\ S_{m+1,n} S_{m-1,n}\ +\ S_{m,n+1} S_{m,n-1}.$$
Unfortunately, formulating a corresponding recursive algorithm, expressing each row in terms of the two previous, $$\begin{aligned}
S_{-2,n}\ &=&\ 0,\quad S_{-1,n}\ =\ 1,\quad S_{0,n} = S_n,\cr
S_{m,n}\ &=&\ \bigl(S_{m-1,n}{}^2\ -\
S_{m-1,n+1}S_{m-1,n-1})\bigr)/S_{m-2,n}\quad\hbox{for $m > 0$}\cr\end{aligned}$$ reveals immediately a major flaw: once a zero has been encountered, computation is unable to proceed beyond the subsequent row, on account of division by $S_{m-2,n} = 0$.
One of the more elementary properties — already familiar in the guise of the [*Padé block theorem*]{} (see [@Gra72]) to Padé table specialists \[who have incidentally been collectively responsible for a remarkable number of bogus proofs of it\] is that zero entries occur only as continuous square regions, surrounded by an [*inner frame*]{} of nonzeros \[easily seen by the Sylvester identity to comprise a geometric sequence along each edge\].
Some time around 1975, I succeeded in generalising the recursion to bypass such zero entries. Ironically (given my original motivation) even the statement of these [*frame theorems*]{} demands sufficient preliminary background to necessitate relegation to appendix A; and their convoluted and technical proof required several attempts, finally involving a combination of methods from ring theory, analysis and algebraic geometry, and sustained over a period of more than a quarter of a century [@Lun01].
There is plainly a close relationship between SLCP’s and number walls — see [@Ste92] for example. However, the more symmetrical definition of the latter considerably facilitates the deployment of genuinely two-dimensional geometry in their investigation, as we shall see later; in contrast, the (diagonal, one-dimensional) generating function technique — encouraged by the LCP paradigm — is for example unable to probe the central diamond of a number wall at all.
D0L and D0LEC systems: Thue-Morse sequence
==========================================
A deterministic context-free Lindenmayer (D0L) system is defined to be a substitution system where there is only one production for each symbol; all productions are applied simultaneously; and production is iterated, starting from some distinguished (stable) symbol, so generating an infinite sequence.
We further define D0LE to mean extended by a final (single-shot) substitution, usually to an alphabet distinct from that used by the generation stage; and D0LEC to mean that both morphisms (sets of production rules) have constant width on the right-hand side — for instance, no sneaky null symbols, mapping to the empty string! \[Much the same idea appears elsewhere under the umbrella of ‘automatic sequences’ — see ‘image, under a coding, of a $k$-uniform morphism’ in sect. 6 of [@All03].\]
Why should D0L (and particularly D0LEC) systems be worthy of study? LFSR systems arise naturally in a number of applications (signal-processing, cryptography), and the number wall is a natural tool with which to investigate them. When we come to study number walls in turn, their extremal behaviour is observed to occur for D0LEC sequences (which incidentally arise in other unrelated applications as well). So D0LEC sequences in some sense constitute a natural third layer in a complexity hierarchy commencing thus: polynomial sequences, LFSR sequences, D0LEC sequences, $\ldots$
The distribution properties of these sequences can easily be established, using classical Markov-process methods [@Fel57]. Another bonus is algorithmic: the D0LEC paradigm permits both the computation of a distant term $S_n$ of a sequence, and furthermore the inversion of this process to recover $n$ from $S_n$ (where this is single-valued), in time of order $\log n$ by means of a finite-state automaton — see [@All03].
In illustration of these ideas, we turn now to consider the Thue-Morse sequence. \[This is conventionally constructed as the fixed point of the morphism $0 \to 01,\ 1 \to 10$; however, the following indirect construction proves more illuminating.\] Recall that a sequence of symbols is [*square-free*]{} when no factor word (of consecutive symbols) is followed immediately by a copy of itself; similarly, a sequence may be [*cube-free*]{}, [*power-free*]{}.
Consider the D0L system on 4-symbols defined by the generating morphism $$\Phi: A \to BC,\ B \to BD,\ C \to CA,\ D \to CB;$$ notice the symmetry of $\Phi$ under the permutation $(AD)(BC)$. Starting from $B$ and applying $\Phi$ repeatedly gives what turns out to be a square-free right-infinite quaternary sequence: $$[V_n]\ =\ BDCBCABD\ CABCBDCB\ CABCBDCA\ BDCBCABD\ \ldots$$ \[This could be made left- and right-infinite by starting with $AB$ or $CB$ and fixing the origin in the centre; but then $\Phi^2$ rather than $\Phi$ would be required to obtain stability.\]
The final morphism $$A \to 0,\ B \to 0,\ C \to 1,\ D \to 1,$$ now yields the classical cube-free binary Thue-Morse sequence $$[T_n]\ =\ 01101001\ 10010110\ 10010110\ 01101001\ \ldots,$$ explicitly $T_n$ equals the sum modulo 2 of the digits of $n$ when expressed in binary. Alternatively, the final morphism $$A \to 0,\ B \to 1,\ C \to 2,\ D \to 0,$$ yields the related ternary sequence $$[U_n]\ =\ 10212010\ 20121021\ 20121020\ 10212012\ \ldots,$$ which can be shown to be square-free. Proofs are given in appendix B; they bear comparison with rather complicated ad-hoc arguments available elsewhere, e.g. [@Lot83].
More significantly, other final morphisms may be tailored to produce new sequences, such as $$A \to 11,\ B \to 01,\ C \to 10,\ D \to 00,$$ yielding a binary sequence which has no squared words of length exceeding 6: $$01001001\ 10110100\ 10110110\ 01001001\ \ldots,$$ and $$A \to 1101,\ B \to 0011,\ C \to 1000,\ D \to 0010,$$ with no squares exceeding length 4 (optimal): $$01110010\ 10000111\ 10001101\ 01110010\ 10001101\ 01111000\ \ldots$$
Average versus Extremal Walls
=============================
We propose to illustrate the discussion using an interactive Java application which displays number walls of various special sequences modulo a given prime. Entries are encoded as coloured pixels: white for 0, black for 1, grey for 2; or red for 2, green for 3, blue for 4, etc. interactively; the sequence runs along two rows from the top edge. Program source [*ScrollWall.java*]{} is available from the author;
The implementation is based on the frame theorems (appendix A), incorporating an enhancement to obviate searching when circumnavigating a large window. \[The binary case is particularly simple, to the extent that an exceptionally efficient implementation is feasible in the form of a 44-state cellular automaton based on the Firing-Squad Synchronisation Problem (FSSP) — see [@Min67]; [@Lun01] sect.7.\] The given finite segment must be extended into a periodic sequence, to avoid algorithmic complications resulting from the presence of a boundary: therefore in general, only the triangular north quarter of a (square) graphical display is significant; although in special cases, intelligent choice of segment length $n$ may improve this situation. Since a sequence with period $r$ is LFSR with order at most $r$, the number of nonzero rows (including the initial row of empty determinants) for a segment of length $n$ columns must be at most $n+1$.
So as to have something for later comparison, we first take a look at a ‘typical’ number wall. When the domain is a finite field with $q$ elements, it can be shown that for a random sequence, the asymptotic mean density of size-$g$ (or $g\times g$) windows exists (in some suitably weak sense), and equals $$(q-1)/(q+1)q\ \cdot\ 1/q^g ;$$ for example, , modulo $q = 2,3.$
It is tempting to employ this result as a test for randomness: for example, counting the numbers of windows of each size in a suitably large portion of the wall, then applying the $\chi^2$ test to the frequencies. A discouraging counterexample is the sum of the Thue-Morse and Rook sequences modulo 2(see ), which passes this test with flying colours, despite being generated by the 8-symbol D0LEC system: $$A\to Ab, B\to Ad, C\to Cb, D\to Cd, a\to aB, b\to aD, c\to cB, d\to cD;$$ $$A\to 0, B\to 0, C\to 1, D\to 1, a\to 1, b\to 1, c\to 0, d\to 0.$$
Our major target in this essay is the investigation of extremal walls: by which is meant, the extent to which a number wall may deviate from typical window distribution. One pretext for this activity is exposure of the limitations of the paradigm; but it might be more honest to prefer the serendipitous justification, that some of the graphic art so produced is simply rather striking \[and might become more so, were the author’s casually primitive palette to be refined!\]
The (not overly impressive) example of the Rueppel sequence makes a point about the limitations of the original LCP concept. Its definition is $$S_n = \cases{
1 &if $n = 2^k - 1$ for some $k$;\cr
0 &otherwise.\cr}$$ It was proposed as an example of a binary sequence having ‘perfect’ LCP, which in number-wall terms implies a continuous nonzero diagonal staggering from one corner to the opposite. Elsewhere though, its wall is perfectly appalling, composed almost exclusively of windows increasing exponentially in size (see ).
But it suggests an analogous though considerably tougher challenge, which we proceed to take up: to determine the extent to which a (binary, say) wall can avoid zero entries. A combinatorial argument based on the frame theorems shows easily that any extended region of the wall has local zero-density at least $1/5$ — the minimal pattern has isolated zero entries occurring a knight’s move apart. Globally, this minimal pattern can occupy at best an infinite central diamond, the rest of the wall comprising a fractal-like pattern of increasing windows and finite minimal diamonds, see .
To explicitly construct the sequence with this wall, first define the [*Rook*]{} sequence $[R_n]$ as the digit preceding the least-significant $1$ in the binary expansion of $n$, or $0$ if $n=0$ — compare with Thue-Morse. E.g. if $n = 104 = 1101000$ in binary, the final $1$ is 3 digits along from the end, and the 4-th digit along is $R_{104} = 0$. $[R_n]$ is a binary sequence, and a recursion for it is $$R_{-n}\ =\ 1-R_n\hbox{ for }n \ne 0;\quad R_{2n}\ =\ R_n;
\quad R_{2n+1}\ =\ n\bmod 2.$$ The first few values for $n\ge0$ are $$[R_n]\ =\ 00010011\ 00011011\ 00010011\ 10011011\ \ldots$$ Finally, define the [*Knight*]{} sequence by $$K_n = R_{n+1} - R_{n-1} \pmod 2 .$$
For ternary walls, the situation is rather similar: an essentially unique nonzero local pattern exists, composed of alternating zigzag stripes of $+1$ and $-1$ resp. Globally, this motif can be replicated only within a central diamond; the remainder of the wall is now rather sparse, not unlike the Rueppel wall, see . An explicit expression for this sequence is clumsy, but it has the D0LEC definition: $$A \to ACB,\ B \to BCB,\ C \to EDF,\ D \to DDD,\ E \to EDD,\ F \to DDF;$$ $$A \to 1,\ B \to 0,\ C \to 1,\ D \to 0, E \to 2,\ F \to 2.$$ Starting from $A$, the first few terms of generated and final sequences are $$ACBEDFBCB\ EDDDDDDDF\ BCBEDFBCB\ \ldots;$$ $$[Z_n]\ =\ 110202010\ 200000002\ 010202010\ 200000000\ 000000000\ \ldots$$
Accepting that a total absence of zeros (on rows $m\ge-1$) is not possible, we can instead attempt in various ways to circumscribe their occurrence. Rather than become involved in somewhat recondite questions regarding what exactly might be meant by the term [*density*]{} in this context, we shall consider the more concrete problem of bounding the size of the windows.
A simple probabilistic argument can be mounted suggesting that, when the domain is a finite field with $q$ elements, the size of the maximum window occurring within the first $m$ rows of a wall will be of the order of $\log_q m$; and more strongly, that the probability of a sequence having no windows larger than this bound is zero. \[This contrasts with the situation for square-free sequences, where the corresponding probability is nonzero for $q > 2$.\]
With this in mind, we conducted a search for binary sequences with the greatest number of rows having no window of size $d$ or greater, for small values of the \[in LCP jargon\] [*deficiency*]{} $d$. This endeavour is highly speculative: first the critical depth $m$ must be established such that no satisfactory sequence exists with greater depth; then a sufficiently long segment constructed for an evident period to become established.
The resulting handful of sequences is shown in the table: all are periodic with period $t$, and the order $r$ equals the final depth $m$ satisfying the deficiency bound — that is, as soon as the bound fails, the entire wall vanishes — and $m$ seems to increase exponentially with $d$ as expected. Confidence in these results is encouraged by the presence of adventitious symmetries, such as the 0-1 alternating subsequence at odd positions of case $d = 4$. See ,,.
[0.9]{}[@ | r | r | r | r | | l | ]{} $d$ & $m$ & $r$ & $t$ & period 1 & 1 & 1 & 1 & \[1\] 2 & 5 & 5 & 6 & \[111010\] 3 & 19 & 19 & 20 & \[1111010100 1111010010\] 4 & 56 & 56 & 60 & \[0001100100 0110110011 0001101100 & & & & 1110110001 1001001100 1110010011\] 5 & 95+ & ? & ? & (none detected in 800 terms)
Now what about ternary walls? Deficiency $d = 1$ is disposed of trivially, by the period-4 sequence \[1122…\] with $m = r = 2$. But when our search program is let loose on $d = 2$, the first of a number of strange things happens — or in this case, fails to happen — the depth goes on increasing indefinitely, while (necessarily) no period ever properly quite stabilises. To cut quite a long story short, the object which eventually emerges is a remarkably simple D0LEC, has deficiency-2 to any depth we care to examine, and turns out to be essentially identical to the Knight $[K_n]$ — seen earlier in an unrelated context!
To be precise, with $R_n$ the Rook sequence as above, the [*Pagoda*]{} sequence is defined by $$P_n = R_{n+1} - R_{n-1} \pmod 3 .$$ The ternary number wall is shown at ; the symmetrical, fractal-like filigree structures for which it was christened are more easily appreciated after rotation through a quarter-turn, the sequence running down the left side.
Examination of substantial portions of the number-walls of this sequence modulo $$p = 3,7,11,19,23,31,43,47,59,67,71,79$$ encourages the conjecture that its deficiency remains equal to 2 modulo [*any*]{} prime $p = 4k-1$; modulo $p = 83$ however, this elegant simplicity is confounded by the presence of numerous windows of size 2, together with what appears to be a splendidly lone specimen of size 3 commencing atentry $m = 105, n = 188$ \[a specimen discovered only during protracted investigation of an apparent compiler bug causing ScrollWall to report spurious runtime errors\].
The Pagoda was not the first, nor the last of its kind to be discovered: but all these, along with the Knight and Rook sequences, are closely intertwined, in a manner notably reminiscent of our earlier analysis of the Thue-Morse family. Consider the 4-symbol D0L system $$A \to AB,\ B \to AD,\ C \to CB,\ D \to CD;$$ applied to $A$ this generates $$[V_n] = ABADABCD\ ABADCBCD\ ABADABCD\ CBADCBCD\ \ldots$$ \[which can be made infinite both ways by starting instead from $DA$ and choosing the origin to be the first symbol of the (inflated) original $A$.\] Applying the final morphism $$A \to 2201,\ B \to 0211,\ C \to 0221,\ D \to 1201$$ yields the Pagoda sequence $[P_n]$, for $n \ge 0$ \[or all $n$\]; $$A \to 1101,\ B \to 0111,\ C \to 0111,\ D \to 1101$$ yields the Knight sequence $[K_n]$.
Other deficiency-2 variations on the Pagoda may be concocted by varying the final morphism. Also notice that the generator is not symmetric under either transposition $(AC)$ or $(BD)$: so these provide a set of 4 distinct generators, each of which could be used to yield an alternative quaternary sequence. Applying the final morphism $$A \to 0,\ B \to 0,\ C \to 1,\ D \to 1$$ to any of these alternatives yields the same binary sequence, the Rook $[R_n]$.
Modulo these variations, and the continuum of variants obtained by shifting the origin repeatedly during generation, it seems quite plausible that the Pagoda is the unique ternary sequence with this deficiency.
Pagoda Tiling Proof
===================
At this point, we have some probabilistic arguments and experimental evidence to support the conjectures that:
- If $p \bmod 4 = 1$ or $p = 2$, then the maximum depth $m$ to which deficiency $d$ can be maintained by any number wall modulo $p$ is finite, bounded by order $\log_p d$;
- If $p \bmod 4 = -1$, then the number wall modulo $p$ of the Pagoda sequence has bounded deficiency (dependent only on $p$) to any depth; in particular, for $p = 3$ we have $d = 2$ (only isolated zeros).
To actually prove any of these claims poses a considerable challenge. A conventional approach to the Pagoda conjecture might involve explicit algebraic evaluation of the Toeplitz determinants, modulo 3, modulo $p$, or over the integers: while there is some numerical structure visible here which might form a basis for an inductive construction, overall this prospect is not promising.
A more unexpected route proves at least partially successful: invoking a two-dimensional geometrical version of the D0LEC paradigm, extending the representation of the sequence via $[V_n]$ above, into one of the entire wall as a plane quasi-crystallographic tiling. In part this is suggested by close visual inspection of the diagram, which reveals (at the cost of substantial hazard to eyesight) that the ‘pagodas’ recurring at various scales throughout the wall are embedded in repetitive [*diamonds*]{}, square regions rotated through a one-eighth turn.
Factors to be taken into account in the formalisation of this concept include:
- Interaction between faces, edges and vertices of tiles;
- Non-trivial point symmetries tiles may possess;
- Choice of an appropriate translation of tiling origin;
- Extent to which tiles are open or closed subsets of the plane;
- Determination of tile size, or D0L extension width;
- Determination of number of distinct tiles, or D0L symbol count.
All these factors, along with other details relevant to implementation only at a detailed level, need be taken into account in the design of a program to (as it were) tile a wall — to specify the precise spatial ‘inflation’ morphism generating it, along with the extension ‘pattern’ on each tile.
It is natural to align the vertices of a tile with entries of the number wall, so that an entry at a vertex is shared between 4 adjacent tiles, at an edge between 2. This presents a conflict between notational clarity and computational simplicity, resolved by including the entire boundary in tile morphism diagrams (appendix C); while to actually apply a morphism, the boundary must be shrunk and displaced by a half-unit along each axis, so that a tile comprises only complete entries.
In order to verify the frame relations between wall entries, as well as to keep track of inflation of vertices and edges along with faces of tiles, the search program actually operates a 4-fold covering of the plane by overlapping [*supertiles*]{} having twice the diameter of the faces. A post-processor extracts the individual inflations of faces etc, possibly resulting in tile extents becoming reducible to smaller diameter. At this stage also, point-group symmetries of ‘fixed’ tiles are extracted; the number of ‘free’ tiles remaining is then substantially reduced.
For the ternary Pagoda, the program successfully finds a tiling comprising:
- Generator inflation diameter 2 (4 subtiles per inflation);
- Point symmetry group of order 16;
- Tiling origin at $S_{-2,0}$;
- Extent diameter of face 4 (partially spanning 25 wall entries);
- Fixed face count 107, reducing to 13 free;
- Every free face occurring within distance 35 from the origin;
- Free vertex count 39, all within distance 165;
The full morphism will be found in appendix C. Point symmetries comprise products of vertical reflection, horizontal reflection, complementation of odd rows, complementation of odd columns.
Apart from two restricted to meeting the upper zero half-plane $m\le -2$, every tile has only isolated zeros: this completes the proof that the deficiency of the ternary Pagoda equals 2.
But of course, the existence of this tiling permits us to investigate the wall in much greater detail. For instance, by selectively expanding the D0LEC, any given entry $S_{mn}$ can now be computed in time of order logarithmic in the distance $|m| + |n|$ from the origin.
Again, the deficiency theorem may be considerably sharpened:
- If $S_{mn} = 0$ in the ternary wall of the Pagoda sequence $S_n = P_n$, then the power of 2 dividing $m+2$ exceeds that dividing $n$.
In particular, no zeros can occur on rows with $m$ odd, nor on column $n = 0$ (for $m \ge -1$, that is).
Again, applying Markov process analysis to the D0LEC, a $13\times13$ matrix eigenvalue computation establishes that
- Zero entries in this wall possess asymptotic density in a strong sense, and this density equals $3/20$.
While the tiling method has successfully been applied in other simple cases, such as the Knight (6 free faces), Rook ($\le 28$), and Thue-Morse, it has not so far succeeded in tiling the Pagoda modulo 7. Neither is it known whether or not the number wall of every D0LEC sequence can be so tiled: a noteworthy test-case in this respect is the ‘quasi-random’ binary Thue-Rook sum $S_n = T_n + R_n \pmod 2$ mentioned earlier, with window size bounded apparently by order $(\log m)$.
Statement of the Frame Theorems.
================================
A zero entry $S_{m,n} = 0$ in a number-wall can occur only within a [*window*]{}, that is a square $g\times g$ zero region surrounded by a nonzero inner frame. The nullity of (the matrix corresponding to) a zero entry equals its distance $h$ from the (nearest) inner frame edge.
The adjacent diagram illustrates a typical window, together with notation employed subsequently: $$\matrix{
& &E_0 &E_1 &E_2 &\ldots &E_k &\ldots&E_g &E_{g+1} & &\cr
&F_0 &B,A_0 &A_1 &A_2 &\ldots &A_k &\ldots&A_g &A,C_{g+1} &G_{g+1}&\cr
&F_1 &B_1 &{\bf 0}&{\bf 0}&\ldots &{\bf 0}&\ldots&{\bf 0}&C_g &G_g &\cr
&F_2 &B_2 &{\bf 0}&\ddots &(P) &\rightarrow & &\vdots
&\vdots &\vdots &\cr
&\vdots &\vdots &\vdots &(Q) &\ddots& &\uparrow &{\bf 0}&C_k &G_k &\cr
&F_k &B_k &{\bf 0}&\downarrow & &\ddots&(R) &\vdots&\vdots &\vdots &\cr
&\vdots &\vdots &\vdots & &\leftarrow &(T) &\ddots&{\bf 0}&C_2 &G_2 &\cr
&F_g &B_g &{\bf 0}&\ldots &{\bf 0}&\ldots&{\bf 0}&{\bf 0}&C_1 &G_1 &\cr
&F_{g+1}&B,D_{g+1} &D_g &\ldots &D_k &\ldots&D_2 &D_1 &D,C_0 &G_0 &\cr
& &H_{g+1} &H_g &\ldots &H_k &\ldots&H_2 &H_1 &H_0 & &\cr
}$$
The inner frame of a $g\times g$ window comprises four geometric sequences, along North, West, East, South edges, with ratios $P,Q,R,T$ resp., and origins at the NW and SE corners. The ratios satisfy $$PT/QR\ =\ (-)^g;$$ and the corresponding inner frame sequences $A_k,B_k,C_k,D_k$ satisfy $$A_kD_k/B_kC_k\ =\ (-)^{gk} \quad\hbox{for $0\le k \le g+1$}.$$
The outer frame sequences $E_k,F_k,G_k,H_k$ lie immediately outside the corresponding inner, and are aligned with them. They satisfy the relation: For $g\ge 0$, $0\le k\le g+1$, $$QE_k/A_k\ +\ (-)^k PF_k/B_k\ =\ RH_k/D_k\ +\ (-)^k TG_k/C_k.$$
Proofs are expounded in [@Lun01] sect. 3–4.
Proofs that $[V_n]$, $[T_n]$, $[U_n]$ are power-free.
=====================================================
We sketch the proofs that these sequences are power-free as claimed. Suppose that $[V_n]$ is not square-free, and let the earliest occurrence of its shortest non-empty square start at $V_n$ for $n\ge 0$, with length $2l>0$. Suppose $l$ is even: if $n$ is odd, by inspection of $\Phi$ there is only one possible value for $V_{n-1} = V_{n+l-1}$ given $V_{n} = V_{n+l}$, so there is an equally short square earlier; if $n$ is even, we can apply $\Phi^{-1}$ to produce a shorter square of length $l/2$. Suppose on the other hand $l$ is odd: then for each $i$ one of $V_{n+i}$ and $V_{n+l+i}$ has an even subscript, so by inspection has to be $B$ or $C$. No new pairs are generated after $\Phi^3B$, so all words of length 4 occur within $\Phi^4B$; the longest composed of $B$ and $C$ only is seen to have length 3. So $2l \le 3$, and the square must be $BB$ or $CC$, which do not occur in $\Phi^3B$. By contradiction, $[V_n]$ is square-free.
The inverse morphism from $[U_n]$ to $[V_{n-1}]$ is uniquely defined for $n \ge 1$, given either of the symbols $U_{n\pm 1}$ adjacent to $U_n$: it is described by the schema $$(2)0(1)\to A,\quad 1\to B,\quad
2\to C,\quad (1)0(2)\to D,$$ where $U_{n\pm 1}$ is parenthesised. If $[U_n]$ had a square factor with $l > 2$, its inverse image would also be a square in $[V_n]$, since $A$ and $D$ in corresponding positions necessarily have an adjacent $B$ and $C$; but $[V_n]$ is square-free. If $l = 2$ the inverse image might be $AD$ or $DA$, but neither occurs in $[V_n]$.
The inverse morphism from $[T_n]$ to $[V_n]$ is uniquely defined for $n \ge 2$, given $T_{n+1}$; it is described by the schema $$0(0)\to A,\quad 0(1)\to B,\quad
1(0)\to C,\quad 1(1)\to D,$$ where $T_{n+1}$ is parenthesised. If $[T_n]$ had a cubic factor, its inverse image would also be a cube in $[V_n]$, except possibly for the final symbol; but $[V_n]$ is square-free.
Pagoda Tiling Morphisms
=======================
Free tiles are numbered 1–13. The ‘gene’ field diagrams the $2\times2$ diamond into which the tile inflates under the generator morphism, each entry comprising a tile number followed by a combined transformation code. The ‘extn’ field diagrams the $4\times4$ ternary number-wall diamond into which the tile finally extends, including boundary shared with neighbouring tiles. The ‘symm’ field notes all transformations which are symmetries of the tile. Transformation encoding is as follows:
[0.5]{}[@ | c | l | ]{} & [*transform*]{} A & identity B & reflection along rows C & reflection along cols D & half-turn rotation I & identity J & complement odd rows K & complement odd cols L & complement odd rows & cols
0
2 0 0 0
Tile 1: gene 1B 1 , extn 0 0 0 0 0, symm AI,BI;
4 1 1 1
1
0
2 0 0 0
Tile 2: gene 2 2 , extn 0 0 0 0 0, symm full;
2 0 0 0
0
0
3 1 1 1
Tile 3: gene 5 7 , extn 1 2 2 0 1; symm AI;
6 2 1 1
1
0
3B 1 1 1
Tile 4: gene 5D 7BJ , extn 1 1 2 0 1; symm AI;
8 2 1 1
1
1
10 1 1 2
Tile 5: gene 9 9BK , extn 1 0 2 0 1, symm AI,BK;
11 1 1 2
1
1
6C 2 1 1
Tile 6: gene 7BK 7 , extn 1 0 2 0 1, symm AI,BK;
10BJ 2 1 1
1
1
3CJ 1 1 1
Tile 7: gene 12 12BL, extn 1 2 0 1 1, symm AI,CI;
3J 1 1 1
1
1
6D 1 1 2
Tile 8: gene 5D 5D , extn 1 1 2 2 1, symm AI,BK;
11C 2 2 1
1
1
4D 1 1 1
Tile 9: gene 13 13BL, extn 1 1 0 2 1, symm AI,CI;
4B 1 1 1
1
1
8C 2 1 1
Tile 10: gene 5B 5B , extn 1 1 2 2 1, symm AI,BK;
10B 1 2 2
1
1
11J 1 1 2
Tile 11: gene 7J 7BL , extn 1 0 2 0 1, symm AI,BK;
8B 1 1 2
1
1
8BJ 1 1 2
Tile 12: gene 9BJ 12 , extn 1 0 2 1 0, symm AI,CI;
8DJ 1 1 2
1
1
6J 2 1 1
Tile 13: gene 9BL 12J , extn 1 0 2 1 0, symm AI,CI;
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Allouche, Jean-Paul & Shallit, Jeffrey *Automatic Sequences Cambridge (2003). Conway, J. H. & Guy, R. K. *The Book of Numbers Springer (1996). Feller, William *An Introduction to Probability Theory and its Applications vol I Wiley (1957). Gragg, W. B. *The Padé Table and its Relation to Certain Algorithms of Numerical Analysis SIAM Review **14 (1972) 1–62. Lidl, R. & Niederreiter, H. *Introduction to Finite Fields and their Applications Cambridge (1997). Lothaire, M. *Combinatorics on Words Addison-Wesley (1983). Lunnon, W. F. *The Number-Wall Algorithm: an LFSR Cookbook Article 01.1.1 Journal of Integer Sequences **4 (2001). Minsky, M. *Computation: Finite and Infinite Machines Prentice-Hall (1967). Stephens, N. M. *The Zero-square Algorithm for Computing Linear Complexity Profiles in Mitchell, Chris (ed.) *Cryptography and Coding II Clarendon press Oxford (1992) 259–272.**************
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abstract: 'Progress in the theory of anomalous diffusion in weakly turbulent cold magnetized plasmas is explained. Several proposed models advanced in the literature are discussed. Emphasis is put on a new proposed mechanism for anomalous diffusion transport mechanism based on the coupled action of conductive walls (excluding electrodes) bounding the plasma drain current (edge diffusion) together with the magnetic field flux “cutting” the area traced by the charged particles in their orbital motion. The same reasoning is shown to apply to the plasma core anomalous diffusion. The proposed mechanism is expected to be valid in regimes when plasma diffusion scales as Bohm diffusion and at high $B/N$, when collisions are of secondary importance.'
address: 'Department of Physics and Center for Plasma Physics,& Instituto Superior Tecnico, Av. Rovisco Pais, & 1049-001 Lisboa, Portugal'
author:
- 'Mario J. Pinheiro'
bibliography:
- 'Doc2.bib'
title: Anomalous Diffusion at Edge and Core of a Magnetized Cold Plasma
---
[^1]
Introduction
============
Any phenomena occurring with interfacial systems has a fundamental importance in science and technology [@Melehy_1] (e.g. electrostatic charging of insulators, surface tension, forward conduction in p-n junctions). Specifically, the problem of the plasma-wall interactions is of major importance in plasma physics.
Historically, the anomalously high diffusion of ions across magnetic field lines in Calutron ion sources (electromagnetic separator used by E. Lawrence for uranium isotopes) gave the firsts indications of the onset of a new mechanism [@Bohm]. It has been noticed that the plasma moves across the magnetic confining field at a much higher average velocity than it is predicted by classical considerations. The classical diffusion coefficient is given by $D_{\perp}=\eta p/B^2$, while the anomalous diffusion coefficient by $D_{\perp}=\alpha kT/B$. Manifestly it is needed a better understanding of the physical laws governing matter in the far nonequilibrium state, and this is a challenging issue for the advancement of this frontier of physics.
General proposed explanations were advanced. The first one was Simon’s “short-circuit” problem, suggesting that the observed losses could be explained by the highly anisotropic medium induced by the magnetic field lines, favoring electron current to the conducting walls [@Simon]. Experiments done by Geissler [@Geissler1] in the 1960’s have shown that diffusion in a plasma across a magnetic field was nearly classical (standard) diffusion when insulating walls impose plasma ambipolarity, but in the presence of conducting walls charged particles diffused at a much higher rate.
This problem of plasma-wall interaction becomes more complex when a complete description is aimed of a magnetized nonisothermal plasma transport in a conducting vessel. Beilinson [*et al.*]{} [@Beilinson] have shown the possibility to control the discharge parameters by applying potential difference to sectioned vessel conducting walls.
In the area of fusion reactors, there is strong indication that for plasmas large but finite Bohm-like diffusion coefficient appears above a certain range of $B$ [@Montgomery2]. Experiments give evidence of transport of particles and energy to the walls [@Luce]. At the end of the 1960s, experimental results obtained in weakly ionized plasma [@Geissler1] and in a hot electron plasma [@Ferrari] (this one proposing a possible mechanism of flute instability) indicated a strong influence that conducting walls have on plasma losses across magnetic field lines. Geissler [@Geissler1] suggested that the most probable explanation was the existence of diffusion-driven current flow through the plasma to the walls. Concerning fusion reactions, Taylor [@Taylor] provided a new interpretation of tokamak fluctuations as due to an inward particle flux resulting from the onset of filamentary currents.
Progress in the understanding of the generation of confinement states in a plasma is fundamental [@Itoh] to pursue the dream of a fusion reactor [@Bickerton; @Shafranov]. Anomalous diffusion is a cornerstone in this quest, as recent research with tokamaks suggest that the containment time is $\tau \approx 10^8 R^2/2D_B$, with $R$ denoting the minor radius of a tokamak plasma and $D_B$ is the Bohm diffusion coefficient [@Rostoker]. Controlled nuclear fusion experiments have shown that transport of energy and particles across magnetic field lines is anomalously large (i.e, not predicted by classical collision theory).
The conjecture made by Bohm is that the diffusion coefficient is $D_B=\alpha kT/eB$, where $T$ is the plasma temperature and $\alpha$ is a numerical coefficient, empirically taken to be $1/16$ [@Bohm]. Initially, the origin of the anomalous diffusion was assumed to be due to the turbulence of small-scale instabilities (see, for example, Refs. [@Montgomery2; @Taylor; @Montgomery1]). However, it is now clear that there is a number of different mechanisms that can lead to anomalous diffusion such as, coherent structures, avalanches type processes and streamers, which have a different character than a purely diffusive transport process. Recent experimental results such as scaling of the confinement time in L-mode plasmas and perturbative experiments undermine the previous paradigm built on the standard transport processes [@Callen; @Berk] showing conclusively that there are many regimes where plasma diffusion does not scale as $B^{-1}$.
This paper put emphasis on a mechanism of wall current drain set up together with the magnetic field “cutting” lines across the area traced by the charged particles trajectories. The proposed mechanism of anomalous diffusion is expected to be valid in purely diffusive regimes when plasma diffusion scales as Bohm diffusion, both in the edge and core of a cold magnetized plasma. At his stage it was considered of secondary importance the role of collisions in randomizing the particle’s distribution function. From collisional low temperature plasmas to a burning fusion plasma subject the plasma confinement vessel to strong wall load, both in stellarator or tokamak operating modes, this explanation could be of considerable interest, particularly when diffusive transport process are dominant.
We would like to stress that this work is not free from omission of important contributions.
Simon’s “short-circuit” theory
==============================
The first attempt to explain why the plasma diffuse at a much higher average velocity than it is predicted by classical theory has been advanced by A. Simon [@Simon]. The magnetic field lines structure a highly anisotropic medium. Any fluctuation of the space charge builds up an electric field, which has a strong effect on the currents parallel to the magnetic field lines. In fact, the classical equation for conductivity across a magnetic field is given by $$\label{}
\sigma_{\perp} = \frac{\sigma_0}{1+\frac{\Omega_c^2}{\nu_e^2}},$$ where $\Omega_c=eB/m$ is the electron cyclotron frequency, $\nu_e$ is the electron collision frequency, and $\sigma_0=e^2 n_e/m\nu_e$ is the conductivity in the absence of a magnetic field. By the contrary, due to $\Omega_c/\nu_e \gg 1$, this electric field is too small to have any importance on the crossed-field conductivity. From this results that there is a strong current to the wall without a concomitant current to different regions of the plasma, making of this situation a kind of circuital “short-circuit” problem. Although Simon attempted to explain the anomalously high rate of diffusion in Calutron ion sources in the frame of the classical diffusion theory calculating the coefficient $D_{\perp}$ as being approximately equal to the transverse diffusion coefficient of the ions. His proposal is not suitable, however, because the experimental determination of $D_{\perp}$ by means of a decaying plasma have shown that according to the magnetic field strengthen the transverse diffusion coefficient can be much higher than the classical one or smaller than the transverse diffusion coefficient of the ions [@Geissler1].
Plasma turbulence and transport
===============================
Purely diffusive transport models cannot give convincing explanations for a variety of experiments in magnetically confined plasmas in fusion engineering devices, particularly the scaling of the confinement time in L-mode plasmas. The assumed underlying instabilities are driven by either the pressure gradient or the ion temperature gradient. It is well established fact that transport in high temperature confined plasmas is driven by turbulence and plasma profiles, and are subject to transition from L-mode to H-mode (characterized by a a very steep gradient near the plasma surface) [@Itoh_03]. The non-linearity in the gradient-flux relation is the source of turbulence and turbulence-driven transport. The fluxes contain all the dynamic information on the transport process. Accordingly, changes in the gradient trigger local instabilities in the plasma. This local instability induces an increase in the nearby gradients, thus causing a propagation of the instability all across the plasma. In particular, an excessive pressure on the core propagates to the edge in a kind of avalanche.
In weakly turbulent cold magnetized plasmas, besides the Calutron ion sources and the magnetron, the study of particle transport in crossed electric and magnetic fields results from applications to electromagnetic space propulsion (Hall thrusters). Those plasma accelerators work with a radial magnetic field that prevents electron flow toward the anode and forcing the electrons closed-loop drift around the axis of the annular geometry. Neutrals coming from the anode are ionized in this rotating electrons cloud, while ions are accelerated by an axial electric field that freely accelerate them out from the device. This effect develops in the so called extended acceleration zone (or electric-magnetic region plasma). In this acceleration zone the electron gyro-radius and the Debye shielding length are small relative to the apparatus dimension, while the ion gyro radius is larger than the apparatus typical length. From these spatial scales results that the electron motions are $[\mathbf{E} \times \mathbf{B}]$ drifts, but ions are accelerated by the electric field that develops in the plasma. The first observations of a large amount of electron transport toward the anode have been noticed in the 60’s (e.g., Ref. [@Janes_66]) and they have been related to electric field fluctuations since they were correlated with the density variations in order to produce anomalous transport. Other possible mechanism that could explain the high electrons transverse conductivity were advanced: collisions with the wall [@Morozov_72; @Morozov_87]. Electrons moving freely along the lines of forces of the magnetic field collide with the wall more frequently than with ions and neutrals, being reflected at the wall and enhancing emission of low-energy secondary electrons from the wall. As referred, the other strong candidate, which could possibly be the source of a higher axial electron current than predicted by the standard classical kinetic theory, is the turbulent plasma fluctuations. But it seems that there is no clear consensus on this issue [@Boeuf_98; @Smirnov_04; @Hofer_06].
The magnetron is a sputtering tool, used for reactive deposition and etching. The magnetron effect is applicable to different geometries and only need a closed-loop $[\mathbf{E} \times \mathbf{B}]$ drift to work. Rossnagel [*et al.*]{} [@Rossnagel_86] have shown that the Hall-to-discharge current ratio measured in those configurations could be explained if the high collision frequencies for electrons were associated to Bohm diffusion. In particular, Kaufman [@Kaufman_85] argues that anomalous diffusion in closed-loop $[\mathbf{E} \times \mathbf{B}]$ thrusters could shift from core diffusion to edge diffusion (or wall effects) with increasing magnetic fields.
Circuital model of anomalous diffusion
======================================
In a seminal paper [@Robertson] a conjecture was proposed based on the principle of minimum entropy-production rate, stating that a plasma will be more stable whenever the internal product of the current density $\textbf{j}$ by an elementary conducting area $d
\mathcal{A}$ at every point of the boundary - excluding the surface collecting the driving current - is null, $(\mathbf{j} \cdot
\mathcal{A})=0$ at any point of the boundary (and excluding the surfaces collecting the discharge current), independently of the resistance $R_i$. The general idea proposed by Robertson [@Robertson] assumes that the plasma boundary is composed of small elements of area $\mathcal{A}_i$, each one isolated from the others, but each one connected to the exterior common circuit through its own resistor $R_i$ and voltage $V_i$. The entropy production rate in the external circuits is given by: $$\label{Eq1}
\frac{d S}{d t} = \sum_i \frac{1}{T} (\mathbf{j}_i \cdot
\mathcal{A}_i)^2 R_i,$$ where $T$ is the temperature of the resistors, supposed to be in thermal equilibrium with all the others. It is important to remark that the summation is over the different conducting areas eventually confining the plasma, [*excluding*]{} the electrodes areas. Fig.\[Fig1\] illustrates this concept.
![Schematic diagram of the plasma boundary connected to the common circuit through conducting walls.[]{data-label="Fig1"}](Fig_AnomDif1.eps "fig:"){width="3.0" height="3.5"}\
We consider a simple axysymmetric magnetic configuration with magnetic field lines parallel to z-axis with a plasma confined between two electrodes (see Fig.1). In general terms, a particle’s motion in the plasma results in a massive flux. As long as the flux is installed, the flux will depend naturally on a force $\mathbf{F}$ - in this case the pressure gradient-driven process of diffusion to the wall - responsible of the wall driven current $\mathbf{j}$. According to the fundamental thermodynamic relation, the plasma internal energy variation $dU$ is related to the amount of entropy supplied or rejected and the work done by the driven force, through the equation: $$\label{Eq5}
\frac{dU}{dt} = (\mathbf{j} \cdot \mathcal{A})^2 R + \left(
\mathbf{F} \cdot \frac{d \mathbf{r}}{dt} \right).$$
The last term we identify with the macroscopic diffusion velocity $\mathbf{v}_d$ depicting the process of plasma expansion to the wall. To simplify somehow the calculations we assume a single plasma fluid under the action of a pressure gradient ($\mathbf{F}=\mathcal{A} L dp/dy \overrightarrow{j}$, where $\overrightarrow{j}$ is a unit vector directed along the Oy axis).
In the presence of steady and uniform magnetic field lines (this simplifies the equations, but do not limit the applicability of the model), the particles stream freely along them. From magnetohydrodynamics we have a kind of generalized Ohm’s law (see, for example, Ref. [@Kadomtsev1]): $$\label{Eq6}
\nabla p = - e n \mathbf{E} - e n [\mathbf{v} \times \mathbf{B}] +
[\mathbf{j} \times \mathbf{B}] - \frac{e n \mathbf{j}}{\sigma},$$ where $\sigma = e^2 n \tau_e/m_e$ is the electric conductivity, with $\tau_e$ denoting the average collision time between electrons and ions. The force balance equation is given by: $$\label{Eq7}
\nabla p = [\mathbf{j} \times \mathbf{B}],$$ valid whenever the Larmor radius is smaller than the Debye radius. This assumption simplifies further the extension of our model to high enough magnetic fields. Therefore, after inserting Eq. \[Eq7\] into Eq. \[Eq6\] the y component of velocity is obtained: $$\label{Eq8}
v_y = - \frac{E_x}{B} - \frac{1}{\sigma B^2} \frac{d p}{d y}.$$ From Eq. \[Eq8\] we have the classical diffusion coefficient scaling with $1/B^2$ and thus implying a random walk of step length $r_L$ (Larmor radius). To get the anomalous diffusion coefficient and as well understand better its related physics, we must consider the process of diffusion to the wall - in the presence of an entropy source - with the combined action of the wall current drain, as already introduced in Eq. \[Eq5\].
Therefore, using the guiding center plasma model the particle motion is made with velocity given by: $$\label{Eq9}
\mathbf{j}=en\mathbf{v}_d=-\frac{[\nabla p \times
\mathbf{B}]}{B^2}.$$ This equation forms the base of a simplified theory of magnetic confinement. In fact, the validity of Eq. \[Eq9\] is restrained to the high magnetic field limit, when the Larmor radius is shorter than the Debye radius.
Considering the motion along only one direction perpendicular to the wall (y-axis), it is clear that $$\label{Eq10}
(\mathbf{j} \cdot \mathcal{A})^2 = \frac{\mathcal{A}^2}{B^2}
\left( \frac{dp}{dy} \right)^2.$$ If we consider a quasi-steady state plasma operation, the plasma total energy should be sustained. Hence, $dU/dt=0$, and the power associated with the driven pressure-gradient is just maintaining the dissipative process of plasma losses on the wall. Eq. \[Eq5\] governs the evolution of the diffusion velocity. Hence, we have $$\label{Eq12}
n v_d = - \frac{n R \mathcal{A}}{L} \frac{kT}{B^2} \frac{dn}{dy} =
-D_{T} \frac{dn}{dy},$$ with $D_T$ denoting the transverse (across the magnetic field) diffusion coefficient given by: $$\label{Eq13}
D_T = \frac{n R \mathcal{A}}{L} \frac{kT}{B^2}.$$ This new result coincides with the classical diffusion coefficient [@Roth] whenever $nR\mathcal{A}/L \equiv m
\nu_{ei}/e^2$, containing a dependence on collision frequency and particle number density. Other theoretical approaches to this problem were advanced by Bohm [@Bohm], who proposed an empirically-driven diffusion coefficient associating plasma oscillations as the source of the enhanced diffusion, while Tonks [@Tonks] have shown that the current density that is present in a magnetically immobilized plasma is only generated by the particle density gradient, not being associated with any drift of matter. Simon electron “short-circuit” [@Simon] scheme attempt to explain the different rates of diffusion, electrons and ions do experiment across the magnetic field. While the ion flux dominates the radial diffusion, the electron flux dominates axial losses, due to an unbalance of currents flowing to the wall.
In the absence of collisions, the guiding centers of charged particles behave as permanently attached to the same lines of force. On the contrary, as a result of collisions with others charged particles the guiding centers shift from one line of force to another resulting in a diffusion of plasma across the field lines. In our model, each orbit constitutes an elementary current $I$ eventually crossing the wall.
However, the particle diffusion coefficient as shown in Eq. \[Eq13\] gives evidence of an interplay with the resistance that the elementary circuit offer when in contact with the walls in the presence of the frozen-in effect. In fact, for sufficiently strong magnetic fields apparently a hydrodynamic behavior of the plasma is installed [@Montgomery2; @Corkum], with the appearance of “convective cells” and the $1/B$ behavior dominates, giving birth to the anomalous diffusion mechanism. The onset of freezing magnetic lines is valid whenever the Lundquist number $\mathrm{S} \gg 1$ (convection of the magnetic field dominated medium). In this case the magnetic field lines are frozen-in in the medium (consequence of a vortex type of character of the magnetic field $\mathbf{B}$) and the flux of them across a given surface is constant: $$\label{Eq14}
\Phi =B \mathcal{A}' = B L^2 \alpha.$$ Remark that $\mathcal{A}'$ is now the surface delimited by the elementary circuit $\gamma$ (see Fig. \[fig1\]) and $\alpha
\lesssim 1$ is just a geometrical factor (e.g. $\alpha=\pi/4$ at the limit of a circular orbit). This situation is fundamental to the onset of anomalous diffusion. Free electrons orbits are helical, but as Fig. \[fig1\] shows, their projections at rigth angles to the field are circular. Each particle orbit constitute an elementary circuit with $B$-field cutting its surface being associated with it an elementary flux $\Phi$. At the same time we can envisage each orbit as constituting by itself an elementary circuit, some of them intersecting the wall and thus the circuit is closed inside the wall. Therefore a resistance $R$ drags the charged flow at the conducting wall. It is therefore plausible to associate to this elementary circuit a potential drop $V$ and all the process being equivalent to a current $I$ flowing through the elementary circuit.
![Schematic of the geometry for the plasma-wall current drain model. The uniform magnetic field points downward along Oz. Particles describe orbits in the plane xOy intersecting the wall (plan xOz). Orbits are represented by a semi-circular line for convenience. $L$ is the maximum distance the trajectory attains from the wall.[]{data-label="fig1"}](Geom_ad2.eps "fig:"){width="3.5" height="4.5"}\
Assuming the plasma is a typical weakly coupled, hot diffuse plasma with a plasma parameter (number of particles in Debye sphere) $\Lambda = n\lambda_{De}^3 \approx 1$, it is more likely to expect nearly equal average kinetic and potential energy. However, the typical plasma parameter encountered in glow discharges or in nuclear fusion is $\Lambda \gg 1$. This means that the average kinetic energy is larger than the average potential energy. To contemplate all range of $\Lambda$ we can relate them through the relationship $$\label{Eq15}
\rho V = (\mathbf{J} \cdot \mathbf{A}) \delta.$$ Here, $\rho$ is the charge density, $\mathbf{A}$ is the vector potential, $\mathbf{J}$ is the current density and $\delta \leq 1$ is just a parameter representing the ratio of potential to kinetic energy. Of course, when $\Lambda \geq 1$, then $\delta \leq 1$. This basic assumption is consistent with the hydrodynamic approximation taken in the development of equations. The limitations of the model are related with the unknowns $\Lambda$ and $\delta$ that can be uncovered only through a self-consistent model of the plasma. However, our analysis of anomalous diffusion remains general and added new insight to the phenomena.
Now suppose that the diffusion current is along y-axis $\mathbf{J}=-J_y \mathbf{u}_y$ (see Fig.1). Consequently, $\mathbf{A}=-A_y \mathbf{u}_y$, and then the potential drop will depend on x-coordinate: $$\label{Eq16}
\rho [V(x_1) - V(x_0)] = J_y [A_y(x_1) - A_y(x_0)] \delta.$$ Multiplying both members by the area $\mathcal{A}'=x_1 z_1$ and length $L=y_1$, we have $$\label{Eq17}
Q \Delta V = I y_1 [A_y(x_1) - A_y(x_0)] \delta =I \Phi \delta.$$ $\Phi=\oint_{\gamma} (\mathbf{A} \cdot d\mathbf{x})$ is the flux of the magnetic field through the closed surface bounded by the line element $d \mathbf{x}$ (elementary circuit $\gamma$, see also Fig.\[fig1\]). By other side, naturally, the total charge present on the volume $\mathcal{V}=x_1 y_1 z_1$ is such as $Q=ie$, with $i$ an integer. This integer must be related to ions charge number. From Eq. \[Eq17\] we obtain $$\label{Eq19}
R = \frac{\Delta V}{I} = \delta \frac{\Phi }{Q} = \alpha \delta
\frac{B L^2 }{i e}.$$ But, the particle density is given by $n=N/L\mathcal{A}$, with $N$ being now the total number of charged particles present in volume $\mathcal{V}=\mathcal{A}L$. Since $i=N$, we retrieve finally the so-called Bohm-diffusion coefficient $$\label{Eq18}
D_B = \alpha \delta \frac{kT}{eB}.$$
So far, our arguments were applied to edge anomalous diffusion. But they can be generalized to the core anomalous diffusion processes, provided that diffusive transport processes are dominant. For this purpose consider instead of a conducting surface a virtual surface delimiting a given volume, as shown in Fig. \[fig3\].
![Volume control and particle’s trajectory submitted to magnetic field. Magnetic field lines point downward.[]{data-label="fig3"}](Geome_AD3.eps "fig:"){width="3.5" height="3.5"}\
Our coefficient is time-dependent and can be written under the form: $$\label{}
D_{\perp}=\delta \frac{kT(t)}{eB(t)^2}\frac{\Phi(t)}{L^2}.$$ The nonrelativistic solutions of dynamical equation of a charged particle in time-dependent but homogeneous electric and magnetic field give the following approximative expressions for the width trajectories along the xOy plane (see, e.g. Ref.[@chandrasekhar_60]) by: $$\label{}
\begin{array}{cc}
\Delta x = \frac{v_{\perp}}{\Omega_c}\cos(\Omega_c t + \chi) & \Delta
y=\mp \frac{v_{\perp}}{\Omega_c} \sin(\Omega_c t + \chi),
\end{array}$$ where $\chi$ is the initial phase, $v_{\perp}$ denotes the component of the velocity perpendicular to the magnetic field and the $\mp$ sign applies to electrons (-) or positive ions (+). From them we can retrieve the flux “cutting” area: $$\label{}
\mathcal{A} \approx \Delta x.\Delta y =-\frac{1}{2} \left(
\frac{\nu_{\perp}}{\Omega_c} \right)^2 \sin(2 \Omega_c t + 2 \chi).$$ Then the anomalous diffusion coefficient is just given by: $$\label{formulafinal}
D_{\perp} \approx \mp \delta
\frac{kT(t)}{eB(t_0)}\frac{1}{2}\frac{\sin(2 \Omega_c t +
2\chi)}{\mid \cos(\omega t) \mid}.$$ As we can see in Fig.\[fig4\] this last expression describes fairly well the diffusion process for high enough $B/N$ values (the magnetic field to gas number density ratio) and explains the main processes building-up such effects as: i) the negative diffusion, which results from the contraction of the flux “cutting” area; ii) the ciclotronic modulation imprint on the transverse diffusion coefficient; iii) and the anomalous diffusion, due to the fast flux rate of the magnetic field through the area $\mathcal{A}$). All this signs can be seen on Fig. \[fig4\] were it is shown a comparison of numerical results (5000 Hx, 1 Hx=$10^{-27}$ T.m$^3$) obtained with Monte Carlo simulations of electron transport in crossed magnetic and electric fields by Petrović [*et al.*]{} [@Zoran; @Zoran1] with the theoretical prediction given by our Eq. \[formulafinal\]. As long as only a self-consistent model could give us an exact value of the ratio of potential to kinetic energy $\delta$, we assume here $\delta=1/40$. The full agreement with the numerical calculations is not obtained due to neglecting effects related to the electric field variation in time and of the assumed collisionless approximation. This explains the big discrepancy shown in Fig. \[fig4\] when compared with the diffusion coefficient at 1000 Hx when collisions begin to be far more important to randomize individual trajectories and our approach is no more valid (at low enough $B/N$ values).
![Comparison between numerical results for 5000 Hx obtained by Monte Carlo simulations Refs.\[32,33\] of electron transport in crossed electric and magnetic rf fields in argon and Eq. \[formulafinal\] -theo. Dashed line: external time-dependent magnetic field. Parameters used: $\delta=\frac{1}{40}$; $\chi$; applied frequency, f=100 MHz; ciclotron frequency, $\Omega_c \approx 10^{10}$ Hz; $\frac{kT}{e}=5.4$ eV; p=1 Torr, $T_g=300$ K, $\chi_=0$.[]{data-label="fig4"}](Anom_Dif_Fig4.eps "fig:"){width="3.5" height="3.0"}\
Discussion and Summary
======================
However, Eq. \[Eq18\] suffers from the indetermination of the geometrical factor $\alpha$. This factor is related to the ions charge number, it depends on the magnetic field magnitude and as well as on the external operating conditions (due to increased collisional processes, for ex.). The exact value of the product $\alpha \delta$ can only be determined through a self-consistent plasma model, but we should expect from the above discussion that $\alpha \delta < 1$. For a 100-eV plasma in a 1-T field, we obtain $D_B=1.67$ m$^2/$s (using the thermal to magnetic energy ratio with particle’s density $n=10^{14}$ cm$^{-3}$). Furthermore, Eq. \[Eq19\] can be used as a boundary condition (simulating an electrically floating surface) imposed when solving Poisson equation.
Also it worth to emphasize that when inserting Eq. \[Eq19\] into Eq. \[Eq13\], and considering the usual definition of momentum transfer cross section, then it can be obtained a new expression for the classical diffusion coefficient as a function of the ratio of collisional $\nu$ and cyclotron frequency $\Omega$, although (and in contrast with the standard expression), now also dependent on the geometrical factor $\alpha$ and energy ratio $\delta$: $$\label{}
D_T = (\alpha \delta) \frac{\nu}{\Omega} \frac{kT}{m}.$$ This explains the strong dependence of the classical diffusion coefficient on $\nu/\Omega$ showing signs of anomalous diffusion as discussed in Ref. [@Zoran] (obtained with a time resolved Monte Carlo simulation in an infinite gas under uniform fields) and, in addition, the strong oscillations shown up in the calculations of the time dependence of the transverse component of the diffusion tensor for electrons in low-temperature rf argon plasma. Those basic features result on one side from its dependence on $R$, which is proportional to the flux. Therefore, a flux variation can give an equivalent effect to the previously proposed mechanism: whenever a decrease (or increase) in the flux is onset through time dependence of electric and magnetic fields, it occurs a strong increase (or decrease) of the diffusion coefficient. By other side, when the resistance increases it occurs a related decrease of charged particles tangential velocity and its mean energy. So far, this model gives a new insight into the results referred in [@Zoran] and also explains why the same effect is not obtained from the solution of the non-conservative Boltzmann equation as applied to an oxygen magnetron discharge with constant electric and magnetic fields [@White].
A further application of Eq. \[Eq1\] to a cold plasma can give a new insight into the “ambipolar-like” diffusion processes. Considering just one conducting surface (besides the electrodes driving the main current into the plasma) and the plasma build-up of electrons and one ion component to simplify matters, we obtain: $$\label{Eq2}
\frac{d S}{dt} = \frac{e^2}{T} (-n_e \mu_e \mathbf{E} + D_e \nabla
n_e + n_i \mu_i \mathbf{E} - D_i \nabla n_i)^2 \mathcal{A}^2 R.$$ Under the usual assumptions of quasi-neutrality and quasi-stationary plasma (see, for example, Ref. [@Roth]), the following conditions must be verified: $$\label{Eq3}
\begin{array}{cc}
\frac{n_i}{n_e}=\epsilon=const. ; & n_e \mathbf{v_e}=n_i
\mathbf{v}_i,
\\
\end{array}$$ and hence, Eq. \[Eq1\] becomes: $$\label{Eq4}
\frac{dS}{dt}=\frac{e^2}{T} [\mathbf{E}(\epsilon \mu_i - \mu_e)n_e +
\nabla n_e (D_e - D_i \epsilon)]^2 \mathcal{A}^2 R.$$ For a stable steady-state plasma with no entropy sources the condition $\dot{S}=0$ prevails and then an “ambipolar-like” electric field is recovered [@Roth]: $$\label{amb1}
\mathbf{E}=\frac{D_e - \epsilon D_i}{\mu_e - \mu_i
\epsilon}\frac{\nabla n_e}{n_e}.$$ It means that the conducting surface must be at its floating potential. Such conceptual formulation provides new insight into “ambipolar-like” diffusion processes. In a thermal equilibrium state, a plasma confined by insulating walls will have an effective coefficient given by the above Eq. \[amb1\], a situation frequently encountered in industrial applications. This example by itself relates ambipolar diffusion with no entropy production in the plasma. However, allowing plasma currents to the walls, entropy production is greatly enhanced generating altogether instabilities and plasma losses [@Robertson]. As long as confined plasmas are in a far-nonequilibrium state (with external surroundings) it is necessary to establish a generalized principle that rule matter, and this circuital model for anomalous diffusion represents some progress in the physics of plasmas as nonequilibrium systems.
To summarize, we introduced in this study a simple circuital mechanism providing an interpretation of the anomalous diffusion in a magnetized confined plasma in a purely diffusive transport regime. The coupled action of the magnetic field “cutting” flux through the areas traced by the charge carriers elementary orbits, together with the elementary electric circuit constituted by the charged particle trajectory itself are at the basis of the anomalous diffusion process. Whenever conducting walls bounding the plasma drain the current (edge diffusion) or, at the plasma core, the magnetic field flux through the areas traced by the charged particles varies, a Bohm-like behavior of the transverse diffusion coefficient can be expected. Eq. \[formulafinal\] can be used as an analytical formula when simulating plasma behavior at high $B/N$. In the near future we hope to generalize this model taking into account random collisions. The suggested mechanism could lead to a better understanding of the mechanism of plasma-wall interaction and help to develop a full-scale numerical modeling of present fusion devices or collisional low-temperature plasmas.
The author gratefully acknowledge the data supplied to us by Zoran Lj. Petrović and Zoran Raspopovic used in our Fig.4 .
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[^1]: We acknowledge partial financial support from Fundação para a Ciência e Tecnologia and the Rectorate of the Technical University of Lisbon.
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abstract: 'We study the effect of imperfect training data labels on the performance of classification methods. In a general setting, where the probability that an observation in the training dataset is mislabelled may depend on both the feature vector and the true label, we bound the excess risk of an arbitrary classifier trained with imperfect labels in terms of its excess risk for predicting a noisy label. This reveals conditions under which a classifier trained with imperfect labels remains consistent for classifying uncorrupted test data points. Furthermore, under stronger conditions, we derive detailed asymptotic properties for the popular $k$-nearest neighbour ($k$nn), support vector machine (SVM) and linear discriminant analysis (LDA) classifiers. One consequence of these results is that the $k$nn and SVM classifiers are robust to imperfect training labels, in the sense that the rate of convergence of the excess risks of these classifiers remains unchanged; in fact, our theoretical and empirical results even show that in some cases, imperfect labels may improve the performance of these methods. On the other hand, the LDA classifier is shown to be typically inconsistent in the presence of label noise unless the prior probabilities of each class are equal. Our theoretical results are supported by a simulation study.'
author:
- |
Timothy I. Cannings$^\ast$, Yingying Fan$^\dag$ and Richard J. Samworth$^\ddag$\
*$^\ast$University of Edinburgh, $^\dag$University of Southern California*\
and *$^\ddag$University of Cambridge*
title: Classification with imperfect training labels
---
Introduction
============
Supervised classification is one of the fundamental problems in statistical learning. In the basic, binary setting, the task is to assign an observation to one of two classes, based on a number of previous training observations from each class. Modern applications include, among many others, diagnosing a disease using genomics data [@Wright:2015], determining a user’s action from smartphone telemetry data [@Lara:2013], and detecting fraud based on historical financial transactions [@Bolton:2002].
In a classification problem it is often the case that the class labels in the training data set are inaccurate. For instance, an error could simply arise due to a coding mistake when the data were recorded. In other circumstances, such as the disease diagnosis application mentioned above, errors may occur due to the fact that, even to an expert, the true labels are hard to determine, especially if there is insufficient information available. Moreover, in modern Big Data applications with huge training data sets, it may be impractical and expensive to determine the true class labels, and as a result the training data labels are often assigned by an imperfect algorithm. Services such as the *Amazon Mechanical Turk*, (see <https://www.mturk.com>), allow practitioners to obtain training data labels relatively cheaply via crowdsourcing. Of course, even after aggregating a large crowd of workers’ labels, the result may still be inaccurate. @Chen:2015 and @Zhang:2016 discuss crowdsourcing in more detail, and investigate strategies for obtaining the most accurate labels given a cost constraint.
The problem of label noise was first studied by @Lachenbruch:1966, who investigated the effect of imperfect labels in two-class linear discriminant analysis. Other early works of note include @Lachenbruch:1974, @Angluin:1988 and @Lugosi:1992.
@Frenay:2014a and @Frenay:2014b provide recent overviews of work on the topic. In the simplest, homogeneous setting, each observation in the training dataset is mislabelled independently with some fixed probability. @vanRooyen:2015a study the effects of homogeneous label errors on the performance of empirical risk minimization (ERM) classifiers, while @Long:2010 consider boosting methods in this same homogeneous noise setting. Other recent works focus on class-dependent label noise, where the probability that a training observation is mislabelled depends on the true class label of that observation; see @Stempfel:2009, @Natarajan:2013, @Scott:2013, @Blanchard:2016, @Liu:2016 and @Patrini:2016. An alternative model assumes the noise rate depends on the feature vector of the observation. @Manwani:2013 and @Ghosh:2015 investigate the properties of ERM classifiers in this setting; see also @Awasthi:2015. @Menon:2016 propose a *generalized boundary consistent* label noise model, where observations near the optimal decision boundary are more likely to be mislabelled, and study the effects on the properties of the receiver operator characteristics curve.
In the more general setting, where the probability of mislabelling is both feature- and class-dependent, @Bootkrajang:2012 [@Bootkrajang:2014] and @Bootkrajang:2016 study the effect of label noise on logistic regression classifiers, while @Li:2017, @Patrini:2017 and @Rolnick:2017 consider neural network classifiers. On the other hand, @Cheng:2017 investigate the performance of an ERM classifier in the feature- and class-dependent noise setting when the true class conditional distributions have disjoint support.
Our first goal in the present paper is to provide general theory to characterize the effect of feature- and class-dependent heterogeneous label noise for an arbitrary classifier. We first specify general conditions under which the optimal prediction of a true label and a noisy label are the same for every feature vector. Then, under slightly stronger conditions, we relate the misclassification error when predicting a true label to the corresponding error when predicting a noisy label. More precisely, we show that the excess risk, i.e. the difference between the error rate of the classifier and that of the optimal, Bayes classifier, is bounded above by the excess risk associated with predicting a noisy label multiplied by a constant factor that does not depend on the classifier used; see Theorem \[thm:hetnoise\]. Our results therefore provide conditions under which a classifier trained with imperfect labels remains consistent for classifying uncorrupted test data points.
As applications of these ideas, we consider three popular approaches to classification problems, namely the $k$-nearest neighbour ($k$nn), support vector machine (SVM) and linear discriminant analysis (LDA) classifiers. In the perfectly labelled setting, the $k$nn classifier is consistent for any data generating distribution and the SVM classifier is consistent when the distribution of the feature vectors is compactly supported. Since the label noise does not change the marginal feature distribution, it follows from our results mentioned in the previous paragraph that these two methods are still consistent when trained with imperfect labels that satisfy our assumptions, which, in the homogeneous noise case, even allow up to 1/2 of the training data to be labelled incorrectly. On the other hand, for the LDA classifier with Gaussian class-conditional distributions, we derive the asymptotic risk in the homogeneous label noise case. This enables us to deduce that the LDA classifier is typically not consistent when trained with imperfect labels, unless the class prior probabilities are equal to 1/2.
Our second main contribution is to provide greater detail on the asymptotic performance of the $k$nn and SVM classifiers in the presence of label noise, under stronger conditions on the data generating mechanism and noise model. In particular, for the $k$nn classifier, we derive the asymptotic limit for the ratio of the excess risks of the classifier trained with imperfect and perfect labels, respectively. This reveals the nice surprise that using imperfectly-labelled training data can in fact improve the performance of the $k$nn classifier in certain circumstances. To the best of our knowledge, this is the first formal result showing that label noise can help with classification. For the SVM classifier, we provide conditions under which the rate of convergence of the excess risk is unaffected by label noise, and show empirically that this method can also benefit from label noise in some cases.
In several respects, our theoretical analysis acts a counterpoint to the folklore in this area. For instance, @Okamoto:1997 analysed the performance of the $k$nn classifier in the presence of label noise. They considered relatively small problem sizes and small values of $k$, where the $k$nn classifier performs poorly when trained with imperfect labels; on the other hand, our Theorem \[thm:knnhet\] reveals that for larger values of $k$, which diverge with $n$, the asymptotic effect of label noise is relatively modest, and may even improve the performance of the classifier. As another example, @Manwani:2013 and @Ghosh:2015 claim that SVM classifiers perform poorly in the presence of label noise; our Theorem \[thm:SVMhet\] presents a different picture, however, at least as far as the rate of convergence of the excess risk is concerned. Finally, in two-class Gaussian discriminant analysis, @Lachenbruch:1966 showed that LDA is robust to homogeneous label noise when the two classes are equally likely [see also @Frenay:2014b Section III-A]. We observe, though, that this robustness is very much the exception rather than the rule: if the prior probabilities are not equal, then the LDA classifier is almost invariably not consistent when trained with imperfect labels; cf. Theorem \[thm:LDAhomo\].
Although it is not the focus of this paper, we mention briefly that another line of work on label noise investigates techniques for identifying mislabelled observations and either relabelling them, or simply removing them from the training data set. Such methods are sometimes referred to as data *cleansing* or *editing* techniques; see for instance @Wilson:1972, @Wilson:2000 and @Cheng:2017; as well as @Frenay:2014a [Section 3.2], who provide a general overview of popular methods for editing training data sets. Other authors focus on estimating the noise rates and recovering the clean class-conditional distributions [@Blanchard:2016; @Northcutt:2017].
The remainder of this paper is organized as follows. In Section \[sec:setting\] we introduce our general statistical setting, while in Section \[sec:finite\], we present bounds on the excess risk of an arbitrary classifier trained with imperfect labels under very general conditions. In Section \[sec:asymptotic\], we derive the asymptotic properties of the $k$nn, SVM and LDA classifiers when trained with noisy labels. Our empirical experiments, given in Section \[sec:sims\], show that this asymptotic theory corresponds well with finite-sample performance. Finally in the appendix we present the proofs underpinning our theoretical results, as well as an illustrative example involving the 1-nearest neighbour classifier.
The following notation is used throughout the paper. We write $\|\cdot\|$ for the Euclidean norm on $\mathbb{R}^d$, and for $r > 0$ and $z \in \mathbb{R}^d$, write $B_z(r) = \{x \in \mathbb{R}^d:\|x-z\| < r\}$ for the open Euclidean ball of radius $r$ centered at $z$, and let $a_d = \pi^{d/2}/\Gamma(1+d/2)$ denote the $d$-dimensional volume of $B_0(1)$. If $A \in \mathbb{R}^{d \times d}$, we write $\|A\|_{\mathrm{op}}$ for its operator norm. For a sufficiently smooth real-valued function $f$ defined on $D \subseteq \mathbb{R}^m$, and for $x \in D$, we write $\dot{f}(x) = (f_1(x),\ldots,f_m(x))^T$ and $\ddot{f}(x) = (f_{jk}(x))_{j,k=1}^m$ for its gradient vector and Hessian matrix at $x$ respectively. Finally, we write $\triangle$ for symmetric difference, so that $\mathcal{A} \triangle \mathcal{B} = (\mathcal{A}^c \cap \mathcal{B}) \cup (\mathcal{A} \cap \mathcal{B}^c)$.
We conclude this section with a preliminary study to demonstrate our new results for the $k$nn, SVM and LDA classifiers in the homogeneous noise case.
\[Ex:prelim\] *In this motivating example, we demonstrate the surprising effects of imperfect labels on the performance of the $k$nn, SVM and LDA classifiers. We generate $n$ independent training data pairs, where the prior probabilities of classes 0 and 1 are $9/10$ and $1/10$ respectively; class 0 and 1 observations have bivariate normal distributions with means $\mu_0 = (-1,0)^T$ and $\mu_1 = (1,0)^T$ respectively, and common identity covariance matrix. We then introduce label noise in the training data set by flipping the true training data labels independently with probability $\rho =$ 0.3. One example of a data set of size $n=1000$ from this model, both before and after label noise is added, is shown in Figure \[fig:dataplot\].*
![[One training dataset from the model in Example \[Ex:prelim\] for $n = 1000$, without label noise (left) and with label noise (right). We plot class 0 in red and class 1 in black.]{}[]{data-label="fig:dataplot"}](PlotPrelimCleanNoisy.eps){width="\textwidth"}
![[Risks ($\%$) of the $k$nn (black), SVM (red) and LDA (blue) classifiers trained using perfect (solid lines) and imperfect labels (dotted lines). The dot-dashed line shows the Bayes risk, which is 7.0%.]{}[]{data-label="fig:errorplot"}](PlotPrelim.eps){width="60.00000%"}
*In Figure \[fig:errorplot\], we present the percentage error rates, both with and without label noise, of the $k$nn, SVM and LDA classifiers. The error rates were estimated by the average over 1000 repetitions of the experiment of the percentage of misclassified observations on a test set, without label noise, of size 1000. We set $k = k_{n} = \lfloor n^{2/3}/2 \rfloor$ for the $k$nn classifier, and set the tuning parameter $\lambda = 1$ for the SVM classifier; see .*
*In this simple setting where the decision boundary of the Bayes classifier is a hyperplane, all three classifiers perform very well with perfectly labelled training data, especially LDA, whose derivation was motivated by Gaussian class-conditional distributions with common covariance matrix. With mislabelled training data, the performance of all three classifiers is somewhat affected, but the $k$nn and SVM classifiers are relatively robust to the label noise, particularly for large $n$. Indeed, we will show that these classifiers remain consistent in this setting. The gap between the performance of the LDA classifier and that of the Bayes classifier, however, persists even for large $n$; this again is in line with our theory developed in Theorem \[thm:LDAhomo\], where we derive the asymptotic risk of the LDA classifier trained with homogeneous label errors. The limiting risk is given explicitly in terms of the noise rate $\rho$, the prior probabilities, and the Mahalanobis distance between the two class-conditional distributions.*
Statistical setting {#sec:setting}
===================
Let $\mathcal{X}$ be a measurable space. In the basic binary classification problem, we observe independent and identically distributed training data pairs $(X_1, Y_1), \ldots, (X_n, Y_n)$ taking values in $\mathcal{X} \times \{0,1\}$ with joint distribution $P$. The task is to predict the class $Y$ of a new observation $X$, where $(X,Y) \sim P$ is independent of the training data.
Define the *prior probabilities* $\pi_1 = \mathbb{P}(Y = 1) = 1- \pi_0 \in (0,1)$ and *class-conditional distributions* $X \mid \{Y=r\} \sim P_r$ for $r=0,1$. The *marginal feature distribution* of $X$ is denoted $P_X$ and we define the *regression function* $\eta(x) = \mathbb{P}\bigl( Y = 1 \mid X = x)$. A *classifier* $C$ is a measurable function from $\mathcal{X}$ to $\{0,1\}$, with the interpretation that a point $x \in \mathcal{X}$ is assigned to class $C(x)$.
The *risk* of a classifier $C$ is $R(C) = \mathbb{P}\{C(X) \neq Y\}$; it is minimized by the *Bayes classifier* $$C^{\mathrm{Bayes}}(x) = \left\{ \begin{array}{ll} 1& \text{\quad if } \eta(x) \geq 1/2
\\ 0 & \text{\quad otherwise.} \end{array} \right.$$ However, since $\eta$ is typically unknown, in practice we construct a classifier $C_{n}$, say, that depends on the $n$ training data pairs. We say $(C_n)$ is *consistent* if $R(C_{n}) - R(C^{\mathrm{Bayes}}) \rightarrow 0$ as $n \rightarrow \infty$. When we write $R(C_n)$ here, we implicitly assume that $C_{n}$ is a measurable function from $(\mathcal{X} \times \{0,1\})^n\times\mathcal{X}$ to $\{0,1\}$, and the probability is taken over the joint distribution of $(X_1,Y_1),\ldots,(X_n,Y_n), (X,Y)$. It is convenient to set $\mathcal{S} = \{x \in \mathcal{X}:\eta(x) = 1/2\}$.
In this paper, we study settings where the true class labels $Y_1, \ldots, Y_n$ for the training data are not observed. Instead we see $\tilde{Y}_1, \ldots, \tilde{Y}_n$, where the noisy label $\tilde{Y}_{i}$ still takes values in $\{0,1\}$, but may not be the same as $Y_i$. The task, however, is still to predict the true class label $Y$ associated with the test point $X$. We can therefore consider an augmented model where $(X, Y, \tilde{Y}), (X_1, Y_1, \tilde{Y}_1), \ldots, (X_n, Y_n, \tilde{Y}_n)$ are independent and identically distributed triples taking values in $\mathcal{X} \times \{0,1\} \times \{0,1\}$.
At this point the dependence between $Y$ and $\tilde{Y}$ is left unrestricted, but we introduce the following notation: define measurable functions $\rho_0, \rho_1 : \mathcal{X} \rightarrow [0,1]$ by $\rho_{r}(x) = \mathbb{P}(\tilde{Y} \neq Y \mid X = x, Y = r)$. Thus, letting $Z \mid \{X = x, Y = r\} \sim \mathrm{Bin}(1, 1- \rho_r(x))$ for $r=0,1$, we can write $\tilde{Y} = Z Y+ (1-Z)(1-Y)$. We refer to the case where $\rho_0(x) = \rho_1(x) = \rho$ for all $x \in \mathcal{X}$ as *$\rho$-homogeneous noise*. Further, let $\tilde{P}$ denote the joint distribution of $(X,\tilde{Y})$, and let $\tilde{\eta}(x) = \mathbb{P}(\tilde{Y} = 1 \mid X = x)$ denote the regression function for $\tilde{Y}$, so that $$\begin{aligned}
\label{eq:tildeeta}
\tilde{\eta}(x) &= \eta(x) \mathbb{P}(\tilde{Y} = 1 \mid X = x, Y = 1) + \{1- \eta(x)\}\mathbb{P}(\tilde{Y} = 1 \mid X = x, Y = 0) \nonumber
\\& = \eta(x) \{1 - \rho_1(x)\} + \{1 - \eta(x)\}\rho_0(x).\end{aligned}$$ We also define the *corrupted Bayes classifier* $$\tilde{C}^{\mathrm{Bayes}}(x) = \left\{ \begin{array}{ll} 1& \text{\quad if } \tilde{\eta}(x) \geq 1/2
\\ 0 & \text{\quad otherwise,} \end{array} \right.$$ which minimizes the *corrupted risk* $\tilde{R}(C) = \mathbb{P}\{C(X) \neq \tilde{Y}\}$.
Excess risk bounds for arbitrary classifiers {#sec:finite}
============================================
A key property in this work will be that the Bayes classifier is preserved under label noise; more specifically, in Theorem \[thm:hetnoise\](i) below, we will provide conditions under which $$\label{eq:symmetric}
P_{X}\bigl( \{x \in \mathcal{S}^c : \tilde{C}^{\mathrm{Bayes}}(x) \neq C^{\mathrm{Bayes}}(x)\} \bigr) = 0.$$ In Theorem \[thm:hetnoise\](ii), we go on to show that, under slightly stronger conditions on the label error probabilities and for an arbitrary classifier $C$, we can bound the excess risk $R(C) - R(C^{\mathrm{Bayes}})$ of predicting the true label by a multiple of the excess risk of predicting a noisy label $\tilde{R}(C) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})$, where this multiple does not depend on the classifier $C$. This latter result is particularly useful when the classifier $C$ is trained using the imperfect labels, that is with the training data $(X_{1}, \tilde{Y}_{1}), \ldots, (X_{n}, \tilde{Y}_{n})$, because, as will be shown in the next section, we are able to provide further control of $\tilde{R}(C) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})$ for specific choices of $C$.
It is convenient to let $\mathcal{B} = \{x \in \mathcal{S}^c : \rho_0 (x) + \rho_1(x) < 1 \}$, and let $$\mathcal{A} =\biggl\{x \in \mathcal{B}:\frac{\rho_1(x) - \rho_0(x)}{\{2\eta(x)-1\}\{1 - \rho_0(x) - \rho_1(x)\}} < 1\biggr\}.$$
\[thm:hetnoise\] (i) We have $$\label{eq:noisecond}
P_X\bigl(\mathcal{A} \, \triangle \, \{x \in \mathcal{B}: \tilde{C}^{\mathrm{Bayes}}(x) = C^{\mathrm{Bayes}}(x) \} \bigr) = 0.$$ In particular, if $P_{X}(\mathcal{A}^c \cap \mathcal{S}^c) = 0$, then holds.
\(ii) Now suppose, in fact, that there exist $\rho^* < 1/2$ and $a^* < 1$ such that $P_{X}(\{x \in \mathcal{S}^c : \rho_0 (x) + \rho_1(x) > 2\rho^* \}) = 0$, and $$P_X \biggl(\biggl\{ x\in \mathcal{B}: \frac{\rho_1(x) - \rho_0(x)}{\{2\eta(x)-1\}\{1 - \rho_0(x) - \rho_1(x)\}} > a^* \biggr\}\biggr) = 0.$$ Then, for any classifier $C$, $$R(C) - R(C^{\mathrm{Bayes}}) \leq \frac{\tilde{R}(C) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})}{(1- 2\rho^*)(1-a^*) }.$$
In Theorem \[thm:hetnoise\](i), the condition $P_X(\mathcal{A}^c \cap \mathcal{S}^c) = 0$ restricts the difference between the two mislabelling probabilities at $P_X$-almost all $x \in \mathcal{S}^c$, with stronger restrictions where $\eta(x)$ is close to $1/2$ and where $\rho_{0}(x) + \rho_1(x)$ is close to 1. Moreover, since $\mathcal{A}\subseteq\mathcal{B}$, we also have $P_X(\mathcal{B}^c \cap \mathcal{S}^c) = 0$, which limits the total amount of label noise at each point; cf. @Menon:2016 [Assumption 1]. In particular, it ensures that $$\mathbb{P}(\tilde{Y} \neq Y \mid X = x) = \eta(x) \rho_{1}(x) + \{1 - \eta(x)\} \rho_{0}(x) < 1,$$ for $P_{X}$-almost all $x \in \mathcal{S}^c$. In part (ii), the requirement on $a^*$ imposes a slightly stronger restriction on the same weighted difference between the two mislabelling probabilities compared with part (i).
The conditions in Theorem \[thm:hetnoise\] generalize those given in the existing literature by allowing a wider class of noise mechanisms. For instance, in the case of $\rho$-homogeneous noise, we have $P_X(\mathcal{A}^c \cap \mathcal{S}^c) = 0$ provided only that $\rho < 1/2$. In fact, in this setting, we may take $a^* = 0$ [@Ghosh:2015 Theorem 1]. More generally, we may also take $a^{*} = 0$ if the noise depends only on the feature vector and not the true class label, i.e. $\rho_{0}(x) = \rho_{1}(x)$ for all $x$ [@Menon:2016 Proposition 4].
The proof of Theorem \[thm:hetnoise\](ii) relies on the following proposition, which provides a bound on the excess risk for predicting a true label, assuming only that holds.
\[prop:prelim\] Assume that holds. Further, for $\kappa > 0$, let $$A_{\kappa} = \Bigl\{x \in \mathcal{X} : |2\eta(x) - 1| \leq \kappa |2\tilde{\eta}(x) - 1|\Bigr\}.$$ Then, for any classifier $C$, $$\label{eq:minbound}
R(C)\! - \!R(C^{\mathrm{Bayes}}) \leq \min\Bigl[\mathbb{P}\{C(X)\! \neq\! \tilde{C}^{\mathrm{Bayes}}(X)\}, \inf_{\kappa > 0} \Bigl\{\kappa \{\tilde{R}(C) \!-\! \tilde{R}(\tilde{C}^{\mathrm{Bayes}})\} \!+\! P_X(A_{\kappa}^c) \Bigr\} \Bigr].$$
Our main focus in this work is on settings where $\tilde{C}^{\mathrm{Bayes}}$ and $C^{\mathrm{Bayes}}$ agree, i.e. holds, because this is where we can hope for classifiers to be robust to label noise. However, in this instance, we present a more general version of Proposition \[prop:prelim\] as Proposition \[prop:prelim2\] in the appendix; this bounds the excess risk of an arbitrary classifier without the assumption that holds. We see in that result, there is an additional contribution to the risk bound of $R(\tilde{C}^{\mathrm{Bayes}}) - R(C^{\mathrm{Bayes}}) \geq 0$. See also, for instance, @Natarajan:2013, who study asymmetric homogeneous noise, where $\rho_0(x) = \rho_0 \neq \rho_1 = \rho_1(x)$, with $\rho_0$ and $\rho_1$ known.
We can regard $|2\eta(x) - 1|$ as a measure of the ease of classifying $x$. Hence, in Proposition \[prop:prelim\], we can interpret $A_{\kappa}$ as the set of points $x$ where the relative difficulty of classifying $x$ in the corrupted problem compared with its uncorrupted version is controlled. The level of this control can then be traded off against the measure of the exceptional set $A_\kappa^c$.
To provide further understanding of Proposition \[prop:prelim\], observe that in general, we have $$\begin{aligned}
\tilde{R}(C) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}}) & = \int_{\mathcal{X}} \bigl[\mathbb{P}\{C(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\bigr] \{2\tilde{\eta}(x) - 1\} \, dP_{X}(x)
\\ & \leq \mathbb{P}\{C(X) \neq \tilde{C}^{\mathrm{Bayes}}(X)\}. \end{aligned}$$ Thus, if $P_X(A_{1}^c) = 0$, then the second term in the minimum in gives a better bound than the first. However, typically in practice, we would have that $P_X(A_{1}^c) \neq 0$, and indeed, in Example \[ex:1nn\] in the appendix, we show that for the 1-nearest neighbour classifier with homogeneous noise, either of the two terms in the minimum in can be smaller, depending on the noise level. As a consequence of Proposition \[prop:prelim\], we have the following corollary.
\[cor:consistent\] Suppose that $(\tilde{C}_n)$ is a sequence of classifiers satisfying $\tilde{R}(\tilde{C}_n) \rightarrow \tilde{R}(\tilde{C}^{\mathrm{Bayes}})$ and assume that holds. Further, let $\tilde{\mathcal{S}} = \{x \in \mathcal{X}:\tilde{\eta}(x) = 1/2\}$. Then $$\limsup_{n \rightarrow \infty} R(\tilde{C}_n) - R(C^{\mathrm{Bayes}}) \leq P_X(\tilde{\mathcal{S}} \setminus \mathcal{S}).$$ In particular, if $P_X(\tilde{\mathcal{S}} \setminus \mathcal{S}) = 0$, then $R(\tilde{C}_n) \rightarrow R(C^{\mathrm{Bayes}})$ as $n \rightarrow \infty$.
The condition $\tilde{R}(\tilde{C}_n) \rightarrow \tilde{R}(\tilde{C}^{\mathrm{Bayes}})$ asks that the classifier is consistent for predicting a corrupted test label. In Section \[sec:asymptotic\] we will see that appropriate versions of the corrupted $k$nn and SVM classifiers satisfy this condition, provided, in the latter case, that the feature vectors have compact support. To understand the strength of Corollary \[cor:consistent\], consider the special case of $\rho$-homogeneous noise, and a classifier $\tilde{C}_n$ that is consistent for predicting a noisy label when trained with corrupted data. Then $\tilde{\mathcal{S}} = \mathcal{S}$ by , so provided only that $\rho < 1/2$, Corollary \[cor:consistent\] ensures that $\tilde{C}_n$ remains consistent for predicting a true label when trained using the corrupted data.
Asymptotic properties {#sec:asymptotic}
=====================
The $k$-nearest neighbour classifier {#sec:knn}
------------------------------------
We now specialize to the case $\mathcal{X} = \mathbb{R}^d$. The $k$nn classifier assigns the test point $X$ to a class based on a majority vote over the class labels of the $k$ nearest points among the training data. More precisely, given $x \in \mathbb{R}^d$, let $(X_{(1)}, Y_{(1)}), \ldots, (X_{(n)}, Y_{(n)})$ be the reordering of the training data pairs such that $$\|X_{(1)} - x \| \leq \ldots \leq \|X_{(n)} - x \|,$$ where ties are broken by preserving the original ordering of the indices. For $k \in \{1, \ldots, n\}$, the *$k$-nearest neighbour classifier* is $$C^{k\mathrm{nn}}(x) = C_n^{k\mathrm{nn}}(x) = \left\{ \begin{array}{ll} 1& \text{\quad if } \frac{1}{k} \sum_{i = 1} ^{k} \mathbbm{1}_{\{Y_{(i)} =1 \} } \geq 1/2
\\ 0 & \text{\quad otherwise.} \end{array} \right.$$
This simple and intuitive method has received considerable attention since it was introduced by @Fix:1951 [@Fix:1989]. @Stone:1977 showed that the $k$nn classifier is universally consistent, i.e., $R(C^{k\mathrm{nn}}) \rightarrow R(C^{\mathrm{Bayes}})$ for any distribution $P$, as long as $k = k_n \rightarrow \infty$ and $k/n \rightarrow 0$ as $n \rightarrow \infty$. For a substantial overview of the early work on the theoretical properties of the $k$nn classifier, see @PTPR:1996. Further recent studies include @Kulkarni:1995, @Audibert:2007, @Hall:2008, @Biau:2010, @Samworth:2012, @Chaudhuri:2014, @Gadat:16, @Celisse:2018 and @CBS:2017.
Here we study the properties of the *corrupted $k$-nearest neighbour classifier* $$\tilde{C}^{k\mathrm{nn}}(x) = \tilde{C}_n^{k\mathrm{nn}}(x) = \left\{ \begin{array}{ll} 1& \text{\quad if } \frac{1}{k} \sum_{i = 1} ^{k} \mathbbm{1}_{\{\tilde{Y}_{(i)} =1 \} } \geq 1/2
\\ 0 & \text{\quad otherwise,} \end{array} \right.$$ where $\tilde{Y}_{(i)}$ denotes the corrupted label of $(X_{(i)},Y_{(i)})$. Since the $k$nn classifier is universally consistent, we have $\tilde{R}(\tilde{C}^{k\mathrm{nn}}) \rightarrow \tilde{R}(\tilde{C}^{\mathrm{Bayes}})$ for any choice of $k$ satisfying Stone’s conditions. Thus, by Corollary \[cor:consistent\], if holds and $P_X(\tilde{\mathcal{S}} \setminus \mathcal{S}) = 0$, then the corrupted $k$nn classifier remains universally consistent. In particular, in the special case of $\rho$-homogeneous noise, provided only that $\rho < 1/2$, this result tells us that the corrupted $k$nn classifier remains universally consistent.
We now show that, under further regularity conditions on the data distribution $P$ and the noise mechanism, it is possible to give a more precise description of the asymptotic error properties of the corrupted $k$nn classifier. Since our conditions on $P$, which are slight simplifications of those used in @CBS:2017 to analyse the uncorrupted $k$nn classifier, are a little technical, we give an informal summary of them here, deferring formal statements of our assumptions A1–A4 to just before the proof of Theorem \[thm:knnhet\] in Section \[sec:knnproofs\]. First, we assume that each of the class-conditional distributions has a density with respect to Lebesgue measure such that the marginal feature density $\bar{f}$ is continuous and positive. It turns out that the dominant terms in the asymptotic expansion of the excess risk of $k$nn classifiers are driven by the behaviour of $P$ in a neighbourhood $\mathcal{S}^\epsilon$ of the set $\mathcal{S}$, which consists of points that are difficult to classify correctly, so we ask for further regularity conditions on the restriction of $P$ to $\mathcal{S}^\epsilon$. In particular, we ask for both $\bar{f}$ and $\eta$ to have two well-behaved derivatives in $\mathcal{S}^\epsilon$, and for $\dot{\eta}$ to be bounded away from 0 on $\mathcal{S}$. This amounts to asking that the class-conditional densities, when weighted by the prior probabilities of each class, cut at an angle, and ensures that the set $\mathcal{S}$ is a $(d-1)$-dimensional orientable manifold. Away from the set $\mathcal{S}^\epsilon$, we only require weaker conditions on $P_{X}$, and for $\eta$ to be bounded away from $1/2$. Finally, we ask for two $\alpha$th moment conditions to hold, namely that $\int_{\mathbb{R}^d} \|x\|^{\alpha} \, dP_{X}(x) < \infty$ and $\int_{\mathcal{S}} \bar{f}(x_0)^{d/(\alpha+d)} \, d\mathrm{Vol}^{d-1}(x_0) < \infty$, where $d\mathrm{Vol}^{d-1}$ denotes the $(d-1)$-dimensional volume form on $\mathcal{S}$.
For $\beta \in (0,1/2)$, let $K_{\beta} = \{\lceil (n-1)^\beta \rceil, \ldots, \lfloor (n-1)^{1-\beta} \rfloor \}$ denote the set of values of $k$ to be considered for the $k$nn classifier. Define $$B_{1} = \int_{\mathcal{S}} \frac{\bar{f}(x_0)}{4\|\dot\eta(x_0)\|} \, d\mathrm{Vol}^{d-1}(x_0),\quad B_{2} = \int_{\mathcal{S}} \frac{\bar{f}(x_0)^{1-4/d}}{\|\dot\eta(x_0)\|}a(x_0)^2 \, d\mathrm{Vol}^{d-1}(x_0),$$ where $$a(x) = \frac{\sum_{j=1}^d \bigl\{\eta_j(x)\bar{f}_j(x) + \frac{1}{2}\eta_{jj}(x)\bar{f}(x)\bigr\}}{(d+2)a_d^{2/d}\bar{f}(x)}.$$ We will also make use of a condition on the noise rates near the Bayes decision boundary:
: There exist $\delta > 0$ and a function $g:(1/2-\delta, 1/2 + \delta) \rightarrow [0,1)$ that is differentiable at $1/2$, with the property that for $x$ such that $\eta(x) \in (1/2-\delta, 1/2 + \delta)$, we have $\rho_0(x) = g(\eta(x))$ and $\rho_1(x) = g(1 - \eta(x))$.
This assumption asks that, when $\eta(x)$ is close to $1/2$, the probability of label noise depends only on $x$ through $\eta(x)$, and moreover, this probability varies smoothly with $\eta(x)$. In other words, Assumption B1 says that the probability of mislabelling an observation with true class label 0 depends only on the extent to which it appeared to be from class 1; conversely, the probability of mislabelling an observation with true label 1 depends only, and in a symmetric way, on the extent to which it appeared to be from class 0. To give just one of many possible examples, one could imagine that the probability that a doctor misdiagnoses a malignant tumour as benign depends on the extent to which it appears to be malignant, and vice versa. We remark that @Menon:2016 [Definition 11] introduce a related probabilistically transformed noise model, where $\rho_0 = g_0 \circ \eta$ and $\rho_1 = g_1 \circ \eta$, but they also require that $g_0$ and $g_1$ are increasing on $[0,1/2]$ and decreasing on $[1/2,1]$; see also @Bylander:1997.
\[thm:knnhet\] Assume *A1*, *A2*, *A3* and *A4*($\alpha$). Suppose that $\rho_0$, $\rho_1$ are continuous, and that both $$\rho^* = \frac{1}{2} \sup_{x \in \mathbb{R}^d} \{\rho_0(x) + \rho_1(x)\} < \frac{1}{2}$$ and $$a^* = \sup_{x \in \mathcal{B}} \frac{\rho_1(x) - \rho_0(x)}{\{2\eta(x)-1\}\{1 - \rho_0(x) - \rho_1(x)\}} < 1.$$ Moreover, assume *B1* holds with the additional requirement that $g$ is twice continuously differentiable, $\dot{g}(1/2) > 2g(1/2) -1$ and that $\ddot{g}$ is uniformly continuous. Then we have two cases:
\(i) Suppose that $d \geq 5$ and $\alpha > 4d/(d-4)$. Then for each $\beta \in (0,1/2)$, $$R(\tilde{C}^{k\mathrm{nn}}) - R(C^{\mathrm{Bayes}}) = \frac{B_1}{k\{1- 2g(1/2) + \dot{g}(1/2)\}^{2}} + B_{2} \Bigl(\frac{k}{n}\Bigr)^{4/d} + o\biggl(\frac{1}{k} + \Bigl(\frac{k}{n}\Bigr)^{4/d}\biggr)$$ as $n \to \infty$, uniformly for $k \in K_{\beta}$.
\(ii) Suppose that either $d \leq 4$, or, $d \geq 5$ and $\alpha \leq 4d/(d-4)$. Then for each $\beta \in (0,1/2)$ and each $\epsilon> 0$ we have $$R(\tilde{C}^{k\mathrm{nn}}) - R(C^{\mathrm{Bayes}}) = \frac{B_1}{k\{1- 2g(1/2) + \dot{g}(1/2)\}^{2}} + o\Bigl(\frac{1}{k} +\Bigl(\frac{k}{n}\Bigr)^{\frac{\alpha}{\alpha+d} - \epsilon}\Bigr)$$ as $n \to \infty$, uniformly for $k \in K_{\beta}$.
The proof of Theorem \[thm:knnhet\] is given in Section \[sec:knnproofs\], and involves two key ideas. First, we demonstrate that the conditions assumed for $\eta$ also hold for the corrupted regression function $\tilde{\eta}$. Second, we show that the dominant asymptotic contribution to the desired excess risk $R(\tilde{C}^{k\mathrm{nn}}) - R(C^{\mathrm{Bayes}})$ is $\{\tilde{R}(\tilde{C}^{k\mathrm{nn}}) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})\}/\{1-2g(1/2) + \dot{g}(1/2)\}$, a scalar multiple of the excess risk when predicting a noisy label. We then conclude the argument by appealing to @CBS:2017 [Theorem 1], and of course, can recover the conclusion of that result for noiseless labels as a special case of Theorem \[thm:knnhet\] by setting $g = 0$.
In the conclusion of Theorem \[thm:knnhet\](i), the terms $B_1/[k\{1- 2g(1/2) + \dot{g}(1/2)\}^{2}]$ and $B_{2}(k/n)^{4/d}$ can be thought of as the leading order contributions to the variance and squared bias of the corrupted $k$nn classifier respectively. It is both surprising and interesting to note that the type of label noise considered here affects only the leading order variance term compared with the noiseless case; the dominant bias term is unchanged. To give a concrete example, $\rho$-homogeneous noise satisfies the conditions of Theorem \[thm:knnhet\], and in the setting of Theorem \[thm:knnhet\](i), we see that the dominant variance term is inflated by a factor of $(1-2\rho)^{-2}$.
We now quantify the relative asymptotic performance of the corrupted $k$nn and uncorrupted $k$nn classifiers. Since this performance depends on the choice of $k$ in each case, we couple these choices together in the following way: given any $k$ to be used by the uncorrupted classifier $C^{k\mathrm{nn}}$, and given the function $g$ from Theorem \[thm:knnhet\], we consider the choice $$\label{Eq:kg}
k_g = \bigl\lfloor \{1-2g(1/2) + \dot{g}(1/2)\}^{-2d/(d+4)} k \bigr\rfloor$$ for the noisy label classifier $\tilde{C}^{k\mathrm{nn}}$. This coupling reflects the ratio of the optimal choices of $k$ for the corrupted and uncorrupted label settings.
\[cor:knnRR\] Under the assumptions of Theorem \[thm:knnhet\](i), and provided that $B_2 > 0$, we have that for any $\beta \in (0,1/2)$, $$\label{eq:RR}
\frac{R(\tilde{C}^{k_g\mathrm{nn}}) - R(C^{\mathrm{Bayes}})}{R(C^{k\mathrm{nn}}) - R(C^{\mathrm{Bayes}})}
\rightarrow \{1 - 2g(1/2) + \dot{g}(1/2)\}^{-8/(d+4)},$$ as $n \rightarrow \infty$, uniformly for $k \in K_\beta$.
If $\dot{g}(1/2) > 2 g(1/2)$, then the limiting regret ratio in is less than 1 – this means that the label noise helps in terms of the asymptotic performance! This is due to the fact that, under the noise model in Theorem \[thm:knnhet\], if $\dot{g}(1/2) > 2 g(1/2)$ then for points $X_i$ with $\eta(X_i)$ close to $1/2$, the noisy labels $\tilde{Y}_{i}$ are more likely than the true labels $Y_i$ to be equal to the Bayes labels, $\mathbbm{1}_{\{\eta(X_{i}) \geq 1/2\}}$. To understand this phenomenon, first note that by rearranging , we have $$\begin{aligned}
\tilde{\eta}(x) - 1/2 = \{\eta(x) - 1/2\}\{1-\rho_0(x) - \rho_1(x)\} + \frac{1}{2}\{\rho_0(x) - \rho_1(x)\}.\end{aligned}$$ Thus $\tilde \eta(x)-1/2 = \eta(x)-1/2$ for $x \in \mathcal{S}$ using B1. On the other hand, for $x\in \mathcal{S}^c$, we have $$\label{eq: 001}
\tilde {\eta}(x) - 1/2 = \{\eta(x) - 1/2\} \biggl(1-\rho_0(x) - \rho_1(x) + \frac{\rho_0(x) - \rho_1(x)}{2\eta(x)-1}\biggr).$$ We next study the term in the second parentheses on the right-hand side above. Write $t = \eta(x) - 1/2$. Then, for $x$ such that $|\eta(x) -1/2| \in (0,\delta)$, we have $\rho_0(x) = g(1/2+t)$ and $\rho_1(x) = g(1/2-t)$. It follows, for such $x$, that $$\begin{aligned}
1 \! - \! \rho_0(x)\! -\! \rho_1(x) \! + \!\frac{\rho_0(x) - \rho_1(x)}{2\eta(x)-1} &= 1-g(1/2+t) - g(1/2-t) + \frac{g(1/2+t)-g(1/2-t)}{2t} \\
&\rightarrow 1 - 2g(1/2) + \dot{g}(1/2)\end{aligned}$$ as $|t|\searrow 0$. Since $1-2g(1/2)+\dot{g}(1/2)>1$, we obtain that for any $\varepsilon \in \bigl(0, \dot{g}(1/2)/2-g(1/2)\bigr)$, there exists $\delta_0 \in (0, \delta)$ such that for all $x$ with $|\eta(x) - 1/2|\in (0,\delta_0)$, we have that $$1-\rho_0(x) - \rho_1(x) + \frac{\rho_0(x) - \rho_1(x)}{2(\eta(x)-1/2)} > 1 -2g(1/2) + \dot{g}(1/2)-\varepsilon>1.$$ This together with ensures that, for all $x$ such that $|\eta(x)-1/2| \in (0,\delta_0)$, we have $$|\tilde\eta(x)-1/2| > |\eta(x)-1/2|.$$
*Suppose that for some $g_0 \in (0,1/2)$ and $h_0 > 2 - 1/g_0$ we have $g(1/2 + t) = g_0(1 + h_0 t)$ for $t \in (-\delta,\delta)$. Then $g(1/2) = g_0$ and $\dot{g}(1/2) = g_0 h_0$, which gives $1 - 2g(1/2) + g'(1/2) = 1 + (h_0 - 2) g_0 $. We therefore see from Corollary \[cor:knnRR\] that if $h_0 < 2$, then the limiting regret ratio is greater than $1$, but if $h_0 > 2$, then the limiting regret ratio is less than one, so the label noise aids performance.*
Support vector machine classifiers {#sec:SVM}
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In general, the term *support vector machines* (SVM) refers to classifiers of the form $$C^{\mathrm{SVM}}(x) = C_n^{\mathrm{SVM}}(x) = \left\{ \begin{array}{ll} 1& \text{\quad if } \hat{f} (x) \geq 0
\\ 0 & \text{\quad otherwise,} \end{array} \right.$$ where the *decision function* $\hat{f}$ satisfies $$\label{eq:SVMopt}
\hat{f} \in \operatorname*{argmin}_{f \in H} \biggl\{\frac{1}{n} \sum_{i = 1}^{n} L(Y_{i}, f(X_{i})) + \Omega(\lambda, \|f\|_{H})\biggr\}.$$ See, for example, @Cortes:1995 and @Steinwart:2008. Here $L: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ is a loss function, $\Omega : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ is a regularization function, $
\lambda > 0$ is a tuning parameter and $H$ is a *reproducing kernel Hilbert space* (RKHS) with norm $\|\cdot\|_{H}$ [@Steinwart:2008 Chapter 4].
We focus throughout on the *L1-SVM*, where $L(y, t) = \max\{0, 1 - (2y-1)t\}$ is the *hinge loss* function and $\Omega(\lambda, t) = \lambda t^{2}$. Let $K : \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ be the positive definite kernel function associated with the RKHS. We consider the Gaussian radial basis function, namely $K(x,x') = \exp(-\sigma^2 \|x - x'\|^{2} )$, for $\sigma > 0$. The corrupted SVM classifier is $$\label{eq:SVMK}
\tilde{C}^{\mathrm{SVM}}(x) = \tilde{C}_n^{\mathrm{SVM}}(x) = \left\{ \begin{array}{ll} 1& \text{\quad if } \tilde{f}(x) \geq 0
\\ 0 & \text{\quad otherwise,} \end{array} \right.$$ where $$\label{eq:SVMopttilde}
\tilde{f} \in \operatorname*{argmin}_{f \in H} \biggl\{\frac{1}{n} \sum_{i = 1}^{n} \max\{0, 1 - (2\tilde{Y}_i-1)f(X_{i})\} + \lambda \|f\|^2_{H}\biggr\}.$$
@Steinwart:2005 [Corollary 3.6 and Example 3.8] show that the uncorrupted L1-SVM classifier is consistent as long as $P_X$ is compactly supported and $\lambda = \lambda_{n}$ is such that $\lambda_{n} \rightarrow 0$ but $n\lambda_{n}/(|\log{\lambda_{n}} |^{d+1}) \rightarrow \infty$. Therefore, under these conditions, provided that holds and $P_X(\tilde{\mathcal{S}} \setminus \mathcal{S}) = 0$, by Corollary \[cor:consistent\], we have that $R(\tilde{C}^{\mathrm{SVM}}) \rightarrow R(C^{\mathrm{Bayes}})$ as $n \rightarrow \infty$.
Under further conditions on the noise probabilities and the distribution $P$, we can also provide more precise control of the excess risk for the SVM classifier. Our analysis will make use of the results in @Steinwart:2007, who study the rate of convergence of the SVM classifier with Gaussian kernels in the noiseless label setting. Other works of note on the rate of convergence of SVM classifiers include @Lin:1999 and @Blanchard:2008; see also @Steinwart:2008 [Chapters 6 and 8].
We recall two definitions used in the perfect labels context. The first of these is the well-known *margin assumption* of, for example, @Audibert:2007. We say that the distribution $P$ satisfies the margin assumption with parameter $\gamma_1 \in [0,\infty)$ if there exists $\kappa_{1}>0$ such that $$P_{X}(\{x \in \mathbb{R}^{d} : 0 < |\eta(x) - 1/2| \leq t\}) \leq \kappa_{1} t^{\gamma_1}$$ for all $t>0.$ If $P$ satisfies the margin assumption for all $\gamma_1 \in [0,\infty)$ then we say $P$ satisfies the margin assumption with parameter $\infty$. The margin assumption controls the probability mass of the region where $\eta$ is close to $1/2$.
The second definition we need is that of the *geometric noise exponent* [@Steinwart:2007 Definition 2.3]. Let $\mathcal{S}_{+} = \{x\in \mathbb{R}^{d} : \eta(x) > 1/2\}$ and $\mathcal{S}_{-} = \{x\in \mathbb{R}^{d} : \eta(x) < 1/2\}$, and for $x \in \mathbb{R}^{d}$, let $\tau_{x} = \inf_{x' \in \mathcal{S} \cup \mathcal{S}_{+}} \|x - x'\| + \inf_{x' \in \mathcal{S} \cup \mathcal{S}_{-}} \|x - x'\|$. We say that the distribution $P$ has geometric noise exponent $\gamma_2 \in [0,\infty)$ if there exists $\kappa_2>0$, such that $$\int_{\mathbb{R}^{d}} |2\eta(x) - 1| \exp\Bigl(-\frac{\tau_{x}^{2}}{t^2}\Bigr) \, dP_{X}(x) \leq \kappa_{2}t^{\gamma_2 d}$$ for all $t>0.$ If $P$ has geometric noise exponent $\gamma_2$ for all $\gamma_2 \in [0,\infty)$ then we say it has geometric noise exponent $\infty$.
Under these two conditions, @Steinwart:2007 [Theorem 2.8] show that, if $P_X$ is supported on the closed unit ball, then for appropriate choices of the tuning parameters, the SVM classifier achieves a convergence rate of $O(n^{-\Gamma + \epsilon})$ for every $\epsilon >0$, where $$\Gamma = \left\{ \begin{array}{ll}
\frac{\gamma_2}{2\gamma_2 + 1} & \text{\quad if } \gamma_2 \leq \frac{\gamma_1 + 2}{2\gamma_1} \\
\frac{2\gamma_2(\gamma_1+1)}{2\gamma_2(\gamma_1 + 2) + 3 \gamma_1 + 4} & \text{\quad otherwise.} \end{array} \right.$$
In the imperfect labels setting, and under our stronger assumption on the noise mechanism when $\eta$ is close to $1/2$, we see that the SVM classifier trained with imperfect labels satisfies the same bound on the rate of convergence as in the perfect labels case.
\[thm:SVMhet\] Suppose that $P$ satisfies the margin assumption with parameter $\gamma_1 \in [0, \infty]$, has geometric noise exponent $\gamma_2 \in (0,\infty)$ and that $P_X$ is supported on the closed unit ball. Assume the conditions of Theorem \[thm:hetnoise\](ii) and B1 holds. Then $$R(\tilde{C}^{\mathrm{SVM}}) - R(C^{\mathrm{Bayes}}) = O(n^{-\Gamma+\epsilon}),$$ as $n \rightarrow \infty$, for every $\epsilon > 0$. If $\gamma_2 = \infty$, then the same conclusion holds provided $\sigma_n = \sigma$ is a constant with $\sigma > 2d^{1/2}$.
Linear discriminant analysis {#sec:LDA}
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If $P_0 = N_d(\mu_0, \Sigma)$ and $P_1 = N_d(\mu_1, \Sigma)$, then the Bayes classifier is $$\label{eq:LDABayes}
C^{\mathrm{Bayes}}(x) = \left\{ \begin{array}{ll} 1 & \mbox{\quad if $\log\bigl(\frac{\pi_1}{\pi_0}\bigr) + \bigl(x - \frac{{\mu}_1 + {\mu}_0}{2}\bigr)^T {\Sigma}^{-1} ({\mu}_1 - {\mu}_0) \geq 0$}
\\ 0 & \mbox{\quad otherwise.} \end{array} \right.$$ The Bayes risk can be expressed in terms of $\pi_0, \pi_1$, and the squared Mahalanobis distance $\Delta^2 = (\mu_1 - \mu_0)^T \Sigma^{-1} (\mu_1 - \mu_0)$ between the classes as $$R(C^{\mathrm{Bayes}}) = \pi_0\Phi\biggl(\frac{1}{\Delta} \log\Bigl(\frac{\pi_1}{\pi_0}\Bigr) - \frac{\Delta}{2}\biggr) + \pi_1\Phi\biggl(\frac{1}{\Delta}\log\Bigl(\frac{\pi_0}{\pi_1}\Bigr) - \frac{\Delta}{2}\biggr),$$ where $\Phi$ denotes the standard normal distribution function.
The LDA classifier is constructed by substituting training data estimates of $\pi_0, \pi_1, \mu_0, \mu_1,$ and $\Sigma$ in to . With imperfect training data labels, and for $r = 0,1$, we define estimates $\hat{\pi}_r = n^{-1}\sum_{i=1}^n \mathbbm{1}_{\{\tilde{Y}_i = r\}}$ of $\pi_r$, as well as estimates $\hat{\mu}_r = \sum_{i = 1}^n X_i \mathbbm{1}_{\{\tilde{Y}_i = r\}}/\sum_{i=1} ^n \mathbbm{1}_{\{\tilde{Y}_i = r\}}$ of the class-conditional means $\mu_r$, and set $$\hat{\Sigma} = \frac{1}{n-2} \sum_{i = 1}^{n} \sum_{r = 0}^1 (X_i - \hat{\mu}_r) ( X_i - \hat{\mu}_r)^T \mathbbm{1}_{\{\tilde{Y}_i = r\}}.$$ This allows us to define the corrupted LDA classifier $$\tilde{C}^{\mathrm{LDA}}(x) = \tilde{C}_n^{\mathrm{LDA}}(x) = \left\{ \begin{array}{ll} 1 & \text{\quad if } \log\bigl(\frac{\hat{\pi}_1}{\hat{\pi}_0}\bigr) + \bigl(x - \frac{\hat{\mu}_1 + \hat{\mu}_0}{2}\bigr)^T \hat{\Sigma}^{-1} (\hat{\mu}_1 - \hat{\mu}_0) \geq 0
\\ 0 & \text{\quad otherwise.} \end{array} \right.$$ Consider now the $\rho$-homogeneous noise setting. In this case, writing $\tilde{P}_{r}$, $r\in \{0,1\}$, for the distribution of $X \mid \{\tilde{Y} = r\}$, we have $\tilde{P}_r = p_r N_d(\mu_r, \Sigma) + (1-p_r) N_d(\mu_{1-r}, \Sigma)$, where $p_r = \pi_r (1-\rho)/\{\pi_r (1-\rho) + \pi_{1-r}\rho\}$. Notice that while $\hat{\pi}_r, \hat{\mu}_r$ and $\hat{\Sigma}$ are intended to be estimators of $\pi_r, \mu_r$ and $\Sigma$, respectively, with label noise these will in fact be consistent estimators of $\tilde{\pi}_r = \pi_r (1-\rho) + \pi_{1-r} \rho$, $\tilde{\mu}_r = p_r \mu_r + (1-p_r) \mu_{1-r}$, and $\tilde{\Sigma} = \Sigma + \alpha(\mu_1-\mu_0)(\mu_1-\mu_0)^T$, respectively, where $\alpha > 0$ is given in the proof of Theorem \[thm:LDAhomo\].
We will also make use of the following well-known lemma in the homogeneous label noise case [e.g. @Ghosh:2015 Theorem 1], which holds for an arbitrary classifier and data generating distribution. We include the short proof for completeness.
\[lem:tilde\] For $\rho$-homogeneous noise with $\rho\in [0,1/2)$ and for any classifier $C$, we have $R(C) = \{\tilde{R}(C) - \rho\}/(1- 2\rho)$. Moreover, $R(C) - R(C^{\mathrm{Bayes}}) = \bigl\{\tilde{R}(C) - \tilde{R}(C^{\mathrm{Bayes}})\bigr\}/(1- 2\rho)$.
The following is the main result of this subsection.
\[thm:LDAhomo\] Suppose that $P_r = N_d(\mu_r, \Sigma)$ for $r = 0,1$ and that the noise is $\rho$-homogeneous with $\rho \in [0,1/2)$. Then $$\lim_{n \rightarrow \infty} \tilde{C}^{\mathrm{LDA}}(x) = \left\{ \begin{array}{ll} 1 & \mbox{\quad if $c_0 + \bigl(x - \frac{\mu_1 + \mu_0}{2}\bigr)^T \Sigma^{-1} (\mu_1 - \mu_0) > 0$}
\\ 0 & \mbox{\quad if $c_0 + \bigl(x - \frac{\mu_1 + \mu_0}{2}\bigr)^T \Sigma^{-1} (\mu_1 - \mu_0) < 0$,} \end{array} \right.$$ where $$c_0 \! = \! \Bigl\{(1-2\rho) + \frac{\rho(1\!-\!\rho)(1 \! + \! \pi_0\pi_1\Delta^2)}{(1-2\rho)\pi_{1}\pi_{0}} \Bigr\} \log\Bigl(\frac{(1-2\rho)\pi_1 + \rho}{(1-2\rho)\pi_0 + \rho} \Bigr) - \frac{(\pi_{1} - \pi_{0})\rho(1-\rho)\Delta^{2}}{2\{(1\!-\!2\rho)^{2} \pi_1 \pi_{0} + \rho(1\!-\!\rho)\}} .$$ As a consequence, $$\label{Eq:LimitingLDARisk}
\lim_{n \rightarrow \infty} R(\tilde{C}^{\mathrm{LDA}}) = \pi_0\Phi\biggl(\frac{c_0}{\Delta} - \frac{\Delta}{2}\biggr) + \pi_1\Phi\biggl(-\frac{c_0}{\Delta} - \frac{\Delta}{2}\biggr) \geq R(C^{\mathrm{Bayes}}).$$ For each $\rho \in (0,1/2)$ and $\pi_0 \neq \pi_1$, there exists a unique value of $\Delta > 0$ for which equality in the inequality in is attained.
The first conclusion of this theorem reveals the interesting fact that, regardless of the level $\rho \in (0,1/2)$ of label noise, the limiting corrupted LDA classifier has a decision hyperplane that is parallel to that of the Bayes classifier; see also @Lachenbruch:1966 and @Manwani:2013 [Corollary 1]. However, for each fixed $\rho \in (0,1/2)$ and $\pi_0 \neq \pi_1$, there is only one value of $\Delta > 0$ for which the offset is correct and the corrupted LDA classifier is consistent.
Numerical comparison {#sec:sims}
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In this section, we investigate empirically how the different types of label noise affect the performance of the $k$-nearest neighbour, support vector machine and linear discriminant analysis classifiers. We consider two different model settings for the pair $(X,Y)$:
Model 1: Let $\mathbb{P}(Y=1) = \pi_1 \in \{$0.5, 0.9$\}$ and $X \mid \{Y = r\} \sim N_{d}(\mu_{r}, I_d)$, where $\mu_{1} = (3/2, 0, \ldots, 0)^T = -\mu_{0} \in \mathbb{R}^{d}$ and $I_d$ denotes the $d$ by $d$ identity matrix.
Model 2: For $d \geq 2$, let $X \sim U([0,1]^{d})$ and $\mathbb{P}(Y =1 \mid X = x) = \eta(x_1, \ldots, x_d) = \min\{ 4(x_1 - 1/2)^2 + 4(x_2 - 1/2)^2, 1\}$.
In each setting, our risk estimates are based on an uncorrupted test set of size 1000, and we repeat each experiment 1000 times. This ensures that all standard errors are less than 0.4$\%$ and 0.14 for the risk and regret ratio estimates, respectively; in fact, they are often much smaller.
![[Risk estimates for the $k$nn (left), SVM (middle) and LDA (right) classifiers. Top: Model 1, $d = 2$, $\pi_{1} =$ 0.5, Bayes risk = 6.68%, shown as the black dotted line. Bottom: Model 2, $d = 2$, Bayes risk = 19.63%. We present the results without label noise (black) and with homogeneous label noise at rate $\rho =$ 0.1 (red) and 0.3 (blue). ]{}[]{data-label="fig:errorplot1a"}](PlotNEWlog_1.eps "fig:"){width="\textwidth"} ![[Risk estimates for the $k$nn (left), SVM (middle) and LDA (right) classifiers. Top: Model 1, $d = 2$, $\pi_{1} =$ 0.5, Bayes risk = 6.68%, shown as the black dotted line. Bottom: Model 2, $d = 2$, Bayes risk = 19.63%. We present the results without label noise (black) and with homogeneous label noise at rate $\rho =$ 0.1 (red) and 0.3 (blue). ]{}[]{data-label="fig:errorplot1a"}](PlotNEWlog_4b.eps "fig:"){width="\textwidth"}
![[Risk estimates for the LDA classifier. Model 1, $d=5$, $\pi_1 =$ 0.9, Bayes risk = 3.37%. We present the estimated error without label noise (black) and with homogeneous label noise at rate $\rho =$ 0.1 (red), 0.2 (blue), 0.3 (green) and 0.4 (purple). The dotted lines show the corresponding asymptotic limit as given by Theorem \[thm:LDAhomo\].]{}[]{data-label="fig:errorplotLDA"}](PlotLDA.eps){width="61.00000%"}
Our first goal is to illustrate numerically our consistency and inconsistency results for the $k$nn, SVM and LDA classifiers. In Figure \[fig:errorplot1a\] we present estimates of the risk for the three classifiers with different levels of homogeneous label noise. We see that for Model 1 when the class prior probabilities are equal, all three classifiers perform well and in particular appear to be consistent, even when as many as 30% of the training data labels are incorrect on average. For the $k$nn and SVM classifiers we observe very similar results for Model 2; the LDA classifier does not perform well in this setting, however, since the Bayes decision boundary is non-linear. These conclusions are in accordance with Corollary \[cor:consistent\] and Theorem \[thm:LDAhomo\].
We further investigate the effect of homogeneous label noise on the performance of the LDA classifier for data from Model 1, but now when $d=5$ and the class prior probabilities are unbalanced. Recall that in Theorem \[thm:LDAhomo\] we derived the asymptotic limit of the risk in terms of the Mahalanobis distance between the true class distributions, the class prior probabilities and the noise rate. In Figure \[fig:errorplotLDA\], we present the estimated risks of the LDA classifier for data from Model 1 with $\pi_1 =$ 0.9 for different homogeneous noise rates alongside the limit as specified by Theorem \[thm:LDAhomo\]. This articulates the inconsistency of the corrupted LDA classifier, as observed in Theorem \[thm:LDAhomo\].
Finally, we study empirically the asymptotic regret ratios for the $k$nn and SVM classifiers. We focus on the noise model in Example 2 in Section \[sec:asymptotic\], where the label errors occur at random as follows: fix $g_0 \in (0,1/2)$, $h_0 > 2 - 1/g_0$, we let $g(1/2 + t) = \max[0, \min\{g_0(1 + h_0 t), 2g_{0}\}]$, then set $\rho_{0}(x) = g(\eta(x))$ and $\rho_{1}(x) = g(1-\eta(x))$. In particular, we use the following settings: (i) $g_{0} = $ 0.1, $h_{0} = 0$; (ii) $g_{0} = $ 0.1, $h_{0} = -1$; (iii) $g_{0} = $ 0.1, $h_{0} = 1$; (iv) $g_{0} = $ 0.1, $h_{0} = 2$; (v) $g_{0} = $ 0.1, $h_{0} = 3$. Noise setting (i), where $h_0 = 0$, corresponds to $g_0$-homogeneous noise.
![[Estimated regret ratios for the $k$nn (left) and SVM (right) classifiers. Top: Model 1, with $d = 5$ and $\pi_{1} = $ 0.5. Bottom: Model 2, with $d = 5$. We present the results with label noise of type (i – red), (ii – blue), (iii – green), (iv – black), and (v – purple).]{}[]{data-label="fig:errorplot4a"}](PlotknnSVMRR_1.eps "fig:"){width="90.00000%"} ![[Estimated regret ratios for the $k$nn (left) and SVM (right) classifiers. Top: Model 1, with $d = 5$ and $\pi_{1} = $ 0.5. Bottom: Model 2, with $d = 5$. We present the results with label noise of type (i – red), (ii – blue), (iii – green), (iv – black), and (v – purple).]{}[]{data-label="fig:errorplot4a"}](PlotknnSVMRR_2.eps "fig:"){width="90.00000%"}
For the $k$nn classifier, where $k$ is chosen to satisfy the conditions of Corollary \[cor:knnRR\], our theory says that when $d=5$ in Models 1 and 2, the asymptotic regret ratios in the five noise settings are 1.22, 1.37, 1.10, 1 and 0.92 respectively. We see from the left-hand plots of Figure \[fig:errorplot4a\] that, for $k$ chosen separately in the corrupted and uncorrupted cases via cross-validation, the empirical results provide good agreement with our theory, especially in the last three settings. Reasons for the slight discrepancies between our asymptotic theory and empirically observed regret ratios in the first two noise settings include the following facts: the choices of $k$ in the noisy and noiseless label settings do not necessarily satisfy exactly; the asymptotics in $n$ may not have fully ‘kicked in’; and Monte Carlo error (when $n$ is large, we are computing the ratio of two small quantities, so the standard error tends to be larger). The performance of the SVM classifier is similar to that of the $k$nn classifier for both models.
Finally, we discuss tuning parameter selection. We have seen that for the $k$nn classifier the the choice of $k$ is important for achieving the optimal bias–variance trade-off; see also @Hall:2008. Similarly, we need to choose an appropriate value of $\lambda$ for the SVM classifier; in practice, this is typically done via cross-validation. When the classifier $\tilde{C}$ is trained with $\rho$-homogeneous noisy labels, we would like to select a tuning parameter to minimize $R(\tilde{C})$, but since the training data is corrupted, a tuning parameter selection method will target the minimizer of $\tilde{R}(\tilde{C})$. However, by Lemma \[lem:tilde\], we have that $R(\tilde{C}) = \{\tilde{R}(\tilde{C}) - \rho\}/(1- 2\rho)$, and it follows that our tuning parameter selection method requires no modification when trained with noisy labels. In the heterogeneous noise case, however, we do not have this direct relationship; see @Inouye:2017 for more on this topic.
In our simulations, we chose $k$ for the $k$nn classifier and $\lambda$ for the SVM classifier via leave-one-out and 10-fold cross-validation respectively, where the cross-validation was performed over the noisy training dataset. Moreover, for the SVM classifier, we used the default choice $\sigma^2= 1/d$ for the hyper-parameter for the kernel function.
Proofs and an illustrative example {#sec:proofs}
==================================
Proofs from Section \[sec:finite\]
----------------------------------
\(i) First, we have that for $P_X$-almost all $x \in \mathcal{B}$, $$\begin{aligned}
\label{eq:etaetatilde}
\tilde{\eta}(x) - 1/2 & = \{\eta(x) - 1/2\} \{1 - \rho_0(x) - \rho_1(x)\} + \frac{1}{2} \{\rho_0(x) - \rho_1(x)\} \nonumber
\\& = \{\eta(x) - 1/2\} \{1 - \rho_0(x) - \rho_1(x)\}\biggl( 1 - \frac{\rho_1(x) - \rho_0(x)}{\{2\eta(x)-1\}\{1 - \rho_0(x) - \rho_1(x)\}} \biggr).
\end{aligned}$$ Thus, for $P_X$-almost all $x \in \mathcal{B}$, we have $\{\rho_1(x) - \rho_0(x)\}/[\{2\eta(x)-1\}\{1 - \rho_0(x) - \rho_1(x)\}] < 1$ if and only if $$\mathrm{sgn}\{\tilde{\eta}(x) - 1/2\} = \mathrm{sgn}\{\eta(x) - 1/2\}.$$ This completes the proof of . It follows that, if $P_X(\mathcal{A}^c \cap \mathcal{S}^c)=0$, then $P_X(\{x \in \mathcal{B} : \tilde{C}^{\mathrm{Bayes}}(x)= C^{\mathrm{Bayes}}(x)\}^c \cap \mathcal{S}^c)=0$. In other words $P_X(\{x \in \mathcal{S}^c: \tilde{C}^{\mathrm{Bayes}}(x)\neq C^{\mathrm{Bayes}}(x) \})=0$, i.e. (2) holds. Here we have used the fact that $\mathcal{A} \subseteq \mathcal{B}$, so if $P_X(\mathcal{A}^c \cap \mathcal{S}^c)=0$, then $P_X(\mathcal{B}^c \cap \mathcal{S}^c)=0$.
\(ii) For the proof of this part, we apply Proposition \[prop:prelim\]. First, since holds, we have $\tilde{R}(C^{\mathrm{Bayes}}) = \tilde{R}(\tilde{C}^{\mathrm{Bayes}})$. From , we have that for $P_X$-almost all $x \in \mathcal{B}$, $$\begin{aligned}
\label{Eq:Akappa}
|2\tilde{\eta}(x) - 1| &= |2\eta(x) - 1| \{1 - \rho_0(x) - \rho_1(x)\}\Bigl( 1 - \frac{\rho_1(x) - \rho_0(x)}{\{2\eta(x)-1\}\{1 - \rho_0(x) - \rho_1(x)\}} \Bigr) \nonumber
\\ & \geq |2\eta(x) - 1| (1 - 2\rho^*) ( 1 - a^*).
\end{aligned}$$ In fact, the conclusion of remains true (trivially) when $x \in \mathcal{S}$. Thus, by Proposition \[prop:prelim\], $$\begin{aligned}
R(C) - R(C^{\mathrm{Bayes}})
&\leq \inf_{\kappa > 0} \Bigl\{\kappa \{\tilde{R}(C) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})\} + P_X(A_{\kappa}^c) \Bigr\}
\\ & \leq \frac{\tilde{R}(C) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})}{ (1 - 2\rho^*) ( 1 - a^*)} + P_X(A_{(1 - 2\rho^*)^{-1} ( 1 - a^*)^{-1}}^c)
= \frac{\tilde{R}(C) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})}{ (1 - 2\rho^*) ( 1 - a^*)},
\end{aligned}$$ since $P_X(A_{(1 - 2\rho^*)^{-1} ( 1 - a^*)^{-1}}^c) \leq P_X(A_{(1 - 2\rho^*)^{-1} ( 1 - a^*)^{-1}}^c \cap \mathcal{B}) + P_X(\mathcal{B}^c) = 0$, by .
Proposition \[prop:prelim\] is a special case of the following result.
\[prop:prelim2\] Let $\mathcal{D} = \bigl\{x \in \mathcal{S}^c : \tilde{C}^{\mathrm{Bayes}}(x) = C^{\mathrm{Bayes}}(x)\bigr\}$, and recall the definition of $A_\kappa$ in Proposition \[prop:prelim\]. Then, for any classifier $C$, $$\begin{aligned}
R(C) - R(C^{\mathrm{Bayes}}) &\leq R(\tilde{C}^{\mathrm{Bayes}}) - R(C^{\mathrm{Bayes}}) + \min\Bigl[\mathbb{P}\bigl\{ \{C(X) \neq \tilde{C}^{\mathrm{Bayes}}(X)\} \cap \{X \in \mathcal{D}\} \bigr\},
\\ & \hspace{80pt} \inf_{\kappa > 0} \bigl\{\kappa \{\tilde{R}(C) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})\} + \mathbb{E}\bigl(|2\eta(X) - 1| \mathbbm{1}_{\{X \in \mathcal{D} \setminus A_{\kappa}\}}\bigr) \bigr\} \Bigr].
\end{aligned}$$
Remark: If holds, i.e. $P_X(\mathcal{D}^c \cap \mathcal{S}^c) = 0$, then $R(\tilde{C}^{\mathrm{Bayes}}) = R(C^{\mathrm{Bayes}})$, and moreover we have that $\mathbb{E}\bigl(|2\eta(X) - 1| \mathbbm{1}_{\{X \in \mathcal{D} \setminus A_{\kappa}\}}\bigr) \leq P_X\bigl(\mathcal{D} \setminus A_{\kappa}\bigr) \leq P_X\bigl(A_{\kappa}^c\bigr)$.
First write $$\begin{aligned}
\label{eq:risk}
R(C) &= \int_{\mathcal{X}} \mathbb{P}\{C(x) \neq Y \mid X=x\} \, dP_{X}(x) \nonumber
\\& = \int_{\mathcal{X}} \bigl[ \mathbb{P}\{C(x) = 0 \} \mathbb{P}(Y =1 \mid X = x) + \mathbb{P}\{C(x) = 1 \}\mathbb{P}(Y =0\mid X = x) \bigr]\, dP_{X}(x) \nonumber
\\& = \int_{\mathcal{X}} \bigl[ \mathbb{P}\{C(x) = 0 \}\{2\eta(x) - 1\} + \{1-\eta(x)\} \bigr] \, dP_{X}(x).
\end{aligned}$$ Here we have implicitly assumed that the classifier $C$ is random since it may depend on random training data. However, in the case that $C$ is non-random, one should interpret $\mathbb{P}\{C(x) = 0\}$ as being equal to $\mathbbm{1}_{\{C(x) = 0\} }$, for $x \in \mathcal{X}$.
Now, for $P_X$-almost all $x \in \mathcal{D}$, $$\begin{aligned}
\bigl[\mathbb{P}\{C(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\bigr] \{2\eta(x) - 1\} & = \bigl|\mathbb{P}\{C(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\bigr| |2\eta(x) - 1|
\\ & \leq \bigl|\mathbb{P}\{C(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\bigr|
\\ & = \mathbb{P}\{C(x) \neq \tilde{C}^{\mathrm{Bayes}}(x)\}.
\end{aligned}$$ Moreover, for $P_X$-almost all $x \in \mathcal{D}^c$, we have $$\label{Eq:NegativeProduct}
\bigl[\mathbb{P}\{C(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\bigr] \{2\eta(x) - 1\} \leq 0$$ It follows that $$\begin{aligned}
R(C) - R(\tilde{C}^{\mathrm{Bayes}}) & = \int_{\mathcal{X}} \bigl[\mathbb{P}\{C(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\bigr] \{2\eta(x) - 1\} \, dP_{X}(x)
\\ & = \int_{\mathcal{D}} \bigl[\mathbb{P}\{C(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\bigr] \{2\eta(x) - 1\} \, dP_{X}(x)
\\ & \hspace {30 pt} + \int_{\mathcal{D}^c} \bigl[\mathbb{P}\{C(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\bigr] \{2\eta(x) - 1\} \, dP_{X}(x)
\\ & \leq \mathbb{P}\bigl(\{C(X) \neq \tilde{C}^{\mathrm{Bayes}}(X)\} \cap \{X \in \mathcal{D}\} \bigr).
\end{aligned}$$
To see the right-hand bound, observe that by , for $\kappa > 0$, $$\begin{aligned}
R(C) - R(\tilde{C}^{\mathrm{Bayes}}) & = \int_{\mathcal{X}} \bigl[\mathbb{P}\{C(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\bigr] \{2\eta(x) - 1\} \, dP_{X}(x)
\\ & \leq \int_{\mathcal{D}} \bigl[\mathbb{P}\{C(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\bigr] \{2\eta(x) - 1\} \, dP_{X}(x)
\\ & \leq \kappa \int_{\mathcal{D} \cap A_\kappa} \bigl[\mathbb{P}\{C(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\bigr] \{2\tilde{\eta}(x) - 1\} \, dP_{X}(x)
\\ & \hspace{180 pt} + \mathbb{E}\bigl(|2\eta(X) - 1| \mathbbm{1}_{\{X \in \mathcal{D} \setminus A_{\kappa}\}}\bigr)
\\ & = \kappa \{\tilde{R}(C) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})\} + \mathbb{E}\bigl(|2\eta(X) - 1| \mathbbm{1}_{\{X \in \mathcal{D} \setminus A_{\kappa}\}}\bigr),
\end{aligned}$$ where the last step follows from .
\[ex:1nn\] *Suppose that $\mathcal{X} \subseteq \mathbb{R}^d$ and that the noise is $\rho$-homogeneous with $\rho \in (0,1/2)$. Consider the corrupted $1$-nearest neighbour classifier $\tilde{C}^{1\mathrm{nn}}(x) = \tilde{Y}_{(1)}$, where $(X_{(1)},\tilde{Y}_{(1)}) = (X_{(1)}(x),\tilde{Y}_{(1)}(x)) = (X_{i^*},\tilde{Y}_{i^*})$ is the training data pair for which $i^* = \operatorname*{sargmin}_{i =1,\ldots,n} \|X_i-x\|$, where $\operatorname*{sargmin}$ denotes the smallest index of the set of minimizers. We first study the first term in the minimum in . Noting that $\tilde{R}(\tilde{C}^{\mathrm{Bayes}}) = \mathbb{E}[\min\{\tilde{\eta}(X), 1-\tilde{\eta}(X) \}]$, we have* $$\begin{aligned}
\label{Eq:LongDisplay}
\bigl|\mathbb{P}\{\tilde{C}^{1\mathrm{nn}}&(X) \neq \tilde{C}^{\mathrm{Bayes}}(X)\} - \tilde{R}(\tilde{C}^{\mathrm{Bayes}}) \bigr| \nonumber
\\ & = \bigl|\mathbb{P}\{\tilde{Y}_{(1)}(X) \neq \tilde{C}^{\mathrm{Bayes}}(X)\} - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})\bigr| \nonumber
\\ & = \bigl| \mathbb{E}[\mathbbm{1}_{\{\tilde{\eta}(X) < 1/2\}}\tilde{\eta}(X_{(1)}(X)) + \mathbbm{1}_{\{\tilde{\eta}(X) \geq 1/2\}}\{1- \tilde{\eta}(X_{(1)}(X))\} ] - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})\bigr| \nonumber
\\ & = \bigl|\mathbb{E}[\mathbbm{1}_{\{\tilde{\eta}(X) < 1/2\}}\{\tilde{\eta}(X_{(1)}(X)) - \tilde{\eta}(X) \} + \mathbbm{1}_{\{\tilde{\eta}(X) \geq 1/2\}}\{\tilde{\eta}(X)- \tilde{\eta}(X_{(1)}(X))\} ] \bigr| \nonumber
\\& \leq \mathbb{E}\bigl|\tilde{\eta}(X_{(1)}(X)) - \tilde{\eta}(X)\bigr | \rightarrow 0,
\end{aligned}$$ *where the final limit follows by @PTPR:1996 [Lemma 5.4].*
*Now focusing on the second term in the minimum in , by @PTPR:1996 [Theorem 5.1], we have* $$\tilde{R}(\tilde{C}^{1\mathrm{nn}}) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}}) \rightarrow 2\mathbb{E}[\tilde{\eta}(X)\{1-\tilde{\eta}(X)\}] - \tilde{R}(\tilde{C}^{\mathrm{Bayes}}).$$ *Moreover, in this case, $P_X(A_{\kappa}^c) = 1$ for all $\kappa \leq (1 - 2\rho)^{-1}$, and 0 otherwise. Therefore, if $\rho$ is small enough that $\rho\tilde{R}(\tilde{C}^{\mathrm{Bayes}}) < \tilde{R}(\tilde{C}^{\mathrm{Bayes}}) - \mathbb{E}[\tilde{\eta}(X)\{1-\tilde{\eta}(X)\}]$, then* $$\begin{aligned}
\label{Eq:TwoMins}
\lim_{n \rightarrow \infty} \inf_{\kappa > 0} \Bigl\{ \kappa \{\tilde{R}(\tilde{C}^{1\mathrm{nn}}) &- \tilde{R}(\tilde{C}^{\mathrm{Bayes}})\} + P_X(A_\kappa^c) \Bigr\} = \lim_{n\rightarrow \infty} \frac{\tilde{R}(\tilde{C}^{1\mathrm{nn}}) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})}{1 - 2\rho} \nonumber \\
&= \frac{2\mathbb{E}[\tilde{\eta}(X)\{1-\tilde{\eta}(X)\}] - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})}{1-2\rho} \nonumber \\
&< \tilde{R}(\tilde{C}^{\mathrm{Bayes}}) = \lim_{n \rightarrow \infty} \mathbb{P}\{\tilde{C}^{1\mathrm{nn}}(X) \neq \tilde{C}^{\mathrm{Bayes}}(X)\},
\end{aligned}$$ *where the final equality is due to . Thus, in this case, the second term in the minimum in is smaller for sufficiently large $n$. However, if $\rho\tilde{R}(\tilde{C}^{\mathrm{Bayes}}) > \tilde{R}(\tilde{C}^{\mathrm{Bayes}}) - \mathbb{E}[\tilde{\eta}(X)\{1-\tilde{\eta}(X)\}]$, the asymptotically better bound is given by the first term in the minimum in the conclusion of Proposition \[prop:prelim\], because then the inequality in is reversed.* $\Box$
Let $\epsilon_n = \max\bigl[\sup_{m\geq n}\{\tilde{R}(\tilde{C}_m) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})\}^{1/2}, n^{-1}\bigr]$. Then, by Proposition \[prop:prelim2\], $$\begin{aligned}
R(\tilde{C}_n) - R(\tilde{C}^{\mathrm{Bayes}}) &\leq \frac{1}{\epsilon_n} \{\tilde{R}(\tilde{C}_n) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})\} + \mathbb{E}\bigl(|2\eta(X) - 1| \mathbbm{1}_{\{X \in \mathcal{D} \setminus A_{\epsilon_n^{-1}}\}}\bigr)
\\ & \leq \{\tilde{R}(\tilde{C}_n) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})\}^{1/2} + P_X(\mathcal{D} \setminus A_{\epsilon_n^{-1}}\bigr).
\end{aligned}$$ Since $(\epsilon_n)$ is decreasing, it follows that $$\limsup_{n \rightarrow \infty} R(\tilde{C}_n) - R(C^{\mathrm{Bayes}}) \leq R(\tilde{C}^{\mathrm{Bayes}}) - R(C^{\mathrm{Bayes}}) + P_X(\tilde{\mathcal{S}} \cap \mathcal{D}).$$ In particular, if holds, then $$\limsup_{n \rightarrow \infty} R(\tilde{C}_n) - R(C^{\mathrm{Bayes}}) \leq P_X(\tilde{\mathcal{S}} \setminus \mathcal{S}),$$ as required.
Conditions and proof of Theorem \[thm:knnhet\] {#sec:knnproofs}
----------------------------------------------
A formal description of the conditions of Theorem \[thm:knnhet\] is given below:
: The probability measures $P_0$ and $P_1$ are absolutely continuous with respect to Lebesgue measure, with Radon–Nikodym derivatives $f_0$ and $f_1$, respectively. Moreover, the marginal density of $X$, given by $\bar{f} = \pi_{0} f_{0} + \pi_{1} f_{1}$, is continuous and positive.
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: The set $\mathcal{S}$ is non-empty and $\bar{f}$ is bounded on $\mathcal{S}$. There exists $\epsilon_0 > 0$ such that $\bar{f}$ is twice continuously differentiable on $\mathcal{S}^{\epsilon_0} = \mathcal{S} + B_{\epsilon_0}(0)$, and $$\label{eq:A2eq}
F(\delta) = \sup_{x_0 \in \mathcal{S}: \bar{f}(x_0) \geq \delta} \max \biggl\{ \frac{\|\dot{\bar{f}}(x_0)\|}{\bar{f}(x_0)}, \frac{\sup_{u \in B_{\epsilon_0}(0)}\|\ddot{\bar{f}}(x_0+u)\|_{\mathrm{op}}}{ \bar{f}(x_0)} \biggr\} = o(\delta^{-\tau})$$ as $\delta \searrow 0$, for every $\tau>0$. Furthermore, recalling $a_d = \pi^{d/2}/\Gamma(1+d/2)$ and writing $p_\epsilon(x) = P_X(B_\epsilon(x))$, there exists $\mu_{0} \in (0,a_d)$ such that for all $x \in \mathbb{R}^{d}$ and $\epsilon \in (0,\epsilon_{0}]$, we have $$p_\epsilon(x) \geq \mu_{0} \epsilon^{d} \bar{f}(x).$$
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: We have $\inf_{x_0\in \mathcal{S}}\|\dot{\eta}(x_0)\| > 0$, so that $\mathcal{S}$ is a $(d-1)$-dimensional, orientable manifold. Moreover, $\sup_{x \in \mathcal{S}^{2\epsilon_0}} \|\dot{\eta}(x)\| < \infty$ and $\ddot{\eta}$ is uniformly continuous on $\mathcal{S}^{2\epsilon_0}$ with $\sup_{x\in \mathcal{S}^{2\epsilon_0}} \|\ddot{\eta} (x)\|_{\mathrm{op}} < \infty$. Finally, the function $\eta$ is continuous, and $$\inf_{x \in \mathbb{R}^d \setminus \mathcal{S}^{\epsilon_0}} |\eta(x) - 1/2| > 0.$$
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: We have that $\int_{\mathbb{R}^d} \|x\|^{\alpha} \, dP_{X}(x) < \infty$ and $\int_{\mathcal{S}} \bar{f}(x_0)^{d/(\alpha+d)} \, d\mathrm{Vol}^{d-1}(x_0) < \infty$, where $d\mathrm{Vol}^{d-1}$ denotes the $(d-1)$-dimensional volume form on $\mathcal{S}$.
*Part 1:* We show that the distribution $\tilde{P}$ of the pair $(X, \tilde{Y})$ satisfies suitably modified versions of Assumptions A1, A2, A3 and A4($\alpha$).
Assumption A1: For $r \in \{0,1\}$, let $\tilde{P}_r$ denote the conditional distribution of $X$ given $\tilde{Y} = r$. For $x \in \mathbb{R}^{d}$, and $r = 0,1$, define $$\tilde{f}_{r}(x) = \frac{\pi_{r}\{1-\rho_r(x)\}f_{r}(x) + \pi_{1-r}\rho_{1-r}(x) f_{1-r}(x)} {\int_{\mathbb{R}^{d}} \pi_{r}\{1-\rho_r(z)\}f_{1-r}(z) + \pi_{1-r}\rho_{1-r}(z)f_{1-r}(z) \, dz}.$$ Now, for a Borel subset $A$ of $\mathbb{R}^{d}$, we have that $$\begin{aligned}
\tilde{P}_{1}(A) & = \mathbb{P}(X \in A \mid \tilde{Y} = 1) = \frac{\mathbb{P}(X \in A ,\tilde{Y} = 1)}{ \mathbb{P} ( \tilde{Y} =1) }
\\& = \frac{\pi_{1}\mathbb{P}(X \in A ,\tilde{Y} = 1 \mid Y = 1) + \pi_{0}\mathbb{P}(X \in A ,\tilde{Y} = 1 \mid Y = 1)}{ \mathbb{P} ( \tilde{Y} =1) }
\\& = \frac{\pi_{1}\int_{A} \{1-\rho_1(x)\} f_{1}(x) \, dx + \pi_{0} \int_{A} \rho_0(x) f_{0}(x) \, dx } { \mathbb{P} ( \tilde{Y} =1) } = \int_{A} \tilde{f}_{1}(x) \, dx.
\end{aligned}$$ Similarly, $\tilde{P}_{0}(A) = \int_{A} \tilde{f}_{0}(x) \, dx$. Hence $\tilde{P}_{0}$ and $\tilde{P}_{1}$ are absolutely continuous with respect to Lebesgue measure, with Radon–Nikodym derivatives $\tilde{f}_{0}$ and $\tilde{f}_{1}$, respectively. Furthermore, $\tilde{f} = \mathbb{P}(\tilde{Y}=0) \tilde{f}_{0} + \mathbb{P}(\tilde{Y}=1)\tilde{f}_{1} = \bar{f}$ is continuous and positive.
Assumption A2: Since A2 refers mainly to the marginal distribution of $X$, which is unchanged under the addition of label noise, this assumption is trivially satisfied for $\tilde{f} = \bar{f}$, as long as $\tilde{\mathcal{S}} = \{ x\in \mathbb{R}^{d} : \tilde{\eta}(x) = 1/2\} = \mathcal{S}$. To see this, let $\delta_0 > 0$ and note that for $x$ satisfying $\eta(x) -1/2 > \delta_0$, we have from that $$\begin{aligned}
\label{eq:etabound}
\tilde{\eta}(x) - 1/2
& = \{\eta(x) - 1/2\}\{1 - \rho_0(x) - \rho_1(x)\}\Bigl\{1 + \frac{\rho_{0}(x) - \rho_{1}(x)}{\{2\eta(x)-1\}\{1 - \rho_0(x) - \rho_1(x)\}}\Bigr\}\nonumber
\\& > \{\eta(x) - 1/2\}(1- 2\rho^*)(1 - a^*) \geq \delta_0 (1- 2\rho^*)(1 - a^*).
\end{aligned}$$ Similarly, if $1/2 - \eta(x) > \delta_0$, then we have that $1/2 - \tilde{\eta}(x) > \delta_0 (1- 2\rho^*)(1 - a^*)$. It follows that $\tilde{\mathcal{S}} \subseteq \mathcal{S}.$ Now, for $x$ such that $|\eta(x) -1/2| < \delta$, we have $$\label{Eq:geq}
\tilde{\eta}(x) - 1/2 = \eta(x) - 1/2 + \{1-\eta(x)\}g(\eta(x)) - \eta(x)g(1-\eta(x)).$$ Thus $\mathcal{S} \subseteq \tilde{\mathcal{S}}$.
Assumption A3: Since $g$ is twice continuously differentiable, we have that $\tilde{\eta}$ is twice continuously differentiable on the set $\{x \in \mathcal{S}^{2\epsilon_0} : |\eta(x) -1/2| < \delta \}$. On this set, its gradient vector at $x$ is $$\begin{aligned}
\dot{\tilde{\eta}}(x)
& = \dot{\eta}(x)\Bigl[1 - g(\eta(x)) - g(1 - \eta(x)) + \{1-\eta(x)\} \dot{g}(\eta(x)) + \eta(x) \dot{g}(1 - \eta(x)) \Bigr].
\end{aligned}$$ The corresponding Hessian matrix at $x$ is $$\begin{aligned}
\ddot{\tilde{\eta}}(x) & = \ddot{\eta}(x)\Bigl[1 - g(\eta(x)) - g(1 - \eta(x)) + \{1-\eta(x)\} \dot{g}(\eta(x)) + \eta(x) \dot{g}(1 - \eta(x))\Bigr ]
\\ & \hspace{30 pt} - \dot{\eta}(x) \Bigl[\dot{\eta}(x)^T \dot{g}(\eta(x)) - \dot{\eta}(x)^T \dot{g}(1 - \eta(x)) + \dot{\eta}(x)^T\dot{g}(\eta(x))
\\ & \hspace{60 pt} - \{1-\eta(x)\} \dot{\eta}(x)^T \ddot{g}(\eta(x)) - \dot{\eta}(x)^T\dot{g}(1-\eta(x)) + \eta(x) \dot{\eta}(x)^T \ddot{g}(1 - \eta(x))\Bigr ].
\end{aligned}$$ In particular, for $x_0 \in \mathcal{S}$ we have $$\label{Eq:etadotddot}
\dot{\tilde{\eta}}(x_0) = \dot{\eta}(x_0) \{1 - 2g(1/2) + \dot{g}(1/2)\}; \quad \ddot{\tilde{\eta}}(x_0) = \ddot{\eta}(x_0)\{1 - 2g(1/2) + \dot{g}(1/2)\}.$$ Now define $$\epsilon_1 = \sup\Bigl\{ \epsilon > 0 : \sup_{x \in \mathcal{S}^{2\epsilon}} |\eta(x) - 1/2| < \delta\Bigr\} > 0,$$ where the fact that $\epsilon_1$ is positive follows from Assumption A3. Set $\tilde{\epsilon}_0 = \min\{\epsilon_0, \epsilon_1\}/2$. Then, using the properties of $g$, we have that $\inf_{x_0\in \mathcal{S}}\|\dot{\tilde{\eta}}(x_0)\| > 0$. Moreover, $\sup_{x \in \mathcal{S}^{2\tilde{\epsilon}_{0}}} \|\dot{\tilde{\eta}}(x)\| < \infty$ and $\ddot{\tilde{\eta}}$ is uniformly continuous on $\mathcal{S}^{2\tilde{\epsilon}_{0}}$ with $\sup_{x\in \mathcal{S}^{2\tilde{\epsilon}_{0}}} \|\ddot{\tilde{\eta}} (x)\|_{\mathrm{op}} < \infty$. Finally, the function $\tilde{\eta}$ is continuous since $\rho_0,\rho_1$ are continuous, and, by , $$\inf_{x \in \mathbb{R}^d \setminus \mathcal{S}^{\tilde{\epsilon}_{0}}} |\tilde{\eta}(x) - 1/2| > 0.$$
Assumption A4($\alpha$): This holds for $\tilde{P}$ because the marginal distribution of $X$ is unaffected by the label noise and $\tilde{\mathcal{S}} = \mathcal{S}$.
*Part 2*: Recall the function $F$ defined in . Let $c_n = F(k/(n-1))$, and set $\epsilon_n = \{c_n\beta^{1/2} \log^{1/2}(n-1)\}^{-1}$, $\Delta_{n}= k(n-1)^{-1} c_{n}^{d}\log^{d}((n-1)/k)$, $\mathcal{R}_n = \{x \in \mathbb{R}^d : \bar{f}(x) > \Delta_n \}$ and $\mathcal{S}_n= \mathcal{S} \cap \mathcal{R}_n$. Then, by and the fact that $\inf_{x_0 \in \mathcal{S}} \|\dot{\tilde{\eta}}(x_0)\| > 0$, there exists $c_0 > 0$ such that for every $\epsilon \in (0,\tilde{\epsilon}_0]$, $$\inf_{x \in \mathbb{R}^d \setminus \mathcal{S}^{\epsilon}} |\tilde{\eta}(x) -1/2| > c_0\epsilon.$$ Now let $\tilde{S}_{n}(x) = k^{-1} \sum_{i = 1}^{k} \mathbbm{1}_{\{\tilde{Y}_{(i)} = 1\}}$, $X^{n} = (X_{1}, \ldots, X_{n})$ and $\tilde{\mu}(x, X^n) = \mathbb{E}\{\tilde{S}_{n}(x) \mid X^{n}\} = k^{-1}\sum_{i=1}^{k} \tilde{\eta}(X_{(i)})$. Define $A_{k} = \bigl\{\|X_{(k)}(x) - x\| \leq \epsilon_{n}/2 \ \mbox{for all} \ x \in \mathcal{R}_{n} \bigr\}$. Now suppose that $z_{1}, \dots, z_{N} \in \mathcal{R}_{n}$ are such that $\|z_{j} - z_{\ell}\| > \epsilon_{n}/4$ for all $j \neq \ell$, but $\sup_{x \in \mathcal{R}_{n}} \min_{j=1, \dots, N} \|x - z_{j}\| \leq \epsilon_{n}/4$. Then by the final part of Assumption A2, for $n \geq 2$ large enough that $\epsilon_n/8 \leq \epsilon_0$, we have $$1 = P_{X}(\mathbb{R}^{d}) \geq \sum_{j=1}^{N} p_{\epsilon_{n}/8}(z_{j}) \geq \frac{N \mu_{0} \beta^{d/2}\log^{d/2}(n-1)}{8^d(n-1)^{1-\beta}}.$$ Then by a standard binomial tail bound [@Shorack:86 Equation (6), p. 440], for such $n$ and any $M > 0$, $$\begin{aligned}
\mathbb{P}(A_{k}^c) & = \mathbb{P}\Bigl\{ \sup_{x \in \mathcal{R}_{n} }\|X_{(k)}(x) - x\| > \epsilon_{n}/2\Bigr\} \nonumber \leq \mathbb{P}\Bigl\{\max_{j=1, \ldots, N} \|X_{(k)}(z_{j}) - z_{j}\| > \epsilon_{n}/4 \Bigr\} \nonumber \\
& \leq \sum_{j=1}^{N} \mathbb{P}\bigl\{\|X_{(k)}(z_{j}) - z_{j}\| > \epsilon_{n}/4 \bigr\} \nonumber \leq N \max_{j=1,\ldots,N} \exp\Bigl(-\frac{1}{2}np_{\epsilon_n/4}(z_j) +k \Bigr) = O(n^{-M}),
\end{aligned}$$ uniformly for $k \in K_{\beta}$.
Now, on the event $A_{k}$, for $\epsilon_n < \tilde{\epsilon}_0$ and $x \in \mathcal{R}_{n} \setminus \mathcal{S}^{\epsilon_{n}}$, the $k$ nearest neighbours of $x$ are on the same side of $\mathcal{S}$, so $$\begin{aligned}
|\tilde{\mu}_n(x,X^n) - 1/2| & = \biggl|\frac{1}{k} \sum_{i=1}^{k} \tilde{\eta}(X_{(i)}) - \frac{1}{2}\biggr| \geq \inf_{z \in B_{\epsilon_n/2}(x)} |\tilde{\eta}(z) - 1/2| \geq c_0 \frac{\epsilon_n}{2}.
\end{aligned}$$ Moreover, conditional on $X^n$, $\tilde{S}_n(x)$ is the sum of $k$ independent terms. Therefore, by Hoeffding’s inequality, $$\begin{aligned}
\label{eq:nM}
\sup_{k \in K_\beta}& \sup_{x\in \mathcal{R}_{n}\setminus \mathcal{S}^{\epsilon_{n}}} \bigl|\mathbb{P}\{\tilde{C}_{n}^{k\mathrm{nn}}(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\bigr| \nonumber
\\ & = \sup_{k \in K_\beta} \sup_{x\in \mathcal{R}_{n}\setminus \mathcal{S}^{\epsilon_{n}}} \bigl|\mathbb{P}\{\tilde{S}_n(x) <1/2\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\bigr| \nonumber
\\ & = \sup_{k \in K_\beta}\sup_{x\in \mathcal{R}_{n}\setminus \mathcal{S}^{\epsilon_{n}}} \bigl|\mathbb{E}\{ \mathbb{P}\{\tilde{S}_n(x) <1/2 \mid X^n\}- \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\}\bigr| \nonumber
\\ & \leq \sup_{k \in K_\beta}\sup_{x\in \mathcal{R}_{n}\setminus \mathcal{S}^{\epsilon_{n}}}\mathbb{E}\bigl[\exp(-2k \{ \tilde{\mu}_n(x, X^n) -1/2\}^2)\mathbbm{1}_{A_{k}} \bigr] + \sup_{k \in K_\beta}\mathbb{P}(A_{k}^c) = O(n^{-M})
\end{aligned}$$ for every $M > 0$.
Next, for $x \in \mathcal{S}^{\epsilon_2}$, we have $|\eta(x) - 1/2| < \delta$, and therefore, letting $t = \eta(x) - 1/2$, from we can write $$\begin{aligned}
2\eta(x) -1 &- \frac{2\tilde{\eta}(x) -1}{1 - 2g(1/2) + \dot{g}(1/2)}
\\ & = \{2\eta(x) - 1\} \Bigl\{1 - \frac{1 - g(\eta(x)) - g(1-\eta(x))}{1 - 2g(1/2) + \dot{g}(1/2)} \Bigr\} - \frac{g(\eta(x)) - g(1 - \eta(x))}{1 - 2g(1/2) + \dot{g}(1/2)}
\\ & = 2t \Bigl\{1 - \frac{1 - g(1/2+t) - g(1/2-t)}{1 - 2g(1/2) + \dot{g}(1/2)} \Bigr\} - \frac{g(1/2 + t) - g(1/2 - t)}{1 - 2g(1/2) + \dot{g}(1/2)} = G(t),
\end{aligned}$$ say. Observe that $$\dot{G}(t) = 2 \Bigl\{1 - \frac{1 - g(1/2+t) - g(1/2-t)}{1 - 2g(1/2) + \dot{g}(1/2)} \Bigr\} + \frac{(2t-1) \dot{g}(1/2+t) - (2t+1) \dot{g}(1/2-t)}{1 - 2g(1/2) + \dot{g}(1/2)};$$ and $$\ddot{G}(t) = \frac{4\{\dot{g}(1/2+t) - \dot{g}(1/2-t)\}}{1 - 2g(1/2) + \dot{g}(1/2)} + \frac{(2t-1) \ddot{g}(1/2+t) + (2t+1) \ddot{g}(1/2-t)}{1 - 2g(1/2) + \dot{g}(1/2)}.$$ In particular, we have $G(0) = 0$, $\dot{G}(0) = 0 $, $\ddot{G}(0) = 0$ and $\ddot{G}$ is bounded on $(-\delta,\delta)$.
Now there exists $n_0$ such that $\epsilon_n < \epsilon_2$, for all $n > n_0$ and $k \in K_\beta$. Therefore, writing $\mathcal{S}_n^{\epsilon_n} = \mathcal{S}^{\epsilon_n} \cap \mathcal{R}_n$, for $n > n_0$, we have that $$\begin{aligned}
&\Biggl|R(\tilde{C}^{k\mathrm{nn}}) - R(C^{\mathrm{Bayes}}) - \frac{\tilde{R}(\tilde{C}^{k\mathrm{nn}}) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}}) }{1- 2g(1/2) + \dot{g}(1/2)} \Biggr |
\\& \hspace{0pt} = \Biggl|\int_{\mathbb{R}^{d}} [\mathbb{P}\{\tilde{C}^{k\mathrm{nn}}(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}] \Bigl\{2\eta(x) -1 - \frac{2\tilde{\eta}(x) -1}{1 - 2g(1/2) + \dot{g}(1/2)} \Bigr\}\, dP_X(x)\Biggr|
\\ & \hspace{0pt} \leq \Biggl|\int_{\mathcal{S}_n^{\epsilon_n}} [\mathbb{P}\{\tilde{C}^{k\mathrm{nn}}(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}] \Bigl\{2\eta(x) -1 - \frac{2\tilde{\eta}(x) -1}{1 - 2g(1/2) + \dot{g}(1/2)} \Bigr\}\, dP_X(x)\Biggr|
\\ & \hspace{60 pt} + \biggl(1+ \frac{1}{1 - 2g(1/2) + \dot{g}(1/2)}\biggr)P_X(\mathcal{R}_n^c) + O(n^{-M}),
\end{aligned}$$ uniformly for $k \in K_\beta$, where the final claim uses . Then, by a Taylor expansion of $G$ about $t = 0$, we have that $$\begin{aligned}
&\Biggl|\int_{\mathcal{S}_n^{\epsilon_n}} [\mathbb{P}\{\tilde{C}^{k\mathrm{nn}}(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}] \Bigl\{2\eta(x) -1 - \frac{2\tilde{\eta}(x) -1}{1 - 2g(1/2) + \dot{g}(1/2)} \Bigr\}\, dP_X(x)\Biggr|
\\& \leq \frac{1}{2}\sup_{t \in (-\delta, \delta)} |\ddot{G}(t)| \int_{\mathcal{S}_n^{\epsilon_n}} |\mathbb{P}\{\tilde{C}^{k\mathrm{nn}}(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}| \{2\eta(x) -1\}^{2} \, dP_X(x)
\\ & \leq \frac{1}{2} \sup_{t \in (-\delta, \delta)} |\ddot{G}(t)| \sup_{x \in \mathcal{S}_n^{\epsilon_n}} |2\eta(x) -1| \int_{\mathcal{S}_n^{\epsilon_n}} \{\mathbb{P}\{\tilde{C}^{k\mathrm{nn}}(x) = 0\} - \mathbbm{1}_{\{\tilde{\eta}(x) < 1/2\}}\} \{2\eta(x) -1\} \, dP_X(x)\
\\ & \leq \frac{1}{2}\sup_{t \in (-\delta, \delta)} |\ddot{G}(t)| \sup_{x \in \mathcal{S}_n^{\epsilon_n}} |2\eta(x) -1| \{ R(\tilde{C}^{k\mathrm{nn}}) - R(C^{\mathrm{Bayes}}) \}
\\ & \leq \frac{1}{2}\sup_{t \in (-\delta, \delta)} |\ddot{G}(t)| \sup_{x \in \mathcal{S}_n^{\epsilon_n}} |2\eta(x) -1| \frac{\tilde{R}(\tilde{C}^{k\mathrm{nn}}) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}}) }{(1-2\rho^*)(1-a^*)} = o\Bigl(\tilde{R}(\tilde{C}^{k\mathrm{nn}}) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}}) \Bigr),
\end{aligned}$$ uniformly for $k \in K_\beta$.
Finally, to bound $P_X(\mathcal{R}_n^c)$, we have by the moment condition in Assumption A4($\alpha$) and Hölder’s inequality, that for any $u \in (0,1)$, and $v > 0$, $$\begin{aligned}
P_X(\mathcal{R}_n^c) &= \mathbb{P}\{ \bar{f}(X) \leq \Delta_n \} \leq (\Delta_n)^\frac{\alpha(1-u)}{\alpha+d} \int_{x: \bar{f}(x) \leq \Delta_n} \bar{f}(x)^{1-\frac{\alpha(1-u)}{\alpha+d}} \, dx \nonumber
\\ & \leq (\Delta_n)^\frac{\alpha(1-u)}{\alpha+d} \Bigl\{\int_{\mathbb{R}^d} (1+\|x\|^\alpha)\bar{f}(x) \, dx\Bigr\}^{1- \frac{\alpha(1-u)}{\alpha+d}} \nonumber
\\ & \hspace{120 pt} \Bigl\{\int_{\mathbb{R}^d} \frac{1}{(1+\|x\|^\alpha)^\frac{d+\alpha u}{\alpha(1-u)} } \, dx \Bigr\}^\frac{\alpha(1-u)}{\alpha+d} = o\biggl(\Bigl(\frac{k}{n}\Bigr)^{\frac{\alpha(1-u)}{\alpha+d} - v}\biggr), \nonumber
\end{aligned}$$ uniformly for $k \in K_{\beta}$.
Since $u \in (0,1)$ was arbitrary, we have shown that, that for any $v > 0$, $$R(\tilde{C}^{k\mathrm{nn}}) - R(C^{\mathrm{Bayes}}) - \frac{\tilde{R}(\tilde{C}^{k\mathrm{nn}}) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}}) }{1- 2g(1/2) + \dot{g}(1/2)} = o\biggl(\tilde{R}(\tilde{C}^{k\mathrm{nn}}) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}}) + \Bigl(\frac{k}{n}\Bigr)^{\frac{\alpha}{\alpha+d} - v}\biggr),$$ uniformly for $k \in K_\beta$. Since Assumptions A1, A2, A3 and A4($\alpha$) hold for $\tilde{P}$, the proof is completed by an application of @CBS:2017 [Theorem 1], together with .
Proofs from Section \[sec:SVM\] {#sec:SVMproofs}
-------------------------------
Before presenting the proofs from this section, we briefly discuss measurability issues for the SVM classifier. Since this is constructed by solving the minimization problem in , it is not immediately clear that it is measurable. It is convenient to let $\mathcal{C}_d$ denote the set of all measurable functions from $\mathbb{R}^d$ to $\{0,1\}$. By @Steinwart:2008 [Definition 6.2, Lemma 6.3 and Lemma 6.23], we have that the function $\tilde{C}_n^{\mathrm{SVM}}: (\mathbb{R}^d \times \{0,1\})^n \rightarrow \mathcal{C}_d$ and the map from $(\mathbb{R}^d \times \{0,1\})^n \times \mathbb{R}^d$ to $\{0,1\}$ given by $\bigl((x_1,\tilde{y}_1),\ldots,(x_n,\tilde{y}_n),x\bigr) \mapsto \tilde{C}_n^{\mathrm{SVM}}(x)$ are measurable with respect to the universal completion of the product $\sigma$-algebras on $(\mathbb{R}^d \times \{0,1\})^n$ and $(\mathbb{R}^d \times \{0,1\})^n \times \mathbb{R}^d$, respectively. We can therefore avoid measurability issues by taking our underlying probability space $(\Omega, \mathcal{F}, \mathbb{P})$ to be as follows: let $\Omega = (\mathbb{R}^d \times \{0,1\} \times \{0,1\})^{n+1}$, and $\mathcal{F}$ to be the universal completion of the product $\sigma$-algebra on $\Omega$. Moreover, we let $\mathbb{P}$ denote the canonical extension of the product measure on $\Omega$. The triples $(X_1, Y_1,\tilde{Y}_1), \ldots, (X_n, Y_n, \tilde{Y}_n),(X,Y,\tilde{Y})$ can be taken to be the coordinate projections of the $(n+1)$ components of $\Omega$.
We first aim to show that $\tilde{P}$ satisfies the margin assumption with parameter $\gamma_1$, and has geometric noise exponent $\gamma_2$. For the first of these claims, by , we have for all $t > 0$ that $$\begin{aligned}
P_{X}(\{x \in \mathbb{R}^{d} : 0 < |\tilde{\eta}(x) - 1/2| \leq t\}) & \leq P_{X}\bigl(\bigl\{x: 0 < |\eta(x) - 1/2|(1-2\rho^{*})(1-a^{*})\leq t\bigr\}\bigr)
\\ & \leq \frac{\kappa_{1}}{(1-2\rho^{*})^{\gamma_1}(1-a^{*})^{\gamma_1}} t^{\gamma_1},
\end{aligned}$$ as required; see also the discussion in Section 3.9.1 of the 2015 Australian National University PhD thesis by M. van Rooyen (<https://openresearch-repository.anu.edu.au/handle/1885/99588>). The proof of the second claim is more involved, because we require a bound on $|2\tilde{\eta}(x) - 1|$ in terms of $|2\eta(x) - 1|$. We consider separately the cases where $|\eta(x)-1/2|$ is small and large, and for $r > 0$, define $\mathcal{E}_r = \{x \in \mathbb{R}^d:|\eta(x)-1/2| < r\}$. For $x \in \mathcal{E}_{\delta} \cap \mathcal{S}^c$, we can write $t_0 = \eta(x) - 1/2 \in (-\delta,\delta)$, so that by again, $$\begin{aligned}
\label{Eq:Step1}
2\tilde{\eta}(x) -1 &= \{2\eta(x) -1\} \Bigl\{ 1 - g(\eta(x)) - g(1-\eta(x)) + \frac{g(\eta(x)) - g(1 - \eta(x))}{2\eta(x) -1}\Bigr\} \nonumber \\
&= \{2\eta(x) -1\}\biggl\{1 - g(1/2 + t_0) - g(1/2-t_0) + \frac{g(1/2+t_0) - g(1/2 - t_0) }{ 2t_0}\biggr\}.
\end{aligned}$$ Now, by reducing $\delta > 0$ if necessary, and since $1 - 2g(1/2) + \dot{g}(1/2) > 0$ by hypothesis, we may assume that $$\label{Eq:Step2}
\biggl|1 - g(1/2 + t_0) - g(1/2-t_0) + \frac{g(1/2+t_0) - g(1/2 - t_0) }{ 2t_0}\biggr| \leq 2\{1 - 2g(1/2) + \dot{g}(1/2)\}$$ for all $t_0 \in [-\delta,\delta]$. Moreover, for $x \in \mathcal{E}_{\delta}^c$, we have $$\begin{aligned}
\label{Eq:Step3}
\Bigl| \{2\eta(x) - 1\} & \{1 - \rho_0(x) - \rho_1(x)\} + \rho_{0}(x) - \rho_{1}(x) \Bigr| \nonumber
\\ & = |2\eta(x) - 1| \biggl|1 - \rho_0(x) - \rho_1(x) + \frac{\rho_{0}(x) - \rho_{1}(x)}{2\eta(x)-1} \biggr| \nonumber
\\ & \leq |2\eta(x) - 1| \biggl\{ 1 + \frac{|\rho_{0}(x) - \rho_{1}(x)|}{2\delta}\biggr\} \leq |2\eta(x) - 1| \Bigl( 1 + \frac{1}{2\delta_0}\Bigr).
\end{aligned}$$ Now that we have the required bounds on $|2\tilde{\eta}(x)-1|$, we deduce from , and that $$\begin{aligned}
\int_{\mathbb{R}^{d}} |2\tilde{\eta}(x) &- 1| \exp\Bigl(-\frac{\tau_{x}^{2}}{t^2}\Bigr) \, dP_{X}(x) \\
&= \int_{\mathbb{R}^{d}} \Bigl| \{2\eta(x) - 1\} \{1 - \rho_0(x) - \rho_1(x)\} + \rho_{0}(x) - \rho_{1}(x) \Bigr| \exp\Bigl(-\frac{\tau_{x}^{2}}{t}\Bigr) \, dP_{X}(x) \\
&\leq \max\Bigl\{2 - 4g(1/2) + 2\dot{g}(1/2), 1 + \frac{1}{2\delta_{0}}\Bigr\}\int_{\mathbb{R}^{d}} |2\eta(x) - 1| \exp\Bigl(-\frac{\tau_{x}^{2}}{t}\Bigr) \, dP_{X}(x) \\
&\leq \max\Bigl\{2 - 4g(1/2) + 2\dot{g}(1/2), 1 + \frac{1}{2\delta_{0}}\Bigr\}\kappa_2 t^{\gamma_2d},
\end{aligned}$$ so $\tilde{P}$ does indeed have geometric noise exponent $\gamma_2$.
Now, for an arbitrary classifier $C$, let $\tilde{L}(C) = \tilde{P}\bigl(\{(x,y) \in \mathbb{R}^d \times \{0,1\}:C(x) \neq y\}\bigr)$ denote the test error. The quantity $\tilde{ L}(\tilde{C}^{\mathrm{SVM}})$ is random because the classifier depends on the training data and the probability in the definition of $\tilde{L}(\cdot)$ is with respect to test data only. It follows by @Steinwart:2007 [Theorem 2.8] that, for all $\epsilon >0$, there exists $M>0$ such that for all $n \in \mathbb{N}$ and all $\tau \geq 1$, $$\mathbb{P}\Bigl(\tilde{ L}(\tilde{C}^{\mathrm{SVM}}) - \tilde{L}(\tilde{C}^{\mathrm{Bayes}}) > M\tau^2 n^{-\Gamma + \epsilon}\Bigr) \leq e^{-\tau}.$$ We conclude by Theorem \[thm:hetnoise\](ii) that $$\begin{aligned}
R(\tilde{C}^{\mathrm{SVM}}&) - R(C^{\mathrm{Bayes}}) \leq \frac{\tilde{R}(\tilde{C}^{\mathrm{SVM}}) - \tilde{R}(\tilde{C}^{\mathrm{Bayes}})}{(1-2\rho^*)(1-a^*)} \\
&= \frac{1}{(1-2\rho^*)(1-a^*)} \int_{0}^{\infty} \mathbb{P}\Bigl(\tilde{ L}(\tilde{C}^{\mathrm{SVM}}) - \tilde{L}(\tilde{C}^{\mathrm{Bayes}}) > u \Bigr) \, du
\\ & = \frac{2M n^{-\Gamma + \epsilon}}{(1-2\rho^*)(1-a^*)} \int_{0}^{\infty} \tau \mathbb{P} \Bigl(\tilde{ L}(\tilde{C}^{\mathrm{SVM}}) - \tilde{L}(\tilde{C}^{\mathrm{Bayes}}) > M\tau^2 n^{-\Gamma + \epsilon} \Bigr) \, d\tau
\\ & \leq \frac{2M n^{-\Gamma + \epsilon}}{(1-2\rho^*)(1-a^*)}\biggl\{\int_{0}^{1} \tau \, d\tau + \int_{1}^\infty \tau \exp(-\tau) \, d\tau\biggr\} = \frac{M n^{-\Gamma + \epsilon}}{(1-2\rho^*)(1-a^*)} \Bigl(1 + \frac{4}{e}\Bigr),
\end{aligned}$$ as required.
Proofs from Section \[sec:LDA\]
-------------------------------
Since, for homogeneous noise, the pair $(X,Y)$ and the noise indicator $Z$ are independent, we have $\mathbb{P}\{C(X) \neq Y \mid Z = r \} = \mathbb{P}\{C(X) \neq Y \}$, for $r = 0,1$. It follows that $$\begin{aligned}
\tilde{R}(C) \! = \! \mathbb{P}\{C(X) \neq \tilde{Y}\} & = \mathbb{P}(Z = 1) \mathbb{P}\{C(X) \neq Y \mid Z = 1 \} + \mathbb{P}(Z = 0) \mathbb{P}\{C(X) = Y \mid Z = 0\}
\\ & = (1 - \rho)\mathbb{P}\{C(X) \neq Y \} + \rho [1 - \mathbb{P}\{C(X) \neq Y\}]\\
& = \rho + (1-2\rho)R(C).
\end{aligned}$$ Rearranging terms gives the first part of the lemma, and the second part follows immediately.
For $r \in \{0,1\}$, we have that ${\hat{\pi}}_r \stackrel{\mathrm{a.s.}}{\rightarrow} (1-\rho) \pi_r + \rho \pi_{1-r} = (1-2\rho) \pi_r + \rho$. Now, writing $${\hat{\mu}}_r = \frac{n^{-1}\sum_{i = 1}^n X_i \mathbbm{1}_{\{\tilde{Y}_i = r\}} }{{\hat{\pi}}_r} = \frac{n^{-1}\sum_{i = 1}^n X_i \mathbbm{1}_{\{\tilde{Y}_i = r\}} (\mathbbm{1}_{\{Y_i = r\}}+ \mathbbm{1}_{\{Y_i = 1 - r\}})} {{\hat{\pi}}_r},$$ we see that $${\hat{\mu}}_r \stackrel{\mathrm{a.s.}}{\rightarrow} \frac{(1-\rho) \pi_{r} \mu_r + \rho \pi_{1-r} \mu_{1-r}}{(1-\rho) \pi_r + \rho \pi_{1-r}}.$$ Hence $$\begin{aligned}
{\hat{\mu}}_1 + {\hat{\mu}}_0 &\stackrel{\mathrm{a.s.}}{\rightarrow} \frac{(1-\rho) \pi_{1} \mu_1 + \rho \pi_{0} \mu_{0}}{(1-\rho) \pi_1 + \rho \pi_{0}}+\frac{(1-\rho) \pi_{0} \mu_0 + \rho \pi_{1} \mu_{1}}{(1-\rho) \pi_0 + \rho \pi_{1}}
\\ & = \mu_{1} \biggl\{\frac{(1-2\rho)^{2}\pi_0\pi_1 + 2\rho(1-\rho)\pi_{1} } {(1-2\rho)^{2} \pi_0\pi_1 + \rho (1-\rho) }\biggr\} + \mu_{0} \biggl\{\frac{ (1-2\rho)^{2} \pi_0\pi_1 + 2\rho(1-\rho) \pi_0 }{(1-2\rho)^{2} \pi_0 \pi_1 + \rho(1-\rho) }\biggr\}.
\end{aligned}$$ Moreover $$\begin{aligned}
{\hat{\mu}}_1 - {\hat{\mu}}_0 &\stackrel{\mathrm{a.s.}}{\rightarrow} \frac{(1-\rho) \pi_{1} \mu_1 + \rho \pi_{0} \mu_{0}}{(1-\rho) \pi_1 + \rho \pi_{0}} - \frac{(1-\rho) \pi_{0} \mu_0 + \rho \pi_{1} \mu_{1}}{(1-\rho) \pi_0 + \rho \pi_{1}}
\\ & = \biggl\{\frac{(1-2\rho) \pi_{0} \pi_{1}} {(1-2\rho)^{2} \pi_0\pi_1 + \rho (1-\rho) }\biggr\} (\mu_{1} - \mu_{0}).
\end{aligned}$$ Observe further that $$\begin{aligned}
{\hat{\Sigma}} &\stackrel{\mathrm{a.s.}}{\rightarrow} \mathrm{cov}\bigl((X_1 - \tilde{\mu}_1)(X_1 - \tilde{\mu}_1)^T\mathbbm{1}_{\{\tilde{Y}_1=1\}} + (X_1 - \tilde{\mu}_0)(X_1 - \tilde{\mu}_0)^T\mathbbm{1}_{\{\tilde{Y}_1=0\}}\bigr) \\
&= \{(1-2\rho) \pi_1 + \rho\} \tilde{\Sigma}_1 + \{(1-2\rho) \pi_0 + \rho\} \tilde{\Sigma}_0,
\end{aligned}$$ where $\tilde{\Sigma}_r = \mathrm{cov}(X \mid \tilde{Y} = r)$, and we now seek to express $\tilde{\Sigma}_0$ and $\tilde{\Sigma}_1$ in terms of $\rho$, $\pi_0$, $\pi_1$, $\mu_0$, $\mu_1$ and $\Sigma$. To that end, we have that $$\tilde{\Sigma}_r = \mathbb{E}\{ \mathrm{cov}(X \mid Y, \tilde{Y} = r)\mid \tilde{Y} = r \} + \mathrm{cov}\{ \mathbb{E}( X \mid Y, \tilde{Y} = r) \mid \tilde{Y} = r\} = \Sigma + \mathrm{cov}\{ \mu_Y \mid \tilde{Y} = r\}.$$ Note that $$\mathbb{P}(Y =1 \mid \tilde{Y} = 1) = \frac{ \mathbb{P}(Y =1 , \tilde{Y} = 1)}{ \mathbb{P}(\tilde{Y} = 1) } = \frac{\pi_1 (1-\rho)}{ \pi_1 (1-\rho) + \pi_0 \rho } = \frac{\pi_1 (1-\rho)}{ \pi_1 (1 - 2\rho) + \rho }.$$ Hence $$\mathbb{E}(\mu_Y \mid \tilde{Y} = 1) = \mu_1 \mathbb{P}(Y =1 \mid \tilde{Y} = 1) + \mu_0 \mathbb{P}(Y =0 \mid \tilde{Y} = 1) = \frac{\pi_1 \mu_1 (1-\rho) + \pi_0 \mu_0 \rho}{ \pi_1 (1-2\rho) + \rho }.$$ It follows that $$\begin{aligned}
\tilde{\Sigma}_1 &= \frac{\pi_1 (1-\rho)}{ \pi_1 (1 - 2\rho) + \rho } \Bigl(\mu_1 - \frac{\pi_1 \mu_1 (1-\rho) + \pi_0 \mu_0 \rho }{ \pi_1 (1 - 2\rho) + \rho } \Bigr) \Bigl(\mu_1 - \frac{\pi_1 \mu_1 (1-\rho) + \pi_0 \mu_0 \rho }{ \pi_1 (1 - 2\rho) + \rho } \Bigr)^T
\\ & \hspace{20pt} + \frac{\pi_0 \rho}{ \pi_1 (1 - 2\rho) + \rho } \Bigl(\mu_0 - \frac{\pi_1 \mu_1 (1-\rho) + \pi_0 \mu_0 \rho }{ \pi_1 (1 - 2\rho) + \rho } \Bigr) \Bigl(\mu_0 - \frac{\pi_1 \mu_1 (1-\rho) + \pi_0 \mu_0 \rho }{ \pi_1 (1 - 2\rho) + \rho } \Bigr)^T
\\ &= \frac{\pi_1 (1-\rho)}{ \pi_1 (1 - 2\rho) + \rho } \Bigl( \frac{\pi_0 \rho (\mu_1 - \mu_0) }{ \pi_1 (1 - 2\rho) + \rho } \Bigr) \Bigl( \frac{\pi_0 \rho (\mu_1 - \mu_0) }{ \pi_1 (1 - 2\rho) + \rho } \Bigr)^T
\\ & \hspace{20pt} + \frac{\pi_0 \rho}{ \pi_1 (1 - 2\rho) + \rho } \Bigl( \frac{\pi_1(1- \rho)(\mu_0 - \mu_1)}{ \pi_1 (1 - 2\rho) + \rho } \Bigr)\Bigl( \frac{\pi_1(1- \rho)(\mu_0 - \mu_1)}{ \pi_1 (1 - 2\rho) + \rho } \Bigr)^T
\\ & = \frac{\pi_0 \pi_1\rho(1-\rho)}{ (\pi_1 (1 - \rho) + \pi_0\rho)^2 } (\mu_1 - \mu_0)(\mu_1 - \mu_0)^T.
\end{aligned}$$ Similarly $$\tilde{\Sigma}_0 = \frac{\pi_0\pi_1 \rho(1-\rho)}{ (\pi_0 (1 - \rho) + \pi_1\rho)^2 } (\mu_1 - \mu_0)(\mu_1 - \mu_0)^T.$$ We deduce that $${\tilde{\Sigma}} \stackrel{\mathrm{a.s.}}{\rightarrow} \Sigma + \frac{\pi_0 \pi_1 \rho(1-\rho)}{\pi_1\pi_0(1-2\rho)^2 + \rho(1-\rho) } (\mu_1 - \mu_0)(\mu_1 - \mu_0)^T = \Sigma + \alpha (\mu_1 - \mu_0)(\mu_1 - \mu_0)^T,$$ where $\alpha = \pi_0 \pi_1 \rho(1-\rho)/\{\pi_0\pi_1(1-2\rho)^2 + \rho(1-\rho)\}$. Now $$\bigl(\Sigma + \alpha (\mu_1 - \mu_0)(\mu_1 - \mu_0)^T\bigr)^{-1} = \Sigma^{-1} - \frac{ \alpha \Sigma^{-1} (\mu_1 - \mu_0) (\mu_1 - \mu_0)^T \Sigma^{-1}}{1 + \alpha \Delta^2},$$ where $\Delta^2 = (\mu_1 - \mu_0)^T\Sigma^{-1} (\mu_1 - \mu_0)$. It follows that there exists an event $\Omega_0$ with $\mathbb{P}(\Omega_0) = 1$ such that on this event, for every $x \in \mathbb{R}^d$, $$\begin{aligned}
\Bigl(x - & \frac{{\hat{\mu}}_1 + {\hat{\mu}}_0}{2}\Bigr)^T {\hat{\Sigma}}^{-1} ({\hat{\mu}}_1 - {\hat{\mu}}_0)
\\&\rightarrow \biggl[x - \frac{\mu_{1}}{2} \Bigl\{\frac{(1-2\rho)^{2}\pi_0\pi_1 + 2\rho(1-\rho)\pi_{1} } {(1-2\rho)^{2} \pi_0\pi_1 + \rho (1-\rho) }\Bigr\} + \frac{\mu_{0}}{2} \Bigl\{\frac{ (1-2\rho)^{2} \pi_0\pi_1 + 2\rho(1-\rho) \pi_0 }{(1-2\rho)^{2} \pi_0 \pi_1 + \rho(1-\rho) }\Bigr\}\biggr]^T
\\& \hspace{30pt} \Bigl(\Sigma^{-1} - \frac{ \alpha \Sigma^{-1} (\mu_1 - \mu_0) (\mu_1 - \mu_0)^T \Sigma^{-1}}{1 + \alpha \Delta^2} \Bigr) \Bigl\{\frac{(1-2\rho) \pi_0 \pi_1} {(1-2\rho)^{2} \pi_0\pi_1 + \rho (1-\rho) }\Bigr\} (\mu_{1} - \mu_{0})
\\& = \biggl[x - \frac{\mu_{1}}{2} \Bigl\{\frac{(1-2\rho)^{2}\pi_0\pi_1 + 2\rho(1-\rho)\pi_{1} } {(1-2\rho)^{2} \pi_0\pi_1 + \rho (1-\rho) }\Bigr\} + \frac{\mu_{0}}{2} \Bigl\{\frac{ (1-2\rho)^{2} \pi_0\pi_1 + 2\rho(1-\rho) \pi_0 }{(1-2\rho)^{2} \pi_0 \pi_1 + \rho(1-\rho) }\Bigr\}\biggr]^T
\\& \hspace{60pt} \Bigl(\frac{ 1}{1 + \alpha\Delta^{2}}\Bigr) \Bigl\{\frac{(1-2\rho) \pi_{0} \pi_{1}} {(1-2\rho)^{2} \pi_0\pi_1 + \rho (1-\rho) }\Bigr\} \Sigma^{-1}(\mu_{1} - \mu_{0})
\\& = \biggl(x - \frac{\mu_{1} + \mu_{0}}{2}\biggr)^T \Bigl(\frac{ 1}{1 + \alpha\Delta^{2}}\Bigr) \Bigl\{\frac{(1-2\rho) \pi_{0} \pi_{1}} {(1-2\rho)^{2} \pi_0\pi_1 + \rho (1-\rho) }\Bigr\} \Sigma^{-1}(\mu_{1} - \mu_{0})
\\ & \hspace{30pt} - \biggl[\frac{\mu_{1}}{2} \Bigl\{\frac{(2\pi_{1} - 1)\rho(1-\rho)} {(1-2\rho)^{2} \pi_0\pi_1 + \rho (1-\rho) }\Bigr\} + \frac{\mu_{0}}{2} \Bigl\{\frac{ (2\pi_{0} - 1)\rho(1-\rho)}{(1-2\rho)^{2} \pi_0 \pi_1 + \rho(1-\rho) }\Bigr\} \biggr]^T
\\& \hspace{ 60pt} \Bigl(\frac{ 1}{1 + \alpha\Delta^{2}}\Bigr) \Bigl\{\frac{(1-2\rho) \pi_{0} \pi_{1}} {(1-2\rho)^{2} \pi_0\pi_1 + \rho (1-\rho) }\Bigr\} \Sigma^{-1}(\mu_{1} - \mu_{0}).
\end{aligned}$$ Hence, on $\Omega_0$, $$\lim_{n \rightarrow \infty} \tilde{C}^{\mathrm{LDA}}(x) = \left\{ \begin{array}{ll} 1 & \text{if } c_0 + \bigl(x - \frac{\mu_1 + \mu_0}{2}\bigr)^T \Sigma^{-1} (\mu_1 - \mu_0) > 0
\\ 0 & \text{if } c_0 + \bigl(x - \frac{\mu_1 + \mu_0}{2}\bigr)^T \Sigma^{-1} (\mu_1 - \mu_0) < 0, \end{array} \right.$$ where $$\begin{aligned}
c_0 & = \frac{(1 + \alpha \Delta^2)\rho(1-\rho)}{\alpha(1-2\rho)} \log\biggl(\frac{(1-2\rho)\pi_1 + \rho}{(1-2\rho)\pi_0 + \rho} \biggr) - \frac{(\pi_{1} - \pi_{0})\alpha\Delta^2}{2\pi_0\pi_1} .
\end{aligned}$$ This proves the first claim of the theorem. It follows that $$\begin{aligned}
\lim_{n \rightarrow \infty} R(\tilde{C}^{\mathrm{LDA}}) & = \pi_0\Phi\biggl(\frac{c_0}{\Delta} - \frac{\Delta}{2}\biggr) + \pi_1\Phi\biggl(-\frac{c_0}{\Delta} - \frac{\Delta}{2}\biggr),
\end{aligned}$$ which proves the second claim. Now consider the function $$\psi(c_0) = \pi_0\Phi\biggl(\frac{c_0}{\Delta} - \frac{\Delta}{2}\biggr) + \pi_1\Phi\biggl(-\frac{c_0}{\Delta} - \frac{\Delta}{2}\biggr).$$ We have $$\dot{\psi}(c_0) = \frac{\pi_0}{\Delta}\phi\biggl(\frac{c_0}{\Delta} - \frac{\Delta}{2}\biggr) - \frac{\pi_1}{\Delta} \phi\biggl(-\frac{c_0}{\Delta} - \frac{\Delta}{2}\biggr) = \frac{\pi_0}{\Delta}\phi\biggl(\frac{c_0}{\Delta} - \frac{\Delta}{2}\biggr) \Bigl\{1 - \frac{\pi_1}{\pi_0} \exp(-c_0)\Bigr\},$$ where $\phi$ denotes the standard normal density function. Since $\mathrm{sgn}\bigl(\dot{\psi}(c_0)\bigr) = \mathrm{sgn}\bigl(c_0 - \log(\pi_1/\pi_0)\bigr)$, we deduce that $$\pi_0\Phi\biggl(\frac{c_0}{\Delta} - \frac{\Delta}{2}\biggr) + \pi_1\Phi\biggl(-\frac{c_0}{\Delta} - \frac{\Delta}{2}\biggr) \geq R(C^{\mathrm{Bayes}}),$$ and it remains to show that if $\rho \in (0,1/2)$ and $\pi_1 \neq \pi_0$, then there is a unique $\Delta > 0$ with $c_0 = \log(\pi_1/\pi_0)$. To that end, suppose without loss of generality that $\pi_1 > \pi_0$ and note that $$\begin{aligned}
\frac{(\pi_{1} - \pi_{0})(1-2\rho)}{(1-2\rho)^{2} \pi_0 \pi_1 + \rho(1-\rho)} &= \frac{\pi_{1} (1-2\rho) + \rho }{(1-2\rho)^{2} \pi_1 \pi_{0} + \rho(1-\rho)} - \frac{\pi_{0} (1-2\rho) + \rho }{(1-2\rho)^{2} \pi_1 \pi_{0} + \rho(1-\rho)}
\\ &= \frac{1 }{(1-2\rho)\pi_{0} + \rho} - \frac{1}{(1-2\rho) \pi_1 + \rho}.
\end{aligned}$$ Hence, writing $t = (1-2\rho)\pi_1 + \rho > 1/2$, we have $$\log\Bigl(\frac{(1-2\rho)\pi_1 + \rho}{(1-2\rho)\pi_0 + \rho} \Bigr) - \frac{(\pi_{1} - \pi_{0})(1-2\rho)}{2\{(1-2\rho)^{2} \pi_1 \pi_{0} + \rho(1-\rho)\}} =
\log\Bigl(\frac{t}{1-t} \Bigr) + \frac{1}{2t} - \frac{1}{2(1-t)} < 0.$$ Next, let $$\begin{aligned}
\chi(\pi_1) &= \log\Bigl(\frac{\pi_1}{\pi_0}\Bigr) - \frac{\rho(1-\rho)}{\alpha(1-2\rho)} \log\Bigl(\frac{(1-2\rho)\pi_1 + \rho}{(1-2\rho)\pi_0 + \rho} \Bigr) \nonumber
\\ &= \log\Bigl(\frac{\pi_1}{1 - \pi_1}\Bigr) - \frac{(1-2\rho)^{2}\pi_{1}(1-\pi_{1}) + \rho(1-\rho)}{(1-2\rho)\pi_1(1-\pi_1)} \log\Bigl(\frac{(1-2\rho)\pi_1 + \rho}{(1-2\rho)(1-\pi_1) + \rho} \Bigr).
\end{aligned}$$ Then $$\dot{\chi}(\pi_1) = \frac{\rho(1-\rho) (1- 2\pi_1) }{(1-2\rho) \pi_1^2(1-\pi_1)^2} \log\Bigl(\frac{(1-2\rho)\pi_1 + \rho}{(1-2\rho)(1-\pi_1) + \rho} \Bigr) < 0,$$ for all $\pi_1 \in (0,1)$. Since $\chi(1/2) = 0$, we conclude that $\chi(\pi_1) < 0$ for all $\pi_1 > \pi_0$. But $$c_0 - \log\Bigl(\frac{\pi_1}{\pi_0}\Bigr) = \frac{\Delta^2 \rho(1-\rho)}{1-2\rho} \Bigl\{\log\Bigl(\frac{(1-2\rho)\pi_1 + \rho}{(1-2\rho)\pi_0 + \rho} \Bigr) - \frac{(\pi_{1} - \pi_{0})(1-2\rho)}{2\{(1-2\rho)^{2} \pi_1 \pi_{0} + \rho(1-\rho)\}} \Bigr\} - \chi(\pi_1),$$ so the final claim follows.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Jinchi Lv for introducing us to this topic, and the anonymous reviewers for helpful and constructive comments. The second author is partly supported by NSF CAREER Award DMS-1150318, and the third author is supported by an EPSRC Fellowship EP/P031447/1 and grant RG81761 from the Leverhulme Trust. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme ‘Statistical Scalability’ when work on this paper was undertaken. This work was supported by Engineering and Physical Sciences Research Council grant numbers EP/K032208/1 and EP/R014604/1.
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abstract: 'We give an introduction to the “stable algebra of matrices” as related to certain problems in symbolic dynamics. We consider this stable algebra (especially, shift equivalence and strong shift equivalence) for matrices over general rings as well as various specific rings. This algebra is of independent interest and can be followed with little attention to the symbolic dynamics. We include strong connections to algebraic K-theory and the inverse spectral problem for nonnegative matrices. We also review key features of the automorphism group of a shift of finite type, and the work of Kim, Roush and Wagoner giving counterexamples to Williams’ Shift Equivalence Conjecture.'
address:
- 'Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA '
- 'Department of Mathematics, University of Denver, Denver, CO 80210, USA'
author:
- Mike Boyle
- Scott Schmieding
bibliography:
- 'BS.bib'
title: Symbolic dynamics and the stable algebra of matrices
---
Introduction
============
We will give an exposition of certain “stable” algebraic invariants and relations of matrices, with connections to symbolic dyamics, linear algebra and algebraic K-theory.
Here, “stable algebra” refers loosely to matrix relations and properties which are invariant under some notion of stabilization. The most fundamental stable relation we consider is strong shift equivalence. Square matrices $A,B$ over a semiring $\mathcal S$ are elementary strong shift equivalent (ESSE-$\mathcal S$) if there exist matrices $R,S$ over $\mathcal S$ such that $A=RS$ and $B=SR$. Strong shift equivalence (SSE-$\mathcal S$) is the equivalence relation on square matrices over $\mathcal S$ generated by ESSE-$\mathcal S$.
The simplicity of the SSE definition is utterly deceptive. The relation SSE-${\mathbb Z}_+$ was introduced by Williams to classify shifts of finite type (the most fundamental systems in symbolic dynamics) up to topological conjugacy. Almost fifty years later, we do not know if the relation is even decidable. Williams also introduced a coarser relation, shift equivalence over $\mathcal S$ (SE-$\mathcal S$). This relation is more complicated to define than SSE-$\mathcal S$, but it is far more tractable. For example, a complete invariant of SE-${\mathbb C}$ is the nonnilpotent part of the Jordan form. For any ring $\mathcal R$, SE-$\mathcal R$ has a formulation, well suited to standard algebraic tools, as isomorphism of associated $\mathcal R[t]$-modules.
Let $\mathcal R$ be a ring. SSE-$\mathcal R$ can be characterized as the equivalence relation generated by (i) similarity over $\mathcal R$ and (ii) “zero extensions”, i.e. $A \sim \left( \begin{smallmatrix} A&X\\0&0 \end{smallmatrix}\right)
\sim \left( \begin{smallmatrix} A&0\\X&0 \end{smallmatrix}\right)$ (Proposition \[ssesubsec\]). SE-$\mathcal R$ can be characterized as the equivalence relation generated by (i) similarity over $\mathcal R$ and (ii) “nilpotent extensions”, i.e. $A \sim \left( \begin{smallmatrix} A&X\\0&N \end{smallmatrix}\right)
\sim \left( \begin{smallmatrix} A&0\\X&N \end{smallmatrix}\right)$, with $N$ nilpotent (Theorem \[thm:endorelations\]). The refinement of SE-$\mathcal R$ by SSE-$\mathcal R$ is subtle. Only recently (decades after Williams) have we learned how to distinguish these relations, using algebraic $K$-theory.
The first section presents basic definitions, invariants and symbolic dynamics background. In Section 2, after brief general remarks about strong shift equivalence, we give an extensive discussion of shift equivalence, including several example cases. The theory of SE and SSE can be entirely recast in terms of polyomial matrices; we do this in Section 3. This is essential for later K-theory connections.
A classical problem of linear algebra, the NIEP, asks which multisets of complex numbers can be the spectrum of a nonnegative matrix. A stable version of the NIEP asks which multisets of nonzero complex numbers can be the nonzero part of the spectum of a nonnegative matrix. We review results and conjectures related to this stable approach in Section 4.
In Section 5, we present the sliver of algebraic $K$-theory needed for later connections. In Section 6, we give the algebraic K-theoretic characterization of the refinement of shift equivalence over a ring by strong shift equivalence.
In Section 7, we give an overview of results on the automorphism group of a shift of finite type: its actions and representations, related problems and (briefly) recently introduced related groups. Two crucial representations (dimension and SGCC) are very concrete, and become key ingredients to the Kim-Roush/Wagoner work giving counterexamples to the Williams Conjecture that SE-${\mathbb Z}_+$ implies SSE-${\mathbb Z}_+$. In Section 8, we give an overview of the counterexample work. This work takes place within the machinery of Wagoner’s CW complexes for SSE. The counterexample invariant of Kim and Roush is a relative sign-gyration number. Wagoner later formulated a different counterexample invariant, with values in a certain group coming out of algebraic $K$-theory.
The eight sections (“lectures”) are an expanded version of our 8-lecture course at the 2019 Yichang G2D2 school/conference. The first four sections are by Boyle, the last four by Schmieding. For the bulk of the lectures, someone disinterested in symbolic dynamics can regard it as background motivation, and simply study the algebraic material. In the spirit of a school, we have tried to keep in mind a nonexpert graduate student audience (although there are new bits even for experts). To focus on communicating ideas and statements, in each lecture we relegate various remarks and proofs to an appendix, as indicated in the table of contents. A numerical reference beginning with Ap is a reference to such an appendix (e.g., [(Ap. ]{}[sePrimitive]{}) is a reference to an appendix item in Section \[sec:sesse\]).
Fundamental problems remain open for strong shift equivalence, and the classification and automorphism groups of shifts of finite type. We wrote these lectures with the hope of encouraging some contribution to their solution.
For feedback and corrections in Yichang, we thank Peter Cameron, Tullio Ceccherini-Silberstein, Alexander Mednykh, Akihiro Munemasa and Yinfeng Zhu. We are also grateful to Sompong Chuysurichay for a reading and corrections.
Finally we thank Yaokun Wu, without whose vision and organization these lectures would not exist.
Basics {#sec:basics}
======
In this first section, we review the fundamentals of shifts of finite type and the algebraic invariants of the matrices which present them.
Topological dynamics
--------------------
By a topological dynamical system (or system), we will mean a homeomorphism of a compact metric space. This is one setting for considering how points can/must/typically behave over time, i.e., how points move under iteration of the homeomorphism. A system/homeomorphism $S: X\to X$ is often written as a pair, $(X,S)$. Formally: we have a category, in which
- an object is a topological dynamical system,
- a morphism $\phi: (X,S) \to (Y,T)$ is a continuous map $\phi: X\to Y$ such that $T\phi = S \phi$, i.e. the following diagram commutes. $$\xymatrix{
X \ar[r]^{S} \ar[d]_{\phi } & X \ar[d]^{\phi} \\
Y \ar[r]^{T} & Y
}$$
The morphism is a topological conjugacy (an isomorphism in our category) if $\phi$ is a homeomorphism.
We can think of a topological conjugacy $\phi: X\to Y$ as follows: $\phi$ renames points without changing the mathematical structure of the system.
- Because $\phi$ is a homeomorphism, it gives new names to points in essentially the same topological space.
- Because $T\phi = \phi S$, the renamed points move as they did with their original names. E.g., with $y=\phi x$, $$\xymatrix{
\cdots \ \ar[r]^{S} & x \ar[r]^{S} \ar[d]_{\phi } &Sx \ar[r]^{S} \ar[d]_{\phi }
& S^2x \ar[r]^{S}\ar[d]_{\phi } & \ \cdots\\
\cdots \ \ar[r]^{T} & y \ar[r]^{T} &Ty \ar[r]^{T}
& T^2y \ar[r]^{T} & \ \cdots
}$$
As $\phi$ respects the mathematical structure under consideration, we see that the systems $(X,S)$ and $(Y,T)$ are essentially the same, just described in a different language. (Perhaps, points of $X$ are described in English, and points of $Y$ are described in Chinese, and $\phi$ gives a translation.)
We are interested in “dynamical” properties/invariants of a topological dynamical system – those which are respected by topological conjugacy. For example, suppose $\phi :(X,S) \to (Y,T)$ is a topological conjugacy, and $x$ is a fixed point of $S$: i.e., $S(x)=x$. Then $\phi (x) $ is a fixed point of $T$. The proof is trivial: $T(\phi (x)) =
\phi (Sx) = \phi (x) $. (We almost don’t need a proof: however named, a fixed point is a fixed point.) So, the cardinality of the fixed point set, $\text{card}(\text{Fix}(S))$, is a dynamical invariant.
For $k\in {\mathbb N}$, $S^k$ is $S$ iterated $k$ times. E.g., $S^2: x\mapsto S(S(x))$.
If $\phi :(X,S) \to (Y,T)$ is a topological conjugacy, then $\phi $ is also is a topological conjugacy $(X,S^k) \to (Y,T^k)$, for all $k$ (because $T\phi = \phi S \implies T^k\phi = \phi S^k$). So, the sequence $(|\text{card}(\text{Fix}(S^k)|)_{k\in {\mathbb N}} $ is a dynamical invariant of the system $(X,S)$.
Symbolic dynamics {#subsec:symbolicdynamics}
-----------------
Let $\mathcal A$ be a finite set. Then $\mathbf{\mathcal A^{\mathbf{{\mathbb Z}}}}$ is the set of functions from ${\mathbb Z}$ to $\mathcal A$. We write an element $x$ of $\mathcal A^{{\mathbb Z}}$ as a doubly infinite sequence, $x= \dots x_{-1}x_0x_1\dots$, with each $x_k$ an element of $\mathcal A$. (The bisequence defines the function $k\mapsto x_k$.) Often $\mathcal A$ is called the alphabet, and its elements are called symbols.
Let $\mathcal A$ have the discrete topology and let $\mathcal A^{{\mathbb Z}}$ have the product topology. Then $\mathcal A^{{\mathbb Z}}$ is a [*compact metrizable space*]{} [(Ap. ]{}[zerodimetc]{}). For one metric compatible with the topology, given $x\neq y$ set ${\textnormal{dist}}(x,y) = 1/(M+1)$, where $M = \min \{|k|: x_k \neq y_k\}$. Points $x,y$ are close when they have the same central word, $x_{-M}\dots x_M = y_{-M}\dots y_M$, for large $M$.
The [*shift map*]{} $\sigma: \mathcal A^{{\mathbb Z}}
\to \mathcal A^{{\mathbb Z}}$ is defined by $(\sigma x)_n = x_{n+1}$. (This is the “left shift”: visually, a symbol in box $n+1$ moves left into box $n$.) The shift map $\sigma: \mathcal A^{{\mathbb Z}} \to \mathcal A^{{\mathbb Z}}$ is easily checked to be a homeomorphism. The system $(\mathcal A^{{\mathbb Z}}, \sigma )$ is called the full shift on $n$ symbols, if $n= |\mathcal A| $. One notation: a dot over a symbol indicates it occurs in the zero coordinate. Then $$\begin{aligned}
\sigma: \ x &\mapsto\ \sigma(x) \\
\sigma :\ \dots x_{-2}x_{-1}\overset{\bullet}{x_0}x_1x_2\dots \ &\mapsto\
\dots x_{-2}x_{-1}x_0\overset{\bullet}{x_1}x_2\dots \ \ .\end{aligned}$$
A [*subshift*]{} is a subsystem $(X,\sigma|_X)$ of some full shift $(\mathcal A^{{\mathbb Z}}, \sigma)$ (i.e. $X$ is a closed subset of $\mathcal A^{{\mathbb Z}}$, and $\sigma (X)=X$). For notational simplicity, we generally write $(X,\sigma|_X)$ as $(X,\sigma)$. Among the subshifts, the subshifts of finite type [(Ap. ]{}[sft]{}) (also called [*shifts of finite type*]{}, or [*SFTs*]{}) are a fundamental class. Every SFT is topologically conjugate to a particular type of SFT, an edge SFT. To reach our goal of relating matrix algebra and dynamics as quickly as possible, we will move directly to edge SFTs.
Edge SFTs {#sec:edgesfts}
---------
An edge SFT is defined by a square matrix over ${\mathbb Z}_+$. We are interested in algebraic properties of the matrix which correspond to dynamical properties of the edge SFT. We are especially interested in the classification problem: when do two matrices define edge SFTs which are topologically conjugate?
For us, always, “[*graph*]{}” means “directed graph”. Given an ordering of the vertices, $v_1, \dots v_n$, the [*adjacency matrix*]{} $A$ of the graph is defined by setting $A(i,j)$ to be the number of edges from vertex $v_i$ to vertex $v_j$. For simplicity we often just refer to vertices $1, \dots , n$.
(Edge SFT) Given a square matrix $A$ over ${\mathbb Z}^+$, we let $ \Gamma_A$ denote a graph with adjacency matrix $A$. Let $\mathcal E$ be the set of edges of $ \Gamma_A$. $X_A$ is the set of doubly infinite sequences $x = \dots x_{-2}x_{-1}x_{0}x_{1}x_{2} \dots$ such that each $x_n$ is in $\mathcal E$, and for all $n$ the terminal vertex of $x_n$ equals the inital vertex of $x_{n+1}$. (So, the points in $X_A$ correspond to doubly infinite walks through $ \Gamma_A$.) The system $(X_A, \sigma)$, is the edge shift, or edge SFT, defined by $A$ [(Ap. ]{}[edgesftnote]{}). We may also use the notation $\sigma_A$ to denote the map $\sigma: X_A\to X_A$.
$(X_A,\sigma)$ is the full shift on the two symbols $a,b$. The graph has a single vertex, denoted as 1. $$A=\begin{pmatrix} 2 \end{pmatrix} \ , \qquad \quad \quad
\Gamma_A \quad = \quad
\xymatrix{ *+[F-:<3pt>]{1} \ar@(lu,ld)_{a} \ar@(ru,rd)^{b}
}
\quad = \quad
\xymatrix{ \cdot \ar@(lu,ld)_{a}\ar@(ru,rd)^{b}
}$$
The edge set $\mathcal E$ is $\{a,b,c,d\}$, and the vertex set is $\{1,2\}$: $$A=\begin{pmatrix} 1& 2 \\ 1 & 0 \end{pmatrix} \ , \qquad \quad \quad
\Gamma_A \quad = \quad
\xymatrix{ *+[F-:<3pt>]{1} \ar@(lu,ld)_{a}
\ar@/^/[r]^{b}
\ar@/^2pc/[r]^{c}
& *+[F-:<3pt>]{2} \ar@/^/[l]^{d} }
\quad =
\quad
\xymatrix{\cdot \ar@(lu,ld)_{a}
\ar@/^/[r]^{b}
\ar@/^2pc/[r]^{c}
&\cdot \ar@/^/[l]^{d}
}$$ Here $ ... aabdc ... $ can occur in a point of $X_A$, but not $ ... bc ... $ .
The continuous shift-commuting maps {#subsec:ctsshiftcommmaps}
-----------------------------------
For a subshift $(X,\sigma)$ and $n\in {\mathbb N}$, $\mathcal W_n(X)$ denotes the set of $X$-words of length $n$: $$\begin{aligned}
\mathcal W_n(X) &=\{x_{0}\dots x_{n-1} : x\in X\} \\
&=
\{x_{i+n}\dots x_{i+n-1} : x\in X\},\quad \text{for every }i\in {\mathbb Z}\ .\end{aligned}$$
### Block codes
Suppose $(X,\sigma)$ and $(Y,\sigma)$ are subshifts. Suppose $\Phi : \mathcal W_N(X) \to \mathcal W_1(Y)$, and $j,k$ are integers, with $j+N-1= k$. Then for $x\in X$, we can define a bisequence $y= \phi (x)$ by the rule $y_n = \Phi (x_{n +j} \dots x_{n+k})$, for all $n$.
For example, with $N=4, j=-1$ and $k=2$: $$\begin{aligned}
{4}
\cdots \ \ &x_{-1} x_0 x_1 x_2 &&\ \ \cdots\ \ && x_{n-1}x_nx_{n+1}x_{n+2}&& \ \ \cdots \\
&\quad \ \ \downarrow && && \quad\quad \ \downarrow \ &&\\
\cdots \ \ & \quad\ \ y_0 &&\ \ \cdots\ \ && \quad \quad \ y_n && \ \ \cdots
\end{aligned}$$ where $y_0 = \Phi(x_{-1}x_0x_1x_2)$ and $y_n = \Phi(x_{n-1}x_nx_{n+1}x_{n+2})$. The point $y$ is defined by “sliding” the rule $\Phi$ along $x$. For some rules $\Phi$, the image of $\phi$ is contained in the subshift $(Y, \sigma)$.
The rule $\Phi$ above is called a [*block code*]{}. The map $\phi$ defined by $j$ and $\Phi$, is called a sliding block code (or a block code, or just a code). The map $\phi$ has range $n$ if $\Phi$ above can be chosen with $(j,k)= (-n,n)$ (i.e., $x_{-n}\cdots x_n$ determines $ (\phi x)_0$).
The following result is fundamental for symbolic dynamics, though it is easy to prove [(Ap. ]{}[chl]{}).
(Curtis-Hedlund-Lyndon)\[thm:chlthm\] Suppose $(X,\sigma)$ and $(Y,\sigma)$ are subshifts, and $\phi : X \to Y$. The following are equivalent.
1. $\phi $ is continuous and $\sigma \phi = \phi \sigma$.
2. $\phi$ is a block code.
The CHL Theorem tells us the morphisms between subshifts are given by block codes.
We’ll define some examples by stating the rule $(\phi x)_0 = \Phi (x_i \dots x_{i+N-1})$.
The shift map $\sigma$ and its powers are sliding block codes. E.g., $(\sigma x)_0 = x_1$, $(\sigma^2 x)_0 = x_2$ and $(\sigma^{-1} x)_0 = x_{-1}$ .
Let $(X,\sigma)$ be the full shift on the two symbols $0,1$.\
Define $\phi:(X,\sigma)\to (X,\sigma)$ by $(\phi x)_0 = x_0 +x_1 $ (mod 2). E.g. if $\ \ x = \dots 11\overset{\bullet}{0}\, 111\, 000\, 1 \dots $, then $\phi (x) = \dots 01\overset{\bullet}{1}\, 001\, 001 \,
\dots $ .
### Higher block presentations
Given a subshift $(X,\sigma)$ and $k\in {\mathbb N}$, we define $X^{[k]}$ to be the image of $X$ under the block code $\phi: x\mapsto y$, where for each $n$, the symbol $y_n $ is $x_n\dots x_{n+k-1} $, the $X$-word (block) of length $k$ beginning at $x_n$. We might put parentheses around this word for visual clarity. E.g., with $k=2$, $$\begin{aligned}
x & = \dots x_{-1} {\overset{\bullet}}{x_0} x_1 x_2 x_3 x_4 \dots \\
\phi (x) & = \dots (x_{-1} x_0 )
{\overset{\bullet}}{(x_{0} x_1 )}
(x_{1} x_2 ) (x_{2} x_3 )
(x_{3} x_4) \dots \end{aligned}$$ For each $k$, the map $\phi : X\to X^{[k]}$ is easily checked to be a topological conjugacy, $(X,\sigma) \to (X^{[k]}, \sigma)$.\
The subshift $(X^{[k]}, \sigma)$ is the $k$-block presentation of $(X,\sigma)$.
Powers of an edge SFT
---------------------
Let $n$ be a positive integer. Then the $n$th power system $(X_A, \sigma^n)$ is topologically conjugate to the edge SFT $(X_{A^n}, \sigma)$ defined by $A^n$.
The proposition holds because in a graph with adjacency matrix $A$, the number of paths of length $n$ from vertex $i$ to vertex $j$ is $A^n(i,j)$ [(Ap. ]{}[pathsandpowers]{}).
E.g. let $n=2$, and let $\mathcal V$ be the vertex set of $\Gamma_A$. Let $\mathcal G$ be the graph with vertex set $\mathcal V$ for which an edge from $i$ to $j$ is a two-edge path $(ab)$ from $i$ to $j$ in $\mathcal G$. Since $A^2$ is an adjacency matrix for $\mathcal G$, we may take for $(X_{A^2}, \sigma)$ the edge SFT on edge paths in $\mathcal G$. We have [(Ap. ]{}[notblock]{}) a topological conjugacy $\phi : (X_A, \sigma^2 ) \to (X_{A^2}, \sigma )$, defined by $(\phi x)_n = x_{2n}x_{2n+1}$ , for $n\in {\mathbb Z}$ :
$$\xymatrix{
\dots x_{-2}x_{-1} {\overset{\bullet}}{x_0} x_1 x_2 x_3 \dots \
\ar[r]^{\sigma^2 } \ar[d]_{\phi } & \
\dots x_{-2}x_{-1} x_0 x_1 {\overset{\bullet}}{x_2} x_3 \dots \ar[d]_{\phi } \\
\dots (x_{-2}x_{-1 }) {\overset{\bullet}}{(x_0 x_1)} (x_2 x_3) \dots \
\ar[r]^{\sigma } &
\ \dots (x_{-2}x_{-1 }) (x_0 x_1) {\overset{\bullet}}{(x_2 x_3)} \dots
}$$ The inverse system $(X_A,\sigma^{-1})$ is conjugate to $(X_{A^T}, \sigma)$, the edge SFT defined by the transpose of $A$ [(Ap. ]{}[TransposeAndInverse]{}).
Periodic points and nonzero spectrum
------------------------------------
Given a subshift, let $\text{Fix}(\sigma^k) = \{x\in X: \sigma^kx=x\}$. We can regard the sequence $(|\text{Fix} (\sigma^k)|)_{k \in {\mathbb N}}$ as the [*periodic data*]{} of the system [(Ap. ]{}[periodicdata]{}). For an edge SFT $(X_A, \sigma_A)$, we will derive from $A$ a complete invariant for the periodic data.
### Periodic data $\ \leftrightarrow\ $ trace sequence of A
$x$ is a fixed point for $\sigma$ iff $x = ... aa{\overset{\bullet}}{a}aaa .... $ for some edge $a$ with terminal vertex =initial vertex. The number of edges from vertex $i$ to vertex $i$ is $A(i,i)$. So, in $X_A$, $$|\text{Fix} (\sigma)| = \sum_i A(i,i) \ = \ {\textnormal{trace}}(A) \ .$$ Likewise, a length $k$ path with initial vertex = terminal vertex gives a fixed point of $\sigma^k$, and $$|\text{Fix} (\sigma)^k| = \text{trace}(A^k)\ .$$ Thus $(|\text{Fix} (\sigma^k)|)_{k \in {\mathbb N}}\ =\
({\textnormal{trace}}(A^k))_{k\in {\mathbb N}}$.
### Trace sequence of A $\ \leftrightarrow\ $ det(I-tA)
There is a standard equation [(Ap. ]{}[zetaeq]{}) $$\frac 1{\det(I-tA)} = \exp \sum_{n=1}^{\infty} \frac 1n {\textnormal{trace}}(A^n) t^n \ .$$ From this, one sees the trace sequence and $\det(I-tA)$ determine each other [(Ap. ]{}[proofwithzeta]{}). (This mutual determination holds for a matrix over any torsion-free commutative ring [(Ap. ]{}[NewtonIdentities]{})).
### det(I-tA) $\ \leftrightarrow\ $ nonzero spectrum of A
If a matrix $A$ has characteristic polynomial $t^k \prod_{i=1}^m (t-\lambda_i)$, with the $\lambda_i$ nonzero, then the [*nonzero spectrum*]{} of $A$ is $(\lambda_1, \dots , \lambda_m)$. Here – by abuse of notation [(Ap. ]{}[abuse]{})) – the $m$-tuple is used as notation for a multiset: the multiplicity of entries of $(\lambda_1, \dots , \lambda_m)$ matters, but not their order. For example, $(2,1,1)$ and $(1,2,1)$ denote the same nonzero spectrum, but $(2,1)$ is different.
If $A$ has nonzero spectrum $\Lambda = (\lambda_1, \dots , \lambda_m)$, then $\det (I-tA) = \prod_{i=1}^m (1-\lambda_i t)$. For example, $$\begin{aligned}
{2}
A &= \begin{pmatrix} 3&0&0&0 \\ 0&3&0&0 \\ 0&0&5&0\\ 0&0&0&0
\end{pmatrix}\ , \qquad &
I-t A&= \begin{pmatrix} 1-3t&0&0&0 \\ 0&1-3t&0&0 \\ 0&0&1-5t&0\\ 0&0&0&1
\end{pmatrix} \\
&&& \\
\Lambda &= (3,3,5) \ , &
\det (I-tA) &= (1-3t)^2(1-5t)\ .\end{aligned}$$ The nonzero spectrum and the polynomial $\det(I-tA)$ determine each other.
Classification of SFTs
----------------------
\[prob:classificationprob\] Given square matrices $A,B$ over ${\mathbb Z}_+$, determine whether they present SFTs which are topologically conjugate.
There are trivial ways to produce infinitely many distinct matrices which define the same SFT. E.g., $$\begin{pmatrix} 2 \end{pmatrix}\ ,
\begin{pmatrix} 2&0\\0&0 \end{pmatrix}\ ,
\begin{pmatrix} 2&0\\1&0 \end{pmatrix}\ ,
\begin{pmatrix} 2&1\\0&0 \end{pmatrix}\ ,
\begin{pmatrix} 2&0&0\\1&0&0\\1&0&0 \end{pmatrix}\ ,
\begin{pmatrix} 2&1&1\\0&0&1\\0&0&0 \end{pmatrix}\ , \dots
$$ Every SFT $(X_A, \sigma)$ equals one which is defined by a matrix which is [*nondegenerate*]{} (has no zero row and no zero column) [(Ap. ]{}[nondegenerate]{}). We can avoid the trivial problem by considering only nondegenerate matrices.\
\
Still, in the nontrivial case (the case that $X_A$ contains infinitely many points), there are nondegenerate matrices of unbounded size which define SFTs topologically conjugate to $(X_A, \sigma)$ [(Ap. ]{}[higherblocksystems]{}).
Strong shift equivalence of matrices, classification of SFTs {#subsec:sseandclassification}
------------------------------------------------------------
A [*semiring*]{} is a set with operations addition and multiplication satisfying all the ring axioms, except that an element is not required to have an additive inverse. In these lectures, the semiring is always assumed to contain a multiplicative identity, 1.
Below, $\mathcal S$ is a subset of a semiring [(Ap. ]{}[boolean]{}) containing 0 and 1. For $\mathcal S$ a subset of ${\mathbb R}$, $\mathcal S_+ $ denotes $\mathcal S \cap \{x\in {\mathbb R}: x\geq 0\}$. We are especially interested in $\mathcal S =
{\mathbb Z}, {\mathbb Z}_+ , {\mathbb R}, {\mathbb R}_+$.
Let $A$ and $B$ be square matrices over $\mathcal S$ (not necessarily of the same size).
$A$ and $B$ are [*elementary strong shift equivalent*]{} over $\mathcal S$ (ESSE-$\mathcal S$) if there exist matrices $R,S$ over $\mathcal S$ such that $A=RS$ and $B=SR$.
Note, if a matrix $R$ is $m\times n$, and $S$ is a matrix such that $RS$ and $SR$ are well defined, then $S$ must be $n\times m$, and the matrices $RS$ and $SR$ must be square.
$A$ and $B$ are [*strong shift equivalent*]{} over $\mathcal S$ (SSE-$\mathcal S$) if there are matrices $A=A_0, A_1, \dots , A_{\ell} =B$ over $\mathcal S$ such that $A_i$ and $A_{i+1}$ are ESSE-$\mathcal S$, $0\leq i < \ell $.
The number $\ell $ above is called the lag of the strong shift equivalence.
The relation ESSE-$\mathcal S$ is reflexive and symmetric. Easy examples [(Ap. ]{}[smallestlag]{}) show ESSE-$\mathcal S$ is not transitive. SSE-$\mathcal S$, the transitive closure of ESSE-$\mathcal S$, is an equivalence relation. Williams introduction of strong shift equivalence in [@Williams73] – the foundation for all later work on the classification of shifts of finite type – is explained by the following theorem.
\[rfwtheorem\] [(Ap. ]{}[liberties]{}) Suppose $A$ and $B$ are square matrices over $\mathcal {\mathbb Z}_+$. The following are equivalent.
1. $A$ and $B$ are SSE-$\mathcal {\mathbb Z}_+$.
2. The SFTs defined by $A$ and $B$ are topologically conjugate.
The difficult implication $(2) \implies (1)$ follows from the Decomposition Theorem [(Ap. ]{}[decomp]{}). We will prove the easy direction, $(1) \implies (2)$.
It suffices to consider an ESSE over ${\mathbb Z}_+$, $A=RS, B=SR$. Define a square matrix $M$ with block form $\begin{pmatrix} 0&R \\ S&0\end{pmatrix}$, and edge SFT $(X_M, \sigma)$. Then $M^2 =
\begin{pmatrix} RS&0 \\ 0&SR\end{pmatrix}
= \begin{pmatrix} A&0 \\ 0&B\end{pmatrix}
$. The system $(X_M, \sigma^2)$ is a disjoint union of two systems, $(X_1, \sigma^2|{X_1})$ and $(X_2, \sigma^2|{X_2})$. The shift map $\sigma: X_1 \to X_2$ gives a topological conjugacy between these subsystems.
For all $i,j$, we have $A(i,j)= \sum_k R(i,k)S(k,j)$. Therefore, we may choose a bijection $\alpha: a\mapsto rs$ from the set of $\Gamma_A$ edges to the set of $R,S $ paths in $\Gamma_M$ (an $R,S$ path is an $R$ edge followed by an $S$ edge) which respects initial and terminal vertex. Similarly we choose a bijection $\beta: b\mapsto sr$ from $\Gamma_B$ edges to $S,R$ paths in $\Gamma_M$. We define a conjugacy $\phi_{\alpha}: (X_A, \sigma ) \to (X_1, \sigma^2|{X_1})$ , $\
\phi_{\alpha} :
\dots x_{-1}x_0x_1 \dots \mapsto
\dots (r_{-1}s_{-1})(r_0s_0)(r_1s_1) \dots \ ,
$ by replacing each $x_n$ with $\alpha (x_n)$. We define a conjugacy $\phi_{\beta}: (X_B, \sigma ) \to (X_2, \sigma^2|{X_2})$ in the same way.
We now have a conjugacy $c(R,S):X_A \to X_B$ as the composition, $c(R,S) = \phi_{\beta}^{-1} \sigma \phi_{\alpha}$, $$\begin{aligned}
c(R,S): \ \dots x_{-1}x_0x_1 \dots & \mapsto
\dots (r_{-1}s_{-1})(r_0s_0)(r_1s_1) \dots \\
& \mapsto
\dots (s_{-1}r_0)(s_0r_1)(s_1r_2) \dots \mapsto
\dots y_{-1}y_0y_1 \dots \ .\end{aligned}$$
The technical statements of the next remark are not needed at all before Sections \[sectionAut\] and \[sectionWagoner\].
\[rem:crsconj\] Let $(R,S)$ be an ESSE-${\mathbb Z}_+$, with $A=RS$ and $B=SR$. Let $c(R,S)$ be a topological conjugacy from $(X_A,\sigma)$ to $(X_B,\sigma)$ defined as in the proof above. The conjugacy $c(R,S)$ is uniquely determined by $(R,S)$ when all entries of $A$ and $B$ are in $\{0, 1\}$ (then, the bijections $\alpha, \beta$ are unique). But in general, the conjugacy depends on the choice of those bijections. With appropriate choice of those bijections, we have the following:
1. $c(S,R)\circ c(R,S) = \sigma_A$, the shift map on $X_A$ .
2. $(c(R,S))^{-1}= \sigma_A^{-1} c(S,R)
= c(S,R) \sigma_B^{-1}$.
3. $c(I,A) = \text{Id}$, and $c(A,I) = \sigma_A$.
Also: with $c(R,S)$, $\ x_0x_1$ determines $y_0$; with $(c(R,S))^{-1}$, $\ y_{-1}y_0$ determines $x_0$.
Shift equivalence
-----------------
Despite the seeming simplicity of its definition, SSE over ${\mathbb Z}_+$ is a very difficult relation to fully understand. Consequently, Williams introduced shift equivalence.
Let $A,B$ be square matrices over a semiring $\mathcal S$. Then $A,B$ are shift equivalent over $\mathcal S$ (SE-$\mathcal S$) if there exist matrices $R,S$ over $\mathcal S$ and a positive integer $\ell$ such that the following hold: $$A^{\ell}=RS\ , \quad B^{\ell}= SR\ , \quad
AR=RB\ ,\quad SA=BS\ .$$ Here, $(R,S)$ is a shift equivalence of lag $\ell $ from $A$ to $B$.
The next proposition is an easy exercise [(Ap. ]{}[easyfacts]{}).
Let $\mathcal S$ be a semiring.
1. SE over $\mathcal S$ is an equivalence relation.
2. SSE over $\mathcal S$ implies SE over $\mathcal S$.
Williams’ Shift Equivalence Conjecture
--------------------------------------
(Williams, 1974) [@Williams73]\[conj:williams\] Suppose $A,B$ are two square matrices which are SE-${\mathbb Z}_+$. Then they are SSE-${\mathbb Z}_+$.
Despite the seeming complexity of its definition, shift equivalence is much easier to understand than strong shift equivalence, as we’ll see. A positive solution to Williams’ Conjecture would have been a very satisfactory solution to the classification problem for SFTs. Alas ... there are counterexamples to the conjecture, due to Kim and Roush (building on work of Wagoner, and Kim-Roush-Wagoner). The first Kim-Roush counterexample was in 1992.
We recall now a definition fundamental for the theory of nonnegative matrices (as we will review in Lecture 4).
\[primitiveDefinition\] A primitive matrix is a square matrix such that every entry is a nonnegative real number and for some positive integer $k$, every entry of $A^k$ is positive.
By far the most important case of Williams’ Conjecture is the case that the matrices $A,B$ are primitive. An edge SFT defined from a nondegenerate matrix $A$ is [*mixing*]{}
[See [@LindMarcus1995] for the definition of the dynamical property “mixing”, which we do not need.]{}
if and only if $A$ is primitive. The mixing SFTs play a role among SFTs very much analogous to the role played by primitive matrices in the theory of nonnegative matrices.
The Kim-Roush counterexample for primitive matrices came in 1999.
Over twenty years later, we have no new theorem or counterexample for primitive matrices over ${\mathbb Z}_+$. The Kim-Roush counterexamples require quite special constructions (reviewed in Section \[sectionWagoner\]). The proof method can work only in special SE-${\mathbb Z}$ classes, and can never show that there is an infinitely family of primitive matrices which are SE-${\mathbb Z}_+$ but are pairwise not SSE-${\mathbb Z}_+$ (see Sec. \[WagonerSubSecRemarks\]).
### The gap between SE-${\mathbb Z}_+$ and SSE-${\mathbb Z}_+$?
How big is the gap between SE-${\mathbb Z}_+$ and SSE-${\mathbb Z}_+$? We really don’t know.
Suppose $A$ is ANY square matrix over ${\mathbb Z}_+$ such that $A$ is primitive (for some $n$, every entry of $A^n$ is positive), and $A\neq (1)$. (The case $A=(1)$ is trivial.) As we approach a half century following Williams’ conjecture, we cannot verify or rule out either of the following statements.
1. There is an algorithm which takes as input any square matrix $B$ over ${\mathbb Z}_+$ and decides whether $A$ and $B$ are SSE-${\mathbb Z}_+$.
2. There are infinitely many matrices which are SE-${\mathbb Z}_+$ to $A$ and which are pairwise not SSE-${\mathbb Z}_+$.
Regarding the first item above: we do not know upper bounds on the lag of a possible SSE or the sizes of the matrices in its chain of ESSEs. (See [(Ap. ]{}[smallestlag]{}) - [(Ap. ]{}[badArithmeticLag]{}) for more on lag issues.) Also, for example, the “$1 \times 1$ case” is completely open. We will see [(Ap. ]{}[1by1]{}) that a square matrix over ${\mathbb Z}_+$ is SE-${\mathbb Z}_+$ to $(k)$ $\iff$ its nonzero spectrum is $(k)$. But, for every positive integer $k>1$, we do not know whether a matrix SE over ${\mathbb Z}_+$ to $(k)$ must be SSE over ${\mathbb Z}_+$ to $(k)$ [(Ap. ]{}[ashley]{}). Remarkably, even for two $2\times 2$ matrices over ${\mathbb Z}_+$, we do not know whether SE-${\mathbb Z}_+$ implies SSE-${\mathbb Z}_+$ (although, here there are significant partial results, e.g. [@baker1983; @baker1987; @CuntzKriegerDicyclic; @Williams1992]).
Nevertheless ... perhaps the situation is not hopeless.
1. If $A$ is a matrix over ${\mathbb R}_+$ with $\det(I-tA) =1-\lambda t$, then $A$ is SSE over ${\mathbb R}_+$ to $(\lambda )$. (Over ${\mathbb R}_+$, the “$1 \times 1$ case” is solved!)
Despite limited progress, I think the proof framework for this result of Kim and Roush is promising for proving SE-${\mathbb R}_+$ implies SSE-${\mathbb R}_+$ for positive matrices [(Ap. ]{}[RealSSEandBKR]{}).
2. In recent years we have (at last) gained a much better (not complete) understanding of strong shift equivalence over a ring, as discussed in Lecture 6. This gives more motivation for investigation, and new ideas to explore.
Appendix 1 {#a1}
----------
For general introduction to basics of symbolic dynamics relevant to our lectures, see the book of Lind and Marcus [@LindMarcus1995]. The book is intended to be widely accessible, and a math graduate student can easily read it without guidance. My old survey [@Boyle91matrices], aimed at matrix theorists, intersects our lectures, and has some complementary material.
The rest of this subsection contains various remarks, proofs and comments referenced in earlier parts of Section \[sec:basics\].
\[zerodimetc\] Let the finite set $\mathcal A$ have the discrete topology. Then compactness of $\mathcal A^{{\mathbb Z}}$ follows from a diagonal argument from the chosen metric, or from Tychonoff’s Theorem (the product topology is the same as the topology coming from the chosen metric).
Suppose $X$ is a closed nonempty subset of $\mathcal A^{{\mathbb Z}}$. A “cylinder set” is a set $C$ in $X$ of the following form: there is a point $x\in X$, and $i\leq j$ in ${\mathbb Z}$, such that $C = \{y \in X: y_n = x_n \text{ if } i\leq n \leq j\}$ . The cylinder sets form a basis for the topology on $X$.
The cylinder sets are closed open. A subset of $X$ is closed open if and only if it is the union of finitely many cyinders. By definition, a metric space is zero dimensional if there is a base for the topology consisting of closed open sets. Therefore $X$ is zero dimensional.
\[sft\] By definition, a subshift $(X, \sigma )$ is SFT if if there is a finite set $\mathcal F$ of words on the alphabet $\mathcal A$ of $X$ such that $X$ is the subset of points $x$ in $\mathcal A^{{\mathbb Z}}$ such that no subword $x_m\cdots x_n$ is in $\mathcal F$.
\[edgesftnote\] To be careful, we’ll be a little pedantic.
Two different but isomorphic graphs define different but isomorphic SFTs. The topological conjugacy of SFTs in this case is rather trivial. If the graph isomorphism gives a map on edges $e\mapsto \overline e$, then the topological conjugacy $\phi$ is defined by $(\phi x)_n = \overline{x_n}$, for all $n$.
In the other direction, given just the matrix $A$, a graph $\mathcal G$ with adjacency matrix $A$ is only defined up to graph isomorphism. If $A$ is $n\times n$, then there is an ordering of the vertices, $\ \nu_1, \nu_2, \dots ,\nu_n$ , such that $A(i,j)$ is the number of edges from $\nu_i$ to $ \nu_j$. For simplicity, we often just regard the vertex set as $\{ 1,2, \dots , n \}$, with $\nu_i =i$.
\[Curtis-Hedlund-Lyndon\] \[chl\] Suppose $(X,\sigma)$ and $(Y,\sigma)$ are subshifts, and $\phi : X \to Y$. TFAE.
\(1) $\phi $ is continuous and shift-commuting.
\(2) There are integers $j,k$ with $j\leq k$, such that for $N=k-j+1$ there is a function $\Phi : \mathcal W_N(X) \to \mathcal W_1(Y)$, such that for all $n$ in ${\mathbb Z}$ and $x$ in $X$, $(\phi x)_n = \Phi (x_{n +j} \dots x_{n+k})$ .
\(1) $\implies $ (2) Suppose $\phi$ is continuous, hence uniformly continuous, on $X$. There is an $\epsilon >0$ such that for $y, y'$ in $Y$, $y_0 \neq (y')_0 \implies {\textnormal{dist}}(y_0, (y')_0) > \epsilon $. By the uniform continuity, there is $m\in {\mathbb N}$ such that for $x,w$ in $X$, $$x_{-m}\dots x_m = w_{-m}\dots w_m
\implies
(\phi x)_0 = (\phi w)_0\ .$$ This gives a rule $\Phi : \mathcal W_{2m+1} (X) \to \mathcal W_1 (Y) $ such that for all $x$ in $X$, $(\phi x)_0 = \Phi (x_{-m} \dots x_m)$. Because $\phi$ is shift commuting, we then get for all $n$ that $$\Phi (x_{n-m} \dots x_{n+m})
= \Phi ((\sigma^n x)_{-m} \dots (\sigma^n x)_m)
= (\phi (\sigma^n x) )_0
= ( \sigma^n(\phi x) )_0
= (\phi x)_n \ .$$
We leave the proof of (2) $\implies $ (1) as an exercise.
There are other, equivalent ways to state the CHL Theorem. (I didn’t copy the original statement.)
\[pathsandpowers\] Let a graph have adjacency matrix $A$. Then the number of paths of length $n$ from vertex $i$ to vertex $j$ is $A^n(i,j)$.
A length 2 path from $i$ to $j$ is, for some vertex $k$, an edge from $i$ to $k$ followed by an edge from $k$ to $j$. The number of such paths is $\sum_k A(i,k) A(k,j) = A^2 (i,j)$. The claim for paths of length $n$ follows by induction, considering paths of length $n-1$ followed by path of length 1.
\[TransposeAndInverse\] Suppose $A$ is a square matrix over ${\mathbb Z}_+$, with transpose $A^T$. From a graph $\mathcal G$ with adjacency matrix $A$, let $\mathcal G^{\text{reversed}} $ be the graph with the same vertex set as $G$, and edges with the same names but with reversed direction (an edge $e$ from $i$ to $j$ in $\mathcal G$ becomes an edge $e$ from $j$ to $i$ in $\mathcal G^{\text{reversed}} $. Then $A^T$ is an adjacency matrix for $\mathcal G^{\text{reversed}} $.
Now, there is a topological conjugacy $\phi: (X_A, \sigma^{-1})\to (X_{A^T} , \sigma)$, defined by the rule $(\phi x)_n = x_{-n}$, for $n\in {\mathbb Z}$.
\[notblock\] The topological conjugacy $\phi : (X_A, \sigma^2 ) \to (X_{A^2}, \sigma )$ is not a block code. This does not contradict the CHL Theorem, because $(X_A, \sigma^2)$ is not a subshift.
\[periodicdata\] Formally, the “periodic data” for a system $(X,S)$ is the isomorphism class of the system $(\text{Per}(S), S)$, with the periodic points, $\text{Per}(S)$, given the discrete topology (i.e., ignore topology). (Here “system” relaxes our terminology in these lectures that the domain must be compact.)
A complete invariant for the periodic data is one such that two systems agree on the invariant if and only if they have the same periodic data.
Now, one complete invariant of the periodic data of a system is simply the function which assigns to $n$ the cardinality of the set of points of least period $n$. (A point has least period $n$ if its orbit is finite with cardinality $n$.) For a subshift $(X,\sigma)$, there is a finite number $q_n$ of points of least period $n$, and the sequence $(q_n)$ is a complete invariant of the periodic data. Let $\tau_n =(|\text{Fix}(\sigma^n)|$. The sequence $(q_n)$ determines the sequence $(\tau_n)_{n=1}^{\infty}$. For our systems, each $\tau_n$ is a nonnegative integer, and in this case the converse holds: the sequence $(\tau_n)$ determines the sequence $(q_n)$. E.g., $q_1=\tau_1,\ q_2 = \tau_2 - \tau_1, \ \dots \ , q_6 = \tau_6
-\tau_3 -\tau_2 + \tau_1 , \ \dots $. (The formal device for producing a systematic formula for this inclusion-exclusion pattern is Mobius inversion.) So, “we may regard” $(\tau_n)$ as the periodic data in the sense that it is a complete invariant for the periodic data.
\[zetaeq\] Suppose $A$ is a matrix with entries in ${\mathbb C}$. Then $$\label{zetaseriesequation}
\frac 1{\det(I-tA)} \ = \ \exp \sum_{n=1}^{\infty} \frac 1n {\textnormal{trace}}(A^n) t^n \ .$$
Recall, $- \log (1-x) = x + \frac{x^2}2+ \frac{x^3}3 + \cdots $ . Let $(\lambda_1, \dots , \lambda_n)$ be the nonzero spectrum of $A$. Then $$\begin{aligned}
& \ \exp \Big(\sum_{n=1}^{\infty} \frac 1n {\textnormal{trace}}(A^n) t^n\Big)
\ = \ \exp \Big(\sum_{n=1}^{\infty} \frac 1n \big(\sum_i \lambda_i^n\big) t^n \Big) \\
=& \ \exp \Big( \sum_i \big(\sum_{n=1}^{\infty} \frac 1n (\lambda_it)^n\big) \Big)
\ = \ \prod_i \exp \Big(\sum_{n=1}^{\infty} \frac{(\lambda_i t)^n}n \Big) \\
=& \ \prod_i \exp \big(- \log (1-\lambda_i t)\big)
\ = \ \prod_i \frac 1{(1- \lambda_i t)}
\ = \ \frac 1{\det (I-tA)} \ .\end{aligned}$$
(The last proposition remains true as an equation in formal power series if ${\mathbb C}$ is replaced by a torsion-free commutative ring $\mathcal R$. In this case, ${\mathbb N}$ is a multiplicative subset of $\mathcal R$ containing no zero divisor, and all the power series coefficients make sense in the localization $\mathcal R[{\mathbb N}^{-1}]$.)
(The zeta function) \[zetafunction\] Suppose $(X,S)$ is a dynamical system such that for all $n$ in ${\mathbb N}$, $|\text{Fix}(S^n)| < \infty $. Then the (Artin-Mazur) zeta function of the system is defined to be $$\zeta (t) = \exp \Big(\sum_{n=1}^{\infty} \frac 1n |\text{Fix}(S^n)|
t^n \Big) \ .$$ This is defined at least as a formal power series; it’s defined as an anaytic function inside the radius of convergence. The zeta function (where it is defined) is the premier complete invariant of the periodic data. For an edge SFT defined from a matrix $A$, we see $\zeta (t) = 1/\det(I-tA)$.
\[proofwithzeta\] Suppose $A$ is a square matrix over ${\mathbb C}$. Then $\det (I-tA)$ and the sequence $({\textnormal{trace}}(A^n))$ determine each other.
The nontrivial implication, that the trace sequence determines $\det (I-tA)$, follows from the proposition. The proposition also is easily used to prove the reverse implication; but we may also simply notice that $\det(I-tA)$ determines the nonzero spectrum $(\lambda_1, \dots , \lambda_n)$ of $A$, which determines ${\textnormal{trace}}(A^k) = \sum_i (\lambda_i)^k$.
\[NewtonIdentities\] For any square matrix $A$ over any commutative ring, the polynomial $\det(I-tA)$ determines the trace sequence $({\textnormal{trace}}(A^k))_{k=1}^{\infty}$; if the ring is torsion-free, then conversely $({\textnormal{trace}}(A^k))_{k=1}^{\infty}$ must determine $\det(I-tA)$. To see this, let us write $\det (I-tA)$ as $1-f(t) = 1 -f_1t -f_2t^2 \dots -f_Nt^N$, and let $\tau_k$ denote ${\textnormal{trace}}(A^k)$. Then the claimed determinations are easily proved by induction from Newton’s identities,
[As $\det (I-tA)$ is the reversed characteristic polynomial, Newton’s identities can alternately be (and usually are) stated in terms of coefficients of the characteristic polynomial.]{}
valid over any commutative ring: $$\begin{aligned}
{2}
\tau_k \ & = \ kf_k +\sum_{i=1}^{k-1}f_i\tau_{k-i}\ , \quad && \text{if } 1\leq k \leq N\ , \\
& =\ \sum_{i=1}^{N}f_i\tau_{k-i}\ , \quad && \text{if } k > N\ .\end{aligned}$$ To see the torsion-free assumption is not extraneous, let $\mathcal R$ be the ring ${\mathbb Z}_2 \times {\mathbb Z}_2$, and consider the matrices $$A = \begin{pmatrix} (0,1)
\end{pmatrix} \ , \qquad
B
= \begin{pmatrix} (0,0) & (1,1) \\ (1,0) & (0,1)
\end{pmatrix}$$ Here, $\det(I-tA) = 1 -t (0,1) \neq 1 -t (0,1) -t^2(1,0) = \det(I-tB) $, but ${\textnormal{trace}}(A^n) = {\textnormal{trace}}(B^n) = (0,1)$ for every positive integer $n$.
One of the ways to prove Newton’s identities is to take the derivative of the log of both sides of , and equate coefficients in the resulting equation of power series. This makes sense at the level of formal power series when the ring is torsion free, in particular for a polynomial ring ${\mathbb Z}[\{x_{ij}\}]$, where $\{ x_{ij} : 1\leq i,j \leq N\}$ is a set of $N^2$ commuting variables. Then, given $A$ over any commutative ring $\mathcal R$, using the ring homomorphism ${\mathbb Z}[\{x_{ij}\}] \to \mathcal R$ induced by $x_{ij} \mapsto A(i,j)$, from the Newton identities over ${\mathbb Z}[\{x_{ij}\}]$ we obtain the Newton identities for $A$.
\[nondegenerate\] A matrix is [*degenerate*]{} if it has a zero row or a zero column. The [*nondegenerate core*]{} of a square matrix is the largest principal submatrix $C$ which is nondegenerate. If row $i$ or column $i$ of $A$ is zero, then remove row $i$ and column $i$. Continue until a nondegenerate matrix $C$ is reached. This matrix is the nondegenerate core of $A$.
When the matrices have all entries in ${\mathbb Z}_+$, $X_C=X_A$, because if an edge occurs as $x_n$ for some point of $X_A$, then the edge must be followed and preceded by arbitrarily long paths in $\Gamma_A$.
\[higherblocksystems\] The $k$-block presentation of an edge SFT $(X_A, \sigma)$ is conjugate to another edge SFT. For $k>1$ its defining matrix $A^{[k]}$ is the adjacency matrix for a graph with vertex set $\mathcal W_k = \{ x_1\dots x_k : x\in X_A \}$ and edge set $\mathcal W_{k+1}$; the edge $x_1\dots x_{k+1}$ runs from vertex $x_1\dots x_k$ to vertex $x_2\dots x_{k+1}$. When $X_A$ is infinite, $\lim_{k\to\infty}|\mathcal W_k|= \infty$, and the size of $A^{[k]}$ goes to infinity.
For example, let $ A=A^{[1]}= (2)$ , with edge set $\mathcal E_1=\{a,b\}$. The vertex sets $\mathcal W_2$ and $ \mathcal W_3$ for 2 and 3 block presentations are $\{ a,b\}$ and $\{aa,ab,ba,bb\}$. With the lexicographic orderings on these sets (the ordering as written), we get the adjacency matrices $$A^{[2]} = \begin{pmatrix} 1&1\\1&1\end{pmatrix} \ , \qquad
A^{[3]} = \begin{pmatrix} 1&1&0&0 \\0&0&1&1
\\ 1&1&0&0
\\0&0&1&1
\end{pmatrix}\ .$$
\[boolean\] The use of SSE over semirings goes beyond the study of SSE over the positive set of an ordered ring. SSE over the Boolean semiring $\{ 0,1 \}$, in which $1+1=1$, ends up being quite relevant to some constructions over ${\mathbb R}_+$ [@BKR2013], and to relating topological conjugacy and flow equivalence of SFTs [@B02posk]. The Boolean semiring cannot be embedded in a ring, as $1+1=1$ would then force $1=0$.
\[liberties\] For simplicity, I take some liberties with the statement of the theorem. “Edge SFTs” don’t appear in Williams’ paper; he used a more abstract approach to associate SFTs to matrices over ${\mathbb Z}_+$.
Williams 1973 paper [@Williams73] contained a “proof” (erroneous) of his conjecture. The 1974 Conjecture appeared in the erratum. One of the most important papers in symbolic dynamics also included perhaps its most famous mistake.
\[decomp\] We say a little about the Decomposition Theorem, even though we won’t have space to explain it well, because it is a very important feature of SSE. Lind and Marcus give a nice presentation of the Decomposition Theorem [@LindMarcus1995].
The Decomposition Theorem tells us that when there is a conjugacy of edge SFTs $\phi: (X_A, \sigma) \to (X_B, \sigma)$, there is another matrix $C$, an SSE-${\mathbb Z}_+$ from $C$ to $A$ given by a string of column amalgamations, and an SSE -${\mathbb Z}_+$ from $C$ to $B$ given by a string of row amalgamations, such that the associated conjugacies $\alpha: (X_C, \sigma)\to (X_A, \sigma)$ and $\beta: (X_C, \sigma)\to (X_A, \sigma)$ give $\phi = \beta \alpha^{-1}$.
For $x$ in $X_C$: $(\alpha x)_0$ and $(\beta x)_0$ depend only on $x_0$.
A column amalgamation $C\to D$ is an ESSE $C=RS $, $D=SR$, such that $S$ is a zero-one matrix with each column containing exactly one nonzero entry. For example, $$\begin{aligned}
C = \begin{pmatrix}1& 1& 5 \\ 2&2&3 \\1&1&2 \end{pmatrix} =
& \begin{pmatrix} 1& 5 \\ 2&3 \\1&2 \end{pmatrix}
\begin{pmatrix} 1& 1& 0 \\ 0&0& 1 \end{pmatrix} =RS \ , \\
D =\begin{pmatrix} 3&8 \\ 1&2 \end{pmatrix} =
& \begin{pmatrix} 1& 1& 0 \\ 0&0& 1 \end{pmatrix}
\begin{pmatrix} 1& 5 \\ 2&3 \\1&2 \end{pmatrix}=SR \ .\end{aligned}$$ Row amalgamations are correspondingly given by amalgamating rows rather than columns.
The Decomposition Theorem, or a relative, is a tool for the characterization of nonzero spectra of primitive real matrices [@BH91]; for Parry’s cohomological characterization of SSE-${\mathbb Z}_+G$ [@BS05]; and for studying SSE over dense subrings of ${\mathbb R}$ [@BKR2013].
\[easyfacts\] Let $\mathcal S$ be a semiring.
1. SE over $\mathcal S$ is indeed an equivalence relation.
2. SSE over $\mathcal S$ implies SE over $\mathcal S$.
\(1) If $(R_1,S_1)$ is a shift equivalence of lag $\ell_1 $ from $A$ to $B$, and $(R_2,S_2)$ is a shift equivance of lag $\ell_2 $ from $B$ to $C$, then $(R_1R_2,S_2S_1)$ satisfies the equations to be a shift equivalence of lag $\ell_1+\ell_2 $ from $A$ to $C$. (For example, $R_1R_2S_2S_1 = R_1B^{\ell_1}S_1 = R_1S_1A^{\ell_1} =
A^{\ell_2}A^{\ell_1} = A^{\ell_1+\ell_2}$.)
\(2) Suppose we are given a lag $\ell $ SSE from $A$ to $B$:\
$A=A_0, A_1, \dots , A_{\ell}=B$; $\quad \quad A_i=R_iS_i$ and $A_{i+1} =S_iR_i$, $\ \ $ for $0\leq i < \ell$ .\
Set $R=R_1R_2\dots R_{\ell}$ , $\ \ S= S_{\ell}\dots S_2 S_1$.\
Then $(R,S)$ is a shift equivalence of lag $\ell$ from $A$ to $B$. $\qed$
Next we state one of the interesting partial results on Williams’ Conjecture, which we will use later.
\[thm:baker\] [@baker1983 K.Baker] Suppose $A,B$ are positive $2\times 2$ integral matrices with nonnegative determinant which are similar over the integers. Then $A,B$ are strong shift equivalent over ${\mathbb Z}_+$.
\[smallestlag\]([*Nilpotence and lag*]{}) Let $A=RS,B=SR$ be an ESSE over a semiring $\mathcal S$. Suppose $m\geq 2$ is the smallest positive integer such that $A^m=0$. Then $B$ is also nilpotent (because $B^{m+1}=SA^mR=0$), but $B^{m-2}\neq 0$ (because $B^{m-2}= 0$ would force $A^{m-1}=RB^{m-2}S=0$). Thus if $\ell$ is the lag of an SSE-$\mathcal S$ from $A$ to a zero matrix, then $\ell \geq m-1$. For example, there is a lag 2 SSE-${\mathbb R}$ from $\left( \begin{smallmatrix} 0&1&0 \\ 0&0&1 \\ 0&0&0
\end{smallmatrix} \right)$ to $(0)$, but there is no ESSE-${\mathbb R}$ from $\left( \begin{smallmatrix} 0&1&0 \\ 0&0&1 \\ 0&0&0
\end{smallmatrix} \right)$ to $(0)$.
For an example involving primitive matrices, consider the matrix $A=(2)$ and its 3-block presentation matrix $B=A^{[3]}$ in Remark \[higherblocksystems\]. There is a lag 2 SSE-${\mathbb Z}_+$ between $(2)$ and $B$. But there cannot be an ESSE-${\mathbb R}$ of $B$ and $(2)$: if $RS= (2)$ and $SR=B$, then $R$ and $S$ have rank 1, so $SR$ has rank at most 1, contradicting $B$ having rank 2.
The next example (extracted from Norbert Riedel’s paper [@riedel1983], which has more) shows that the lag of an SSE-${\mathbb Z}_+$ is not just a matter of nilpotence.
\[riedelexample\] ([*Bad lag at size 2 from geometry.*]{}) For each positive integer $k$, set $A_k= \left( \begin{smallmatrix} k&2\\1&k \end{smallmatrix} \right)$ and $B_k= \left( \begin{smallmatrix} k-1&1\\1&k+1 \end{smallmatrix} \right)$. For each $k$, the matrices $A_k$, $B_k$ are SSE over ${\mathbb Z}_+$. However, the minimum lag of an SE-${\mathbb Z}_+$ between $A_k,B_k$ (and therefore the minimum lag of an SSE-${\mathbb Z}_+$ between $A_k,B_k$) goes to infinity as $k\to \infty$.
First, $A_k$ and $B_k$ have the same nonzero spectrum $(k+ \sqrt 2, k-\sqrt 2)$, and ${\mathbb Z}[k+\sqrt 2]={\mathbb Z}[\sqrt 2]$, and ${\mathbb Z}[k+\sqrt 2]={\mathbb Z}[\sqrt 2]$. ${\mathbb Z}[\sqrt 2]$ is the ring of algebraic integers in ${\mathbb Q}[\sqrt 2]$, and this ring is well known to have class number 1. By Theorem \[thm:taussky\], $A_k$ and $B_k$ are similar over ${\mathbb Z}$. Then, by Theorem \[thm:baker\], $A_k$ and $B_k$ are SSE over ${\mathbb Z}_+$. By induction one checks that for each $n$, there are polynomials $P^{(n)}_1, P^{(n)}_2$ with positive integral coefficients such that $\text{deg}( P^{(n)}_1) = \text{deg}( P^{(n)}_2) +1$ and for all $k,n$ $$(A_k)^n =
\begin{pmatrix}
P^{(n)}_1(k) & 2P^{(n)}_2(k) \\
P^{(n)}_2(k) & P^{(n)}_1(k)
\end{pmatrix} \ .$$
Now suppose $R,S$ are matrices over ${\mathbb Z}_+$ and $\ell \in {\mathbb N}$ such that $AR=RB, SA=BS, RS=A^{\ell}$. The first two equations force $R,S$ to have the forms $$\begin{aligned}
R &= \begin{pmatrix}
b-a & a+b \\ a & b
\end{pmatrix} ;
\ \ \ \quad
a,b,b-a \in {\mathbb Z}_+
\\
S &=\begin{pmatrix}
b-a & 2a-b \\ a & b
\end{pmatrix}; \ \quad
a,b,b-a, 2a-b \in {\mathbb Z}_+\end{aligned}$$ and from this one can check that $RS$ has the form $$RS = \begin{pmatrix}
a&2b \\ b & a
\end{pmatrix},
\ \ \ \quad
a,b,2b-a \in {\mathbb Z}_+ \ .$$ For fixed $n$, $\lim_k 2P^{(n)}_2(k)/P^{(n)}_1(k) = \infty $ . Thus given $\ell_0 \in {\mathbb N}$, for all sufficiently large $k$ we have for $n\leq \ell_0$ that $P^{(n)}_1(k) > 2P^{(n)}_2(k)$. Thus, for such $k$ the lag of an SE-${\mathbb Z}_+$ between $A_k$ and $B_k$ is greater than $\ell_0$.
It is worth noting that Riedel’s argument showing the smallest lag of an SE-${\mathbb Z}_+$ goes to infinity with $k$ works just as well with ${\mathbb Q}_+$ or ${\mathbb R}_+$ in place of ${\mathbb Z}_+$: bad lags can happen for “geometric” reasons, without nilpotence or arithmetic issues. On the other hand, bad lags can happen for strictly arithmetic reasons, as the next example shows.
\[badArithmeticLag\] ([*Bad lag at size 2 from arithmetic.*]{}) Given $\ell \in {\mathbb N}$, there are $2\times 2 $ positive integral matrices $A,B$ such that (i) $A,B$ are SE-${\mathbb Z}_+$, with minimum lag at least $\ell$, and (ii) $A,B$ are SE-${\mathbb Q}_+$ with lag 2.
[In Example \[badArithmeticLag\], I don’t know any obstruction to existence of an example for which condition (ii) is replaced by $A,B$ are ESSE-${\mathbb Q}_+$ and SSE-${\mathbb Z}_+$.]{}
We list steps to check. Given a prime $q$, and positive integer $x$, set $A_x=
\left( \begin{smallmatrix} q & x \\ 0 & 1
\end{smallmatrix}\right) $.
[*Step 1.*]{} Suppose $(R,S)$ gives an SE-${\mathbb Z}$ from $A_x$ to $A_y$: $A_xR=RA_y$, etc. Then (perhaps after replacing $R,S$ with $-R,-S$) $R$ has the form $\left( \begin{smallmatrix}\pm q^k & z \\ 0 & 1
\end{smallmatrix}\right) $, where $k$ is a nonnegative integer. It follows that $\pm x \equiv q^ky \mod (q-1)$.
[*Step 2.*]{} Suppose there is a smallest positive integer $k$ such that $q^kx \equiv \pm y \mod (q-1)$. Then $A_x$, $A_y$ are SE-${\mathbb Z}$, but any such shift equivalence has lag at least $k$.
[*Step 3.*]{} Choose $p$ prime such that $p-1> 2(2\ell +5)$. Then choose $q$ prime such that $p$ divides $q-1$ (this is possible by Dirichlet’s Theorem [@marcusDaniel]). Because $p-1 \geq 2\ell +5$, by the Pigeonhole Principle we may choose $j$ a positive integer such that $1\leq j \leq 2\ell +5$ and also for $1\leq k \leq \ell +2$ we have $j\not\equiv \pm q^k \mod p$ . Define $x=(q-1)/p$ and $y=j(q-1)/p$. Then $A_x$ and $A_y$ are SE-${\mathbb Z}$ with minimum lag at least $\ell +2$. Also, $0< x< y < (1/2)q$ and $y< qx$.
[*Step 4.*]{} For $z\in \{x,y \}$, define the positive integral matrix $$M_z = \begin{pmatrix} 1&0\\1&1 \end{pmatrix}
\begin{pmatrix} q&z\\0&1 \end{pmatrix}
\begin{pmatrix} 1&0\\-1&1 \end{pmatrix}
= \begin{pmatrix} q-z&z \\ \ q-z-1&1+z \end{pmatrix} \ ,$$ This SIM-$Z$ gives a lag 1 SE-${\mathbb Z}$ between $A_z$ and $M_z$. If follows that there can be no SE-${\mathbb Z}$ from $M_x$ to $M_y$ with lag smaller than $\ell$.
[*Step 5.*]{} It remains to produce the lag 2 SE-${\mathbb Q}_+$ between $M_x$ and $M_y$. For the eigenvalues $q$ and $1$, $M_z$ has right eigenvectors $v=(1,1)^{\text tr}$ and $w_z= (-z,q-z-1)^{\text{tr}} $. Let $U$ be the $2\times 2$ matrix such that $Uv=v$ and $Uw_x=w_y$. Then $R=M_yU, S=M_xU^{-1}$ gives a lag 2 SE-${\mathbb Q}$ between $M_x$ and $M_y$. It remains to check $R,S$ are nonnegative. We have $$\begin{aligned}
R=M_yU &=
\begin{pmatrix} q-y&y \\ \ \ q-y-q & y+1
\end{pmatrix}
\frac 1{q-1}
\begin{pmatrix} q-x-1+y & x-y \\
-x+y & \ \ q+x-y-1
\end{pmatrix}
\\
&= \frac 1{q-1}
\begin{pmatrix}
q^2 - q(x+1) -xy
&
qx-y
\\
q^2 -q(x+2) -xy+1
&
\ \ q(x+1) - (y+1)
\end{pmatrix}\end{aligned}$$ From the last sentence of Step 3, we see the entries of $M_yU$ are positive. The matrix $S$ is obtained from $R$ by interchanging the roles of $x$ and $y$, and $S$ is likewise positive.
\[ashley\] By the way, here is an example (“Ashley’s eight by eight”) of a primitive matrix $A$ SE-${\mathbb Z}$ to $(2)$, but not known to be SSE-${\mathbb Z}_+$ to $(2)$. $A$ is the $8\times 8$ matrix which is the sum of the permutation matrices for the permutations (12345678) and (1)(2)(374865).
\[RealSSEandBKR\] For more on the problem of SSE over ${\mathbb R}$, focused on the case of positive matrices, see [@BKR2013]. (Kim and Roush proved that primitive matrices over ${\mathbb R}_+$ are SSE-${\mathbb R}_+$ to positive matrices. So, the case of SSE-${\mathbb R}_+$ of positive matrices handles the primitive positive trace case.) The method here, due to Kim and Roush, is to derive from a path of similar positive matrices an SSE-${\mathbb R}_+$ between the endpoints. Kim and Roush were able to reduce to considering positive matrices of equal size, similar over ${\mathbb R}$; and in the “$1\times 1$ case”, to produce such a path.
However, even when both $A$ and $B$ are $2\times 2$ positive real matrices, the problem of when they are SSE-${\mathbb R}_+$ is open. It is embarassing that we are not more clever.
\[ssez2ssezplus\] To understand when SE-${\mathbb Z}_+$ matrices $A,B$ are SSE-${\mathbb Z}_+$, it is best to focus on the fundamental case that $A$ and $B$ are primitive. (Then consider irreducible matrices, then general matrices, modulo a solution of the primitive case.) For primitive matrices over ${\mathbb Z}_+$, SE-${\mathbb Z}_+$ is equivalent to SSE-${\mathbb Z}$ (Proposition \[primiiveSE\]). For primitive matrices over ${\mathbb Z}_+$, it has been important to study a reformulation of the problem: when does SSE-${\mathbb Z}$ imply SSE-${\mathbb Z}_+$? This formulation was essential for the Wagoner complex setting for the Kim-Roush counterexamples [@S13] to Williams’ Conjecture, and for some arguments for a general subring $R$ of ${\mathbb R}$ (see [@BKR2013]). For some subrings $R$ of ${\mathbb R}$, SE-$R_+$ does not even imply SSE-${\mathbb R}$, as we will see.
Shift equivalence and strong shift equivalence over a ring {#sec:sesse}
===========================================================
In this section, we present basic facts about shift equivalence and strong shift equivalence over rings, with various example classes.
SE-${\mathbb Z}_+$: dynamical meaning and reduction to SE-${\mathbb Z}$
-----------------------------------------------------------------------
First we give the dynamical meaning of SE-${\mathbb Z}_+$.
Homeomorphisms $S$ and $T$ are eventually conjugate if $S^n, T^n$ are conjugate for all but finitely many positive integers $n$.
Let $A,B$ be square matrices over ${\mathbb Z}_+$. The following are equivalent [(Ap. ]{}[seeventual]{}).
1. $A,B$ are shift equivalent over ${\mathbb Z}_+$.
2. The SFTs $(X_A,\sigma)$, $(X_B,\sigma)$ are eventually conjugate.
Next we consider how SE-${\mathbb Z}$ and SE-${\mathbb Z}_+$ are related. Recall Definition \[primitiveDefinition\]: a primitive matrix is a square nonnegative real matrix such that some power is positive.
The matrices $\begin{pmatrix} 1 \end{pmatrix} $ and $\begin{pmatrix} 1&1\\ 1&0 \end{pmatrix}$ are primitive.\
The matrices $\begin{pmatrix} 1&1\\ 0&0 \end{pmatrix}$, $\begin{pmatrix} 1&1\\ 0&1 \end{pmatrix}$ and $\begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}$ are not primitive.
\[primiiveSE\] [(Ap. ]{}[sePrimitive]{}) Suppose two primitive matrices over a subring $\mathcal R$ of the reals are SE over $\mathcal R$. Then they are SE over $\mathcal R_+$. (Recall, $\mathcal R_+ = \mathcal R \cap \{x\in {\mathbb R}: x\geq 0\}$.)
For primitive matrices, the classification up to SE-${\mathbb Z}_+$ reduces to the tractable problem of classifying up to SE-${\mathbb Z}$. The Proposition becomes false if the hypothesis of primitivity is removed [(Ap. ]{}[bkaplansky]{}).
Strong shift equivalence over a ring {#ssesubsec}
------------------------------------
Let $\mathcal R$ be a ring. Recall, ${\textnormal{GL}}(n, \mathcal R)$ is the group of $n\times n$ matrices invertible over $\mathcal R$; $U\in {\textnormal{GL}}(n,\mathcal R)$ if there is a matrix $V$ over $\mathcal R$ with $UV=VU=I$. This matrix $V$ is denoted $U^{-1}$. If $\mathcal R$ is commutative, then $U\in {\textnormal{GL}}(n,\mathcal R)$ iff $\det U$ is a unit in $\mathcal R$.
Square matrices $A,B$ are similar over $\mathcal R$ (SIM-$\mathcal R$) if there exists $U$ in ${\textnormal{GL}}(n,\mathcal R)$ such that $B = U^{-1}AU$.
Our viewpoint: SE and SSE of matrices over a ring $\mathcal R$ are [*stable versions of similarity*]{} of matrices over $\mathcal R$.
By a “stable version of similarity” we mean an equivalence relation on square matrices which coarsens the relation of similiarity, and is obtained by allowing some kind of neglect of the nilpotent part of the matrix multiplication. (This will be less vague soon.)
\[prop:MallerShub\] SSE over a ring $\mathcal R$ is the equivalence relation on square matrices over $\mathcal R$ generated by the following relations on square matrices $A,B$ over $\mathcal R$.
1. [*(Similarity over $\mathcal R$)*]{} For some $n$, $A$ and $B$ are $n\times n$ and there is a matrix $U$ in $\text{GL}(n,\mathcal R)$ such that $A=U^{-1}BU$ .\
2. [*(Zero extension)*]{} There exists a matrix $X$ over $\mathcal R$ such that in block form, $B= \begin{pmatrix} A&X\\ 0&0 \end{pmatrix} $ or $B= \begin{pmatrix} A&0\\ X&0 \end{pmatrix} $ .
A similarity or a zero extension produces an ESSE:\
\
If $A=U^{-1}BU$, then $A= (U^{-1}B)\, U$ and $B = U\,(U^{-1}B)$.\
\
If $B= \begin{pmatrix} A&X\\ 0&0 \end{pmatrix} $, then $B= \begin{pmatrix} I\\ 0 \end{pmatrix}
\begin{pmatrix} A&X \end{pmatrix} $ and $A=
\begin{pmatrix} A&X \end{pmatrix}
\begin{pmatrix} I\\ 0 \end{pmatrix}$ .\
\
If $B= \begin{pmatrix} A&0\\ X&0 \end{pmatrix} $, then $B= \begin{pmatrix} A\\ X \end{pmatrix}
\begin{pmatrix} I&0 \end{pmatrix} $ and $A=
\begin{pmatrix} I&0 \end{pmatrix}
\begin{pmatrix} A\\ X \end{pmatrix}$ .
Conversely, if $A=RS$ and $B=SR$, then there is a similarity of zero extensions:\
\
$\begin{pmatrix} I&0\\ S&I\end{pmatrix}
\begin{pmatrix} A&R\\ 0&0 \end{pmatrix} =
\begin{pmatrix} 0&R\\ 0&B \end{pmatrix}
\begin{pmatrix} I&0\\ S&I \end{pmatrix} \ \qed $ .
SSE-$\mathcal R$ coarsens SIM-$\mathcal R$ by allowing “zero extensions”. What coarsening could be more mild than this? We might allow only zero extensions of the form $A \to \begin{pmatrix} A&0\\ 0&0 \end{pmatrix}$ . Under this stabilization, matrices would be equivalent if they are similar, modulo enlarging the kernel. by a direct summand isomorphic to $\mathcal R^k $, for some $k$. For example, all square zero matrices would be equivalent, but would not be equivalent to $ \begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}$.
SSE-$\mathcal R$ can be viewed as the second mildest “natural” nontrivial stabilization of SIM-$\mathcal R$. (Here “natural” is intuitive, not rigorous.) The relation SSE-$\mathcal R$ can be very subtle indeed, as we will see. Fortunately, if $\mathcal R$ is ${\mathbb Z}$, or a field, then SSE-$\mathcal R$ = SE-$\mathcal R$.
SE, SSE and det(I-tA) {#detI-tAsubsection}
---------------------
Let $\mathcal R$ be a commutative ring, and $A$ a square matrix over $\mathcal R$. As explained in Remark \[NewtonIdentities\], the polynomial $\det(I-tA)$ determines the trace sequence $({\textnormal{trace}}(A^n))_{n=1}^{\infty}$, and that sequence determines $\det(I-tA)$ if $\mathcal R$ is torsion-free.
If $A$ and $B$ are SSE over $\mathcal R$, then one easily sees $({\textnormal{trace}}(A^n))_{n=1}^{\infty}=({\textnormal{trace}}(B^n))_{n=1}^{\infty}$, simply because ${\textnormal{trace}}(RS)= {\textnormal{trace}}(SR)$. To see that in addition $\det(I-tA) = \det(I-tB)$, apply the Maller-Shub characterization Proposition \[prop:MallerShub\].
If there is a lag $\ell$ shift equivalence over $\mathcal R$ between $A$ and $B$, then $({\textnormal{trace}}(A^k))_{k=\ell}^{\infty} = ({\textnormal{trace}}(B^k))_{k=\ell}^{\infty}$. We shall see below that if $\mathcal R$ is an integral domain, then $\det (I-tA)$ is also an invariant of SE-$\mathcal R$ (because it is an invariant of shift equivalence over the field of fractions of $\mathcal R$).
But in some cases, the trace of a matrix need not be an invariant of SE-$\mathcal R$. Suppose $\mathcal R$ is a ring with a nilpotent element $a$ (i.e., $a\neq 0$ and $a^k=0$ for some positive integer $k$). For example, let $\mathcal R = {\mathbb Z}[t]/(t^2)$ and $a=t$. Consider the $1\times 1$ matrices $A=(a)$ and $B=(0)$. Then $A$ and $B$ are SE-$\mathcal R$ but ${\textnormal{trace}}(A) \neq {\textnormal{trace}}(B)$. This by the way gives an easy example of a ring $\mathcal R$ for which SE-$\mathcal R$ and SSE-$\mathcal R$ are not the same relation.
[Somehow this easy example was missed for many years, perhaps because the rings arising in symbolic dynamics have been rings without nilpotents.]{}
If a commutative ring $\mathcal R$ has no nilpotent element, then $\det(I-tA) $ will be an invariant of the SE-$\mathcal R$ class of $A$ [(Ap. ]{}[reducedrings]{}).
Shift equivalence over a ring $\mathcal R$
------------------------------------------
We consider shift equivalence over a ring $\mathcal R$, by cases.
### $\mathcal R$ is a field.
Suppose $A$ is a square matrix over $\mathcal R$. There is an invertible $U$ over $\mathcal R$ such that $U^{-1}AU$ has the form $\begin{pmatrix} A' & 0 \\ 0 &N \end{pmatrix}$, where $A'$ is invertible and $N$ is triangular with zero diagonal. (For $\mathcal R={\mathbb C}$, use the Jordan form.)
$\mathcal R={\mathbb R}$, $\ \ \ U^{-1}AU =
\begin{pmatrix} \sqrt 2 &0&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \\
0&0&0&0 \end{pmatrix}$, $\ \ \ A'=(\sqrt 2)$.
$A'$ (as a vector space endomorphism) is isomorphic to the restriction of $A$ to the largest invariant subspace on which $A$ acts invertibly. Abusing notation, we call $A'$ [**the nonsingular part**]{} of $A$ (keeping in mind that $A'$ is only well defined up to similarity over the field $\mathcal R$).
A square matrix $A$ over a field $\mathcal R$ is SSE over $\mathcal R$ to its nonsingular part, $A'$.
$A'$ reaches $U^{-1}AU$ by a string of zero extensions.
Suppose $\det A =0$, and $U^{-1}AU
=\begin{pmatrix} A' & 0 \\ 0 &N \end{pmatrix}$, and $\ell$ is the smallest positive integer such that $N^{\ell}=0$. Then the smallest lag of an SSE over $\mathcal R$ from $A$ to $A'$ is $\ell $ [(Ap. ]{}[smallestlag]{}).
From the Proposition, square matrices $A,B$ are SSE-$\mathcal R$ if and only if their nonsingular parts $A',B'$ are SSE-$\mathcal R$. Likewise for SE-$\mathcal R$.
Suppose $A,B$ are square nonsingular matrices over the field $\mathcal R$. The following are equivalent.
1. $A$ and $B$ have the same size and are similar over $\mathcal R$.
2. $A$ and $B$ are SE-$\mathcal R$.
3. $A$ and $B$ are SSE-$\mathcal R$.
$(1) \implies (3) \implies (2)$. Clear.
$(2) \implies (3)$. Suppose $(R,S)$ is a lag $\ell $ SE over $\mathcal R$ from $A$ to $B$: $$A^{\ell} =RS\ ,\quad B^{\ell} =SR\ ,\quad AR=RB\ ,\quad BS=SA\ .$$ Suppose $A$ is $m\times m$ and $B$ is $n\times n$. Then $R$ is $m\times n$. Hence $m=n$, because $$m \ \ = \ \ \text{rank}(RS) \ \ \leq \ \ \text{rank}(R) \ \
\leq \ \ \min \{ m,n \} \ \ \leq n$$ and likewise $n \leq m$.
Now $\det(A^{\ell}) =(\det R)( \det S)$, hence $\det R \neq 0$.\
Then $AR=RB$ gives $B=R^{-1}AR$ .
Suppose matrices $A,B$ are SE over a field $\mathcal R$. Then $\det(I-tA) = \det (I-tB)$ .
By the proposition, the nonsingular parts $A',B'$ of $A,B$ are similar over $\mathcal R$. Therefore they have the same spectrum, which is the nonzero spectrum of $A$ and $B$. Therefore $\det (I-tA)= \det(I-tB)$.
When matrices $A,B$ have entries in a field $\mathcal R$ contained in ${\mathbb C}$, similarity over $\mathcal R$ is equivalent to similarity over ${\mathbb C}$, and the Jordan form of the nonsingular part is a complete invariant for similarity of $A,B$ over $\mathcal R$.
### $\mathcal R$ is a Principal Ideal Domain
The principal ideal domain of greatest interest to us is $\mathcal R = {\mathbb Z}$, the integers. The PID case is like the field case, but with more arithmetic structure. In place of the Jordan form, we use a classical fact. Recall, an [*upper triangular*]{} matrix is a square matrix with only zero entries below the diagonal (i.e., $i > j \implies A(i,j) = 0$). A lower triangular matrix is a square matrix with only zero entries above the diagonal. A matrix is triangular if it is upper or lower triangular. Block triangular matrices are defined similarly, for block structures on square matrices which use the same index sets for rows and columns.
\[PID Block Triangular Form\] [@Newman]\[thm:PIDblocktri\] Suppose $\mathcal R$ is a principal ideal domain (e.g., $\mathcal R={\mathbb Z}$ or a field). Suppose $A$ is a square matrix over $\mathcal R$ and $p_1, \dots , p_k$ are monic polynomials with coefficients in $\mathcal R$ such that the characteristic polynomial of $A$ is $\chi_A(t) = \prod_i p_i(t)$.
Then $A$ is similar over $\mathcal R$ to a block triangular matrix, with diagonal blocks $A_i$, $1\leq i \leq k$ , such that $p_i$ is the characteristic polynomial of $A_i$.
Suppose $A$ is $6\times 6$ and $\chi_A(t)=(t-3)(t^2+5)(t+1)(t)(t)$. Then there is some $U$ in $\text{GL}(6, {\mathbb Z})$ such that $U^{-1}AU$ has the form $$\left( \begin{smallmatrix}
3&*&* &*&*&* \\
0&a&b &*&*&* \\
0&c&d &*&*&* \\
0&0&0 &-1&*&* \\
0&0&0 &0&0&* \\
0&0&0 &0&0&0
\end{smallmatrix}
\right)$$ in which $\begin{pmatrix} a&b\\c&d \end{pmatrix}$ has characteristic polynomial $t^2 +5$ .
For $A$ square over the PID $\mathcal R$: $A$ is similar over $\mathcal R$ to a matrix with block form $\begin{pmatrix} A' & X \\ 0 & N \end{pmatrix}$, where $\det (A') \neq 0$ and $N$ is upper triangular with zero diagonal.
As in the field case, we call $A'$ [*the nonsingular part*]{} of $A$ ($A'$ is defined up to similarity over $\mathcal R$).
For $\mathcal R$ a principal ideal domain, any nonnilpotent square matrix over $\mathcal R$ is SSE-$\mathcal R$ to its nonsingular part (hence, SE-$\mathcal R$ to its nonsingular part).
$A'$ reaches $U^{-1}AU$ by a string of zero extensions.
The Corollary can easily fail even for a Dedekind domain, such as the algebraic integers in a number field [@BH93].
### $\mathcal R$ is ${\mathbb Z}$.
[(Ap. ]{}[1by1]{}) Suppose $A$ is square over ${\mathbb Z}_+$ and $\det (I-tA) = 1-nt$, where $n$ is a positive integer. Then $A$ is SE over ${\mathbb Z}_+$ to the $1\times 1$ matrix $(n)$.
The classification of matrices over ${\mathbb Z}$ up to SE-${\mathbb Z}$ reduces to the classification of nonsingular matrices over ${\mathbb Z}$ up to SE-${\mathbb Z}$. If $A,B$ are SE-${\mathbb Z}$, then $A,B$ are SE-${\mathbb R}$, so their nonsingular parts are similar over ${\mathbb R}$; in particular $A,B$ have the same nonzero spectrum.
It is NOT true that a square matrix over ${\mathbb Z}_+$ must be SE-${\mathbb Z}_+$ to a nonsingular matrix.
[(Ap. ]{}[nonplussed]{}) The primitive matrix $A= \left(\begin{smallmatrix} 1&0&0&1 \\ 0&1&0&1 \\ 0&1&1&0 \\ 1&0&1&0
\end{smallmatrix}\right) $ has nonzero spectrum $(2,1)$. If $A$ is SE-${\mathbb Z}_+$ to a nonsingular matrix $B$, then $B$ must be a primitive $2\times 2$ with nonzero spectrum $(2,1)$. Prove that no such $B$ exists.
Suppose $A$, $B$ are nonsingular matrices over ${\mathbb Z}$ with $|\det (A)|=1$. The following are equivalent.
\(1) $A,B$ are SE-${\mathbb Z}$.
\(2) $A,B$ are SIM-${\mathbb Z}$.
$(1) \implies (2)$ Clear.
$(2) \implies (1)$ An SE $(R,S)$ over ${\mathbb Z}$ from $A$ to $B$ is also an SE $(R,S)$ over the field ${\mathbb Q}$. Therefore $A,B$ have the same size, $n\times n$. Now $A^{\ell} = RS$ forces $\det A$ to divide $\det A^{\ell}$, so $|\det R| =1$. This implies $R\in {\textnormal{GL}}( n, {\mathbb Z})$. Then $AR=RB$ gives $B=R^{-1}AR$.
Let $A= \begin{pmatrix} 3&4\\1&1 \end{pmatrix}$ and $B= \begin{pmatrix} 3&2\\2&1 \end{pmatrix}$. $A$ and $B$ have the same characteristic polynomial, $p(t)= t^2 -4t-1$; as $p$ has no repeated root, $A$ and $B$ are similar over ${\mathbb Q}$.
Now suppose $A,B$ are SE-${\mathbb Z}$. Because $\det A = -1$, they are SIM-${\mathbb Z}$: there is $R$ in $\text{GL}(2,{\mathbb Z})$ such that $B=R^{-1}AR$. Therefore $(B-I)=R^{-1}(A-I)R$. This is a contradiction, because $B-I= \begin{pmatrix} 2&2\\2&0 \end{pmatrix}$ is zero mod 2, but $A-I= \begin{pmatrix} 2&4\\1&0 \end{pmatrix}$ is not.
What else? Here is a quick overview.
Let $p$ be a monic polynomial in ${\mathbb Z}[t]$ with no zero root. Let $\mathcal M(p)$ be the set of matrices over ${\mathbb Z}$ with characteristic polynomial $p$.
If $p$ has no repeated root, then the following hold.
1. All matrices in $\mathcal M(p)$ are SIM-${\mathbb Q}$ (and therefore SE-${\mathbb Q}$).
2. $\mathcal M(p) $ is the union of finitely many SIM-${\mathbb Z}$ classes (hence finitely many SE-${\mathbb Z}$ classes).
3. It is can happen (depending on $p$) that in $\mathcal M(p)$, SIM-${\mathbb Z}$ properly refines SE-${\mathbb Z}$.
If $p$ has a repeated root, then $\mathcal M(p) $ contains infinitely many SE-${\mathbb Z}$ classes, but only finitely many SE-${\mathbb Q}$ classes.
(Easily checked.) For $n\in {\mathbb N}$, the matrices $\begin{pmatrix} 1&n\\0&1 \end{pmatrix} $ are similar over ${\mathbb Q}$, but pairwise not similar over ${\mathbb Z}$.
Lastly, we will report on some decidablity issues for shift equivalence over ${\mathbb Z}$.
Suppose $A,B$ are square matrices over ${\mathbb Z}$.
1. (Grunewald [@grunewaldZDecide]; see also [@grunewaldSegalI].) There is an algorithm to decide whether $A,B$ are SIM-${\mathbb Z}$. The general algorithm is not practical.
2. (Kim and Roush [@S31]) There is an algorithm to decide whether $A,B$ are SE-${\mathbb Z}$. The general algorithm is not practical.
SIM-${\mathbb Z}$ and SE-${\mathbb Z}$: some example classes
------------------------------------------------------------
The proof of the next result, from [@BH93], is an exercise.
\[thm:axb\] Suppose $a,b$ are integers and $a> |b| > 0$. Let $\mathcal M$ be the set of $2\times 2$ matrices over ${\mathbb Z}$ with eigenvalues $a,b$. Then the following hold.
1. Every matrix in $\mathcal M$ is SIM-${\mathbb Z}$ to a triangular matrix $M_x=\begin{pmatrix}a & x\\ 0 &b\end{pmatrix}$.
2. $M_x$ and $M_y$ are SIM-${\mathbb Z}$ iff $x = \pm y$ mod ($a-b$).
3. $M_x$ and $M_y$ are SE-${\mathbb Z}$ iff $x\sim y$, where $\sim$ is the equivalence relation generated by $x\sim y$ if $x = \pm qy$ mod ($a-b$) for a prime $q$ dividing $a$ or $b$.
Suppose $a=6,b=1$.\
Then $\mathcal M$ is the union of three SIM-${\mathbb Z}$ classes and two SE-${\mathbb Z}$ classes.
Suppose $a=6, b=2$.\
Then $\mathcal M$ is the union of two SIM-${\mathbb Z}$ classes and one SE-${\mathbb Z}$ class.
\[transposeexercise\] [(Ap. ]{}[transposeexerciseproof]{}) Use Theorem \[thm:axb\] to prove the following: the matrix $\begin{pmatrix} 256 & 7\\ 0 &1 \end{pmatrix}$ is not SE-${\mathbb Z}$ to its transpose. Then show $\begin{pmatrix} 256 & 7\\ 0 &1 \end{pmatrix}$ is SE-${\mathbb Z}$ to a primitive matrix, which cannot be SE-${\mathbb Z}$ to its transpose.
The next theorem states a result relating a matrix similarity problem to algebraic number theory, and the analagous result for shift equivalence. The similarity result is a special case of a theorem of Latimer and MacDuffee; Olga Taussky-Todd provided a simple proof in this special case, which generalizes nicely to the SE-${\mathbb Z}$ situation [(Ap. ]{}[tausskytoddcase]{}). In the next theorem, for $\mathcal R={\mathbb Z}[\lambda]$ or $\mathcal R={\mathbb Z}[1/\lambda]$, $\mathcal R$-ideals $I,I'$ are equivalent if they are equivalent as $\mathcal R$-modules, which in this case means there is a nonzero $c$ in ${\mathbb Q}[\lambda ]$ such that $cI=I'$. By an ideal class of $\mathcal R$ we mean an equivalence class of nonzero $\mathcal R$-ideals.
\[thm:taussky\] Suppose $p$ is monic irreducible in ${\mathbb Z}[t]$, and $p(\lambda)=0$, where $0\neq \lambda \in {\mathbb C}$. Let $\mathcal M$ be the set of matrices over ${\mathbb Z}$ with characteristic polynomial $p$. There are bijections:
1. $\mathcal M/(\text{SIM}-{\mathbb Z})$ $\leftrightarrow $ Ideal classes of ${\mathbb Z}[\lambda ]$ [@latimerMacDuffee; @tausskyOnLatimerTheorem]
2. $\mathcal M/(\text{SE}-{\mathbb Z}) \ \ \leftrightarrow $ Ideal classes of ${\mathbb Z}[1/\lambda ]$ [@BMT1987].
[(Ap. ]{}[finiteclass]{}) Let $\lambda$ be a nonzero algebraic integer, and let $\mathcal O_{\lambda}$ be the ring of algebraic integers in the number field ${\mathbb Q}[\lambda ]$. It is a basic (and “surprisingly easy to establish” [@marcusDaniel Ch.5]) fact of algebraic number theory that the class number of $\mathcal O_{\lambda}$ is finite. Use this fact to show that ${\mathbb Z}[\lambda ]$ also has finite class number.
The number theory connection is useful. For example, it follows from the exercise that $\mathcal M$ in the theorem contains only finitely many SIM-$Z$ classes ([@Newman Theorem III.14]). In the case that ${\mathbb Z}[\lambda ]$ is a full ring of quadratic integers, one can often simply look up the class number of ${\mathbb Z}[\lambda ]$ in a table.
SE-${\mathbb Z}$ via direct limits {#sec:SEZdirectlimits}
----------------------------------
Let $A$ be an $n\times n$ matrix over ${\mathbb Z}$. We choose to let $A$ act on row vectors. From the action $A: {\mathbb Z}^n \to{\mathbb Z}^n $ one can form the direct limit group, on which there is a group automorphism $\hat A: G_A \to G_A$ induced by $A$.
We will take a very concrete presentation, $\hat A: G_A \to G_A$, for the induced automorphism of the direct limit group [(Ap. ]{}[directlimits]{}).
### The eventual image $V_A$
Define rational vector spaces $W_k = \{ vA^k: v \in {\mathbb Q}^n\}$ and $V_A = \cap_{k\in {\mathbb N}} W_k$ . Then $$\begin{aligned}
& {\mathbb Q}^n \supset W_1 \supset W_{1}\supset W_2 \supset \cdots \\
&\dim (W_{k+1})=\dim (W_k)\implies W_{k+1}=W_k \\
& V_A = W_n \ .\end{aligned}$$ $V_A$ is the “eventual image” of $A$ as an endomorphism of the rational vector space ${\mathbb Q}^n$. $V_A$ is the largest invariant subspace of ${\mathbb Q}^n$ on which $A$ acts as a vector space isomorphism.
### The pair $(G_A, \hat A)$
$G_A$ is the subset of $V_A$ eventually mapped by $A$ into the integer lattice: $G_A := \{ v\in V_A: \exists k\in {\mathbb N}, vA^k \in {\mathbb Z}^n\}$ . The automorphism $\hat A$ of $G_A$ is defined by restriction, $\ \ \hat A : v\mapsto vA $ .
Suppose $|\det A| =1$. Then $V_A= {\mathbb Q}^n$, $G_A = {\mathbb Z}^n$.
$A = (2)$. Then $V_A$ = ${\mathbb Q}$, and\
$G_A$ is the group of dyadic rationals: $\ G_A = {\mathbb Z}[1/2] = \{ n/2^{k}: n\in {\mathbb Z}, k \in {\mathbb Z}_+\}$.
Similarly, for a positive integer $k$, if $A=(k)$ then $G_A = {\mathbb Z}[1/k]$. For positive integers $k$ and $ m$, TFAE:
$\bullet \ $ $ {\mathbb Z}[1/k] = {\mathbb Z}[1/m]$ .
$\bullet \ $ $ k$ and $m$ are divisible by the same primes.
$\bullet \ $ $ {\mathbb Z}[1/k]$ and $ {\mathbb Z}[1/m]$ are isomorphic groups.
$A= \begin{pmatrix} 2 & 0\\ 0& 5 \end{pmatrix}$. $\ G_A = \{ (x,y): \ x\in {\mathbb Z}[1/2], \ y\in {\mathbb Z}[1/5] \} $.
$B= \begin{pmatrix} 2 & 1\\ 0 &5 \end{pmatrix}$.\
The groups $G_B$ and $G_A$ are not isomorphic. ($G_B$ is not the sum of a 2-divisible subgroup and a 5-divisible subgroup [(Ap. ]{}[nonisomorphicpair]{}).)
Two pairs $(G_A,\hat A), (G_B, \hat B)$ are isomorphic if there is a group isomorphism $\phi : G_A \to G_B$ such that $\hat B \phi (x) = \phi \hat A (x)$. (In other words, $\hat A$ and $\hat B$ are isomorphic, in the category of group automorphisms; or, equivalently, in the category of group endomorphisms.)
\[prop:sezdirectlimit\] Let $A,B$ be square matrices over ${\mathbb Z}$. The following are equivalent [(Ap. ]{}[seandpairiso]{}).
1. $A$ and $B$ are SE-${\mathbb Z}$.
2. There is an isomorphism of direct limit pairs $(G_A, \hat A)$ and $(G_B, \hat B)$.
There is a natural way to make $G_A$ above an ordered group (in an important class of ordered groups, the [*dimension groups*]{} [(Ap. ]{}[dimensiongroupsremark]{})). Then, analogous to Proposotion \[prop:sezdirectlimit\], there an ordered group characterization of SE-${\mathbb Z}_+$ [(Ap. ]{}[SEZ+andgroupsremark]{}).
Let $A = (2)$ and $B=\begin{pmatrix} 1&1 \\ 1&1 \end{pmatrix}$.\
These matrices are SE-${\mathbb Z}$ (and even ESSE-${\mathbb Z}_+$).
- $V_A$ = ${\mathbb Q}$ and $G_A= {\mathbb Z}[1/2]$.
- $V_B = \{ (x,x): x\in {\mathbb Q}\}$, the eigenline for eigenvalue 2, and $G_B = \{ (x,x): x\in {\mathbb Z}[1/2] \}$.
- $\phi \colon x\mapsto (x,x)$ defines a group isomorphism $G_A \to G_B$ such that $\hat B \phi (x) = \phi \hat A (x)$ .
SE-${\mathbb Z}$ via polynomials
--------------------------------
It will be important for us to put everything we’ve done into a polynomial setting. We use ${\mathbb Z}[t]$, the ring of polynomials in one variable with integer coefficients.\
\
Let $A$ be an $n\times n$ matrix over ${\mathbb Z}$. Recall
- $V_A = \cap_{k\in {\mathbb N}}W_k = \cap_{k\in {\mathbb N}}\{ vA^k: v \in {\mathbb Q}^n\}$,
- $G_A := \{ v\in V_A: \exists k\in {\mathbb N}, vA^k \in {\mathbb Z}^n\}$,
- $\hat A$ is the automorphism of $G_A$ given by $\hat A: x\mapsto xA$.
We regard the direct limit group $G_A$ as a ${\mathbb Z}[t] $-module, by letting $t$ act by $(\hat A)^{-1}$. (This choice of $t$ action will match ${\textnormal{cok}}(I-tA)$ below.) Isomorphism of the pairs $(G_A,\hat A), (G_B, \hat B)$ is equivalent to isomorphism of $G_A, G_B$ as ${\mathbb Z}[t]$-modules. So, we sometimes simply refer to a pair $(G_A, \hat A)$ as a ${\mathbb Z}[t]$-module. To summarize, we have the following.
Supfpose $A,B$ are square matrices over ${\mathbb Z}$. The following are equivalent.
1. $A$, $B$ are SE over ${\mathbb Z}$.
2. $(G_A,\hat A)$ and $ (G_B,\hat B)$ are isomorphic ${\mathbb Z}[t]$-modules.
Next we get another presentation of these ${\mathbb Z}[t]$-modules.
Cokernel (I-tA), a ${\mathbb Z}[t]$-module
------------------------------------------
Given $A$ $n\times n$ over ${\mathbb Z}$, let $I$ be the $n\times n$ identity matrix. View ${\mathbb Z}[t]^n $ as a ${\mathbb Z}[t]$-module: for $v$ in ${\mathbb Z}[t]^n $ and $c\in {\mathbb Z}[t]$, the action of $c$ is to send $v$ to $cv$, where $cv= c(v_1, \dots , v_n)=(cv_1, \dots , cv_n)$. The map $(I-tA): \mathcal R^n \to \mathcal R^n$, by $\ v\mapsto v(I-tA)\ $, is a ${\mathbb Z}[t]$-module homomorphism, as $(cv)(I-tA) = c(v(I-tA))$.
Now define $ \text{cokernel}(I-tA) := {\mathbb Z}[t]^n / \text{Image} (I-tA) $, where $\text{Image} (I-tA) =
\{v(I-tA)\in {\mathbb Z}[t]^n:
v\in {\mathbb Z}[t]^n\} $ . An element of $\text{cokernel}(I-tA)$ is a coset, $v+ \text{Image} (I-tA)$, denoted $[v]$. $ \text{Cokernel}(I-tA)$ is a ${\mathbb Z}[t]$-module, with $ c: [v] \mapsto [cv]$ .
NOTE: we use row vectors to define the module.
Let $A$ be a square matrix over ${\mathbb Z}$. The ${\mathbb Z}[t]$-modules $\text{cok}(I-tA)$ and $(G_A, \hat A)$ are isomorphic.
Define $$\begin{aligned}
\phi : G_A & \to \text{cokernel}(I-tA) \\
x &\mapsto [x(tA)^k]\end{aligned}$$ where $k$ (dependng on $x$) is any nonnegative integer large enough that $xA^k\in {\mathbb Z}^n$. For a proof, check that this $\phi$ is a well-defined isomorphism of ${\mathbb Z}[t]$-modules.
Let us see how this works out in a concrete example.
$A=(2)$. Here $G_A = {\mathbb Z}[1/2]
= \cup_{k\geq 0}\, (\frac 12)^k{\mathbb Z}\ $ and $\ {\mathbb Z}[t] = \cup_{k\geq 0}\, t^k {\mathbb Z}$ . The isomorphism $\phi :G_A \to \text{cokernel}(I-tA)\quad $ is defined by $$\begin{aligned}
\phi : {\mathbb Z}[1/2] & \to {\mathbb Z}[t]/(1-2t){\mathbb Z}[t] \ \\
(1/2)^kn & \mapsto [t^kn]\ , \qquad \qquad \ \
\text{for }n \text{ in }{\mathbb Z}, \ k \in {\mathbb Z}_+ \ .
\end{aligned}$$ The isomorphism $\phi$ takes $(1/2)^k {\mathbb Z}$ to $[t^k{\mathbb Z}]$. The cokernel relation mimics the $G_A$ relation $(1/2)^kn = (1/2)^{k+1}(2n)$. In more detail, to check that $\phi$ in this example is a ${\mathbb Z}[t]$-module isomorphism, check the following (some details are provided).
- $\phi$ is well defined.\
Because $[x] \in \text{cokernel}(1-2t)$, we have $[x] = [2tx]$, so $$\begin{aligned}
(1/2)^k n \ &\ \mapsto [t^k n] \\
(1/2)^{k+1}(2n)\ &\ \mapsto [t^{k+1} (2n)]
= [(t^kn)(2t)] = [t^kn] \ .\end{aligned}$$
- $\phi$ is a group homomorphism.
- $\phi$ is a ${\mathbb Z}[t]$-module homomorphism : $$\begin{aligned}
t\phi ((1/2)^kn) \ &=\ t[t^kn] = [t^{k+1}n] \ , \\
\phi (t((1/2)^kn))\ & =\ \phi (\hat A^{-1} ((1/2)^kn)
= \phi ( (1/2)((1/2)^kn)) = [ t^{k+1}n] \ .\end{aligned}$$
- $\phi$ is surjective.
- $\phi$ is injective.\
Given $\phi ((1/2)^kn) = [t^kn]=[0]$, there exists $p$ in ${\mathbb Z}[t]$ such that $t^kn = (1-2t)p$. This forces $n=0$. (Otherwise $p\neq 0$, and then $(1-t)p=p-tp$ with nonzero coefficients at different powers of $t$, contradicting $t^kn = (1-2t)p$.)
For square matrices $A,B$ over ${\mathbb Z}$, the following are equivalent.
\(1) The matrices $A,B$ are SE-${\mathbb Z}$.
\(2) $\text{cok}(I-tA), \text{cok}(I-tB)$ are isomorphic ${\mathbb Z}[t]$ modules.
Consider now ${\mathbb Z}[t,t^{-1}]$, the ring of Laurent polynomials in one variable. Given the ${\mathbb Z}[t]$-module $G_A$, with $t$ acting by $\hat A^{-1}$, there is a unique way to extend the ${\mathbb Z}[t]$-module action on $G_A$ to a ${\mathbb Z}[t,t^{-1}]$-module action ($t^{-1}$ must act by $\hat A$). A map $\phi: G_A\to G_B$ is a ${\mathbb Z}[t]$-module isomorphism if and only if it is a ${\mathbb Z}[t,t^{-1}]$-module isomorphism.
Consequently, SE-${\mathbb Z}$ can be (and has been) characterized using ${\mathbb Z}[t,t^{-1}]$-modules above in place of the ${\mathbb Z}[t]$-modules.
Other rings for other systems {#subsec:otherrings}
-----------------------------
We’ve looked at SFTs presented by matrices over ${\mathbb Z}_+$, and considered algebraic invariants in terms of these matrices (e.g.SE-${\mathbb Z}$, SSE-${\mathbb Z}$). There are cases [(Ap. ]{}[othercases]{}) of SFTs with additional structure, or SFT-related systems, for which there is very much the same kind of theory, but with ${\mathbb Z}$ replaced by an integral group ring ${\mathbb Z}G$, and ${\mathbb Z}_+$ replaced by ${\mathbb Z}_+G$. We will say a little about one case, to indicate the pattern, and help motivate our interest in SSE over more general rings.
Let $G$ be a finite group, and let ${\mathbb Z}_+G
= \{ \sum_{g\in G} n_g g: n_g \in {\mathbb Z}_+ \}$, the “positive” semiring in ${\mathbb Z}G$. By a $G$-SFT we mean an SFT together with a free, continuous shift-commuting $G$-action. A square matrix over ${\mathbb Z}_+G$ can be used to define an SFT $T_A$ with such a $G$-action. Two $G$-SFTs are isomorphic if there is a topological conjugacy between them intertwining the $G$-actions. Every $G$-SFT is isomorphic to some $G$-SFT $T_A$.
\[correspremark\] We list below some correspondences [(Ap. ]{}[zgsse]{}).
1. \[corresp1\]SSE-${\mathbb Z}_+G$ of matrices is equivalent to conjugacy of their $G$-SFTs.
2. \[corresp2\]If $n$ is a positive integer, then $(T_A)^n$ and $T_{A^n}$ are conjugate $G$-SFTs.
3. \[corresp3\] SE-${\mathbb Z}_+G$ of matrices is equivalent to eventual conjugacy of their $G$-SFTs. [@BoSc2 Prop. B.11].
4. \[corresp4\] If $G$ is abelian, then the polyomial $\det (I-tA)$ encodes the periodic data.
5. \[corresp5\] If $A$ is a square nondegenerate matrix over ${\mathbb Z}_+G$, then the SFT $T_A$ is mixing if and only if $A$ is $G$-primitive [@BoSc2 Prop. B.8].
6. \[corresp6\] $G$-primitive matrices are SE-${\mathbb Z}G$ if and only if they are SE-${\mathbb Z}_+ G$ [@BoSc2 Prop. B.12].
We add comments for some items in Remark \[correspremark\].
The $G$-action for $(T_A)^n$ above is the $G$-action given for $T_A$.
The determinant is defined for commutative rings, and ${\mathbb Z}G$ is commutative iff the group $G$ is abelian. Above, the polynomial $\det (I-tA)$ has coefficients in the ring ${\mathbb Z}G$ . For abelian $G$, by definition two $G$-SFTs have the same “periodic data” if there is a shift-commuting – not necessarily continuous – bijection between their periodic points which respects the $G$-action.
(\[corresp5\], \[corresp6\]) By definition, a $G$-primitive matrix is a square matrix $A$ over ${\mathbb Z}_+G$ such that for some positive integer $k$, every entry of $A^k$ has the form $\sum_{g\in G} n_g g$ with every $n_g$ a positive integer.
We note one feature of the ${\mathbb Z}$ situation which does NOT translate to ${\mathbb Z G}$. Recall, $\text{SE}-{\mathbb Z}\implies \text{SSE}-{\mathbb Z}$. In contrast, for many $G$, the relationship of SE-${\mathbb Z}G$ and SSE-${\mathbb Z}G$ is highly nontrivial, as we will see.
Appendix 2 {#a2}
----------
This subsection contains various remarks, proofs and comments referenced in earlier parts of Section \[sec:sesse\].
\[seeventual\] For square matrices $A,B$ over ${\mathbb Z}_+$, The following are equivalent.
1. $A$ and $B$ are SE-${\mathbb Z}_+$
2. $A^k$ and $B^k$ are SSE-${\mathbb Z}_+$, for all but finitely many $k$.\
(So, the SFTs defined by $A$ and $B$ are eventually conjugate.)
3. $A^k$ and $B^k$ are SE-${\mathbb Z}_+$, for all but finitely many $k$.
\(1) $\implies$ (2) Suppose matrices $R,S$ give a lag $\ell$ SE-${\mathbb Z}_+$ from $A$ to $B$. Because $AR=RB$ and $SA=BS$, we have for $k $ in ${\mathbb Z}_+$ that $$\begin{aligned}
(A^kR)(S)&=A^k(RS)=A^{k+\ell} \\
(S)(A^kR) &=S(RB^k) = (SR)B^k=B^{k+\ell} \ .\end{aligned}$$
$(2) \implies (3)$ This is trivial.
$(3) \implies (1)$ This argument, due to Kim and Roush, is not so trivial; see [@LindMarcus1995]. SE-${\mathbb Z}$ of $A^k$ and $B^k$ does not always imply SE-${\mathbb Z}$ of $A$ and $B$, because there are different choices of $k$th roots of eigenvalues. For example, consider $A=(3)$, $B= (-3)$ and $k=2$. The very rough idea of the Kim-Roush argument is that when $k$ is a prime very large (with respect to every number field generated by the eigenvalues), then the implication does reverse.
\[reducedrings\]
If $A$ and $B$ are SE over a ring $\mathcal R$, then by Theorem \[thm:ssesefibers\] there is a nilpotent matrix $N$ over $\mathcal R$ such that $B$ is SSE over $\mathcal R$ to $\left( \begin{smallmatrix} A & 0 \\ 0 & N \end{smallmatrix} \right)$. For $\mathcal R$ commutative, it follows that $\det(I-tA)$ fails to be an invariant of SE-$\mathcal R$ if and only if there is a nilpotent matrix $N$ over $\mathcal R$ such that $\det(I-tN) \neq 1$. We check next that this requires $\mathcal R$ to contain a nilpotent element.
\[nilpotentelementsprop\] Suppose $N$ is a nilpotent matrix over a commutative ring $\mathcal R$ and $\det (I-tN) \neq 1$. Then $\mathcal R$ contains a nilpotent element.
Let $\det(I-tN) = 1 + \sum_{i=1}^k c_it^i$, with $c_k \neq 0$. Suppose $N$ is $n\times n$, and take $m$ in ${\mathbb N}$ such that $N^m = 0$. Then the polynomial $\det((I-tN)^m)$ has degree at most $n(m-1)$. For any $r$, $\det(I-tN)^m = (\det(I-tN))^m = (1 + c_1t+ \dots + c_kt^k)^r$. This polynomial equals $(c_k)^rt^{kr}$ plus terms of lower degree. So, for $r > n(m-1)$, we must have $(c_k)^r=0$.
By the way, it can happen that matrices $A,B$ shift equivalent over a commutative ring $\mathcal R$ have ${\textnormal{trace}}(A^n)={\textnormal{trace}}(B^n)$ for all $n$ while $\det (I-tA) \neq \det (I-tB)$. For example, let $\mathcal R$ be ${\mathbb Z}\cup \{a\}$, with $a^2=2a=0$. Then set $A= \left( \begin{smallmatrix} 0 & 1 \\ a & 0 \end{smallmatrix} \right)$ and $B=(0)$.
\[nonplussed\] The primitive matrix $A= \left(\begin{smallmatrix} 1&0&0&1 \\ 0&1&0&1 \\ 0&1&1&0 \\ 1&0&1&0
\end{smallmatrix}\right) $ has nonzero spectrum $(2,1)$. Prove that $A$ is not SE-${\mathbb Z}_+$ to a nonsingular matrix.
Such a matrix $B$ would be $2\times 2$ primitive with diagonal entries $1,2$ (a diagonal entry 3 would force $B$ to have spectral radius greater than 2). But, then $B$ has spectral radius at least as large as the spectral radius of $\left(\begin{smallmatrix} 1&1 \\ 1&2
\end{smallmatrix} \right) $, which is greater than 2. (A more informative obstruction, due to Handelman, shows that $A$ is not SE-${\mathbb Z}_+$ to a matrix of size less than 4 [@BH93 Cor. 5.3].)
In the proof above, we used the following corollary of Theorem \[PerronTheorem\]: for nonnegative square matrices $C,B$, with $C\leq B$ and $C\neq B$ and $B$ primitive, the spectral radius of $B$ is stricty greater than that of $C$.
\[1by1\] Suppose $A$ is square over ${\mathbb Z}$ and $\det (I-tA) = 1-nt$, with $n$ a positive integer. Then $A$ is SE over ${\mathbb Z}$ to the $1\times 1$ matrix $(n)$.
(Here we use some basic theory of nonnegative matrices reviewed in Lecture IV.) There is a permutation matrix $P$ such that $P^{-1}AP$ is block triangular with each diagonal block either $(0)$ or an irreducible matrix. Because the nonzero spectrum is a singleton $(n)$, only one of these blocks is not zero, and this block $B$ must be primitive. There is an SSE-${\mathbb Z}_+$ by zero extensions from $A$ to $B$. Now there is an SE-${\mathbb Z}$ from $B$ to $(n)$. Because $B$ is primitive, this implies there is an SE-${\mathbb Z}_+$ from $B$ to $(n)$.
\[sePrimitive\] Suppose two primitive matrices over a subring $\mathcal R$ of the reals are SE over $\mathcal R$. Then they are SE over $\mathcal R_+$.
See [@LindMarcus1995] for a proof. With matrices $R,S$ giving a lag $\ell$ SE over $\mathcal R$ from $A$ to $B$, the basic idea is to use linear algebra and the Perron Theorem (Lecture 4) to show (possibly after replacing $(R,S)$ with $(-R, -S)$) that for large $n$, the matrices $A^nR$ and $SA^n$ will be positive. Then the pair $RA^n, A^nS$ implements an SE over $\mathcal R_+$ with lag $\ell +2n$.
\[bkaplansky\] An example of myself and Kaplansky, recorded in [@BoyleJordanForm1984], shows that two irreducible nonnegative matrices can be SE-${\mathbb Z}$ but not SE-${\mathbb Z}_+$. The example corrects [@ParryWilliams Remark 4, Sec.5] and shows that [@CuntzKriegerDicyclic Lemma 4.1] should be stated for primitive rather than irreducible matrices (the proof is fine for the primitive case).
\[transposeexerciseproof\] The matrix $\begin{pmatrix} 256 & 7\\ 0 &1 \end{pmatrix}$ is not SE-${\mathbb Z}$ to its transpose.
First, suppose $a,b$ are integers such that $a> |b| >0$. Let $M_x$ denote the matrix $\left( \begin{smallmatrix} a & x \\ 0
&b \end{smallmatrix} \right)$. Now suppose $x,y$ are integers such that $xy = 1 \mod (a-b)$. Then the matrices $(M_x)^{\text{tr}}$ and $M_y$ are SIM-${\mathbb Z}$: $$\begin{pmatrix} a&0 \\ x&b \end{pmatrix}
\begin{pmatrix} a-b&y \\ x&(1-xy)/(b-a) \end{pmatrix}
=
\begin{pmatrix} a-b&y \\ x&(1-xy)/(b-a) \end{pmatrix}
\begin{pmatrix} a&y \\ 0&b \end{pmatrix} \ .$$ Thus $M_x$ and $(M_x)^{\text{tr}}$ are SE-${\mathbb Z}$ if and only if $M_x$ and $M_y$ are SE-${\mathbb Z}$. Fix $a=256=2^8, b=1$. Theorem \[thm:axb\] implies that $M_x$ and $M_y$ are SE-${\mathbb Z}$ if and only if there are integers $j,m$ such that $2^mx = \pm 2^jy \mod 255$. Because $2$ is a unit in ${\mathbb Z}/255{\mathbb Z}$, and $xy = 1 \mod 255$, this holds if and only if there is a nonnegative integer $n$ such that $x^2= \pm 2^n \mod 255$. Because 2 and -2 are not squares mod 5, they are not squares mod 255. Because $2^8=1 \mod 255$, the only squares mod 255 in $\{ \pm 2^n: n\geq 0\}$ are 1,4,16 and 64. The square 49 is not on this list. Therefore the matrix $\left( \begin{smallmatrix} 256 & 7 \\ 0
&1 \end{smallmatrix} \right)$ and its transpose are not SE-${\mathbb Z}$.
The following fact from [@BH93] facilitates constructions of primitive matrices realizing the algebraic invariants above: any $2\times 2$ matrix over ${\mathbb Z}$ with integer eigenvalues $a,b$ with $a>|b|$ is SE-${\mathbb Z}$ to a primitive matrix. In our example, $$\begin{pmatrix} 1 & 0\\ 1 &1 \end{pmatrix}
\begin{pmatrix} 256 & 7\\ 0 &1 \end{pmatrix}
\begin{pmatrix} 1 & 0\\ -1 &1 \end{pmatrix}
\ =
\begin{pmatrix} 249 & 7\\ 248 & 8 \end{pmatrix} \ := B\ .$$ It is an easy exercise to show that when matrices $A,B$ are shift equivalent, if one of $A,B$ is shift equivalent to its transpose then so is the other. Consequently, the matrix $B$ displayed above cannot be SE-${\mathbb Z}$ to its transpose.
I haven’t seen the method of Proposition \[transposeexerciseproof\] used to distinguish the SE-${\mathbb Z}$ classes of a primitive matrix and its transpose, but examples of such were produced long ago. The matrix $A=\left( \begin{smallmatrix} 19&5\\4&1 \end{smallmatrix} \right)$ is an early example, due to Köllmer, of a primitive matrix not SIM-${\mathbb Z}$ (hence not SE-${\mathbb Z}$, as $|\det (A)|=1$) to its transpose (for an elementary proof, see [@ParryTuncel1982 Ch.V,Sec.4]). The connection of $\text{SL}(2,{\mathbb Z})$ to continued fractions leads to a computable characterization of SIM-${\mathbb Z}$ for $2\times 2$ unimodular matrices (see [@CuntzKriegerDicyclic Corollary 2.2]). Lind and Marcus use another connection to ${\mathbb Z}[\lambda ]$ ideal classes to give an example of a primitive integral matrix not SE-${\mathbb Z}$ to its transpose, [@LindMarcus1995 Example 12.3.2]. There are much earlier papers which give many cases in which a square integer matrix and its transpose must correspond to inverse ideal classes of an associated ring (see [@tausskyInverseIdeal] and its connections in the literature), and these ideal classes may differ. However, this still leaves the issue of realizing the algebraic invariants in primitive matrices.
Next, we restate Theorem \[thm:taussky\] and sketch the proof coming out of Taussky-Todd’s work [@tausskyOnLatimerTheorem].
\[tausskytoddcase\] Suppose $p$ is monic irreducible in ${\mathbb Z}[t]$, and $p(\lambda)=0$, with $0\neq \lambda \in {\mathbb C}$. Let $\mathcal M$ be the set of matrices over ${\mathbb Z}$ with characteristic polynomial $p$. Then there are bijections $$\begin{aligned}
\mathcal M/(\textnormal{SIM}-{\mathbb Z})
& \to
\textnormal{ Ideal classes of }{\mathbb Z}[\lambda ]\ ,
\quad \quad \textnormal{ and } \\
\mathcal M/(\textnormal{SE}-{\mathbb Z})
& \to
\textnormal{ Ideal classes of }{\mathbb Z}[1/\lambda ] \ .\end{aligned}$$
If $A$ is in $\mathcal M$, then $A$ has a right eigenvector $r_A$ for $\lambda$. The eigenvector can be chosen with entries in the field ${\mathbb Q}[ \lambda ]$ (solve $(\lambda I- A)r=0$ using Gaussian elimination). Then, after multipying $r$ by a suitable element of ${\mathbb Z}$ to clear denominators, we may assume the entries of $r_A$ are in ${\mathbb Z}[\lambda ]$. Let $I(r_A)$ be the ideal of the ring ${\mathbb Z}[\lambda ]$ generated by the entries of $r_A$. Let $\mathcal I (r_A)$ be the ideal class of ${\mathbb Z}[\lambda ]$ which contains $ I (r_A)$. Now it is routine to check that the map $A\mapsto \mathcal I (r_A)$ is well defined and induces the first bijection.
For the second bijection, just repeat this Taussky-Todd argument, with the ring ${\mathbb Z}[1/\lambda ]$ in place of ${\mathbb Z}[\lambda ]$, and say $\mathcal J(r_A)$ denoting the ${\mathbb Z}[1/\lambda ]$ ideal generated by the entries of $r_A$. The rule $\mathcal I(r_A)\mapsto
\mathcal J(r_A)$ induces a surjective map from the set of ideal classes of ${\mathbb Z}[ \lambda ]$ to those of ${\mathbb Z}[ 1/\lambda ]$, which corresponds to the lumping of SIM-${\mathbb Z}$ classes to SE-${\mathbb Z}$ classes.
There is more detail and comment on this in [@BMT1987].
It is important to note above that the ring ${\mathbb Z}[\lambda]$ is not in general equal to $\mathcal O_{\lambda}$, the full ring of algebraic integers in ${\mathbb Q}[\lambda ]$. When ${\mathbb Z}[\lambda]$ is a proper subset of $\mathcal O_{\lambda}$, its class number will strictly exceed that of $\mathcal O_{\lambda}$ (in this case, a principal ${\mathbb Z}[\lambda ]$ ideal cannot be an $\mathcal O_{\lambda}$ ideal).
\[finiteclass\] Suppose $\lambda$ is a nonzero algebraic integer. Then the class number of ${\mathbb Z}[\lambda ]$ is finite.
Let $n$ be the the dimension of ${\mathbb Q}[\lambda ]$ as a rational vector space. As free abelian groups, $\mathcal O_{\lambda}$ and ${\mathbb Z}[\lambda ] $ (and all of their nonzero ideals) have rank $n$. For $R$ equal to ${\mathbb Z}[\lambda ]$ or $\mathcal O_{\lambda}$, the following are equivalent conditions on $R$-ideals $I,I'$.
- $I,I'$ are equivalent as $R$-ideals.
- there is a nonzero $c\in {\mathbb Q}[\lambda] $ such that $cI=I'$.
- $I,I'$ are isomorphic as $R$-modules.
Because the class number of $\mathcal O_{\lambda}$ is finite, there is a finite set $\mathcal J$ of $\mathcal O_{\lambda}$ ideals such that every nonzero $\mathcal O_{\lambda}$ ideal is equivalent to an element of $\mathcal J$. Let $N$ be a positive integer such that $N\mathcal O_{\lambda}
\subset {\mathbb Z}[\lambda ] $.
Now suppose $I$ is a ${\mathbb Z}[\lambda ]$ ideal, with $\{\gamma_1, \dots , \gamma_n\}$ a ${\mathbb Z}$-basis of $I$. Set $J=\{ \sum_{i=1}^n r_i\gamma_i: r_i \in \mathcal O_{\lambda}, 1\leq i \leq n \}$. There is a nonzero $c$ in ${\mathbb Q}[\lambda ]$ such that $cJ \in \mathcal J$. The ${\mathbb Z}[\lambda ]$ modules $I, cI$ are isomorphic. We have $NJ \subset I \subset J$, and therefore $|J/I| \leq n^N$. There are only finitely many abelian subgroups of $J$ with index at most $n^N$ in $J$. It follows that there are only finitely many possibilities for $cI$ as a ${\mathbb Z}[\lambda ]$ module, and this finishes the proof.
Proposition \[finiteclass\] is a (very) special case of the Jordan-Zassenhaus Theorem (see [@reinerMaximalOrders]).
\[directlimits\] We’ll recall the general notion of direct limit of a group endomorphism, and note how in the ${\mathbb Z}$ case that our concrete presentation really is isomorphic to the general vesion. The concrete version has its merits, but the general version is essential.
For a group endomorphism $\phi : G\to G$, take the union of the disjoint sets $(G,n)$, $n \in {\mathbb Z}_+$. Define an equivalence relation on $\cup_{n\in {\mathbb Z}_+}(G,n)$: $(g,m) \sim (h,n)$ if there exist $j,k$ in ${\mathbb Z}_+$ such that $(\phi^j(g),j+m) = \phi^k(h),n+k)$. Define $\varinjlim_{\phi} G$ to be the quotient set $\big( \cup_{n\in {\mathbb Z}_+}(G,n) \big) /\sim $ . The operation on $\varinjlim_{\phi} G$ given by $[(g,m)] + [(h,n)] = [(\phi^n(g) + \phi^m (h),m+n)]$ is well defined and makes $\varinjlim_{\phi} G$ a group. The endomorphism $\phi$ induces a group automorphism $\widehat{\phi} $ given by $\widehat{\phi}: [(g,n)]\mapsto [(\phi(g), n)]$. The inverse of $\widehat{\phi}$ is defined by $ [(g,n)]\mapsto [(g, n+1)]$.
In our case, $A : {\mathbb Z}^n \mapsto {\mathbb Z}^n$ by $x\mapsto xA$, we may define a map $\psi: G_A \to
\varinjlim_{\phi} G$ by $x\mapsto [(xA^m,m)]$ where $m=m(x)$ is sufficiently large that $xA^m \in {\mathbb Z}^n$. One can check that $\psi$ is a well defined group automorphism, with $\psi \circ A = \widehat A \circ \psi$.
\[nonisomorphicpair\] $A= \left(\begin{smallmatrix} 2&0 \\ 0&5
\end{smallmatrix}\right)$ and $B = \left(\begin{smallmatrix} 2&1 \\ 0&5
\end{smallmatrix}\right)$, we will show that $G_A$ is the sum of a 2-divisible group and a 5-divisible group, but $G_B$ is not.
For $M=A \text{ or }B$, and $\lambda =2\text{ or } 5 $, let $H_{M,\lambda } =
\{v \in G_M: \lambda^{-k}v\in G_M,\text{for all } k\in {\mathbb N}\}$. An isomorphism $G_B\to G_A$ must send $H_{B,\lambda }$ to $H_{A,\lambda }$, for $\lambda =2,5$. For $M=A $ or $M=B$, because the eigenvalues $2,5$ are relatively prime, we can check $H_{M,\lambda } = G_M \cap \{ v\in {\mathbb Q}^2: vM= \lambda v \} $. Clearly $G_A = H_{A,2 }\oplus H_{A, 5 }$. In contrast, $G_B \neq H_{B,2 }\oplus H_{B, 5 }$. For example, $(1,0)\in G_B$, and $(1,0)$ is uniquely a sum of vectors on the two eigenlines, $(1,0) = (1/3) (3,-1) +(1/3)(0,1) $. But $(1/3)(0,1) \notin H_{B,5}$. $\qed$
\[seandpairiso\] Let $A,B$ be square matrices over ${\mathbb Z}$. Then the following are equivalent.
1. $A$ and $B$ are SE-${\mathbb Z}$.
2. There is an isomorphism of direct limit pairs $(G_A, \hat A)$ and $(G_B, \hat B)$.
We will give a proof with the general direct limit definition in Remark \[directlimits\], rather than using the more concrete version of the group involving eventual images. The general proof is easier.
\(1) $\implies $ (2) Suppose $R,S$ gives the lag $\ell$ shift equivalence: $A^{\ell}=RS$, etc. First note that the rule $[(x,n)]\to [(xR,n)]$ gives a well defined map $\phi: G_A \to G_B$, because $ [(xAR,n+1)] =[ (xRB, n+1)] = [(xR,n)]$. Check this is a group homomorphism. Similarly, define $\psi :G_B \to G_A$ by $[(y, m)]\mapsto [(yS,m+\ell)]$.
Then $\psi \circ\phi ([(x,n)]) = \psi ([(xR,n)])
=[(xRS,n+\ell)] = [(xA^{\ell}, n+\ell)] = [(x,n)]$. Similarly, $\phi \circ\psi ([(y,n))] = \psi ([(yS,n+\ell)])
=[(ySR,n+\ell)] = [(yB^{\ell}, n+\ell)] = [(y,n)]$. Therefore the homomorphism $\phi$ is an isomorphism.
\(2) $\implies $ (1) Suppose $\phi : G_A \to G_B$ gives the isomorphism of pairs. Check that there must be $N >0$ and a matrix $R$ such that $\phi: [(x,0)]\to [(xR, N)]$. After postcomposing with the automorphism $[(y,N)]\mapsto [(y,0)]$, we may suppose $N=0$. There must be a matrix $S$ and $\ell >0$ such that the inverse map is $[(y,0)] \mapsto [(yS, \ell )]$. Then $AR=RB$ and $SA=BS$ because $\phi$ and its inverse interwine the actions of $\hat A $ and $\hat B$.
(Dimension groups) \[dimensiongroupsremark\] The dimension groups are an important class of ordered groups arising from functional analysis [@Goodearl1986book], with important applications in $C^*$-algebras [@Effros1981] and topological dynamics [@GPS95; @GMPS10]. We consider only countable groups. As a group, a dimension group is a direct limit of the form $$\xymatrix{
{\mathbb Z}^{n_1} \ \ar[r]^{A_1} &
{\mathbb Z}^{n_2} \ \ar[r]^{A_2} &
{\mathbb Z}^{n_3} \ \ar[r]^{A_3} & \ \cdots
}$$ for which nonnegative integral matrices $A_n$ defined the bonding homomorphisms. For $v$ in ${\mathbb Z}^{n_k}$ (we use row vectors), the element $[(v,n_k)]$ of the group is in the positive set if $vA^{n_1}A^{n_2}\cdots A^{n_j} \in {\mathbb Z}_+$ for some (hence for every large) nonnegative integer $j$. Every torsion free countable abelian group is isomorphic as an unordered group to a dimension group. Effros, Handelman and Shen have given an elegant and important abstract characterization of the ordered groups which are isomorphic to dimension groups [@EHS1980].
\[SEZ+andgroupsremark\] For a square matrix $A$ over ${\mathbb Z}_+$, the group $G_A$ above becomes an ordered group, $(G_A, G_A^+)$, by defining the positive set $G_A^+= \{ x\in G_A: \exists k\in {\mathbb N}, xA^k \geq 0 \}$. The ordered group $(G_A, G_A^+)$ is a dimension group (set every bonding map $A_n$ equal to $A$). Now $(G_A, G_A^+,\hat A)$ is an ordered ${\mathbb Z}[t]$ module (the action of $t$ takes $G_A^+$ to $G_A^+$), and is sometimes called a dimension module. (Sometimes the unordered group $G_A$ is referred to as a dimension group. We have tried to avoid this.)
For $A,B$ over ${\mathbb Z}_+$, SE-${\mathbb Z}_+$ of $A,B$ is equivalent to existence of an isomorphism $G_A\to G_B$ which intertwines $\hat A$ and $\hat B$ and sends $G_A^+$ onto $G_B^+$. For more on this, see Lind and Marcus [@LindMarcus1995].
\[othercases\] Parry and Tuncel made the first beyond-${\mathbb Z}$ connection of this sort in [@ParryTuncel1982stoch], as they studied conjugacies of SFTs taking one Markov measure to another. The matrices they considered are not taken explicitly from a group ring, but the connection to an integral group ring of a finitely generated free abelian group emerges in [@MarcusTuncelwps].
\[zgsse\] It was Bill Parry who introduced the presentation of $G$-SFTs by matrices over ${\mathbb Z}_+G$, and the conjugacy/SSE-${\mathbb Z}_+G$ correspondence. Parry never published a proof (although one can see the ideas emerging from the earlier paper with Tuncel, [@ParryTuncel1982stoch]). For an exposition with proofs, see [@BS05] and [@BoSc2 Appendices A,B]. The items (\[corresp2\], \[corresp4\]) the list in Remark \[correspremark\] are not proved explicitly in [@BS05], but they should not be difficult to verify following the exposition of [@BS05]. For further development of relations between the ${\mathbb Z}_+G$ matrices and their $G$-SFTs, see [@bce:gfe Appendix]. The exposition in [@BS05] includes Parry’s connection between SSE-${\mathbb Z}_+G$ and cohomology of functions [@BS05 Theorem 2.7.1], which is the heart of the matter. When $G$ is not abelian, one needs to be careful about left vs. right actions; [@BoSc2 Appendix A] explains this, and corrects a left/right error in the presentation in [@BS05].
Polynomial matrices {#sec:polynomial}
===================
We will define SFTs, and the algebraic and classification structures around them, using polynomial matrices. This is essential for the K-theory connections to come.
Background {#subsec:backgroundstuff}
----------
Before we move on to the polynomial matrices, we review background on flow equivalence and vertex SFTs. Later, this will be context for the polynomial approach.
### Flow equivalence of SFTs
Two homeomorphisms are [*flow equivalent*]{} if there is a homeomorphism between their mapping tori which takes orbits onto orbits preserving the direction of the suspension flow [(Ap. ]{}[floweqbackground]{}). Roughly speaking: two homeomorphisms are flow equivalent if their suspension flows move in the same way, but at different speeds. If SFTs are topologically conjugate, then they are flow equivalent, but the converse is not true.
An $n\times n$ matrix $C$ over ${\mathbb Z}$ defines a map ${\mathbb Z}^n \to {\mathbb Z}^n$, $v\mapsto vC$, with $\text{Image} (C) = \{vC: v\in {\mathbb Z}^n\}$, and cokernel group ${\textnormal{cok}}_{{\mathbb Z}}(C) =
{\mathbb Z}^n / \text{Image} (C)$.
If SFTs are defined by ${\mathbb Z}_+$ matrices $A,B$ are flow equivalent, then
1. $\det (I-A) = \det (I-B)$ .
2. ${\textnormal{cok}}_{{\mathbb Z}}(I-A)$ and ${\textnormal{cok}}_{{\mathbb Z}}(I-B)$ are isomorphic abelian groups.
Above, (1) is due to Bill Parry and Dennis Sullivan [@parrysullivan]; (2) is due to Rufus Bowen and John Franks [@BowenFranks1977]. The group ${\textnormal{cok}}_{{\mathbb Z}}(I-A)$ is called the Bowen-Franks group of the SFT defined by ${\mathbb Z}_+$-matrix $A$. The group ${\textnormal{cok}}_{{\mathbb Z}}(I-A)$ determines $|\det (I-A)|$, except for the sign of $|\det (I-A)|$ in the case $|\det (I-A)|\neq 0$ [(Ap. ]{}[bfgroup]{}).
When $A$ is irreducible and $A$ is not a permutation matrix, the converse of the theorem holds (John Franks, [@franksFlowEq1983]). So, in this case the Bowen-Franks group determines the flow equivalence class, up to knowing the sign of $\det (I-A)$.
### Vertex SFTs
Once upon a time, before edge SFTs, SFTs were presented only by matrices with entries in $\{0,1\}$. Such a matrix can be viewed as the adjacency matrix of a graph without parallel edges (i.e., for each vertex pair $(i,j)$, there is at most one edge from $i$ to $j$). We can then define a “vertex SFT” as we defined edge SFT, but using bisequences of vertices rather than bisequences of edges to describe infinite walks through the graph.
A vertex SFT is quite natural, especially if one starts from subshifts. A subshift $(X, \sigma)$ is a “topological Markov shift” if whenever points $x,y$ satisfy $x_0=y_0$, the bisequence $z= \dots x_{-3}x_{-2}x_{-1}x_0y_2y_2y_3 \dots $ is also a point in $X$. (That is, the past of $x$ and the future of $y$ can be glued together at their common present to form a point. This is a topological analogue of the independence property of a Markov measure.) One can check that a topological Markov shift is the same object as a vertex SFT, with the alphabet of the subshift being the vertex set [(Ap. ]{}[topmarkovshift]{}).
Defining SFTs (as edge SFTs) with matrices over ${\mathbb Z}_+$ has some significant advantages over defining SFTs (as vertex SFTs) with matrices over $\{0,1\}$, as follows.
- [*Functoriality.*]{} Recall, $(X_A, (\sigma_A)^n)$ is conjugate to the edge SFT defined by $A^n$, whereas $A^n$ cannot define a vertex SFT if $A^n$ has an entry greater than 1.
- [*Conciseness.* ]{} E.g., an edge SFT defined by the perfectly transparent $2\times 2 $ matrix $ A= \begin{pmatrix} 1 & 4 \\ 4& 15 \end{pmatrix}$ has a (rather large) alphabet of 24 symbols; as a vertex SFT, it would be defined by a $24 \times 24$ zero-one matrix. And while $A^n$ is $2\times 2$ for all $n$, the size of the matrix presenting the vertex SFT $(X_A, (\sigma_A)^n)$ goes to infinity as $n\to \infty$.
- [*Proof techniques.*]{} Defining the SFTs directly with matrices over ${\mathbb Z}_+$ allows other proof techniques [(Ap. ]{}[prooftechniquesforedgesft]{}).
We’ll see that some advantages of defining SFTs with ${\mathbb Z}_+$ rather than $\{0,1\}$ matrices are repeated, as we compare defining SFTs with polynomial rather than ${\mathbb Z}_+$ matrices.
Presenting SFTs with polynomial matrices
----------------------------------------
The [*length*]{} of a path $e_1 \dots e_n$ of $n$ edges in a graph is $n$. (We also think of $n$ as the time taken at unit speed to traverse the path.) An $n\times n$ matrix $A$ with polynomial entries in $t{\mathbb Z}_+[t]$ presents a graph $\Gamma_A$ as follows.
- $\{ 1, \dots , n\}$ is a subset of the vertex set of $\Gamma_A$.
- For each monomial entry $t^k$ of $A(i,j)$, there is a distinct path of $k$ edges from vertex $i$ to vertex $j$. We call such a path an [*elementary path*]{} in $\Gamma_A$. (E.g. if $A(i,j) = 2t^3$, then from $i$ to $j$ there are two elementary paths of length 3.)
- There are no other edges, and distinct elementary paths do not intersect at intermediate vertices.
Above, the vertex set $\{ 1, \dots , n\}$ is a [*rome*]{} [(Ap. ]{}[rome]{}) for the graph $\Gamma_A$: every sufficiently long path hits the rome. (All roads lead to Rome …)
\[Asharpgraphexample\] Below, the rome vertex set is $\{1,2\}$; the additional vertices are unnamed black dots; and there are five elementary paths in $\Gamma_A$. $$A= \begin{pmatrix} 2t & t^2 + t^3 \\ t^2&0 \end{pmatrix}\ ,
\quad \quad
\Gamma_A \ = \quad \xymatrix{
& \bullet \ar@/^1pc/[rr]
& \bullet \ar@/^/[rrd]
& \bullet \ar@/^/[rd]
& \\
*+[F-:<2pt>]{1} \ar@(ul,u) \ar@(ul,l) \ar@/^/[ru]\ar@/^/[rru]
& & \bullet \ar[ll]
& & *+[F-:<2pt>]{2} \ar[ll]
}$$
Given $A$ over $t{\mathbb Z}_+[t]$, let $A^{\sharp}$ be the adjacency matrix for the graph $\Gamma_A$. In Example \[Asharpgraphexample\], $A^{\sharp}$ would be $6\times 6$. (The vertex set of the graph is the rome, together with $k-1$ additional vertices for each monomial $t^k$.) We can think of $A$ as being a way to present the edge SFT defined by the matrix $A^{\sharp}$.
[*Conciseness.*]{} Obviously, we can present many SFTs (and, various interesting families of SFTs) much more concisely with polynomial matrices than with matrices over ${\mathbb Z}_+$. For example, a theorem of D. Perrin shows that any number which can be the entropy of an SFT is the entropy of an SFT defined by a $2\times 2$ matrix over $t{\mathbb Z}_+[t]$. [(Ap. ]{}[perrin]{})
An elementary matrix is a square matrix equal to the identity except in at most a single offdiagonal entry.
The polynomial presentation offers more than conciseness. To see this, we need a little preparation. $I_k$ denotes the $k\times k$ identity matrix.
Suppose $\mathcal R$ is a ring. [*Stabilized elementary equivalence*]{} is the equivalence relation $\sim$ on square matrices $C$ over $\mathcal R$ generated by the following two relations.
1. $C \sim C\oplus I_k$ , for $k\in {\mathbb N}.\ \ $ (E.g., $(2)\sim \begin{pmatrix} 2&0\\0&1
\end{pmatrix}$ . )
2. $C \sim D$ if there is an elementary matrix $E$ such that $ D= CE$ or $D=EC$.
Above, condition (1) is the “stabilized” part. A stabilized elementary equivalence from $C$ to $D$ is a finite sequence of the elementary matrix moves, taking $C$ to $D$.
Given $C\sim D$, for either type of relation, we have
1. $\det C = \det D$, if $\mathcal R$ is commutative, and
2. the $\mathcal R$-modules ${\textnormal{cok}}(C) $, ${\textnormal{cok}}(D)$ are isomorphic [(Ap. ]{}[fromequivtocok]{}).
When working in a stable setting, we often say just “elementary equivalence” instead of “stabilized elementary equivalence”.
Algebraic invariants in the polynomial setting
----------------------------------------------
If all nonzero entries of $A$ have degree one, then the relation of $A$ and $A^{\sharp} $ is obvious: for example, $$A=\begin{pmatrix} t& 2t \\ t & 0 \end{pmatrix} \
= \ t \begin{pmatrix} 1& 2 \\ 1 & 0 \end{pmatrix} \
= \ t A^{\sharp}
, \qquad \quad \quad
\Gamma_A \quad =
\quad
\quad
\xymatrix{\cdot \ar@(lu,ld)
\ar@/^/[r]
\ar@/^2pc/[r]
&\cdot \ar@/^/[l]
}$$ Here, $I-A$ equals $I-tA^{\sharp}$. It follows, of course, that $$\begin{aligned}
\det (I-A) &= \det (I-tA^{\sharp}) \ , \quad \text{and} \\
{\textnormal{cok}}_{{\mathbb Z}[t]} (I-A) &\cong {\textnormal{cok}}_{{\mathbb Z}[t]} (I-tA^{\sharp}) \ .\end{aligned}$$
These two statements hold for general $A$ over $t{\mathbb Z}_+[t]$, for the following reason.
There is a stabilized elementary equivalence over the ring ${\mathbb Z}[t]$ from $I-A$ to $I-tA^{\sharp}$.
Next we’ll see the essential ideas of the proof of the proposition. Given $n\times n$ $A$ over $t{\mathbb Z}_+[t]$, let $\mathcal H_A$ be the $n\times n$ labeled graph in which a monomial $t^k$ of $A(i,j)$ gives rise to an edge from $i$ to $k$ labeled $t^k$.
$$A\ =\ \begin{pmatrix} 2t & t + t^4 \\ t^2&0 \end{pmatrix}\ ,
\qquad \qquad
\mathcal H_A\ =\
\xymatrix{
*+[F-:<2pt>]{1}
\ar@(u,ul)_t \ar@(d,dl)^t
\ar@/^/[rr]^{t}
\ar@/^2pc/[rr]^{t^4} & &
*+[F-:<2pt>]{2} \ar@/^/[ll]^{t^2 }
}$$ Note: the graph $\Gamma_A$ with adjacency matrix $A^{\sharp}$ is obtained from $\mathcal H_A$ by replacing each path labeled $t^k$ with a path of length $k$. The graph $\mathcal H_{tA^{\sharp}}$ is the graph $\Gamma_A$ with each edge labeled by $t$.
We can decompose the graph move $\mathcal H_A \to \mathcal H_{tA^{\sharp}}$ into steps, $\mathcal H_0 \to \mathcal H_1 \to \cdots \to \mathcal H_4$, with one vertex added at each step. The labeled graph $\mathcal H_{i+1}$ is obtained from $\mathcal H_{i}$ by replacing some edge labeled $t^k$ with a path of two edges: an edge labeled $t$ followed by an edge labeled $t^{k-1}$. There will be matrices $A_i$ over $t{\mathbb Z}_+[t]$ such that $\mathcal H_i = \mathcal H_{A_i}$, with $A_0=A$ and $A_4=tA^{\sharp}$. Here is the data for the step $\mathcal H_0 \to\mathcal H_1$: $$A=A_0\ =\ \begin{pmatrix} 2t & t + t^4 \\ t^2&0 \end{pmatrix}\ ,
\qquad \qquad
\mathcal H_A\ =\ \mathcal H_0 \ = \
\xymatrix{
*+[F-:<2pt>]{1}
\ar@(u,ul)_t \ar@(d,dl)^t
\ar@/^/[rr]^{t}
\ar@/^2pc/[rr]^{t^4} & &
*+[F-:<2pt>]{2} \ar@/^/[ll]^{t^2 }
}$$ $$B= A_1\ =\ \begin{pmatrix} 2t & t & t \\ t^2&0&0 \\ 0 &t^3 &0
\end{pmatrix}\ ,
\qquad \qquad
\mathcal H_{A_1}\ =\
\mathcal H_{1}\ =\
\xymatrix{
& *+[F-:<2pt>]{3} \ar@/^/[dr]^{t^3} & \\
*+[F-:<2pt>]{1}
\ar@(u,ul)_t \ar@(d,dl)^t
\ar@/^/[rr]^{t}
\ar@/^/[ru]^{t} & &
*+[F-:<2pt>]{2} \ar@/^/[ll]^{t^2 }
}$$
Let us see how the move $A \to A_1$ in the example above is accomplished at the matrix level, by a stabilized elementary equivalence over the ring ${\mathbb Z}[t]$.
First, define the matrix $A\oplus 0 = \begin{pmatrix} 2t & t + t^4 & 0\\
t^2&0& 0\\ 0&0&0 \end{pmatrix}$. The move $A \to A\oplus 0$ is the same as the elementary stabilization move $(I-A) \to (I-A)\oplus 1$. Then multiply $(I-A)\oplus 1$ by elementary matrices to get $(I-A_1)$. This is a small computation: $$\begin{aligned}
(I-B)&=
\begin{pmatrix} 1 & 0 & 0 \\ 0&1&0 \\ 0 & 0 & 1
\end{pmatrix}
- \begin{pmatrix} 2t & t & t \\ t^2&0&0 \\ 0 &t^3 & 0
\end{pmatrix} \\
(I-B)E_1 &=
\begin{pmatrix} 1-2t & -t & -t \\ -t^2&1&0 \\ 0 &-t^3 & 1
\end{pmatrix}
\begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0& t^3 & 1 \end{pmatrix} \\
& = \begin{pmatrix} 1-2t & -t -t^4 & -t \\ -t^2&1&0 \\ 0 &0 & 1
\end{pmatrix}
= I - \begin{pmatrix} 2t & t +t^4 & t \\ t^2&0&0 \\ 0 &0 & 0
\end{pmatrix}
:= I-C
\\
E_2 (I-C) &=
\begin{pmatrix} 1&0&t \\ 0&1&0 \\ 0& 0 & 1 \end{pmatrix}
\begin{pmatrix} 1-2t & -t -t^4 & -t \\ -t^2&1&0 \\ 0 &0 & 1
\end{pmatrix} \\
& =
\begin{pmatrix} 1-2t & -t -t^4 & 0 \\ -t^2&1&0 \\ 0 &0 & 1
\end{pmatrix}
\\
&=
\begin{pmatrix} 1 & 0 & 0 \\ 0&1&0 \\ 0 & 0 & 1 \end{pmatrix}
-
\begin{pmatrix} 2t & t +t^4 & 0 \\ t^2&0&0 \\ 0 &0 & 0
\end{pmatrix}
= (I-A) \oplus 1 \ .\end{aligned}$$
The example computation above contains the ideas of the general proof that there is a stabilized elementary equivalence from $(I-A)$ to $(I-tA^{\sharp})$.
Let $A$ be a square matrix over $t{\mathbb Z}_+[t]$, with $A^{\sharp}$ the adjacency matrix of $\Gamma_A$. Then
1. $\det (I-A) = \det (I-tA^{\sharp})$.
2. The ${\mathbb Z}[t]$-modules ${\textnormal{cok}}(I-A) \ ,\ {\textnormal{cok}}(I-tA^{\sharp} )$ are isomorphic.
The claim follows because the matrices $(I-A)$, $(I-tA^{\sharp})$ are related by a string of the two relations $\sim$ generating stabilized elementary equivalence.
Thus algebraic data of the polynomial matrix $(I-A)$ captures
1. the nonzero spectrum (by $\det (I-A)$), and
2. the SE-${\mathbb Z}$ class of $A^{\sharp}$ (by the isomorphism class of the ${\mathbb Z}[t]$-module ${\textnormal{cok}}(I-A)$).
Polynomial matrices: from elementary equivalence to conjugate SFTs
------------------------------------------------------------------
For square matrices $A,B$ over ${\mathbb Z}_+$, the SFTs $(X_A, \sigma), (X_B, \sigma)$ are topologically conjugate if and only if $A,B$ are SSE-${\mathbb Z}_+$. We will find a relation on polynomial matrices corresponding to topological conjugacy of the SFTs they define.
With $i\neq j$, let $E_{ij}(x)$ denote the elementary matrix with $(i,j)$ entry defined to be $x$, and other entries matching the identity. The size of the square matrix $E_{ij}(x)$ is suppressed from the notation (but evident in context). E.g., $E_{12}(t^2)$ could denote $ \begin{pmatrix} 1&t^2\\0&1 \end{pmatrix}$ or $ \begin{pmatrix} 1&t^2&0\\0&1&0 \\ 0&0&1 \end{pmatrix}$.
There is now a very pleasant surprise.
\[thm:ElEqtoConj\] Suppose $A,B$ are square matrices over $t{\mathbb Z}_+[t]$, with $E(I-A) = (I-B)$ or $(I-A)E=(I-B)$, where $E=E_{ij}(t^k)$.
Then $A,B$ define topologically conjugate SFTs (i.e., $B^{\sharp}$ and $A^{\sharp}$ define topologically conjugate edge SFTs).
Suppose $E=E_{ij}(t^k)$, $A$ is square with entries in $t{\mathbb Z}_+[t]$, and $(I-B)=E(I-A)$ or $(I-B)=(I-A)E$. Then one easily checks (it will be obvious from the next example) that the following are equivalent:
1. The entries of $B$ are in $t{\mathbb Z}_+[t]$.
2. $A(i,j) -t^k \in t{\mathbb Z}_+[t]$.
The ideas of the proof of Theorem \[thm:ElEqtoConj\] should be clear from the next example.
Suppose $A$ is matrix over $t{\mathbb Z}_+[t]$, $A=\begin{pmatrix} a&b+t^3 &c \\ d&e&f \\ g&h&i \end{pmatrix} $, with $b\in t{\mathbb Z}_+[t]$ (i.e., not only $A(1,2)$, but also $A(1,2) -t^3$, is in $t{\mathbb Z}_+[t]$). Now multiply $I-A$ from the left by the elementary matrix $E=E_{12}(t^3)$, $$\begin{aligned}
E(I-A) & =
\begin{pmatrix} 1&t^3&0 \\ 0&1&0\\ 0&0&1 \end{pmatrix}
\begin{pmatrix} 1-a&-b-t^3 &-c \\ -d&1-e&-f \\
-g&-h&1-i \end{pmatrix} \\
&= \begin{pmatrix} 1-a \mathbf{+ t^3(-d)}
&-b -t^3 \mathbf{+t^3(1-e)}
&-c \mathbf{+t^3(-f)} \\ -d&1-e&-f \\
-g&-h&1-i \end{pmatrix}
\\
&=
\begin{pmatrix} 1&0&0 \\ 0&1&0\\ 0&0&1 \end{pmatrix} -
\begin{pmatrix} a +t^3d & b +t^3e &c +t^3f \\ d&e&f \\
g&h&i \end{pmatrix} \ .\end{aligned}$$
We then define a matrix $B$ over $t{\mathbb Z}_+[t]$ by setting $I-B=E(I-A)$, so, $$A=\begin{pmatrix} a&b+t^3 &c \\ d&e&f \\ g&h&i \end{pmatrix} , \quad
B=\begin{pmatrix} a +t^3d &b+t^3e &c +t^3f\\ d&e&f \\ g&h&i \end{pmatrix}\ .$$
[*Producing $\Gamma_A$ from $\Gamma_B$.*]{} Suppose $\tau=\tau_1\tau_2\tau_3$ is the elementary path in $\Gamma_A$ from vertex 1 to vertex 2 corresponding to the term $t^3$ above. Let $|p|$ denote the length (number of edges) in a graph path $p$. We obtain $\Gamma_B$ from $\Gamma_A$ as follows.
1. Remove the elementary path $\tau$ from $\Gamma_A$;
2. For each elementary path $\nu$ of $\Gamma_A$ beginning at vertex 2, put in an elementary path $\widetilde{\nu}$ beginning at vertex 1, such that
1. $|\widetilde{\nu}| = |\tau| + |\nu| = 3+ |\nu| $,
2. the terminal vertices of $\nu$ and $\widetilde{\nu}$ agree.
For example, $$\xymatrix{
& *+[F-:<2pt>]{2} \ar@/^/[dr]^{\nu} & \\
*+[F-:<2pt>]{1}
\ar@/^/[ru]^{\tau}
& &
*+[F-:<2pt>]{j}
}
\qquad \qquad
\begin{matrix} \\ \\ \\ \text{produces} \end{matrix}
\qquad \qquad
\xymatrix{
& *+[F-:<2pt>]{2} \ar@/^/[dr]^{\nu} & \\
*+[F-:<2pt>]{1}
\ar@/^/[rr]^{\widetilde{\nu}}
& &
*+[F-:<2pt>]{j}
}$$ with $|\widetilde{\nu}| = |\tau| + |\nu|$ .
[*Defining the conjugacy $\phi: X_{A^{\sharp}} \to X_{B^{\sharp}}$.*]{} Wherever the elementary path $\tau$ occurs in a point $x$ of $X_A$, it must be followed by an elementary path $\nu$. Now define $ \phi (x)$ be replacing each path $\tau \nu$ with the elementary path $\widetilde{\nu}$:
- If $x_{k+1}\dots x_{k+|\tau\nu|} = \tau \nu$ , $\ \ \ \text{with } \nu$ an elementary path in $\Gamma_A$,\
then $(\phi x)_{k+1}\dots (\phi x)_{k+|\tau\nu|} = \widetilde{\nu}$.
- Otherwise, $(\phi x)_n=x_n$ .
If we look at a succession of elementary paths $\tau$ and $\nu_i$, the code looks like: $$\xymatrix{
\dots \ \nu_{-1} \ \
\tau \nu_1 \ \nu_2 \ \tau \nu_3 \ \nu_4\ \tau \nu_5 \
\ \nu_6 \ \nu_7 \ \dots
\ar[d]_{\phi } \\
\dots \ \nu_{-1} \ \ \
\widetilde{\nu_1} \ \ \nu_2
\ \ \widetilde{\nu_3} \ \ \nu_4\ \
\widetilde{\nu_5}\ \ \ \nu_6 \ \nu_7 \ \dots
}$$ This map $\phi$ is well defined because an elementary path $\nu$ following $\tau$ has no edge in common with $\tau$ (because the initial and terminal vertices of $\tau$ are different) [(Ap. ]{}[tautau]{}). Given that $\phi$ is well defined, it is straightforward to check that $\phi$ defines a topological conjugacy $(X_{A^{\sharp}}, \sigma)\to (X_{B^{\sharp}}, \sigma) $.
Above, we considered $E(I-A) = (I-B)$. Suppose instead we define a matrix $C$ by $(I-A)E = (I-C)$. No surprise: the matrix $C$ also defines an SFT conjugate to that defined by $A$. In this case, instead of a conjugacy $X_A\to X_B$ based on $\tau \nu\mapsto \widetilde{\nu}$ as above, we have a conjugacy $X_A\to X_C$ based on $\nu \tau \mapsto \widetilde{\nu}$, where $\nu$ is an elementary path in $\Gamma_A$ with [*terminal*]{} vertex 1.
If $C$ is a square matrix, then $C\oplus 1$ is the square matrix with block form $\begin{pmatrix} C&0 \\ 0&1 \end{pmatrix}$.
(Positive equivalence) [(Ap. ]{}[poseqterminology]{}) Suppose $\mathcal P$ is a subset of a ring $\mathcal R$. Let $\mathcal S$ be a set of square matrices over a ring $\mathcal R$ which is “1-stabilized” : $$C\in \mathcal S \implies (C \oplus 1) \in \mathcal S\ .$$ Positive equivalence of matrices in $\mathcal S$ (with respect to $\mathcal P$) is the equivalence relation on $\mathcal S$ generated by the following relations (where $C,D$ must [*both*]{} be in $\mathcal S$):
1. $C \sim C\oplus 1$ .
2. $EC=D$ or $ CE=D$ , where $E=E_{ij}(r)$, with $i\neq j$ and $r\in \mathcal P$.
If $\mathcal P$ is not specified, then by default we assume $\mathcal P= \mathcal R$.
For $I-A$ in $\mathcal S$, the requirement that $\mathcal S$ is closed under the move $(I-A) \to (I-A)\oplus 1 $ is equivalent to the requirement that the set $\{A \colon I-A \in \mathcal{S}\}$ is closed under the move $A\to A\oplus 0$.
Now suppose $\mathcal R = {\mathbb Z}[t]$; $\mathcal S$ is the set of square matrices of the form $I-A$, with $A$ over $t{\mathbb Z}_+[t]$; and $\mathcal P = \{t^k: k \in {\mathbb N}\}$. Clearly $A$ and $A \oplus 1$ define conjugate SFTs. It then follows from Theorem \[thm:ElEqtoConj\] that positive equivalent matrices in $\mathcal S$ define topologically conjugate SFTs. Moreover, matrices in $\mathcal S$ are positive equivalent with respect to $\mathcal P =\{t^k: k \in {\mathbb N}\}$ if and only if they are positive equivalent with respect to $\mathcal P = \mathcal R$. To summarize, this gives the following, where we define $$I- \mathcal M (t{\mathbb Z}_+[t]) = \{ I-A : A\textnormal{ is a square
matrix over } t{\mathbb Z}_+[t] \} \ .$$
\[tMatricesAndTopConjug\] Suppose matrices $(I-A)$ and $(I-B)$ are positive equivalent in $I -\mathcal M(t{\mathbb Z}_+[t])$. Then $A,B$ define topologically conjugate SFTs.
The converse of Theorem \[tMatricesAndTopConjug\] is “true up to a technicality” [(Ap. ]{}[exampleforneednzc]{}). For a true converse, we expand the collection of matrices allowed to present SFTs, from $\mathcal M(t{\mathbb Z}_+[t])$ to a slightly larger class, NZC. (On first exposure, it is fine to pretend $\text{NZC}=\mathcal M(t{\mathbb Z}_+[t])$. But we’ll give statements for NZC, just to tell the truth.)
Classification of SFTs by positive equivalence in I-NZC
-------------------------------------------------------
For a matrix $M$ over ${\mathbb Z}[t]$, let $M_0$ be $M$ evaluated at $t=0$.
Let NZC be the set of square matrices $A$ over ${\mathbb Z}_+[t]$ such that $A_0$ is nilpotent.
$A$ and $B$ are in NZC; $C$ and $D$ are not: $$\begin{aligned}
{4}
A& = \begin{pmatrix} t^3 + t & 3t^5 \\ t & 3t^5 \end{pmatrix}\ , \quad
& B& = \begin{pmatrix} t^3 & 1 \\ t & 3t^5 \end{pmatrix}\ , \quad
& C& = \begin{pmatrix} 1 \end{pmatrix} \ , \quad
& D & = \begin{pmatrix} t^3 & 5t^2 +2 \\ 1+t^7 & 3t^5 \end{pmatrix} \ , \\
A_0 & = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\ , \quad
& B_0 & = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\ , \quad
& C_0 & = \begin{pmatrix} 1 \end{pmatrix} \ , \quad
& D_0 & = \begin{pmatrix} 0 & 2 \\ 1 & 0 \end{pmatrix} \ .\end{aligned}$$
If $A$ is in $\mathcal M(t{\mathbb Z}_+[t])$, then $A_0=0$. A matrix in NZC can have some entries with nonzero constant term, but not too many.\
\
Why the term NZC ? Here is the heuristic.\
\
If e.g. $A(i,j) =t^4$, then in the graph with adjacency matrix $A^{\sharp}$, there is an elementary path, from $i$ to $j$, of 4 edges. We consider this a path taking 4 units of time to traverse. The time to traverse concatenations of elementary paths is the sum of the times for its elementary paths. A nonzero term $1$ in $A_0$ is considered as $1=t^0$, giving a path taking zero time to traverse. “NZC” then refers to “No Zero Cycles”, where a zero cycle is a cycle taking zero time to traverse.\
\
In the case NZC, one can make good sense of this heuristic, and everything works [(Ap. ]{}[evenmore]{}). But for a matrix $A$ over ${\mathbb Z}_+[t]$ with zero cycles, we can’t make sense of how $A$ defines an SFT (let alone how multiplication by elementary matrices might induce topological conjugacies).
We do get a classification statement parallelling the SSE-${\mathbb Z}_+$ setup of Williams.
For matrices $A,B$ in $ \text{NZC}$, The following are equivalent.
1. $(I-A)$ and $(I-B)$ are positive equivalent in $I-\text{NZC}$.
2. $A,B$ define topologically conjugate SFTs.\
$(1)\implies (2)$ We have seen this for positive equivalence in $I-\mathcal M(t{\mathbb Z}_+[t])$. This works similarly for matrices in $I-\text{NZC}$ [@BW04; @B02posk].
$(1)\implies (2)$ E.g., $(I-A)$ is positive equivalent in I-NZC to the matrix $I-tA^{\sharp}$, likewise $(I-B)$. So it suffices to get the positive equivalence for matrices $(I-tA^{\sharp})$, $(I-tB^{\sharp})$, assuming the edge SFTS for $A^{\sharp}, B^{\sharp}$ are conjugate, i.e. the ${\mathbb Z}_+$ matrices $A^{\sharp}, B^{\sharp}$ are SSE over ${\mathbb Z}_+$. It suffices to show the positive equivalence given an elementary SSE, $A^{\sharp}=RS, B^{\sharp}=SR$. For this, define matrices $A_0, A_1, \dots ,A_4$ in NZC : $$\begin{pmatrix} tRS & 0 \\ 0 & 0 \end{pmatrix}
\ , \
\begin{pmatrix} tRS & 0 \\ tS & 0 \end{pmatrix}
\ , \
\begin{pmatrix} 0 & R \\ tS & 0 \end{pmatrix}
\ , \
\begin{pmatrix} 0 & 0 \\ tS & tSR \end{pmatrix}
\ , \
\begin{pmatrix} 0 & 0 \\ tS & tSR \end{pmatrix} \ .$$ (Notice, $A_2$ is in NZC, but is not in $\mathcal M(t{\mathbb Z}_+[t])$.) The following Polynomial Strong Shift Equivalence Equations (PSSE Equations), taken from [@BW04], give a positive equivalence in $I-NZC$ between $(I-tA_i)$ and $(I-tA_{i+1})$, for $0\leq i < 4$. $$\begin{aligned}
{2}
\begin{pmatrix} I-tRS & 0 \\ -tS & I \end{pmatrix}
\begin{pmatrix} I & 0 \\ tS & I \end{pmatrix}
& =
\begin{pmatrix} I-tRS & 0 \\ 0 & I \end{pmatrix}
\quad &\text{ is }\quad
(I-A_1)E_1&=(I-A_0) \ ,
\\
\begin{pmatrix} I & R \\ 0 & I \end{pmatrix}
\begin{pmatrix} I & -R \\ -tS & I \end{pmatrix}
&=
\begin{pmatrix} I-tRS & 0 \\ -tS & I \end{pmatrix}
\quad &\text{ is }\quad
E_2(I-A_2)&=(I-A_1) \ ,
\\
\begin{pmatrix} I & -R \\ -tS & I \end{pmatrix}
\begin{pmatrix} I & R \\ 0 & I \end{pmatrix}
&=
\begin{pmatrix} I & 0 \\ -tS & I -tSR \end{pmatrix}
\quad &\text{ is }\quad
(I-A_2)E_3 &=(I-A_3) \ ,
\\
\begin{pmatrix} I & 0 \\ tS & I \end{pmatrix}
\begin{pmatrix} I & 0 \\ -tS & I -tSR \end{pmatrix}
&=
\begin{pmatrix} I & 0 \\ 0 & I -tSR \end{pmatrix}
\quad &\text{ is }\quad
E_4(I-A_3) &=(I-A_4) \ .\end{aligned}$$ One can check that each of the four equivalences given by the PSSE equations is a composition of basic positive equivalences in NZC. That finishes the proof.
[*Tools for construction.*]{} One way to construct a conjugacy between SFTs defined by matrices $A,B$ over ${\mathbb Z}_+$ is to find an SSE over ${\mathbb Z}_+$ from $A$ to $B$. The polynomial matrix setting gives another way: find a chain of elementary positive equivalences from $I-tA$ to $I-tB$. This is not a strict advantage; it’s an alternative tool. There are results for which the only known proof uses this tool [(Ap. ]{}[polytool]{}).
Functoriality: flow equivalence in the polynomial setting
---------------------------------------------------------
We will consider one satisfying feature of presenting SFTs by matrices in NZC (or, just in $\mathcal M(t{\mathbb Z}_+[t])$) [(Ap. ]{}[otherpolyfeatures]{}). With the ${\mathbb Z}_+$ matrix presentation, the algebraic invariant for conjugacy, SE-${\mathbb Z}$, does not have an obvious natural relationship to algebraic invariants for flow equivalence (e.g. Bowen-Franks group, $\det(I-A)$). In the polynomial setting, we do see that natural relationship.
Let $\mathcal M$ be the set of matrices $I-A$ with $A$ in NZC. Say matrices $(I-A), (I-B)$ are related by changing positive powers, $(I-A) \sim_+ (I-B)$, if they become equal after changing positive powers of $t$ to other positive powers. For example, $$\begin{pmatrix} 1-t^2-t^5 & -t - t^3 \\ -t^2&1 \end{pmatrix}
\sim_+
\begin{pmatrix} 1-t^2-t^3 & -t^4 -t^5 \\ -t^7&1 \end{pmatrix}
\sim_+
\begin{pmatrix} 1-2t & -2t \\ -t&1 \end{pmatrix} \ .$$ The next result is one version for SFTs of the Parry-Sullivan characterization of flow equivalence of subshifts.
Suppose $A,B$ are matrices in NZC. The following are equivalent.
1. $A,B$ define flow equivalent SFTs.
2. $(I-A),(I-B)$ are equivalent, under the equivalence relation generated by (i) positive equivalence in $I$-NZC and (ii) $\sim_+$ .
We won’t give a proof for this theorem [(Ap. ]{}[parrysullianetc]{}). But, it is intuitive: flow equivalence arises from conjugacy and time changes, and the time changes are addressed by the $\sim_+$ relation.
Given a matrix $A=A(t)$ in $\mathcal M(t{\mathbb Z}_+[t])$, or in NZC, let $A(1)$ be the matrix defined entrywise by the (augmentation) homomorphism $ {\mathbb Z}[t] \to {\mathbb Z}$ which sends $t$ to 1. For example, $$A= A(t)=\begin{pmatrix} 3t \end{pmatrix} \ , \quad
B= B(t)=\begin{pmatrix} t^2+2t^3 \end{pmatrix} \ , \quad
A(1) = \begin{pmatrix} 3 \end{pmatrix} = B(1) \ .$$ From the Theorem, one can check for SFTs defined by $A,B$ from $\text{NZC}$:
1. Flow equivalent SFTs defined by $A(t),B(t)$ from NZC produce isomorphic groups ${\textnormal{cok}}_{{\mathbb Z}} (I-A(1)), {\textnormal{cok}}_{{\mathbb Z}} (I-B(1))$.
2. ${\textnormal{cok}}_{{\mathbb Z}} (I-A(1))$ is the [*Bowen-Franks group*]{} of the SFT defined by $A$. [(Ap. ]{}[bfgroup]{})
We sometimes use notation ${\textnormal{cok}}_{\mathcal R}$ to emphasize that a cokernel is an $\mathcal R$-module. (A ${\mathbb Z}$-module is just an abelian group.)
Recall, for $A=A(t)$ in NZC, the isomorphism class of ${\textnormal{cok}}_{{\mathbb Z}[t]} (I-A(t))$ is the SE-${\mathbb Z}$ class of the SFT. There is a functor, induced by $t\mapsto 1$: $$\begin{aligned}
{\mathbb Z}[t]\text{-modules } &\to \ {\mathbb Z}[1]\text{-modules} ={\mathbb Z}\text{-modules} =
\text{abelian groups} \\
{\textnormal{cok}}_{{\mathbb Z}[t]} (I-A(t)) &\mapsto {\textnormal{cok}}_{{\mathbb Z}} (I-A(1))\ .\end{aligned}$$ So, this functor gives a presentation of $$\begin{aligned}
\text{SE}-{\mathbb Z}\text{ class } & \to \text{ Bowen-Franks group}\ .
\qquad \qquad \qquad \qquad \\end{aligned}$$ This shows us how algebraic invariants of flow equivalence and topological conjugacy are naturally related in the polynomial setting.
Let $ A=\begin{pmatrix} 3t \end{pmatrix}$ and $ B=\begin{pmatrix} t^2+2t^3 \end{pmatrix} $. The ${\mathbb Z}[t]$-modules ${\textnormal{cok}}(I-A)$ and $ {\textnormal{cok}}(I-B)$ are not isomorphic. (For example, $\det(I-A) \neq \det(I-B)$.) However they do define SFTs which are flow equivalent, with Bowen-Franks group $${\textnormal{cok}}(I-A(1)) = {\textnormal{cok}}(I-B(1)) =
{\textnormal{cok}}\begin{pmatrix} -2 \end{pmatrix} =
{\mathbb Z}/(-2){\mathbb Z}= {\mathbb Z}/2{\mathbb Z}\ . \\$$
There is a useful analog of positive equivalence for constructing maps which give a flow equivalence, using multiplications by elementary matrices over ${\mathbb Z}$ rather than ${\mathbb Z}[t]$ [(Ap. ]{}[feposeq]{}). Also, the passage from SSE-${\mathbb Z}_+$ of matrices $A,B$ to positive equivalence of matrices $(I-tA), (I-tB)$ works with an integral group ring ${\mathbb Z}_+ G$ in place of ${\mathbb Z}_+$, as noted in [@BW04; @B02posk].
Appendix 3 {#a3}
----------
This subsection contains various remarks, proofs and comments referenced in earlier parts of Section \[sec:polynomial\].
\[floweqbackground\] It takes more space than we will spend to give a reasonably understandable introduction to flow equivalence; see e.g. [@bce:fei; @bce:fei:corr] for background and definitions for flow equivalence of subshifts. However, the description to come of the Parry-Sullivan Theorem [@parrysullivan] for SFTs will be quite adequate for our purposes, as a description of what flow equivalence is equivalent to.
Unexpectedly, tools developed for flow equivalence of SFTs turned out to be quite useful for certain classification problems in $C^*$-algebras (see e.g. [@errs:complete; @Restorff2006; @Rordam1995] and their references).
\[bfgroup\] For a square matrix $C$ over ${\mathbb Z}$, one can check that the group ${\textnormal{cok}}_{{\mathbb Z}}(C)$ is infinite when $\det C=0$, and $|{\textnormal{cok}}_{{\mathbb Z}}(C)|= |\det C|$ when $\det C\neq 0$. The groups arising as ${\textnormal{cok}}_{{\mathbb Z}}(C)$ are the finitely generated abelian groups. The group ${\textnormal{cok}}_{{\mathbb Z}}(C)$ may be determined algorithmically by computing the Smith normal form of $C$.
We refer to “the” Bowen-Franks group class associated to an SFT. Formally, the group depends on the presentation; really, we are talking about “the” group up to isomorphism. Americans of a certain age may remember Bill Clinton being mocked for a reply, it depends on what you mean by the word is. In math, we really do need to keep track.
\[topmarkovshift\] Topological Markov shifts were defined as “intrinsic Markov chains” by Bill Parry in the 1964 paper [@Parry1964]. Parry’s paper has the independence of past and future conditioned on the present, and this being presented by a zero-one transition matrix, essentially as a vertex shift.
But before Parry, there was Claude Shannon’s astonishing monograph [@ShannonWeaver1949] in the 1940s, which launched information theory. Shannon already was looking at something we could understand as a Markov shift, with half of the variational principle proved in Parry’s paper. Shannon even used polynomials to present those Markov shifts, just as we describe.
A zero-one matrix can be used to define an edge SFT or a vertex SFT. Yes, they are topologically conjugate SFTs. (The two block presentation of the vertex SFT is the edge SFT.)
\[prooftechniquesforedgesft\] As a postdoc, I heard a talk of John Franks on his classification of irreducible SFTs up to flow equivalence. Edge SFTs were a bit new; he announced for the suspicious that for his proofs, zero-one matrices just weren’t enough.
\[rome\] The “rome” term was introduced in the paper [@bgmy], which also gave a proof that $\det(I-A)=I-tA^{\sharp}$.
\[perrin\] The entropy of an SFT defined by a matrix $B$ over ${\mathbb Z}_+$ is the log of the spectral radius $\lambda$ of $B$. Given $\lambda >1 $ the spectral radius of a primitive matrix over ${\mathbb Z}$, Perrin constructs a $2\times 2$ $A$ over $t{\mathbb Z}_+[t]$ such that $A^{\sharp}$ is primitive with spectral radius $\lambda$ [@Perrin1992]. (The condition that $A^{\sharp}$ is primitive is a significant part of the result.)
\[fromequivtocok\] Suppose $U,C,V$ are matrices over a ring $\mathcal R$; $U$ and $V$ are invertible over $\mathcal R$; and $D=UCV$. Then ${\textnormal{cok}}_{\mathcal R} C$ and ${\textnormal{cok}}_{\mathcal R} D$ are isomorphic as $\mathcal R$-modules.
We consider $C,D$ acting by matrix multiplication on row vectors; of course, the same fact holds for the action on column vectors. Corresponding to the action being on row vectors, we are considering left $\mathcal R$-modules ($c$ in $\mathcal R$ sends $v$ to $cv$), so that matrix multiplication gives an $\mathcal R$-module homomorphism (e.g. $(cv)D=c(vD)$).
Let $C$ be $j\times k$, and let $D$ be $m\times n$. Then $$\begin{aligned}
{\textnormal{cok}}C &=\mathcal R^k / \text{image}(C )=
\mathcal R^k / \{ vC: v\in \mathcal R^j \} \ , \\
{\textnormal{cok}}D &=\mathcal R^n / \text{image}(D )=
\mathcal R^n / \{ vD: v\in \mathcal R^m \} \ .\end{aligned}$$ Define an $\mathcal R$-module isomorphism $\phi : \mathcal R^k \to \mathcal R^n$ by $\phi : w \mapsto wV$. (For most rings of interest, necessarily $j=m$ and $k=n$.) To show $\phi$ induces the isomorphism ${\textnormal{cok}}C \to {\textnormal{cok}}D$, it suffices to show $\phi: \text{image}(C) \to \text{image}(D)$ and $\phi^{-1}: \text{image}(D) \to \text{image}(C)$ . For $xC \in \text{image}(C)$, $$\phi (xC) = xCV= (xU^{-1})(UCV) = (xU^{-1}) D \in
\text{image}(D) \ .$$ For $yD \in \text{image}(D)$, $$\phi^{-1} (yD) = yDV^{-1} = y(UCV)V^{-1} = yUC \in \text{image}(C) \ .$$
Recall, for $A=A(t)$ in NZC, the isomorphism class of ${\textnormal{cok}}_{{\mathbb Z}[t]} (I-A(t))$ determines the SE-${\mathbb Z}$ class of the SFT, and conversely.
\[tautau\] If the initial and terminal vertices of $\tau$ were the same, then we could apply the $\phi$ “rule” to a point $x= \dots \tau\tau{\overset{\bullet}}{\tau}\tau\tau \dots$ (with $\tau$ beginning at $x_0$) in contradictory ways, according to the two groupings $$\begin{aligned}
\dots\ (\tau \tau) (\tau \tau)(& {\overset{\bullet}}{\tau} \tau) (\tau \tau) (\tau \tau)
(\tau \tau) (\tau \tau) \dots \\
\dots\ (\tau \tau) (\tau \tau) (\tau
& {\overset{\bullet}}{\tau}) (\tau \tau) (\tau \tau)
(\tau \tau) (\tau \tau) \dots \ \ .\end{aligned}$$
\[poseqterminology\] The move to polynomial algebraic invariants was pushed by Wagoner, who wanted to exploit analogies between SFT invariants and algebraic K-theory. Positive equivalence was born in the Kim-Roush-Wagoner paper [@KRWJams] as a tool, and taken further in [@BW04] (see also [@B02posk]). The framework developed from considering conjugacy of SFTs via positive equivalence is called “Positive K-theory” (or, Nonnegative K-theory). This reflects the heuristic connection to algebraic K-theory. We will see that the connection is more than heuristic.
The term “positive equivalence” arises from its genesis in our application. We defined positive equivalence rather generally; there is nothing a priori about $\mathcal M$ which must involve positivity. Also, if $E_{ij}(-t^k)(I-A)=(I-B)$, then $E_{ij}(t^k)(I-B)=(I-A)$ – so, multiplications by elementary matrices $E_{ij}(-t^k)(I-A)$ are allowed. If $U$ is a product of elementary matrices over $\mathcal R [t]$, such that $U(I-A)=(I-B)$, with $A,B$ in $I-\text{NZC}$, it need not be the case $I-A$ and $I-B$ are positive equivalent. Each elementary step must be from a matrix in $\mathcal M$ to a matrix in $\mathcal M$.
\[exampleforneednzc\] If $A,B$ in $\mathcal M (t{\mathbb Z}_+[t])$ have polynomial entries with all coefficients in $\{0,1\}$, and define topologically conjugate SFTs, then one can show that $I-A$ and $I-B$ are positive equivalent in $I- \mathcal M(t{\mathbb Z}_+[t])$. But in general, the converse of the theorem is not true; for example, the matrices $\begin{pmatrix} 1-2t \end{pmatrix}$ and $\begin{pmatrix} 1-t& -t \\ -t &1-t \end{pmatrix}$ define SFTs which are conjugate; but, there is not a string of elementary positive equivalences of square matrices over $t{\mathbb Z}_+[t]$, from one to the other. To see this, check the following claim: if $E=E_{ij}(t^k)$ with $k\geq 0$, and $A,B$ are positive equivalent in $\mathcal M(t{\mathbb Z}_+[t])$, and $A(i,i) = 2t + \sum_{k\geq 2} a_kt^k$, then $B(i,i) = 2t + \sum_{k\geq 2} b_kt^k$.
\[evenmore\] We can expand NZC further, and consider matrices $A$ over ${\mathbb Z}[t,t^{-1}]$ with no cycles taking zero time or negative time, and make good sense of their presenting SFTs, and positive equivalence of these matrices $I-A$ as classifying SFTs. This isn’t necessary for classification of SFTs, but might be convenient for some construction.
\[polytool\] For example, constructions of SFTs and topological conjugacies between them, using polynomial matrices and basic positive equivalences, were the proof method for the result in [@KRWForumI; @KRWForumII] of Kim, Roush and Wagoner (a result quite important for SFTs). The hardest step was a construction of brutal complication. But without their proof, we would have no proof at all.
\[otherpolyfeatures\] Edge SFTs are related in a simple and transparent way to their defining matrices over ${\mathbb Z}_+$. When using a matrix $A$ in NZC, or even just in $\mathcal M(t{\mathbb Z}_+[t])$, to define an SFT–we did it by way of the edge SFT defined from $A^{\sharp}$. The relationship between $A$ and $A^{\sharp}$ is not very tight – there is some freedom about what matrix $A^{\sharp}$ is produced. That can be eliminated by precise choices, but these in generality become complicated and rather artificial.
So, for $A$ in ZNC, one would like to have a presentation of an SFT more simply and transparently related to $A$, and with an elementary positive equivalence presented transparently. There is such a presentation – the “path SFT” presented by $A$ (see [@B02posk]).
For a square matrix $A$ over ${\mathbb Z}_+$, and a positive integer $n$, the systems $(X_A, \sigma^n)$ and $(X_{A^n}, \sigma)$ are topologically conjugate. For a polynomial matrix $A$, $A^n$ generally does not define an SFT conjugate to the $n$th power system of the SFT defined by $A$. But, in the path SFT presentation, we recover a natural way to pass to powers of the SFT, which works equally well for negative powers (no passage to transposes needed).
[*Caveat.*]{} We considered three matrix presentations of SFTs: by matrices over $\{0,1\}$, ${\mathbb Z}_+$ and $t{\mathbb Z}_+[t]$. The polynomial presentations have the greatest scope. But we certainly still need edge SFTs – usually the most convenient choice, sometimes the only choice, as for Wagoner’s SSE-${\mathbb Z}_+$ complex.
We also need vertex SFTs. Every topological Markov shift in the sense of Parry (also know as a 1-step shift of finite type) is a vertex SFT, up to naming of symbols. But, not every topological Markov shift is equal to an edge SFT up to naming of symbols. For an example, consider the vertex SFT with adjacency matrix $\left(\begin{smallmatrix} 1&1\\ 1&0 \end{smallmatrix}\right) $. This vertex SFT cannot be an edge SFT after renaming symbols as edges in some directed graph, because a nondegenerate adjacency matrix for a graph with exactly two edges is either $(2)$, $\left( \begin{smallmatrix} 1&0 \\ 0&1 \end{smallmatrix} \right)$ or $\left( \begin{smallmatrix} 0&1 \\ 1&0 \end{smallmatrix} \right)$.
\[parrysullianetc\] See [@B02posk] for a proof of this version of the Parry-Sullivan result [@parrysullivan]. For a careful discussion of flow equivalence for subshifts, and related issues, see [@bce:fei; @bce:fei:corr], which includes references and a detailed proof of the Parry-Sullivan result.
\[feposeq\] For this version of positive equivalence, see the paper [@B02posk] and papers citing it.
Inverse problems for nonnegative matrices {#sec:inverseprobs}
=========================================
In this section, we study certain inverse spectral problems, and related problems, for nonnegative matrices. We are especially interested in inverse problems which involve the realization of stable algebra invariants, such as the nonzero spectrum.
The NIEP
--------
A matrix is nonnegative if every entry is in ${\mathbb R}_+$. A matrix is positive if every entry is positive.
We recall some definitions. If $A$ has characteristic polynomial $\chi_A(t)=\prod_{i=1}^n(t-\lambda_i)$, then the [*spectrum*]{} of $A$ is $(\lambda_1, \dots , \lambda_n)$. We refer to the spectrum as an $n$-tuple by abuse of notation [(Ap. ]{}[abuse]{}): the ordering of the $\lambda_i$ does not matter but the multiplicity does matter. The $\lambda_i$ are in ${\mathbb C}$. Similarly, if $\chi_A(t)= t^j\prod_{i=1}^k(t-\lambda_i)$, with the $\lambda_i$ nonzero, then the [*nonzero spectrum*]{} of $A$ is $(\lambda_1, \dots , \lambda_k)$.
[*The NIEP*]{} (nonnegative inverse eigenvalue problem): What can be the spectrum of an $n\times n$ nonnegative matrix $A$ over ${\mathbb R}$?
Work on the NIEP goes back to (at least) the following result.
Suppose $\Lambda = (\lambda_1, \dots , \lambda_n)$ is a list of real numbers; $\sum_i \lambda_i >0$; and $i>1 \implies \lambda_i <0$. Then $\Lambda$ is the spectrum of a nonnegative matrix.
(In fact, under the assumptions of Suleimanova’s Theorem, the companion matrix of the polynomial $\prod_i(t-\lambda_i)$ is nonnegative [(Ap. ]{}[friedland]{}).)
There is a huge and active literature on the NIEP; see the survey [@NIEP2018] for an overview and extensive bibliography. Despite a rich variety of interesting results, a complete solution is not known at size $n$ if $n > 4$.
\[Johnson-Loewy-London Inequalities\] Suppose $A$ is an $n\times n$ nonnegative matrix. Then for all $k,m$ in ${\mathbb N}$, $${\textnormal{trace}}(A^{mk})\geq \frac{\big({\textnormal{trace}}(A^m)\big)^k}{n^{k-1}} \ .$$
The JLL inequalities, proved independently by Johnson and by Loewy and London, give a quantitative version of an easy compactness result: for $n\times n$ nonnegative matrices $A$ with $\text{trace}(A)\geq \tau >0$, there is a positive lower bound to $\text{trace}(A^k)$ which depends only on $\tau , n, k$. We will use the JLL inequalities later.
Stable variants of the NIEP
---------------------------
Throughout this lecture, $\mathcal R$ denotes a subring of ${\mathbb R}$.
\[inversespecprob\] What can be the nonzero spectrum of a nonnegative matrix $A$ over $\mathcal R$? What can be the nonzero spectrum of an irreducible or primitive matrix over $\mathcal R$?
The case $\mathcal R = {\mathbb Z}$ asks, what are the possible periodic data for shifts of finite type? This is the connection to “stable algebra” for symbolic dynamics (and the original impetus for the paper [@BH91]). Later, we will also consider the realization in nonnegative matrices of more refined stable algebra structure.
To begin we review relevant parts of the Perron-Frobenius theory of nonnegative matrices [(Ap. ]{}[pfnotes]{}). This will let us reduce the different flavors of Problem \[inversespecprob\] to the primitive case.
Primitive matrices
------------------
Recall Definition \[primitiveDefinition\]: a primitive matrix is a square nonnegative matrix $A$ such that for some positive integer $k$, $A^k$ is positive. (Then, $A^n$ is positive for all $n\geq k$.) The next theorem is the heart of the theory of nonnegative matrices [(Ap. ]{}[whyperron]{}). Recall, the [*spectral radius*]{} of a square matrix with real (or complex) entries is the maximum of the moduli of the eigenvalues (i.e., the radius of the smallest circle in ${\mathbb C}$ with center 0 which contains the spectrum).
\[PerronTheorem\] Suppose $A$ is primitive, with spectral radius $\lambda$. Then the following hold.
1. $\lambda$ is a simple root of the characteristic polynomial $\chi_A$.
2. If $\nu$ is another root of $\chi_A$, then $|\nu|< \lambda$.
3. There are left and right eigenvectors $\ell , r$ of $A$ for $\lambda$ which have all entries positive.
4. The only nonnegative eigenvectors of $A$ are the eigenvectors for the spectral radius.\
We list three nonprimitive nonnegative matrices for which a conclusion of the Perron Theorem fails. $$A=\begin{pmatrix} 0&-1 \\ 1& 0 \end{pmatrix} \ , \qquad
B=\begin{pmatrix} 1&0 \\ 0& 1 \end{pmatrix}\ , \qquad
C = \begin{pmatrix} 0&1 \\ 1& 0 \end{pmatrix} \ .$$ $A$ has spectrum $(i, -i)$; the spectral radius of $A$ is 1, but 1 is not an eigenvalue of $A$. $B$ has spectrum $(1,1)$; the spectral radius of $B$ is 1, and 1 is a repeated root of $\chi_B$. $C$ has spectrum $(1, -1)$; the spectral radius 1 is an eigenvalue, but $1 = |-1|$.
The matrix $A=\left(\begin{smallmatrix} 0&3 \\ 4& 1 \end{smallmatrix}\right)$ is primitive with spectrum $(4 ,-3)$. There is a positive left eigenvector for eigenvalue $4$, but not for $3$: $$(1,1)\begin{pmatrix} 0&3 \\ 4& 1 \end{pmatrix} =
4(1,1) \quad \quad
\text{and}
\quad \quad
(-4,3)\begin{pmatrix} 0&3 \\ 4& 1 \end{pmatrix} = -3(-4,3) \ .$$
Irreducible matrices
--------------------
An irreducible matrix is an $n\times n$ nonnegative matrix $A$ such that $$\{ i,j\} \subset \{1, \dots, n\}
\ \ \implies \ \
\exists k>0 \text{ such that }
A^k(i,j) >0 \ .$$
Every primitive matrix is irreducible.
$$A = \begin{pmatrix} 1&1&1 \\ 1& 1& 1 \\ 0&0&0\end{pmatrix} \quad \quad
B= \begin{pmatrix} 1&1 \\ 0& 1 \end{pmatrix} \quad \quad
C=\begin{pmatrix} 0&1&0 \\ 0& 0& 1 \\ 1&0&0\end{pmatrix} \quad \quad
D=\begin{pmatrix} 0&0&1 \\ 0& 0& 1 \\ 1&1&0\end{pmatrix} \ .$$ For all $n\in {\mathbb N}$, we see sign patterns:
$$A^n = \begin{pmatrix} +&+&+ \\ +& +& + \\ 0&0&0\end{pmatrix} \ \
B^n= \begin{pmatrix} +&+ \\ 0& + \end{pmatrix} \ \
C^{3n}=\begin{pmatrix} +&0&0 \\ 0& +& 0 \\ 0&0&+\end{pmatrix} \ \
D^{2n}=\begin{pmatrix} +&+&0 \\ +& +& 0 \\ 0&0&+\end{pmatrix} \ .$$ $A$ and $B$ are not irreducible. $C$ and $D$ are irreducible, but not primitive.
### Block permutation structure
If $n>1$, then an $n\times n$ cyclic-permutation matrix is irreducible but not primitive. This is representative of the general irreducible case.
For a square nonnegative matrix $A$, the following are equivalent.
1. $A$ is irreducible.
2. There is a permutation matrix $Q$ and a positive integer $p$ such that $Q^{-1}AQ$ has the block structure of a cyclic permutation, $$Q^{-1} A Q =
\begin{pmatrix}
0 & A_1 & 0 & 0 &\dots & 0 \\
0 & 0 & A_2 & 0 &\dots & 0 \\
& &\dots &&& \\
0 & 0 & 0 & 0 &\dots & A_{p-1} \\
A_{p} & 0 & 0 & 0 &\dots & 0
\end{pmatrix}$$ such that each of the cyclic products $\ D_1=A_1A_2\cdots A_p$, $\ D_2= A_2A_3\cdots A_1$, $\ \ \dots \ \ $, $\ D_p= A_pA_1\cdots A_{p-1} \ \ $ is a primitive matrix.
The integer $p$ above is called the period of the irreducible matrix $A$. (If $p=1$, then $A$ is primitive.) For $A$ above, $A^p$ is block diagonal, with diagonal blocks $D_1, \dots , D_p$.
From the block permutation structure, one can show the following (in which $D$ could be any of the matrices $D_i$ above).
Suppose $A$ is an irreducible matrix with period $p$. Then there is a primitive matrix $D$ such that $\det (I-tA) =\det (I-t^pD)$.
It is not hard to check that the converse of this theorem is also true [(Ap. ]{}[AforDproof]{}).
### Reduction in terms of nonzero spectrum
Recall, $A$ has nonzero spectrum $ (\lambda_1, \dots , \lambda_k)$ if and only if $\det (I-tA) = \prod_{i=1}^k (1-\lambda_i t)$. The statement $\det (I-tA) =\det (I-t^pD)$ has an equivalent description [(Ap. ]{}[nzspecequiv]{}):\
if $\Lambda$ is the nonzero spectrum of $D$, then $\Lambda^{1/p}$ is the nonzero spectrum of $A$. Here, $ \Lambda^{1/p}$ is defined by replacing each entry of $\Lambda$ with the list of its $p$th roots in ${\mathbb C}$. If $\Lambda$ is $k$ entries, then $\Lambda^{1/p}$ has $pk$ entries.
Suppose $ \det(I-tD) = (1-8t)(1-7t)^2$ and $\det(I-tA) = \det (I-t^3D)$. Let $\xi = e^{2\pi i/3}$. The nonzero spectrum of $D$ is $\Lambda = (8,7,7)$. The nonzero spectrum of $A$ is $$\Lambda^{1/3} = \big(\ \, 2,\, \xi 2, \, \xi^2 2,\, \ \ \
7^{1/3} ,\ \xi 7^{1/3} ,\xi^2 7^{1/3} , \, \ \ \
7^{1/3} ,\ \xi 7^{1/3} ,\xi^2 7^{1/3} \, \ \ \big)
\ .$$
### Multiplicity of zero in the spectrum
Apart from one exception: if a nonzero spectrum is realized by an irreducible matrix over $\mathcal R$ of size $n\times n$, then it can also be realized at any larger size, by an irreducible matrix over $\mathcal R$ of the same period.
The one exception: if $\mathcal R = {\mathbb Z}$, then an irreducible matrix with spectral radius 1 can only be a cyclic permutation matrix.
Also: if $n\times n$ is the smallest size primitive matrix realizing a nonzero spectrum $\Lambda$, then $pn\times pn$ is the smallest size irreducible matrix realizing $\Lambda^{1/p}$.
[*Conclusion.*]{} Knowing the possible spectra of irreducible matrices over a subring $\mathcal R$ of ${\mathbb R}$ reduces to knowing the possible nonzero spectra of primitive matrices over $\mathcal R$, and the smallest dimension in which they can be realized.
Nonnegative matrices
--------------------
[(Ap. ]{}[pfnotes]{}) Suppose $A$ is a square nonnegative matrix. Then there is a permutation matrix $P$ such $P^{-1}AP$ is block triangular, such that each diagonal block is either irreducible or $(0)$.
For $A$ nonnegative as above, let $A_i$ be the $i$th diagonal block, with characteristic polynomial $p_i$. Then the characteristic polynomial of $A$ is $\chi_A(t)=\prod_i p_i(t)$, and the nonzero spectrum is given by $\det (I-tA) = \prod_i \det (I-tA_i)$.
So, the spectrum of a nonnegative matrix is an arbitrary disjoint union of spectra of irreducible matrices, together with an arbitrary repetition of 0.
There are constructions and constraints which work best at the level of nonnegative matrices (e.g., JLL). Still, one approach to the NIEP is to focus on the primitive case (which gives the irreducible case, and then the general case). Obstructions might be more simply formulated in this case. Moreover, in applications a nonnegative matrix must often be irreducible or primitive. (For symbolic dynamics: definitely.) A realization statement for nonnegative matrices does not give a realization statement for irreducible or primitive matrices. So, we focus on primitive matrices. But even in this restricted case, no satisfactory general characterization is known or conjectured.
[*Conclusion.*]{} We will focus on the nonzero spectrum of primitive matrices. And here, at last, we find simplicity.
The Spectral Conjecture
-----------------------
Let $\Lambda= (\lambda_1, \dots , \lambda_k)$ be a $k$-tuple of nonzero complex numbers. We will give three simple conditons $\Lambda$ must satisfy to be the nonzero spectrum of a primitive matrix over $\mathcal R$.\
\
For a tuple $\Lambda= (\lambda_1, \dots , \lambda_k) $ of complex numbers:
- $\lambda_i$ is a [*Perron value*]{} for $\Lambda$ if $\lambda_i$ is a positive real number and $i\neq j \implies
\lambda_i > |\lambda_j|$ .
- ${\textnormal{trace}}(\Lambda) = \sum_{i=1}^k \lambda_i $ .
- $\Lambda^n = \big( (\lambda_1)^n, \dots , (\lambda_k)^n )\ ,\quad$ if $n\in {\mathbb N}$.
Suppose $\Lambda $ is the nonzero spectrum of a primitive matrix over a subring $\mathcal R$ of ${\mathbb R}$. Then the following hold.
1. [Perron Condition]{.nodecor}:\
$\Lambda$ has a Perron value.
2. [Coefficients Condition [(Ap. ]{}[coeff]{})]{.nodecor}:\
The polynomial $p(t)= \prod_{i=1}^k (t-\lambda_i)$ has all its coefficients in $\mathcal R$.
3. [Trace Condition]{.nodecor}:\
If $\mathcal R \neq {\mathbb Z}$, then for all positive integers $n,k$:
1. \(i) ${\textnormal{trace}}\Lambda^n \geq 0$, and
2. \(ii) ${\textnormal{trace}}\Lambda^n > 0 \implies {\textnormal{trace}}\Lambda^{nk} > 0 $ .
If $\mathcal R={\mathbb Z}$, then for all positive integers $n$, ${\textnormal{trace}}_n(\Lambda) \geq 0\ . $
(We define ${\textnormal{trace}}_n(\Lambda)$, the $n$th net trace of $\Lambda$, below.)
\(1) By the Perron Theorem, $\Lambda$ has a Perron value.
\(2) The characteristic polynomial of a matrix over a ring has coefficients in the ring. For some $k\geq 0$, the characteristic polynomial of $A$ is $t^kp(t)$. So, $p$ has coefficients in the ring.
\(3) (i) ${\textnormal{trace}}(\Lambda)= {\textnormal{trace}}(A)$, and ${\textnormal{trace}}(\Lambda^n)={\textnormal{trace}}(A^n)$. The trace of a nonnegative matrix is nonnegative. (ii) Suppose ${\textnormal{trace}}(\Lambda^n)>0$. Then ${\textnormal{trace}}(A^n)>0 $ and $A^n \geq 0$. Therefore ${\textnormal{trace}}{(A^n)^k}>0$ . But, $ {\textnormal{trace}}{(A^n)^k} = {\textnormal{trace}}(A^{nk}) ={\textnormal{trace}}(\Lambda^{nk}) $ .
\(3) Suppose $\mathcal R = {\mathbb Z}$. Conditions (i) and (ii) hold, but a stronger condition holds.
Consider $A$ as the adjacency matrix of a graph. A loop is a path with the same minimal and terminal vertex. The number of loops of length $n$ is trace($A^n$).
A loop is minimal if it is not a concatenation of copies of a shorter loop. So, for example, $$\begin{aligned}
\text{number of minimal loops of length }1 & =
\text{trace} (A) \\
\text{number of minimal loops of length }2 &
= \text{trace}(A^2) - \text{trace} (A) \ .\end{aligned}$$ For example, let $\Lambda = (2,i,-i,i,-i,1)$. Then ${\textnormal{trace}}( \Lambda^2 )- {\textnormal{trace}}(\Lambda) = 1 -3 = -2 <0$. This $\Lambda$ cannot be the nonzero spectrum of a matrix over ${\mathbb Z}_+$, even though $\Lambda$ satisfies conditions 1,2,3(i) and 3(ii).
The number of minimal loops of length $n$, ${\textnormal{trace}}_n(\Lambda)$, can be expressed as a function of the traces of powers of $\Lambda$ using Mobius inversion: $${\textnormal{trace}}_n(\Lambda):= \sum_{d|n}
\mu (n/d)\, {\textnormal{trace}}( \Lambda^d) \ ,$$ where $\mu $ is the Mobius function, $$\begin{aligned}
\mu : {\mathbb N}&\to \{-1,0,1\} \\
: n & \mapsto 0 \quad \text{if }n\text{ is not squarefree} \\
: n & \mapsto (-1)^e \quad \text{if }n\text{ is the product of }e
\text{ distinct primes. }\end{aligned}$$
\[spectralconjecture\] Let $\mathcal R$ be a subring of ${\mathbb R}$. Suppose $\Lambda
= (\lambda_1, \dots , \lambda_k)$ is an $k$-tuple of complex numbers. Then $\Lambda$ is the nonzero spectrum of some primitive matrix over $\mathcal R$ if and only the above conditions (1), (2), (3) hold.
(unbounded realization size) Suppose $\mathcal R={\mathbb R}$. Given $0< \epsilon < (1/2)$, set $$\Lambda_{\epsilon} \ =\
\Big(\ 1\ ,\ i\sqrt{(1-\epsilon )/2}\ ,\ -i\sqrt{(1-\epsilon )/2} \
\Big) \ .$$ This $\Lambda_{\epsilon} $ satisfies the conditions of the Spectral Conjecture.
But, if a nonnegative $n\times n$ matrix $A$ has nonzero spectrum $\Lambda_{\epsilon}$, then $$\begin{aligned}
{\textnormal{trace}}(\Lambda_{\epsilon}^2) &\geq \frac{({\textnormal{trace}}\Lambda_{\epsilon})^2}n
\ , \quad
\text{by the JLL inequality, and therefore }
\\
\epsilon &\geq \frac{1^2}n = 1/n \ .\end{aligned}$$ So, as $\epsilon$ goes to zero, the size of $A$ must go to infinity.
A matrix $A$ is [**eventually positive (EP)**]{} if for all large $k>0$, $A^k$ is positive.
[(Ap. ]{}[HIEP]{}) Suppose $A$ is a square matrix over $\mathcal R$ whose spectrum has a Perron value.
1. If $\mathcal R \neq {\mathbb Z}$, then $A$ is similar over $\mathcal R$ to an EP matrix [@HandelmanJOpTh1981].
2. If $\mathcal R = {\mathbb Z}$, then $A$ is SSE over $\mathcal R$ to an EP matrix [@HandelmanEvPosRational1987].
In particular, the Spectral Conjecture would be true if we were allowed to replace $\Lambda$ with $\Lambda^k$, $k$ large. With $\mathcal R\neq {\mathbb Z}$, and $\Lambda$ an $n$-tuple, we could even realize $\Lambda^k$ with a positive matrix which is $n\times n$.
Let’s consider existing results on the Spectral Conjecture.
Boyle-Handelman Theorem
-----------------------
The Spectral Conjecture is true if $\mathcal R={\mathbb R}$.
The problem of determining the possible nonzero spectra of primitive symmetric matrices is quite different. If an $n$-tuple is the nonzero spectrum of a [**nonnegative symmetric matrix**]{}, then it is achieved by a matrix whose size is bounded above by a function of $n$ [@JohnsonLaffeyLoewySymmetric1996]. Adding more zeros to the spectrum doesn’t help.
The Boyle-Handelman Theorem is a corollary of a stronger result.
Suppose $\Lambda $ satisfies the conditions of the Spectral Conjecture, and a subtuple of $\Lambda $ containing the Perron value of $\Lambda$ is the nonzero spectrum of a primitive matrix over $\mathcal R$. (For example, this holds if the Perron value is in $\mathcal R$.) Then $\Lambda $ is the nonzero spectrum of a primitive matrix over $\mathcal R$.
The proof of the Suptuple Theorem uses ideas from symbolic dynamics. The proof is constructive, in the sense that one could make it a formal algorithm. But the construction is very complicated, and uses matrices of enormous size. It has no practical value as a general algorithm.
\[Boyle-Handelman-Kim-Roush [@BH91]\] Suppose $\Lambda $ satisfies the conditions of the Spectral Conjecture, ${\textnormal{trace}}(\Lambda )>0$ and $\mathcal R\neq {\mathbb Z}$. Then $\Lambda $ is the nonzero spectrum of a primitive matrix over $\mathcal R$.
We will outline the proof.
1. By the B-H Theorem, there is a primitive matrix $A$ over ${\mathbb R}$ with nonzero spectrum $\Lambda $.
2. Given $A$ primitive with positive trace, a theorem of Kim and Roush produces a positive matrix $B$ which is SSE-${\mathbb R}_+$ to $A$ (hence, has the same nonzero spectrum as $A$).
3. There are matrices $U,C$ over $\mathcal R$ such that $U^{-1}CU = A$ and $\det U =1$.
4. $U$ is a product of elementary matrices over ${\mathbb R}$, equal to $I$ except in a single off diagonal entry. By density of $\mathcal R$ in ${\mathbb R}$, these can be perturbed to elementary matrices over $\mathcal R$. Thus $U$ can be perturbed to a matrix $V$ over $\mathcal R$ with determinant 1.
5. Because $U^{-1}CU >0$, if $V$ is close enough to $U$ then $V^{-1}CV>0$.
Suppose $\mathcal R \neq {\mathbb Z}$. It would be very satisfying to see the Spectral Conjecture proved in the remaining case, ${\textnormal{trace}}(\Lambda) =0$, by some analogous perturbation argument. I have no idea how to do this, or if it can be done.
The Kim-Ormes-Roush Theorem
---------------------------
\[Kim-Ormes-Roush\] [@S8] For $\mathcal R = {\mathbb Z}$, the Spectral Conjecture is true.
Let us note an immediate corollary.
For $\mathcal R = {\mathbb Q}$, the Spectral Conjecture is true.
Polynomial matrices and formal power series play a fundamental role in the KOR proof. The KOR Theorem gives us a complete understanding of the possible periodic data for SFTs. The proof, though quite complicated, is much more tractable than the proof of the B-H Theorem. The use of power series leads to an interesting analytical approach to the NIEP [@LaffeyLoewySmigocPowerSeries2016].
Status of the Spectral Conjecture
---------------------------------
The conjecture is true for ${\mathbb R}$, ${\mathbb Q}$ and ${\mathbb Z}$; in the positive trace case; under the Subtuple Theorem assumption; and in other special cases. It is very hard to doubt the conjecture.
One expects the case $\mathcal R={\mathbb Z}$ to be the hardest case. Perhaps it is feasible to prove the Spectral Conjecture by adapting the Kim-Ormes-Roush proof.
Laffey’s Theorem
----------------
Laffey [@Laffey2012BHTheorem] gave a constructive version of the Boyle-Handelman Theorem in the case that $\mathcal R= {\mathbb R}$ and the candidate spectrum $\Lambda$, satisfying the necessary conditions of the Spectral Conjecture for ${\mathbb R}$, also satisfies $${\textnormal{trace}}(\Lambda^k)>0\ , \quad k\geq 2 \ .$$ The primitive matrix which Laffey constructs to realize $\lambda$ has a rather classical form, and there is a comprehensible formula giving an upper bound on the size of the smallest $N$ given by the construction. From here, we give some remarks on Laffey’s theorem.
[*The Coefficients Condition.*]{} In the case of ${\mathbb R}$, the Coefficients Condition of the Spectral Conjecture follows automatically from the Trace Conditions [(Ap. ]{}[coeff]{}),
[*Laffey’s upper bound on the smallest size $N$ of a primitive matrix $A$ realizing a given $\Lambda = ( \lambda_1, \dots, \lambda_n)$.*]{} If $A$ is primitive with nonzero spectrum $\Lambda$ and $c>0$, then $cA$ is primitive with nonzero spectrum $c\Lambda
= (c\lambda_1, \dots, c\lambda_n)$. So, to consider an upper bound $N$, for simplicity we consider just the special case that the Perron value of $\Lambda$ is $\lambda_1 =1$.
Laffey’s explicit, computable formula giving an upper bound for $N$ is rather complicated. But, using the Perron value $\lambda_1=1$, and considering only the nontrivial case $n\geq 2$, it can be shown that Laffey’s bound implies $$\label{LaffeysN}
N \leq \kappa_n \Bigg(\frac{1}{MG}\Bigg)^n $$ where $\ \kappa_n $ depends only on $n$, $$\begin{aligned}
G \ &=\ 1 - \max \{|\lambda_i|: 2\leq i \leq n\}\ , \quad
\\
M\ &=\ \min \{ {\textnormal{trace}}(\Lambda )^n : n \geq 2 \ \} \ .\end{aligned}$$ The numbers $\kappa_n$ obtained from the estimate grow very rapidly; e.g. $\kappa_n \geq n^n$ .
This bound is certainly nonoptimal! For example, suppose $0< \epsilon <1$. The nonzero spectrum $(1,\, -1+\epsilon )$ is realized by the $2\times 2$ primitive matrix $\begin{pmatrix} 0&1 \\ 1-\epsilon & \epsilon
\end{pmatrix} $. But here, $\epsilon =G$, and as $\epsilon$ goes to zero the upper bound in [(Ap. ]{}[LaffeysN]{}) goes to infinity.
Nevertheless: this is a transparent and meaningful bound. The bound involves only $n$, $M$ and $G$. The spectral gap $G$ appears repeatly in the use of primitive matrices (and more generally), e.g. for convergence rates. Also, the terms $1/M$ and $1/G$ cannot simply be deleted, as we note next.
[*The term $1/M$.*]{} If Laffey’s formula for an upper bound on $N$ were replaced by a formula of the form $N \leq f(G,n)$, then even at $n=3$ the formula could not give a correct bound, on acount of the JLL Inequalities [(Ap. ]{}[reflectjll]{}).
[*The spectral gap term $(1/G)$*]{} If Laffey’s formula for an upper bound on $N$ were replaced by a formula of the form $N \leq f(M,n)$, then even at $n=4$, the formula could not give a correct bound [(Ap. ]{}[gaprefl]{}).
Let $\Lambda = (\, 1.1,\, \xi ,\, \overline{\xi}\, )$, where $\xi = \exp{(\pi i/10)}$ and $\overline{\xi}$ is its complex conjugate. Laffey stated there is a $128\times 128$ primitive matrix realizing this nonzero spectrum.
[*The matrix form.*]{} The primitive matrix with nonzero spectrum $\Lambda$ has (for sufficiently large $k$) the banded form $$\begin{pmatrix}
x_1 & 1 & 0 & 0 &\cdots & 0 & 0 & 0 \\
x_2 & x_1 & 2 & 0 &\cdots & 0 & 0 & 0 \\
x_3 & x_2 & x_1 & 3 &\cdots & 0 & 0 & 0 \\
x_4 & x_3 & x_2 & x_1 &\cdots & 0 & 0 & 0 \\
\cdots &\cdots &\cdots &\cdots &\cdots &\cdots &\cdots &\cdots \\
x_{k-2} & x_{k-3} &x_{k-4} &\cdots &\cdots & x_1 & k-2 & 0 \\
x_{k-1} & x_{k-2} &x_{k-3} & \cdots &\cdots & x_2 & x_1 & k-1 \\
x_k & x_{k-1} & x_{k-2} &\cdots &\cdots & x_3 & x_2 & x_1
\end{pmatrix} \ .$$ For the relation of the matrix entries to $\Lambda$, see Laffey’s paper [@Laffey2012].
[*Limits of the argument.*]{} A lot of the complication of the B-H proof involves complications of ${\textnormal{trace}}(\Lambda^n) = 0 $ for a variety of sets of $n$. These general difficulties aren’t addressed in Laffey’s result. Laffey’s argument also proves the Spectral Conjecture over any subfield $\mathcal R$ of ${\mathbb R}$, under the restriction ${\textnormal{trace}}(\Lambda^k) >0$ for $k>1$. But it does not work for all $\mathcal R$. The argument uses division by integers in $\mathcal R$.
The Generalized Spectral Conjectures
------------------------------------
The NIEP refines to an even harder question: what can be the Jordan form of a square nonnegative matrix over ${\mathbb R}$? We refer to [@NIEP2018 Sec.9] for a discussion. A rather sobering example of Laffey and Meehan [@laffeyMeehan1999] shows that $(3+t, 3-t,-2,-2,-2)$ is the spectrum of a $5\times 5$ nonnegative matrix if $t> (16\sqrt{6})^{1/2} -39 \approx 0.437\dots$, but it is the spectrum of a diagonalizable nonnegative matrix if and only if $t\geq 1$.
Suppose $A$ is a nonnilpotent square matrix over ${\mathbb R}$. The [*nonsingular part*]{} of $A$ is a nonsingular matrix $A'$ over ${\mathbb R}$ such that $A$ is similar to the direct sum of $A'$ and a nilpotent matrix. ($A'$ is only defined up to similarity over ${\mathbb R}$.) Analagous to the Spectral Conjecture , we have the following.
\[realgsc\] If $B$ is a square real matrix satisfying the necessary conditions of the Spectral Conjecture, then $B$ is the nonsingular part of some primitive matrix over ${\mathbb R}$.
Let $A,B$ be square matrices over ${\mathbb R}$, with nonsingular parts $A',B'$. Recall, the following are equivalent:
1. $A'$ and $B'$ are SIM-${\mathbb R}$ (similar over ${\mathbb R}$).
2. $A$ and $B$ are SE-${\mathbb R}$ (shift equivalent over ${\mathbb R}$).
3. $A$ and $B$ are SSE-${\mathbb R}$ (strong shift equivalent over ${\mathbb R}$).
So, the conjecture above is a special case of either of the following conjectures [(Ap. ]{}[gscrefs]{}).
Suppose $A$ is a square matrix over a subring $\mathcal R$ of ${\mathbb R}$, and the nonzero spectrum of $A$ satisfies the necessary conditions of the Spectral Conjecture.
Then $A$ is SE-$\mathcal R$ to a primitive matrix.
\[(Strong) Generalized Spectral Conjecture, Boyle-Handelman 1993\] Suppose $A$ is a square matrix over a subring $\mathcal R$ of ${\mathbb R}$, and the nonzero spectrum of $A$ satisfies the necessary conditions of the Spectral Conjecture.
Then $A$ is SSE-$\mathcal R$ to a primitive matrix.
The Strong GSC is the strongest viable conjecture we know which reflects the idea that the only obstruction to expressing stable algebra in a primitive matrix is the nonzero spectrum obstruction.
In the next result, a nontrivial unit is a unit in the ring not equal to $\pm 1$. (The assumption of a nontrivial unit is probably an artifact of the proof.)
[@BH93 Theorem 3.3] Let $\mathcal R$ be a unital subring of ${\mathbb R}$. Suppose that either $\mathcal R= {\mathbb Z}$ or $\mathcal R$ is a Dedekind domain with a nontrivial unit. Let $A$ be a square matrix with entries from $\mathcal R$ whose nonzero spectrum $\Lambda$ satisfies the necessary conditions of the Spectral Conjecture and consists of elements of $\mathcal R$. Then $A$ is algebraically shift equivalent
[Algebraically shift equivalent over $\mathcal R $ was the notation in [@BH93] for what we are calling SE-$\mathcal R$, shift equivalence over the ring $\mathcal R$. Also, [@BH93 Prop.2.4] established that SE and SSE are equivalent over a Dedekind domain, so the conclusion could have been stated for strong shift equivalence.]{}
over $\mathcal R $ to a primitive matrix.
The following corollary is immediate.
Conjecture \[realgsc\] is true under the additional assumption that the spectrum of $B$ is real.
For example, the corollary covers the case that $B$ in Conjecture \[realgsc\] is a diagonal matrix. (For example, if $B$ is diagonal with a Laffey-Meehan spectrum $(3+t,3-t,-2,-2,-2)$, for any $t>0$). On the other hand, the following (embarassing) open problem indicates how little we know.
Suppose $A$ is a $2\times 2$ matrix over ${\mathbb Z}$ with irrational eigenvalues satisfying the conditions of the Spectral Conjecture. Prove that $A$ is SE-${\mathbb Z}$ to a primitive matrix.
When the Generalized Spectral Conjectures were made, it was not known whether SE-$\mathcal R$ implied SSE-$\mathcal R$ for every ring $\mathcal R$. We now know that there are many rings over which SSE properly refines SE [@BoSc1], including some subrings of ${\mathbb R}$. So, the weak and strong conjectures are not [*a priori*]{} equivalent. Nevertheless, it can be proved, for every subring $\mathcal R$ of ${\mathbb R}$, that if any matrix in a given SE-$\mathcal R$ class is primitive, then every matrix in that SE-$\mathcal R$ class is SSE-$\mathcal R$ to a primitive matrix [@BoSc3]. So, we now know the weak and strong conjectures are equivalent.
Appendix 4 {#a4}
----------
This subsection contains various remarks, proofs and comments referenced in earlier parts of Section \[sec:inverseprobs\].
\[abuse\] “Abuse of notation” is a use of notation to mean something it does not literally represent, for simplicity. For example, describing the spectrum correctly as a multiset (set with multiplicities) seems to divert more mental energy than one uses to be aware that an $n$-tuple is not literally a multiset.
\[friedland\] The companion matrix characterization for Suleimanova’s Theorem is attributed in [@Laffey2012] to Shmuel Friedland.
\[jll\] ([JLL Inequalities]{.nodecor}) Let $A$ be an $n\times n$ nonnegative matrix. Then for all $k,m$ in ${\mathbb N}$ : $$\textnormal{trace} (A^{mk}) \geq \frac{\big(
\textnormal{trace} (A^m)\big)^k}{n^{k-1}} \ .$$
This result was proved independently by Loewy and London [@LoewyLondon], and by Johnson [@Johnson1981]. The proof of this insightful result is not difficult.
1. Note, if $B$ is an $n\times n$ nonnegative matrix, and $k\in {\mathbb N}$, then ${\textnormal{trace}}(B^k) \geq \sum_{i=1}^n \big(B(i,i)\big)^k$.\
(Because: the $B(i,i)^k $ are some of the terms contributing to ${\textnormal{trace}}(B^k)$, and the other terms are nonnegative.)
2. Now suppose $\tau = {\textnormal{trace}}(B) >0$, and solve the problem: if $x_1, \dots, x_n$ are nonnegative numbers with positive sum $\tau$, what is the minimum possible for the sum $s_k= \sum_{i=1}^n (x_i)^k$ ?
You can check (with Lagrange multipliers, say, or Hölder’s inequality) that the minimum is achieved at $(x_1, \dots , x_n)=(\tau /n, \tau /n, \dots , \tau /n)$. (Intuitively, there is no other candidate, because there is a minimum and the minimum is not achieved at $(x_1, \dots , x_n)
=(\tau_1, 0, \dots , 0)$.) Then, for $B$ and for $B=A^m$, $$\begin{aligned}
{\textnormal{trace}}(B^k) \geq s_k & \geq \sum_{i=1}^n (\tau/n)^k
= n (\tau/n)^k = \tau^k / n^{k-1} \\
{\textnormal{trace}}(B^k) & \geq \tau^k / n^{k-1} \\
{\textnormal{trace}}(A^{mk}) ={\textnormal{trace}}((A^m)^k) & \geq ({\textnormal{trace}}(A^m))^k/n^{k-1} \ . \qed\end{aligned}$$
\[pfnotes\] There are a number of excellent works on the Perron-Frobenius theory of nonnegative matrices. My own exposition [@boylepfnotes] covers the heart of the theory (statements in this chapter), but not all parts of it.
\[whyperron\] Briefly: why is the Perron theorem so important?
Suppose $A$ is primitive with spectral radius $\lambda$. Let $ \ell, r$ be be positive left, right eigenvectors for $\lambda$, such that $\ell r = (1)$. The Perron Theorem implies that for many purposes, for large $n$, $A^n$ is very well approximated by the rank one positive matrix $\lambda^n r\ell$.
What is “very well approximated”? Let $\mu $ be the second highest eigenvalue modulus. There is a matrix $B$ with spectral radius $\mu$ such that $A = (\lambda r\ell) + B$, with $(\lambda r\ell) B =0 = B(\lambda r\ell)$, so $A^n = (\lambda^n r\ell) + B^n$. Entries of $B^n$ cannot grow at an exponential rate greater than $\mu^n$; but every entry of $A^n $ grows at the exponentially greater rate $\lambda^n$.
\[AforDproof\] Suppose $D$ is a primitive matrix over a subring $\mathcal R$ of ${\mathbb R}$, and $p$ is a positive integer. Then there is an irreducible matrix $A$ over $\mathcal R$ with period $p$ such that $\det (I-tA) =\det (I-t^pD)$.
We give a proof for $p=4$ (which should make the general case obvious). Define $A = \left(\begin{smallmatrix} 0 & D & 0 & 0 \\ 0 & 0 & I& 0 \\
0&0 & 0 & I \\ I & 0 & 0 & 0
\end{smallmatrix} \right)$. We compute a product $$(I-tA)U =
\left(\begin{smallmatrix} I & -tD & 0 & 0 \\ 0 & I & -tI& 0 \\
0&0 & I & -tI \\ -tI & 0 & 0 & I
\end{smallmatrix} \right)
\left(\begin{smallmatrix} I & 0 & 0 & 0 \\
t^3I & I & 0 & 0 \\
t^2I&0 & I & 0 \\
tI &0 & 0 & I
\end{smallmatrix}\right)
=
\left(\begin{smallmatrix} I -t^4D & -tD & 0 & 0 \\ 0 & I & -tI& 0 \\
0&0 & I & -tI \\ 0 & 0 & 0 & I
\end{smallmatrix} \right)$$ Noting $\det U = 1$, we see $\det(I-tA) = \det\big((I-tA)U\big) =\det(I-t^4D)$ .
\[nzspecequiv\] Suppose $A,D$ are square real matrices, $\det(I-tA) = \det (I-t^pD)$, and the nonzero spectrum of $D$ is $\Lambda=(\lambda_1, \dots , \lambda_k)$.
Then the nonzero spectrum of $A$ is $\Lambda^{1/p}$.
Because the nonzero spectrum of $D$ is $\Lambda=(\lambda_1, \dots , \lambda_k)$, we have $\det (I-tD) = \prod_{i=1}^k(1-\lambda_it)$. Therefore, $\det (I-t^pD) = \prod_{i=1}^k(1-\lambda_it^p)$. Given $\lambda_i$, let $\mu_{i1}, \dots , \mu_{ip}$ be a list of its $p$th roots in ${\mathbb C}$. Then, $$(1-\lambda_it^p) = \prod_{j=1}^p(1-\mu_{ij} t) \ .$$ Thus, $\det (I-tA) =\prod_{i=1}^k \prod_{i=1}^p(1-\mu_{ij} t) $, and it follows that the nonzero spectrum of $A$ is $\Lambda^{1/p}$.
\[coeff\] The Coefficients Condition of the Spectral Conjecture holds if the ring $\mathcal R$ contains ${\mathbb Q}$ and if ${\textnormal{trace}}(\Lambda^n) \in \mathcal R$ for all positive integers $n$. For a self contained proof of this, consider the companion matrix $C$ to the the polynomial $$p(x)
= \prod_{i=1}^k(t-\lambda_i) = t^k -c_1t^{k-1} -c_2t^{k-2}- \dots \ \ .$$ Clearly $c_1\in \mathcal R$ iff ${\textnormal{trace}}(\Lambda) \in \mathcal R$. Now suppose $c_1, \dots , c_{j-1}$ are in $\mathcal R$, and $j\leq k$. From this assumption and the form of $C$, we have that $jc_j$ equals an element of $\mathcal R$ plus ${\textnormal{trace}}(\Lambda^j)$.
In particular, the Coefficients Condition is redundant if $\mathcal R={\mathbb R}$, because ${\textnormal{trace}}(\Lambda^n) \geq 0$ implies ${\textnormal{trace}}(\Lambda^n) \in {\mathbb R}$
\[HIEP\] Given $A$ over $\mathcal R\neq {\mathbb Z}$ with a Perron value $\lambda$, Handelman finds $U$ invertible over $\mathcal R$ such that $U^{-1}AU$ has positive left and right eigenvectors for $\lambda$. This matrix $U^{-1}AU$ must be eventually positive. He also exhibits an obstruction to this in the case $\mathcal R={\mathbb Z}$: if $\ell, r$ are left, right integral eigenvectors for $\lambda$, then the minimum inner product $\ell \cdot r$ does not improve with similarity, and if it is smaller than the size of $A$ then it is impossible to find $U$ invertible over ${\mathbb Z}$ such that $U^{-1}AU$ has the positive left, right eigenvectors. But, if needed, Handelman produces an SSE-${\mathbb Z}$ to a larger matrix (of smallest size possible) for which he produces the desired $U$.
\[reflectjll\] If Laffey’s formula for an upper bound on $N$ were replaced by a formula of the form $N \leq f(G,n)$, then even at $n=3$ the formula could not give a correct bound.
To show this, it suffices to exhibit a family $\{ \Lambda_{\epsilon}: 0 <
\epsilon < 1/2 \} $ of 3-tuple nonzero spectra of primitive matrices, with spectral gaps bounded away from zero, which cannot be realized by matrices of bounded size.
Set $ \Lambda_{\epsilon}
= (1, i\sqrt{(1-\epsilon)/2}, -i\sqrt{(1-\epsilon)/2})$. Each $ \Lambda_{\epsilon} $ satisfies the conditions of the Spectral Conjecture for ${\mathbb R}$, with spectral gap greater than 1/2. But, if $\Lambda_{\epsilon} $ is the nonzero spectrum of an $N\times N$ matrix, we have already seen from the JLL inequalities that $N \geq 1/\epsilon$.
\[gaprefl\] If Laffey’s formula for an upper bound on $N$ were replaced by a formula of the form $N \leq f(M,n)$, then even at $n=4$ the formula could not give a correct bound.
It suffices to find a family $\{ \Lambda_{\epsilon}: 0 < \epsilon <\epsilon_0\} $ of 4-tuple nonzero spectra of primitive matrices, with $$\inf_{\epsilon} \inf
\{ {\textnormal{trace}}( (\Lambda_{\epsilon})^k) : k\in {\mathbb N}\} \ > \ 0 \ ,$$ such that the $\Lambda_{\epsilon}$ cannot be the nonzero spectra of matrices of bounded size.
Let $\Lambda_{\epsilon} = (1, 1-\epsilon , .9i, -.9i)$, with $0< \epsilon < \epsilon_0 = . 0001$ (to avoid computation). Each $\Lambda_{\epsilon}$ is the nonzero spectrum of a primitive matrix over ${\mathbb R}$. For $\Lambda= (1,1, .9i , - .9i)$, for $n\in {\mathbb N}$, ${\textnormal{trace}}(\Lambda^{2n}) = 2$ and ${\textnormal{trace}}(\Lambda^{2n+1}) = 2 -2(.9)^n$ and therefore ${\textnormal{trace}}(\Lambda^{n}) \geq 2 -2(.9) = .2 $. With $\epsilon_0$ small enough, likewise $ \inf_{\epsilon} \inf
\{ {\textnormal{trace}}( (\Lambda_{\epsilon})^k) : k\in {\mathbb N}\} > 0 $ .
Suppose for some positive integer $K$, for each $\Lambda_{\epsilon}$ there is a nonnegative matrix $A_{\epsilon}$ of size $K\times K$ with nonzero spectrum $\Lambda_{\epsilon}$. Then by compactness, there is a subsequence of the sequence $(A_{1/n})$ which converges to a nonnegative matrix $A$. The spectrum is a continuous function of the matrix entries, so $A$ has nonzero spectrum $\Lambda= (1,1, .9i , - .9i)$. By the Perron-Frobenius spectral constraints, $A$ cannot be irreducible, and $\Lambda$ is the union of nonzero spectra of irreducible matrices, $(1)$ and $(1, .9i, -.9i)$. But $1^2 + (.9i)^2 + ( -.9i)^2= - 1.8 < 0$, a contradiction.
\[gscrefs\] The Weak Generalized Spectral Conjecture was stated in the 1991 publication [@BH91]. The Strong Generalized Spectral Conjecture was stated in the 1993 publication [@Boyle91matrices]. Although Handelman was not a coauthor of the latter paper, the Strong conjecture was a conjecture by both of us.
A brief introduction to algebraic K-theory
==========================================
Shift equivalence and strong shift equivalence are relations on sets of matrices over a semiring. When the semiring is actually a ring, this naturally lies in the realm of linear algebra over the ring. Algebraic K-theory offers many tools for such a setting[^1], so from this viewpoint, it seems natural to suspect algebraic K-theory might be useful for studying the relations of shift and strong shift equivalence. This suspicion is correct, and we will present two cases where this happens:
1. For a general ring ${\mathcal{R}}$, the refinement of SE-${\mathcal{R}}$ by SSE-${\mathcal{R}}$.
2. Wagoner’s obstruction map detecting a difference between SE-$\mathbb{Z}_{+}$ and SSE-$\mathbb{Z}_{+}$
The first is a purely algebraic problem, motivated by applications to symbolic dynamics, and to topics in algebra. The second, Wagoner’s obstruction map, is concerned with an “order” problem, and is one of two known methods to produce counterexamples to Williams’ Conjecture (discussed in Lecture 1).\
Lectures 5 and 6 will focus on addressing the first item above. Lecture 7 will discuss automorphisms of shifts of finite type, an important topic in its own right. Lecture 7 is also used partly to prepare for Lecture 8, which addresses the second item above.\
To begin, we introduce some necessary background from algebraic K-theory, relevant for Lecture 6.
$K_{1}$ of a ring ${\mathcal{R}}$ {#k1ofring}
---------------------------------
Given a ring ${\mathcal{R}}$, consider the group $GL_{n}({\mathcal{R}})$ of invertible $n \times n$ matrices over ${\mathcal{R}}$. If one wishes to understand the structure of this group, a natural question one may ask is: what is the abelianization of $GL_{n}({\mathcal{R}})$? While the answer may be fairly complicated depending on $n$ and ${\mathcal{R}}$, Whitehead, in 1950 in [@Whitehead1950], made a beautiful observation: by stabilizing, the commutator subgroup becomes more accessible.\
To describe Whitehead’s result, first let us say that by stabilizing, we mean the following.
\[def:stabilization\] For any $n$, there is a group homomorphism $$\begin{gathered}
GL_{n}({\mathcal{R}}) \hookrightarrow GL_{n+1}({\mathcal{R}})\\
A \mapsto \begin{pmatrix} A & 0 \\ 0 & 1 \end{pmatrix}\end{gathered}$$ and we define $$GL({\mathcal{R}}) = \varinjlim GL_{n}({\mathcal{R}}).$$
The group $GL({\mathcal{R}})$ is often called the *stabilized general linear group* (over the ring ${\mathcal{R}}$).\
An important collection of invertible matrices are the *elementary matrices*. A matrix $E \in GL_{n}({\mathcal{R}})$ is an elementary matrix if $E$ agrees with the identity except in at most one off-diagonal entry. The following observation may be familiar from linear algebra: if $E$ is an $n \times n$ elementary matrix and $B$ is any $n \times n$ matrix then
1. $EB$ is obtained from $B$ by an elementary row operation (adding a multiple of one row of $B$ to another row of $B$).
2. $BE$ is obtained from $B$ by an elementary column operation (adding a multiple of one column of $B$ to another column of $B$).
We define $El_{n}({\mathcal{R}})$ to be the subgroup of $GL_{n}({\mathcal{R}})$ generated by $n \times n$ elementary matrices.\
Like $GL({\mathcal{R}})$, we can also stabilize the elementary subgroups. The homomorphisms in Definition \[def:stabilization\] map $El_{n}({\mathcal{R}})$ to $El_{n+1}({\mathcal{R}})$, and we define $$El({\mathcal{R}}) = \varinjlim El_{n}({\mathcal{R}}).$$
If $X \in El({\mathcal{R}})$, then $X$ can be written as a product of elementary matrices $$X = \prod_{i=1}^{k}E_{i}.$$ It follows that, for any matrix $A \in GL({\mathcal{R}})$, $XA$ is obtained from $A$ by performing a sequence of row operations, and $AX$ is obtained from $A$ by performing a sequence of column operations.\
Note that when we write $AX$ and $XA$, $A$ and $X$ may be of different sizes. However, the process of stabilization allows us replace $A$ with $A \oplus I$ or $X$ with $X \oplus I$ as necessary to carry out the multiplication.\
The group $El({\mathcal{R}})$ turns out to be the key to analyzing the abelianization of $GL({\mathcal{R}})$.
\[thm:whitehead\] For any ring ${\mathcal{R}}$, $[GL({\mathcal{R}}),GL({\mathcal{R}})] = El({\mathcal{R}})$.
A proof of this can be found in a number of places; for example, see [@WeibelBook Chapter III]. To see why the commutator $[GL({\mathcal{R}}),GL({\mathcal{R}})]$ is contained in $El({\mathcal{R}})$, one can check that if $A \in GL_{n}({\mathcal{R}})$, then $$\begin{pmatrix} A & 0 \\ 0 & A^{-1} \end{pmatrix} = \begin{pmatrix} 1 & A \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -A^{-1} & 1 \end{pmatrix} \begin{pmatrix} 1 & A \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$ and that the last matrix in the above lies in $El({\mathcal{R}})$, so that $\begin{pmatrix} A & 0 \\ 0 & A^{-1} \end{pmatrix}$ is always in $El({\mathcal{R}})$. Now observe that we have $$\begin{pmatrix} ABA^{-1}B^{-1} & 0 \\ 0 & I \end{pmatrix} = \begin{pmatrix} A & 0 \\ 0 & A^{-1} \end{pmatrix} \begin{pmatrix} B & 0 \\ 0 & B^{-1} \end{pmatrix} \begin{pmatrix} (BA)^{-1} & 0 \\ 0 & BA \end{pmatrix}$$ so any commutator lies in $El({\mathcal{R}})$.
For a ring ${\mathcal{R}}$, the first algebraic K-group (of ${\mathcal{R}}$) is defined by $$K_{1}({\mathcal{R}}) = GL({\mathcal{R}})_{ab} = GL({\mathcal{R}}) / El({\mathcal{R}}).$$
We use $[A]$ to refer to the class of a matrix $A$ in $K_{1}({\mathcal{R}})$.\
The second equality in the above definition is precisely Whitehead’s Theorem. We note a few things regarding $K_{1}$:
1. $K_{1}({\mathcal{R}})$ is always an abelian group.
2. As noted before, multiplying a matrix $A$ by an elementary matrix from the left (resp. right) corresponds to performing an elementary row (resp. column) operation on $A$. Thus the group $K_{1}({\mathcal{R}})$ coincides with equivalence classes of (stabilized) invertible matrices over ${\mathcal{R}}$, where two matrices are equivalent if one can be obtained from the other by a sequence of elementary row and column operations.
3. The group operation in $K_{1}({\mathcal{R}})$ is, by definition, $$[A][B] = [AB]$$ where again the product $AB$ is defined because we have stabilized. However, the group operation is equivalently defined by $$[A] + [B] = \left[\begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}\right].$$ To see this, as we noted before, for any $A \in GL({\mathcal{R}})$, the matrix $\begin{pmatrix} A & 0 \\ 0 & A^{-1} \end{pmatrix}$ is in $El({\mathcal{R}})$. Since we have stabilized, we may assume that $A$ and $B$ are the same size, and $$[AB] = \left[\begin{pmatrix} A & 0 \\ 0 & I\end{pmatrix} \begin{pmatrix} B & 0 \\ 0 & I \end{pmatrix} \right]\left[\begin{pmatrix} B^{-1} & 0 \\ 0 & B \end{pmatrix}\right] = \left[\begin{pmatrix} A & 0 \\ 0 & I\end{pmatrix} \begin{pmatrix} B & 0 \\ 0 & I \end{pmatrix} \begin{pmatrix} B^{-1} & 0 \\ 0 & B \end{pmatrix}\right] = \left[\begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}\right].$$
Historically, one of Whitehead’s main motivations was to define what is now called *Whitehead torsion*. If $f \colon X \to Y$ is a homotopy equivalence between two finite CW complexes, Whitehead showed how to define a certain *torsion* class $\tau(f)$ in $K_{1}(\mathbb{Z}\pi_{1}(X))$. He showed that $f$ is a simple homotopy equivalence (one obtained through some finite sequence of elementary moves) if and only if $\tau(f) = 0$. For more on this, see [@RosenbergBook Section 2.4].\
What about computing $K_{1}({\mathcal{R}})$? In general this is a difficult problem, but there are many cases where the answer is accessible, and we’ll give some examples shortly.\
Before discussing these examples, suppose now that ${\mathcal{R}}$ is commutative. Then there is a determinant homomorphism $$\begin{gathered}
{\textnormal{det}}\colon K_{1}({\mathcal{R}}) \to {\mathcal{R}}^{\times}\\
{\textnormal{det}}([A]) = {\textnormal{det}}(A).\end{gathered}$$ The kernel of the determinant map is denoted by $$SK_{1}({\mathcal{R}}) = \ker {\textnormal{det}}.$$ Since the determinant map is surjective and right split (by identifying ${\mathcal{R}}^{\times}$ with $GL_{1}({\mathcal{R}})$), we get an exact sequence of abelian groups $$0 \to SK_{1}({\mathcal{R}}) \longrightarrow K_{1}({\mathcal{R}}) \stackrel{{\textnormal{det}}}\longrightarrow {\mathcal{R}}^{\times} \to 0$$ and $$K_{1}({\mathcal{R}}) \cong SK_{1}({\mathcal{R}}) \oplus {\mathcal{R}}^{\times}.$$
The determinant map turns out to be very useful in actually computing $K_{1}({\mathcal{R}})$; often, it is actually an isomorphism.\
Here are a few examples of $K_{1}$ for some rings.
1. When ${\mathcal{R}}$ is a field, or even a Euclidean domain, the group $SK_{1}({\mathcal{R}})$ is trivial, and $K_{1}({\mathcal{R}}) \cong {\mathcal{R}}^{\times}$. When ${\mathcal{R}}$ is a field, this is just the classical fact that, over a field, any invertible matrix $A$ can be row and column reduced to the matrix $\det A \oplus 1$. When ${\mathcal{R}}$ is a Euclidean domain, $SK_{1}({\mathcal{R}}) = 0$ as well (see [@WeibelBook Ex. 1.3.5]). Thus for example $$K_{1}(\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z} = \{1,-1\}$$ where we’ve identified $\{1,-1\}$ with the group of units in $\mathbb{Z}$.
2. If ${\mathcal{R}}$ is an integrally closed subring of a finite field extension $E$ of $\mathbb{Q}$, then $SK_{1}({\mathcal{R}}) = 0$ (this is a deep theorem of Bass, Milnor, and Serre; see [@BMS67 4.3]).
3. When $G$ is an abelian group, the integral group ring $\mathbb{Z}G$ is commutative, so $SK_{1}(\mathbb{Z}G)$ is defined. There are finite abelian groups $G$ for which $SK_{1}(\mathbb{Z}G) \ne 0$; for example, if $H = \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, then $SK_{1}(\mathbb{Z}H) \cong \mathbb{Z}/2\mathbb{Z}$ [@OliverBook Example 5.1]). In general, the calculation of $SK_{1}(\mathbb{Z}G)$ is very nontrivial (see [@OliverBook]).
This last example is especially important in topology (see [@RosenbergBook Section 4] for a brief discussion of this), and in addition, has applications to symbolic dynamics; see [@BoSc2].\
$NK_{1}({\mathcal{R}})$
-----------------------
We introduce now a certain algebraic $K$-group called $NK_{1}({\mathcal{R}})$. This group will play a key role for us later, when we discuss strong shift equivalence and shift equivalence over a ring ${\mathcal{R}}$.\
Any homomorphism of rings $f \colon {\mathcal{R}}\to {\mathcal{S}}$ induces, for each $n$, a homomorphism of groups $GL_{n}({\mathcal{R}}) \to GL_{n}({\mathcal{S}})$ and hence a group homomorphism $GL({\mathcal{R}}) \to GL({\mathcal{S}})$. The homomorphism $f$ then induces a group homomorphism on $K_{1}$ $$f_{*} \colon K_{1}({\mathcal{R}}) \to K_{1}({\mathcal{S}})$$ In fact, the assignment ${\mathcal{R}}\to K_{1}({\mathcal{R}})$ defines a functor from the category of rings to the category of abelian groups. For any ring ${\mathcal{R}}$, we may consider the ring of polynomials ${\mathcal{R}}[t]$ over ${\mathcal{R}}$, and there is a ring homomorphism $$\begin{gathered}
ev_{0} \colon {\mathcal{R}}[t] \to {\mathcal{R}}\\
p(t) \mapsto p(0).\end{gathered}$$ This induces a homomorphism on $K_{1}$ $$(ev_{0})_{*} \colon K_{1}({\mathcal{R}}[t]) \to K_{1}({\mathcal{R}})$$ and the kernel of this map is denoted by $$NK_{1}({\mathcal{R}}) = \ker \left(K_{1}({\mathcal{R}}[t]) \stackrel{(ev_{0})_{*}}\longrightarrow K_{1}({\mathcal{R}})\right).$$
Thus by definition, $NK_{1}({\mathcal{R}})$ is a subgroup of $K_{1}({\mathcal{R}}[t])$. In particular, it is always an abelian group.\
The group $NK_{1}({\mathcal{R}})$ is important in algebraic K-theory. It appears (among other places) in the Fundamental Theorem of Algebraic K-theory, relating the K-groups of ${\mathcal{R}}[t]$ and ${\mathcal{R}}[t,t^{-1}]$ to the K-groups of ${\mathcal{R}}$ (see [@WeibelBook Chapter III].\
Here are a few facts about $NK_{1}({\mathcal{R}})$:
1. If ${\mathcal{R}}$ is a Noetherian regular ring (see [@RosenbergBook Chapter 3], then $NK_{1}({\mathcal{R}}) = 0$. In particular, if ${\mathcal{R}}$ is a field, a PID, or a Dedekind domain, then $NK_{1}({\mathcal{R}}) = 0$ (see [@WeibelBook III.3.8]).
2. A theorem of Farrell [@Farrell] shows that if $NK_{1}({\mathcal{R}}) \ne 0$, then it is not finitely generated as an abelian group.
Thus, to summarize the above two items: $NK_{1}({\mathcal{R}})$ very often vanishes, but when it doesn’t vanish, it’s large (as an abelian group).\
There are rings ${\mathcal{R}}$ for which $NK_{1}({\mathcal{R}}) \ne 0$. For an easy example, take any commutative ring ${\mathcal{R}}$, and let ${\mathcal{S}}= {\mathcal{R}}[s] / (s^{2})$. Then $NK_{1}({\mathcal{S}}) \ne 0$. Indeed, over the ring ${\mathcal{S}}[t]$, the matrix $(1+st)$ is invertible, and hence we can consider its class $[(1+st)] \in K_{1}({\mathcal{S}}[t])$. Clearly $[(1+st)]$ lies in $NK_{1}({\mathcal{S}})$, and the class $[1+st]$ is nontrivial in $K_{1}({\mathcal{S}}[t])$ since $\det(1+st) \ne 1$.\
Here are some more interesting examples:
1. $NK_{1}(\mathbb{Q}[t^{2},t^{3},z,z^{-1}]) \ne 0$ (see [@SchmiedingExamples] for details on this calculation). This is a nontrivial fact: since the ring $\mathbb{Q}[t^{2},t^{3},z,z^{-1}]$ is reduced (has no nontrivial nilpotent elements), we have $NK_{1}({\mathcal{R}}) \subset SK_{1}({\mathcal{R}}[t])$ (see Exercise \[exer:reducednk1sk1\] below), and often it is not easy to determine whether $SK_{1}$ vanishes[^2].
2. There are finite groups $G$ for which $NK_{1}(\mathbb{Z}G) \ne 0$; for example, for $G = \mathbb{Z}/4\mathbb{Z}$, $NK_{1}(\mathbb{Z}[\mathbb{Z}/4\mathbb{Z}]) \ne 0$ (details for this particular $G$ can be found in [@Weibelnk1calc]).
See [@BoSc3] for an application of the example $(1)$ above. The example $(2)$ above of integral group rings of finite groups is relevant for applications to symbolic dynamics (see [@BoSc2]). In general, the calculation of $NK_{1}(\mathbb{Z}G)$ for $G$ a finite group is complicated, and not fully known (see e.g. [@Harmon1987], [@Weibelnk1calc]).\
The following is a very useful tool for studying $NK_{1}({\mathcal{R}})$. The result is often referred to as Higman’s Trick.
\[lemma:higmantrick\] Let ${\mathcal{R}}$ be a ring and let $A$ be a matrix in $GL({\mathcal{R}}[t])$ such that $[A] \in NK_{1}({\mathcal{R}})$. Then there exists a nilpotent matrix $N$ over ${\mathcal{R}}$ such that $[A] = [I-tN]$ in $NK_{1}({\mathcal{R}})$.
Use the fact that we are in the stabilized setting to kill off powers of $t$ from $A$ using elementary operations, arriving at a matrix of the form $A_{0}+A_{1}t$. Since $[A] \in NK_{1}({\mathcal{R}})$, $[A_{0}] = 0 \in K_{1}({\mathcal{R}})$, so $[A] = [I+B_{1}t]$ for some $B_{1}$ over ${\mathcal{R}}$. Since the matrix $I+B_{1}t$ is invertible over ${\mathcal{R}}[t]$, $B_{1}$ must be nilpotent.
A more detailed proof of Theorem \[lemma:higmantrick\] may be found in [@WeibelBook III.3.5.1].
\[exer:reducednk1sk1\] [(Ap. ]{}[appexer:reducednk1sk1]{}) Suppose ${\mathcal{R}}$ is a commutative ring which is reduced, i.e. ${\mathcal{R}}$ has no nontrivial nilpotent elements. Then $NK_{1}({\mathcal{R}}) \subset SK_{1}({\mathcal{R}}[t])$.
\[exer:nk1vanishpid\] [(Ap. ]{}[appexer:nk1vanishpid]{}) If ${\mathcal{R}}$ is a principal ideal domain, then $NK_{1}({\mathcal{R}}) = 0$.
$Nil_{0}({\mathcal{R}})$ {#sec:subsecnil0R}
------------------------
Higman’s Trick suggests there is a connection between the group $NK_{1}({\mathcal{R}})$ and the structure of nilpotent matrices over the ring ${\mathcal{R}}$. This is indeed the case, and we’ll describe this relationship quite explicitly in this subsection [(Ap. ]{}[rem:nilviewlocalization]{}). To begin, we first define another group coming from algebraic K-theory. This group is often called the *class group of the category of nilpotent endomorphisms over ${\mathcal{R}}$*. That’s quite a long name, and we usually just call it “nil zero (of ${\mathcal{R}}$)”, since it’s denoted by $Nil_{0}({\mathcal{R}})$.\
\[def:nil0def1\] Let ${\mathcal{R}}$ be a ring. Define $Nil_{0}({\mathcal{R}})$ to be the free abelian group on the set of generators $$\{[N] \mid N \emph{ \textnormal{is a nilpotent matrix over} } {\mathcal{R}}\}$$ together with the following relations:
1. $[N_{1}] = [N_{2}]$ if $N_{1} = P^{-1}N_{2}P$ for some $P \in GL({\mathcal{R}})$.
2. $[N_{1}] + [N_{2}] = \left[ \begin{pmatrix} N_{1} & B \\ 0 & N_{2} \end{pmatrix} \right]$ for any matrix $B$ over ${\mathcal{R}}$.
3. $[0] = 0$.
Where does the group $Nil_{0}({\mathcal{R}})$ come from? First let us recall some definitions. Consider the category $\textbf{Nil}{\mathcal{R}}$ whose objects are pairs $(P,f)$ where $P$ is a finitely generated projective ${\mathcal{R}}$-module and $f$ is a nilpotent endomorphism of $P$, and where a morphism from $(P,f)$ to $(Q,g)$ is given by an ${\mathcal{R}}$-module homomorphism $\alpha \colon P \to Q$ for which the square $$\xymatrix{
P \ar[r]^{f} \ar[d]_{\alpha} & P \ar[d]^{\alpha} \\
Q \ar[r]^{g} & Q \\
}$$ commutes. The category $\textbf{Nil}{\mathcal{R}}$ has a notion of exact sequence by defining $$(P_{1},f_{1}) \to (P_{2},f_{2}) \to (P_{3},f_{3})$$ to be exact if the corresponding sequence of ${\mathcal{R}}$-modules $$P_{1} \to P_{2} \to P_{3}$$ is exact, i.e. ${\textnormal{Image}}(P_{1} \to P_{2}) = \ker(P_{2} \to P_{3})$ (see [(Ap. ]{}[rem:exactcategory]{}) regarding how $\textbf{Nil}{\mathcal{R}}$ with this notion of exact sequence fits into a more general setting). Given this, define $K_{0}(\textbf{Nil}{\mathcal{R}})$ to be the free abelian group on isomorphism classes of objects $(P,f)$ in $\textbf{Nil}{\mathcal{R}}$, together with the relation: $$\begin{gathered}
[(P_{1},f_{1})] + [(P_{3},f_{3})] = [(P_{2},f_{2})] \\
\textnormal{ whenever } \\
0 \to (P_{1},f_{1}) \to (P_{2},f_{2}) \to (P_{3},f_{3}) \to 0\\
\textnormal{ is exact. }\end{gathered}$$ Let $\textbf{Proj}{\mathcal{R}}$ denote the category of finitely generated projective ${\mathcal{R}}$-modules and consider the standard notion of an exact sequence in $\textbf{Proj}{\mathcal{R}}$. We can likewise define the group $K_{0}(\textbf{Proj}{\mathcal{R}})$ to be the free abelian group on isomorphism classes of objects in $\textbf{Proj}{\mathcal{R}}$ with the similar relations: $$\begin{gathered}
[P_{1}] + [P_{3}] = [P_{2}] \\
\textnormal{ whenever } \\
0 \to P_{1} \to P_{2} \to P_{3} \to 0 \\
\textnormal{ is exact in } \textbf{Proj}{\mathcal{R}}.\end{gathered}$$ These relations are equivalent to the set of relations $$[P_{1}] + [P_{2}] = [P_{1} \oplus P_{2}], \qquad P_{1},P_{2} \textnormal{ in } \textbf{Proj}{\mathcal{R}}$$ since any exact sequence of projective ${\mathcal{R}}$-modules splits. Thus $K_{0}(\textbf{Proj}{\mathcal{R}})$ is isomorphic to the group completion of the abelian monoid of isomorphism classes of finitely generated projective ${\mathcal{R}}$-modules under direct sum, which is often given as the definition of the group $K_{0}({\mathcal{R}})$.\
There is a functor $\textbf{Nil}{\mathcal{R}}\to \textbf{Proj}{\mathcal{R}}$ given by $(P,f) \mapsto P$, and this functor respects exact sequences, so there is an induced map on the level of the $K_{0}$ groups defined above $$K_{0}(\textbf{Nil}{\mathcal{R}}) \to K_{0}(\textbf{Proj}{\mathcal{R}}).$$ The kernel of this map is isomorphic to $Nil_{0}({\mathcal{R}})$ (details of this isomorphism can be found in [@WeibelBook Chapter II]).\
The following formalizes the connection between $NK_{1}({\mathcal{R}})$ and nilpotent matrices over ${\mathcal{R}}$.
\[thm:nil0nk1iso\] The map $$\label{eqn:nil0nk1iso}
\begin{gathered}
\Psi \colon Nil_{0}({\mathcal{R}}) \to NK_{1}({\mathcal{R}})\\
\Psi \colon [N] \mapsto [I-tN]
\end{gathered}$$ is an isomorphism of abelian groups.
[(Ap. ]{}[appexer:psiwelldefined]{}) Show the map $\Psi$ defined in is a well-defined group homomorphism.
Towards showing $\Psi$ is an isomorphism, given Higman’s Theorem \[lemma:higmantrick\] above, one obvious thing to try is to define an inverse map $$\label{eqn:nk1nil0}
\begin{gathered}
NK_{1}({\mathcal{R}}) \to Nil_{0}({\mathcal{R}})\\
[I-tN] \mapsto [N].
\end{gathered}$$
This in fact works: this map turns out to be well-defined, and is an inverse to the map $\Psi$. This is classically done, in algebraic K-theory, using a fair amount of machinery and long exact sequences coming from localization results (e.g. [@WeibelBook III.3.5.3]). Later we will see there is an alternative, more elementary, proof using strong shift equivalence theory.\
We will make frequent use of the isomorphism above in later lectures.
[(Ap. ]{}[appexer:uppertrivanish]{}) Consider an upper triangular matrix $N$ over ${\mathcal{R}}$ with zero diagonal. Then $I-tN$ lies in $El({\mathcal{R}}[t])$, and hence $[I-tN] = 0$ in $NK_{1}({\mathcal{R}})$. Using the relations defining $Nil_{0}({\mathcal{R}})$, show the class of such an $N$ must be zero in $Nil_{0}({\mathcal{R}})$.
The isomorphism $NK_{1}({\mathcal{R}}) \cong Nil_{0}({\mathcal{R}})$ is only one instance of a larger phenomenon, which, loosely speaking, relates the K-theory of polynomial rings ${\mathcal{R}}[t]$ (in fact, certain localizations of them) to the K-theory of endomorphisms over the ring ${\mathcal{R}}$ [(Ap. ]{}[rem:ktheoryendomorphisms]{}). The strong shift equivalence theory also fits nicely into this framework, and we’ll describe this in a little more detail later.
$K_{2}$ of a ring ${\mathcal{R}}$ {#subsec:k2ofaring}
---------------------------------
This short subsection gives a definition and a few very basic properties of the group $K_{2}$ of a ring, motivated by its appearance later in Lecture 8. For a more thorough introduction to $K_{2}$, see either [@MilnorBook] or [@WeibelBook III. Sec. 5].\
Roughly speaking, $K_{2}({\mathcal{R}})$ measures the existence of “extra relations” among elementary matrices over ${\mathcal{R}}$. We’ll make this more formal below, but the idea is that elementary matrices always satisfy a certain collection of relations which do not depend on the ring. The group $K_{2}({\mathcal{R}})$ is a way to detect additional relations coming from the ring.\
Let ${\mathcal{R}}$ be a ring. Given $n \ge 1$ and $1 \le i \ne j \le n$, let $e_{i,j}(r)$ denote the matrix which has $r$ in the $i,j$ entry, and agrees with the identity matrix everywhere else. Recall the group $El_{n}({\mathcal{R}})$ of $n \times n$ elementary matrices over ${\mathcal{R}}$ is generated by matrices $e_{i,j}(r)$, $i \ne j$. It is straightforward to check that $El_{n}({\mathcal{R}})$ always satisfies certain relations: for any $r, s \in {\mathcal{R}}$, we have
1. $e_{i,j}(r)e_{i,j}(s) = e_{i,j}(r+s)$.
2. $[e_{i,j}(r),e_{k,l}(s)] =
\begin{cases}
1 & \mbox{if } i \ne l \mbox{ and } j \ne k\\
e_{i,l}(rs) & \mbox{if } i \ne l \mbox{ and } j = k\\
e_{k,j}(-sr) & \mbox{if } j \ne k \mbox{ and } i = l.
\end{cases}$
The key here is that these relations are satisfied by $El_{n}({\mathcal{R}})$ for *every* ring. This perhaps motivates defining the following group:
Let ${\mathcal{R}}$ be a ring and $n \ge 3$. The $n$th Steinberg group $St_{n}({\mathcal{R}})$ has generators $x_{i,j}(r)$, where $1 \le i \ne j \le n$ and $r \in {\mathcal{R}}$, and relations:
1. $x_{i,j}(r)x_{i,j}(s) = x_{i,j}(r+s)$.
2. $[x_{i,j}(r),x_{k,l}(s)] =
\begin{cases}
1 & \mbox{if } i \ne l \mbox{ and } j \ne k\\
x_{i,l}(rs) & \mbox{if } i \ne l \mbox{ and } j = k\\
x_{k,j}(-sr) & \mbox{if } j \ne k \mbox{ and } i = l.
\end{cases}$
The map $$x_{i,j}(r) \mapsto e_{i,j}(r)$$ defines a surjective group homomorphism $$\theta_{n} \colon St_{n}({\mathcal{R}}) \to El_{n}({\mathcal{R}}).$$
The relations for $St_{n}({\mathcal{R}})$ and $St_{n+1}({\mathcal{R}})$ imply there is a well-defined group homomorphism $$\begin{gathered}
St_{n}({\mathcal{R}}) \to St_{n+1}({\mathcal{R}})\\
x_{ij}(r) \mapsto x_{ij}(r)\\
\end{gathered}$$ and we define $$St({\mathcal{R}}) = \varinjlim St_{n}({\mathcal{R}})$$ and assemble the $\theta_{n}$’s to get a group homomorphism $$\theta \colon St({\mathcal{R}}) \to El({\mathcal{R}}).$$
Finally, we define $$K_{2}({\mathcal{R}}) = \ker \theta.$$
It turns out the sequence $$K_{2}({\mathcal{R}}) \to St({\mathcal{R}}) \to El({\mathcal{R}})$$ is the universal central extension of the group $El({\mathcal{R}})$. The group $K_{2}({\mathcal{R}})$ is precisely the center of $St({\mathcal{R}})$, and so is always abelian. Furthermore, the assignment ${\mathcal{R}}\to K_{2}({\mathcal{R}})$ is functorial; see [@WeibelBook III, Sec.5] for more details on this.\
An observation we’ll make use of later is the following. An expression of the form $$\prod_{i=1}^{k}E_{i} = 1$$ where $E_{i}$ are elementary matrices can be used to produce an element of $K_{2}({\mathcal{R}})$: lift each $E_{i}$ to some $x_{i}$ in $St({\mathcal{R}})$ and consider $$x = \prod_{i}^{k}x_{i} \in St({\mathcal{R}}).$$ Then $x \in K_{2}({\mathcal{R}})$, although in general this element may depend on the choice of lifts.\
Let ${\mathcal{R}}= \mathbb{Z}$, and consider $$E = e_{1,2}(1)e_{2,1}(-1)e_{1,2}(1) = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.$$ One can check directly that $$E^{4} = I$$ so we can consider the element of $K_{2}(\mathbb{Z})$ $$x = \big(x_{1,2}(1)x_{2,1}(-1)x_{1,2}(1)\big)^{4}.$$ Milnor in [@MilnorBook Sec. 10] proves that $x$ is nontrivial in $K_{2}(\mathbb{Z})$, $x^{2} = 1$, and $x$ is actually the only nontrivial element of $K_{2}(\mathbb{Z})$. Thus we have $$K_{2}(\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}.$$
It turns out (see [@WeibelBook Chapter V]) that $K_{2}(\mathbb{Z}[t]) \cong K_{2}(\mathbb{Z})$. Given $m \ge 1$, there is a split surjection $K_{2}(\mathbb{Z}[t]/(t^{m})) \to K_{2}(\mathbb{Z})$, and we can define the group $K_{2}(\mathbb{Z}[t]/(t^{m}),(t))$ to be the kernel of this split surjection. In [@vdK], van der Kallen proved that $K_{2}(\mathbb{Z}[t]/(t^{2}),(t)) \cong \mathbb{Z}/2\mathbb{Z}$, a fact which will prove to be useful later in Lecture 8. More generally, the following was proved by Geller and Roberts.
\[thm:k2calctruncatedpoly\] For any $m \ge 2$, the group $K_{2}(\mathbb{Z}[t]/(t^{m}),(t))$ is isomorphic to $\bigoplus_{k=2}^{m}\mathbb{Z}/k\mathbb{Z}$.
Appendix 5
----------
This appendix contains some remarks, proofs, and solutions of exercises for Lecture 5.
\[appexer:reducednk1sk1\] Suppose ${\mathcal{R}}$ is a commutative ring which is reduced, i.e. ${\mathcal{R}}$ has no nontrivial nilpotent elements. Then $NK_{1}({\mathcal{R}}) \subset SK_{1}({\mathcal{R}}[t])$.
If ${\mathcal{R}}$ is commutative and reduced, the only units in ${\mathcal{R}}[t]$ are degree zero. Thus for a nilpotent matrix $N$ over ${\mathcal{R}}$, since $I-tN$ is invertible, $\det(I-tN)$ must be 1, so together with Higman’s Trick (Theorem \[lemma:higmantrick\]), we have $NK_{1}({\mathcal{R}}) \subset SK_{1}({\mathcal{R}}[t])$.
\[appexer:nk1vanishpid\] If ${\mathcal{R}}$ is a principal ideal domain, then $NK_{1}({\mathcal{R}}) = 0$.
By Higman’s Trick (Theorem \[lemma:higmantrick\]), it suffices to show that if $N$ is a nilpotent matrix over ${\mathcal{R}}$ then $[I-tN] = 0$ in $K_{1}({\mathcal{R}}[t])$. Given $N$ nilpotent, by Theorem \[thm:PIDblocktri\] from Lecture 2, there exists some $P \in GL({\mathcal{R}})$ such that $P^{-1}NP$ is upper triangular with zero diagonal. Then $$[I-tN] = [P^{-1}(I-tN)P] = [I-t(P^{-1}NP)].$$ in $K_{1}({\mathcal{R}}[t])$. Since $P^{-1}NP$ is upper triangular with zero diagonal, $I-t(P^{-1}NP)$ lies in $El({\mathcal{R}}[t])$, and hence $[I-t(P^{-1}NP)] = 0$ in $K_{1}({\mathcal{R}}[t])$.
\[appexer:psiwelldefined\] The map $$\begin{gathered}
\Psi \colon Nil_{0}({\mathcal{R}}) \to NK_{1}({\mathcal{R}}[t])\\
\Psi \colon [N] \mapsto [I-tN]
\end{gathered}$$ is a well-defined group homomorphism.
Since $[I-tN_{1}] + [I-tN_{2}] = [I-t(N_{1} \oplus N_{2})] = [(I-tN_{1}) \oplus (I-tN_{2})]$ in $K_{1}({\mathcal{R}}[t])$, $\Psi$ respects the group operations. To see it is well-defined it suffices to check $\Psi$ on the relations for $Nil_{0}({\mathcal{R}})$. For the first relation of $Nil_{0}({\mathcal{R}})$, if $N$ is a nilpotent matrix over ${\mathcal{R}}$ and $P \in GL({\mathcal{R}})$ then $$[I-tN] = [P^{-1}(I-tN)P] = [I-t(P^{-1}NP)]$$ in $NK_{1}({\mathcal{R}})$. For the second relation, suppose $N_{1},N_{2}$ are nilpotent matrices and $B$ is some matrix over ${\mathcal{R}}$ and consider $$\begin{pmatrix} I-tN_{1} & -tB \\ 0 & I - tN_{2} \end{pmatrix}.$$ Since $I-tN_{2}$ is invertible over ${\mathcal{R}}[t]$, we can consider the block matrix in $El({\mathcal{R}}[t])$ given by $$E = \begin{pmatrix} I & tB(I-tN_{2})^{-1} \\ 0 & I \end{pmatrix}.$$ Then $$E \begin{pmatrix} I-tN_{1} & -tB \\ 0 & I - tN_{2} \end{pmatrix} = \begin{pmatrix} I-tN_{1} & 0 \\ 0 & I-tN_{2} \end{pmatrix}$$ so the second relation is preserved by $\Psi$. The third relation is obvious.
\[appexer:uppertrivanish\] Consider an upper triangular matrix $N$ over ${\mathcal{R}}$ with zero diagonal. Then $I-tN$ lies in $El({\mathcal{R}}[t])$, and hence $[I-tN] = 0$ in $NK_{1}({\mathcal{R}})$. Using the relations defining $Nil_{0}({\mathcal{R}})$, show the class of such an $N$ must be zero in $Nil_{0}({\mathcal{R}})$.
If $N$ is size one or two then this is immediate from relation (2) in the definition of $Nil_{0}({\mathcal{R}})$. Now if $N$ is upper triangular of size $n \ge 2$ with zero diagonal, then there is some matrix $B$ such that $$N = \begin{pmatrix} N_{1} & B \\ 0 & 0 \end{pmatrix}$$ where $N_{1}$ is upper triangular of size $n-1$ with zero diagonal. Now use relation $(2)$ of $Nil_{0}({\mathcal{R}})$ and induction.
\[rem:nilviewlocalization\] For a more abstract viewpoint, the connection between $NK_{1}$ and the class group $Nil_{0}({\mathcal{R}})$ of nilpotent endomorphisms over ${\mathcal{R}}$ essentially comes from the localization sequence in algebraic K-theory, together with identifying the category of ${\mathcal{R}}[t]$-modules of projective dimension less than or equal to 1 which are $t$-torsion (i.e. are annihilated by $t^{k}$ for some $k$) with the category of pairs $(P,f)$ where $P$ is a finitely generated projective ${\mathcal{R}}$-module and $f$ is a nilpotent endomorphism of $P$; see [@WeibelBook Chapter III] for more on this viewpoint.
\[rem:exactcategory\] The category $\textbf{Nil}{\mathcal{R}}$ equipped with the notion of exact sequence as defined here is a particular case of the more general concept, introduced by Quillen, of an *exact category*, a category equipped with some notion of exact sequences which satisfy some conditions. Such a category has enough structure to define $K$-groups of the category; our definition of $K_{0}(\textbf{Nil}{\mathcal{R}})$ coincides with $K_{0}$ of the exact category $\textbf{Nil}{\mathcal{R}}$. See [@WeibelBook II Sec. 7] for details regarding this viewpoint.
\[rem:ktheoryendomorphisms\] One may also define a class group for endomorphisms over a ring ${\mathcal{R}}$. Define $\textbf{End}{\mathcal{R}}$ to be the category whose objects are pairs $(P,f)$ where $P$ is a finitely generated projective ${\mathcal{R}}$-module and $f \colon P \to P$ is an endomorphism, and a morphism $(P,f) \to (Q,g)$ is given by an ${\mathcal{R}}$-module homomorphism $h \colon P \to Q$ such that $hf = gh$. Analogous to $\textbf{Nil}{\mathcal{R}}$, we call a sequence $$(P_{1},f_{1}) \to (P_{2},f_{2}) \to (P_{3},f_{3})$$ in $\textbf{End}{\mathcal{R}}$ exact if the associated sequence of ${\mathcal{R}}$-modules $$P_{1} \to P_{2} \to P_{3}$$ is exact. Then $K_{0}(\textbf{End}{\mathcal{R}})$ is defined to be the free abelian group on isomorphism classes of objects $(P,f)$ in $\textbf{End}{\mathcal{R}}$ together with the relations $$\begin{gathered}
{[(P_{1},f_{1})]} + [(P_{3},f_{3})] = [(P_{2},f_{2})]\\
\textnormal{whenever}\\
0 \to (P_{1},f_{1}) \to (P_{2},f_{2}) \to (P_{3},f_{3}) \to 0\\
\textnormal{is exact in } \textbf{End}{\mathcal{R}}.
\end{gathered}$$ There is a forgetful functor $\textbf{End}{\mathcal{R}}\to \textbf{Proj}{\mathcal{R}}$ given by $(P,f) \mapsto P$ and an induced group homomorphism on the level of $K_{0}$ $$\begin{gathered}
K_{0}(\textbf{End}{\mathcal{R}}) \to K_{0}(\textbf{Proj}{\mathcal{R}})\\
[(P,f)] \mapsto [P].
\end{gathered}$$ Now define $End_{0}({\mathcal{R}})$ to be the kernel of this homomorphism. The group $End_{0}({\mathcal{R}})$ has a presentation analogous to the one given in Definition \[def:nil0def1\]: $End_{0}({\mathcal{R}})$ is the free abelian group on the set of generators $$\{[A] \mid A \textnormal{ is a square matrix over } {\mathcal{R}}\}$$ together with the relations
1. $[A_{1}] = [A_{2}]$ if $A_{1} = P^{-1}A_{2}P$ for some $P \in GL({\mathcal{R}})$.
2. $[A_{1}] + [A_{2}] = \left[ \begin{pmatrix} A_{1} & B \\ 0 & A_{2} \end{pmatrix} \right]$ for any matrix $B$ over ${\mathcal{R}}$.
3. $[0] = 0$.
There is an equivalence relation on square matrices over ${\mathcal{R}}$ defined by $A \sim_{end} B$ if $[A]=[B]$ in $End_{0}({\mathcal{R}})$. A natural question is how this relation compares to the relations of strong shift equivalence and shift equivalence over ${\mathcal{R}}$. In fact, this is settled in the commutative case by the following theorem of Almkvist (which was also proved, and greatly generalized, by Grayson in [@Grayson1977]). In the theorem, for ${\mathcal{R}}$ commutative we let $\tilde{{\mathcal{R}}}$ denote the multiplicative subgroup of $1+t{\mathcal{R}}[[t]]$ given by $$\tilde{{\mathcal{R}}} = \left\{\frac{p(t)}{q(t)} \mid p(t), q(t) \in {\mathcal{R}}[t] \textnormal{ and } p(0)=q(0)=1 \right\}.$$
\[thm:almkvistend0\] Let ${\mathcal{R}}$ be a commutative ring. The map $$\label{eqn:end0iso}
\begin{gathered}
End_{0}({\mathcal{R}}) \to \tilde{{\mathcal{R}}}\\
[A] \mapsto \det(I-tA)
\end{gathered}$$ is an isomorphism.
There is an extension of Theorem \[thm:almkvistend0\] to general (i.e. not necessarily commutative) rings due to Sheiham [@SheihamWhitehead], but we do not have room to state it here.\
As a consequence of the theorem, if $\mathcal R$ is an integral domain then the relation $\sim_{end}$ is coarser than shift equivalence over ${\mathcal{R}}$. For example, when ${\mathcal{R}}= \mathbb{Z}$ and $A$ over $\mathbb{Z}_{+}$ presents a shift of finite type $(X_{A},\sigma_{A})$, knowing the class $[A]$ in $End_{0}(\mathbb{Z})$ is the same as knowing the zeta function $\zeta_{\sigma_{A}}(t)$. Also (see Section \[detI-tAsubsection\]), $\det(I-tA)$ is an invariant of SSE-$\mathcal R$ for any commutative ring $\mathcal R$, but there are comutative rings for which the trace is not an invariant of shift equivalence, and for such a ring $\mathcal R$, SE-$\mathcal R$ does not refine $\sim_{end}$.\
For a symbolic system presented by a matrix $A$ over a noncommutative ring (for example, the integral group ring $\mathbb{Z}G$ where $G$ is nonabelian), Theorem \[thm:almkvistend0\] suggests the class $[A]$ in $End_{0}({\mathcal{R}})$ can serve as an analogue of the zeta function of the symbolic system presented by $A$.
The algebraic K-theoretic characterization of the refinement of strong shift equivalence over a ring by shift equivalence
=========================================================================================================================
Let ${\mathcal{R}}$ be a semiring. Recall that square matrices $A,B$ are *elementary strong shift equivalent over ${\mathcal{R}}$* (ESSE-${\mathcal{R}}$ for short, denoted ${{A} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} esse-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} esse-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} B \end{tiny}}$) if there exists matrices $R,S$ over ${\mathcal{R}}$ such that $$A=RS, \hspace{.1in} B=SR.$$
Recall also from Lecture 2 the following two equivalence relations defined on the collection of square matrices over ${\mathcal{R}}$:
1. Square matrices $A$ and $B$ are *strong shift equivalent over ${\mathcal{R}}$* (SSE-${\mathcal{R}}$ for short, denoted ${{A} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} sse-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} sse-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} B \end{tiny}}$) if there exists a chain of elementary strong shift equivalences over ${\mathcal{R}}$ from $A$ to $B$: $$A = {{A_{0}} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} esse-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} esse-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} A_{1} \end{tiny}} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} esse-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} esse-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} \end{tiny} \cdots \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} esse-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} esse-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} \end{tiny} {{A_{n-1}} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} esse-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} esse-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} A_{n} \end{tiny}} = B.$$
2. Square matrices $A$ and $B$ are *shift equivalent over ${\mathcal{R}}$* (SE-${\mathcal{R}}$ for short, denoted ${{A} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} se-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} se-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} B \end{tiny}}$) if there exists matrices $R,S$ over ${\mathcal{R}}$ and a number $l \in \mathbb{N}$ such that $$\begin{gathered}
A^{l} = RS, \hspace{.1in} B^{l} = SR\\
AR = RB, \hspace{.1in} BS = SA.\end{gathered}$$
For an $n \times n$ square matrix $A$ over ${\mathcal{R}}$ there is an ${\mathcal{R}}$-module endomorphism ${\mathcal{R}}^{n} \to {\mathcal{R}}^{n}$ given by $x \mapsto xA$ and we can form the direct limit ${\mathcal{R}}$-module $$G_{A} = \varinjlim\{{\mathcal{R}}^{n},x \mapsto xA\}.$$ This was introduced in the case ${\mathcal{R}}= \mathbb{Z}$ in Section \[sec:SEZdirectlimits\] of Lecture 2. The ${\mathcal{R}}$-module $G_{A}$ becomes an ${\mathcal{R}}[t,t^{-1}]$-module by defining $x \cdot t^{-1} = xA$. The following result was given in Proposition \[prop:sezdirectlimit\] of Lecture 2 in the case ${\mathcal{R}}= \mathbb{Z}$; the proof there (which is given in [(Ap. ]{}[seandpairiso]{})) carries over to here.
For square matrices $A,B$ over ${\mathcal{R}}$, we have $$\begin{gathered}
{{A} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} se-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} se-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} B \end{tiny}}\\
\textnormal{ if and only if }\\
G_{A} \textnormal{ and } G_{B} \textnormal{ are isomorphic as } {\mathcal{R}}[t,t^{-1}]-\textnormal{modules}.\end{gathered}$$
This proposition shows that shift equivalence over a ring ${\mathcal{R}}$ has a nice classical algebraic interpretation.
Comparing shift equivalence and strong shift equivalence over a ring {#subsec:comparingseandsse}
--------------------------------------------------------------------
Recall that for any semiring ${\mathcal{R}}$ and square matrices $A,B$ over ${\mathcal{R}}$, $${{A} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} sse-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} sse-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} B \end{tiny}} \Longrightarrow {{A} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} se-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} se-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} B \end{tiny}}.$$
Lectures 1 and 2 discussed various aspects of both shift equivalence and strong shift equivalence, especially in the central case of ${\mathcal{R}}= \mathbb{Z}_{+}$ and $\mathbb{Z}$. Recall Conjecture \[conj:williams\] from Lecture 1:
If $A$ and $B$ are square matrices over $\mathbb{Z}_{+}$ which are shift equivalent over $\mathbb{Z}_{+}$, then $A$ and $B$ are strong shift equivalent over $\mathbb{Z}_{+}$.
There are counterexamples to Williams’ Conjecture; we’ll discuss some of this in Lecture 8. We can generalize the conjecture in the obvious way to arbitrary semirings, and rephrase as a more general problem:
Suppose ${\mathcal{R}}$ is a semiring, and $A,B$ are square matrices over ${\mathcal{R}}$. If $A$ and $B$ are shift equivalent over ${\mathcal{R}}_{+}$, must $A$ and $B$ be strong shift equivalent over ${\mathcal{R}}_{+}$?
Williams’ original Shift Equivalence Conjecture concerns the case ${\mathcal{R}}= \mathbb{Z}_{+}$, and is most immediately linked to shifts of finite type, through its relation to topological conjugacy (as discussed in Lecture 1). It turns out, even in the case ${\mathcal{R}}= \mathbb{Z}_{+}$ the answer to Williams Problem is ‘not always’. We will talk more about this in Lecture 4, but for now let us consider the following picture, which outlines how the General Williams’ Problem can be approached:
Looking at the picture above, Williams’ Problem concerns the top arrow. The picture describes how the problem can be broken down into a few parts: an ‘algebra’ part (${\tikz[baseline=(char.base)]{
\node[shape=circle, draw, inner sep=1pt,
minimum height=12pt] (char) {2};}}$ in the picture), and two ‘order’ parts (${\tikz[baseline=(char.base)]{
\node[shape=circle, draw, inner sep=1pt,
minimum height=12pt] (char) {1};}}$ and ${\tikz[baseline=(char.base)]{
\node[shape=circle, draw, inner sep=1pt,
minimum height=12pt] (char) {3};}}$ in the picture). In key cases, the answer to ${\tikz[baseline=(char.base)]{
\node[shape=circle, draw, inner sep=1pt,
minimum height=12pt] (char) {1};}}$ is yes for a fundamental subclass of matrices over ${\mathcal{R}}_{+}$. Recall, for ${\mathcal{R}}\subset \mathbb{R}$, a matrix $A$ is primitive if there exists $k$ such that $A^{k}$ has all positive entries. Then as shown in Proposition \[sePrimitive\] in Lecture 2, we have:
Suppose ${\mathcal{R}}\subset \mathbb{R}$ and ${\mathcal{R}}_{+} = {\mathcal{R}}\cap \mathbb{R}_{+}$. If $A$ and $B$ are primitive matrices over ${\mathcal{R}}$, then $A$ and $B$ are SE-${\mathcal{R}}$ if and only if they are shift equivalent over ${\mathcal{R}}_{+}$.
This result says that, when the ring is a subring of $\mathbb{R}$, we can reduce the question of SE-${\mathcal{R}}_{+}$ of primitive matrices to the purely algebraic question of SE-${\mathcal{R}}$.\
Part ${\tikz[baseline=(char.base)]{
\node[shape=circle, draw, inner sep=1pt,
minimum height=12pt] (char) {2};}}$ is the main topic of this and the next lecture. Part ${\tikz[baseline=(char.base)]{
\node[shape=circle, draw, inner sep=1pt,
minimum height=12pt] (char) {3};}}$, in the case of ${\mathcal{R}}= \mathbb{Z}_{+}$, we will discuss in Lecture 4, and contains the remaining core of Williams’ Problem.
The algebraic shift equivalence problem
---------------------------------------
We consider now ${\tikz[baseline=(char.base)]{
\node[shape=circle, draw, inner sep=1pt,
minimum height=12pt] (char) {2};}}$, which we can restate as:
\[algseproblem\] Let ${\mathcal{R}}$ be a ring and $A,B$ be square matrices over ${\mathcal{R}}$. If $A$ and $B$ are shift equivalent over ${\mathcal{R}}$, must $A$ and $B$ be strong shift equivalent over ${\mathcal{R}}$?
Williams gave an argument in [@Williams1970 Lemma 4.6] (which needed an additional step, later given in [@Williams1992]) showing that, when ${\mathcal{R}}= \mathbb{Z}$, the answer to Problem \[algseproblem\] is yes. Effros also gave a similar argument, in an unpublished work, in the case ${\mathcal{R}}= \mathbb{Z}$ , and it was observed in [@BH93] that both arguments work in the case ${\mathcal{R}}$ is a principal ideal domain. It was then shown by Boyle and Handelman [@BH93] that the answer to Problem \[algseproblem\] is also yes when ${\mathcal{R}}$ is a Dedekind domain. The Boyle-Handelman paper [@BH93] was published in 1993, and after that point no further progress was made; in fact, it was still not known whether the answer to Problem \[algseproblem\] might be yes for *every* ring. Now, from recent work [@BoSc1], we know the answer to Problem \[algseproblem\] is not always yes, and we have a pretty satisfactory characterization (Corollary \[cor:nk1vanishandsse\]) of the rings ${\mathcal{R}}$ for which the relations SE-${\mathcal{R}}$ and SSE-${\mathcal{R}}$ are the same. It turns out to depend on some K-theoretic properties of the ring ${\mathcal{R}}$ in question, and we’ll spend the remainder of the lecture discussing how this works.\
In short, the answer to Problem \[algseproblem\] turns out to depend on the group $NK_{1}({\mathcal{R}})$. Before getting into the precise statements, recall from Proposition \[prop:MallerShub\] in Lecture 2 that SSE-${\mathcal{R}}$ is the relation generated by similarity and extensions by zero. Since the direct limit module associated to a nilpotent matrix is clearly trivial, it is reasonable to suspect that determining the strong shift equivalence classes of nilpotent matrices is connected to determining which nilpotent matrices over the ring can be obtained from the zero matrix (up to similarity) by extensions by zero. In fact this is the case, and the question of which nilpotent matrices over the ring can be obtained from the zero matrix (up to similarity) by extensions by zero turns out to be governed by $Nil_{0}({\mathcal{R}})$.\
Fix now a ring ${\mathcal{R}}$. For a matrix $A$ over ${\mathcal{R}}$, we let $[A]_{sse}$, $[A]_{se}$ denote the strong shift equivalence (respectively shift equivalence) class of $A$ over ${\mathcal{R}}$ (we suppress the ${\mathcal{R}}$ in the notation, as it is cumbersome). We define the following sets $$\begin{gathered}
SSE({\mathcal{R}}) = \{[A]_{sse} \mid A \textnormal{ is a square matrix over } {\mathcal{R}}\}\\
SE({\mathcal{R}}) = \{[A]_{se} \mid A \textnormal{ is a square matrix over } {\mathcal{R}}\}.\end{gathered}$$
Since matrices which are strong shift equivalent over ${\mathcal{R}}$ must be shift equivalent over ${\mathcal{R}}$, there is a well-defined map of sets $$\label{eqn:ssetosemap}
\begin{gathered}
\pi \colon SSE({\mathcal{R}}) \to SE({\mathcal{R}})\\
\pi \colon [A]_{sse} \mapsto [A]_{se}.
\end{gathered}$$
Problem \[algseproblem\] is equivalent to determining whether $\pi$ is injective. We’ll discuss when this happens, and in fact, we will do much more: we will describe the fiber over a class $[A]_{se}$ in terms of some K-theoretic data involving $NK_{1}({\mathcal{R}})$.\
Strong shift equivalence and elementary equivalence
---------------------------------------------------
From here on, we identify a square matrix $M$ over ${\mathcal{R}}[t]$ with its class in the stabilization of matrices given by $$\begin{gathered}
M_{n}({\mathcal{R}}[t]) \hookrightarrow M_{n+1}({\mathcal{R}}[t])\\
M \mapsto \begin{pmatrix} M & 0 \\ 0 & 1 \end{pmatrix}.\end{gathered}$$
Let ${\mathcal{R}}$ be a ring. We say matrices $M, N$ over ${\mathcal{R}}[t]$ are *elementary equivalent over ${\mathcal{R}}[t]$*, denoted ${{M} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} El-$\mathcal{R}[t]$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} El-$\mathcal{R}[t]$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} N \end{tiny}}$, if there exist $E,F \in El({\mathcal{R}}[t])$ such that $$EMF = N.$$
Note that, as when we first met the definition of $K_{1}$ of a ring, matrices $M,N$ over ${\mathcal{R}}[t]$ are elementary equivalent over ${\mathcal{R}}[t]$ if and only if they (after stabilizing!) can be transformed into each other through a sequence of elementary row and column operations.\
Given square matrices $M,N$ over ${\mathcal{R}}[t]$, it may be tempting to ask why we don’t just define ${{M} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} El-$\mathcal{R}[t]$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} El-$\mathcal{R}[t]$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} N \end{tiny}}$ if and only if $[M] = [N]$ in $K_{1}({\mathcal{R}}[t])$, but this doesn’t make sense: since $M,N$ may not be invertible over ${\mathcal{R}}[t]$, we can’t consider their class in $K_{1}({\mathcal{R}}[t])$.
The following is one of the key results for studying strong shift equivalence over a ring ${\mathcal{R}}$.
\[thm:sseeleeq\] Let ${\mathcal{R}}$ be a ring. For any square matrices $A,B$ over ${\mathcal{R}}$, we have $${{A} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} sse-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} sse-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} B \end{tiny}} \hspace{.1in} \textnormal{ if and only if } \hspace{.1in} {{I-tA} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} El-$\mathcal{R}[t]$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} El-$\mathcal{R}[t]$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} I-tB \end{tiny}}.$$
To see how this fits into the endomorphism $\leftrightarrow$ polynomial philosophy, consider a finitely generated free ${\mathcal{R}}$-module $P$ and an endomorphism $f \colon P \to P$ (one may allow more generally $P$ to be finitely generated projective; see [(Ap. ]{}[apprem:projectivetofree]{})). The endomorphism $f$ gives $P$ the structure of an ${\mathcal{R}}[t]$-module with $t$ acting by $f$, and the similarity class of $f$ (over ${\mathcal{R}}$) corresponds to the isomorphism class of the ${\mathcal{R}}[t]$-module. The direct limit ${\mathcal{R}}[t,t^{-1}]$-module $\mathcal{M}_{f} = \varinjlim \{P,v \mapsto f(v)\}$ is isomorphic as an ${\mathcal{R}}[t,t^{-1}]$-module to $P \otimes_{{\mathcal{R}}[t]} {\mathcal{R}}[t,t^{-1}]$, and it follows that passing from the similarity class of $f$ to the shift equivalence class of $f$ is, in the polynomial world, the same as ‘localizing at $t$’ (note that, besides here, our convention for the direct limit modules is that $t^{-1}$ acts by $f$). On the endomorphism side, the strong shift equivalence relation lies between the similarity relation and the shift equivalence relation, and Theorem \[thm:sseeleeq\] tells us the meaning of the strong shift equivalence relation in the polynomial world.\
We can summarize the above in the following chart [(Ap. ]{}[apprem:endtopolyeq]{}), where $A_{f}$ denotes a matrix over ${\mathcal{R}}$ representing $f \colon P \to P$ in a chosen basis for $P$ as a free ${\mathcal{R}}$-module:\
-------------------------------------- ------------------------------------------- --------------------------------------
Endomorphisms over ${\mathcal{R}}$ ${\mathcal{R}}[t]$-endomorphism relation ${\mathcal{R}}[t]$-module relation
Similarity class of $A_{f}$ $Gl$-${\mathcal{R}}[t]$-conjugacy class Isomorphism class of
of $t-A_{f}$ the ${\mathcal{R}}[t]$-module $P$
SSE-${\mathcal{R}}$ class of $A_{f}$ $El$-${\mathcal{R}}[t]$-equivalence class ??
of $1-tA_{f}$
SE-${\mathcal{R}}$ class of $A_{f}$ $Gl$-${\mathcal{R}}[t]$-equivalence class Isomorphism class of
of $1-tA_{f}$ the ${\mathcal{R}}[t,t^{-1}]$-module
$P \otimes {\mathcal{R}}[t,t^{-1}]$
-------------------------------------- ------------------------------------------- --------------------------------------
In the chart, $GL-{\mathcal{R}}[t]$-equivalence of (stabilized) matrices $C$ and $D$ means there exists $U,V \in GL({\mathcal{R}}[t])$ such that $UCV = D$. The ?? entry indicates that we do not have a good intrinsic interpretation of SSE-${\mathcal{R}}$ at the ${\mathcal{R}}[t]$-module level.\
Theorem \[thm:sseeleeq\] determines the algebraic relation in the polynomial world corresponding to SSE-${\mathcal{R}}$. It also gives another idea of how strong shift equivalence arises algebraically in a natural way. Recall for $A$ and $B$ invertible over ${\mathcal{R}}[t]$ we have $$[A] = [B] \textnormal{ in } K_{1}({\mathcal{R}}[t]) \textnormal{ if and only if } A \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} El-$\mathcal{R}[t]$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} El-$\mathcal{R}[t]$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} \end{tiny} B.$$ In light of Theorem \[thm:sseeleeq\], if one tries to naively extend the K-theory of ${\mathcal{R}}[t]$ to not necessarily invertible matrices over ${\mathcal{R}}[t]$, then strong shift equivalence naturally appears.\
See [(Ap. ]{}[apprem:endo0intable]{}) for a discussion of how $\det(1-tf)$ fits into the above table in the case ${\mathcal{R}}$ is commutative.\
As a nice corollary of Theorems \[thm:sseeleeq\] and \[thm:nil0nk1iso\], we have the following:
\[cor:nilsse\] Let ${\mathcal{R}}$ be a ring, and let $N$ be a nilpotent matrix over ${\mathcal{R}}$. Then $$0 = [N] \textnormal{ in } Nil_{0}({\mathcal{R}}) \hspace{.11in} \textnormal{ if and only if } \hspace{.11in} [N]_{sse} = [0]_{sse}.$$
In other words, a nilpotent matrix is strong shift equivalent over ${\mathcal{R}}$ to the zero matrix if and only if its class in $Nil_{0}({\mathcal{R}})$ is trivial. One can use this together with Theorem \[thm:sseeleeq\] to show that the map defined in is injective.
The refinement of shift equivalence over a ring by strong shift equivalence
---------------------------------------------------------------------------
Theorem \[thm:sseeleeq\] gives us a key tool to understand the refinement of shift equivalence by strong shift equivalence over ${\mathcal{R}}$, obtaining a description of the fibers of the map $\pi$ above. We do this as follows.\
In light of Corollary \[cor:nilsse\] above, there is a well-defined action $\mathfrak{N}$ of the group $Nil_{0}({\mathcal{R}})$ on the set $SSE({\mathcal{R}})$ by $$\mathfrak{N}([N]) \colon [A]_{sse} \mapsto \left[ \begin{pmatrix} A & 0 \\ 0 & N \end{pmatrix} \right]_{sse}, \hspace{.11in} [N] \in Nil_{0}({\mathcal{R}}), \hspace{.11in} [A] \in SSE({\mathcal{R}}).$$
The following gives a description of the fibers of the map $$\pi \colon SSE({\mathcal{R}}) \to SE({\mathcal{R}})$$ defined in .
\[thm:ssesefibers\] Let ${\mathcal{R}}$ be a ring, and let $A$ be a square matrix over ${\mathcal{R}}$. There is a bijection $$\begin{gathered}
\pi^{-1}([A]_{se}) \stackrel{\cong}\longrightarrow \mathfrak{N}\textnormal{-orbit of } [A]_{sse}.\end{gathered}$$ In other words, there is a bijection between the set of strong shift equivalence classes of matrices which are shift equivalent to $A$, and the orbit of $[A]_{sse}$ under the action of $Nil_{0}({\mathcal{R}})$.
As a corollary, we get the following.
\[cor:nk1vanishandsse\] Let ${\mathcal{R}}$ be a ring. Then $NK_{1}({\mathcal{R}}) = 0$ if and only if, for all square matrices $A,B$ over ${\mathcal{R}}$, $${{A} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} se-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} se-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} B \end{tiny}} \textnormal{ if and only if } {{A} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} sse-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} sse-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} B \end{tiny}}.$$
First suppose $NK_{1}({\mathcal{R}})=0$. By Theorem \[thm:nil0nk1iso\], this implies $Nil_{0}({\mathcal{R}}) = 0$, so the action $\mathfrak{N}$ is trivial. Thus if $A$ is any square matrix over ${\mathcal{R}}$, by Theorem \[thm:ssesefibers\], the fiber $\pi^{-1}([A]_{se})$ is also trivial. It follows that if $B$ is any square matrix over ${\mathcal{R}}$, then ${{A} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} se-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} se-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} B \end{tiny}} \Leftrightarrow {{A} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.03in} sse-$\mathcal{R}$ }}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.03in} sse-$\mathcal{R}$ }}$}{O}{c}{F}{T}{S}\mkern1mu}}} B \end{tiny}}$ as desired.\
Now suppose matrices which are SE-${\mathcal{R}}$ must be SSE-${\mathcal{R}}$. Given $N$ nilpotent over ${\mathcal{R}}$, $N$ is clearly shift equivalent over ${\mathcal{R}}$ to the zero matrix, and hence by assumption, strong shift equivalent over ${\mathcal{R}}$ to the zero matrix. By Corollary \[cor:nilsse\], this implies $[N]=0$ in $Nil_{0}({\mathcal{R}})$. Since $N$ was a general nilpotent matrix, it follows that $Nil_{0}({\mathcal{R}})=0$.
So what is the behavior of the action $\mathfrak{N}$? In general, its orbit structure is far from trivial. Given $A$ over ${\mathcal{R}}$, define the $\mathfrak{N}$-stabilizer of $[A]_{sse}$ to be $${\textnormal{St}}_{\mathfrak{N}}(A) = \{[N] \in Nil_{0}({\mathcal{R}}) \mid [A \oplus N]_{sse} = [A]_{sse}\}.$$ Note the $\mathfrak{N}$-stabilizer depends only on the SSE-${\mathcal{R}}$ class of a matrix $A$, but to avoid cumbersome notation, we write simply ${\textnormal{St}}_{\mathfrak{N}}(A)$ instead of ${\textnormal{St}}_{\mathfrak{N}}([A]_{sse})$.\
The notation used here differs from what is used in [@BoSc1]; see [(Ap. ]{}[apprem:elstnotation]{}).\
There is a bijection between the $\mathfrak{N}$-orbit of $[A]_{sse}$ and the quotient $Nil_{0}({\mathcal{R}}) / {\textnormal{St}}_{\mathfrak{N}}(A)$ given by mapping a coset of $[N]$ in $Nil_{0}({\mathcal{R}}) / {\textnormal{St}}_{\mathfrak{N}}(A)$ to $[A \oplus N]_{sse}$.\
For a commutative ring ${\mathcal{R}}$, we define $$SNil_{0}({\mathcal{R}}) = \{[N] \in Nil_{0}({\mathcal{R}}) \mid {\textnormal{det}}(I-tN) = 1\}.$$ It is straightforward to check that $SNil_{0}({\mathcal{R}})$ is a subgroup of $Nil_{0}({\mathcal{R}})$, and that $SNil_{0}({\mathcal{R}})$ is precisely the pullback via the isomorphism of the subgroup $NK_{1}({\mathcal{R}}) \cap SK_{1}({\mathcal{R}}[t])$ in $NK_{1}({\mathcal{R}})$.
\[thm:stabilizerthm1\] For any ring ${\mathcal{R}}$, both of the following hold:
1. If $A$ is nilpotent or invertible over ${\mathcal{R}}$, then ${\textnormal{St}}_{\mathfrak{N}}(A)$ is trivial.
2. If ${\mathcal{R}}$ is commutative, then $$\bigcup_{[A]_{sse} \in SSE({\mathcal{R}})} {\textnormal{St}}_{\mathfrak{N}}(A) = SNil_{0}({\mathcal{R}}).$$
[(Ap. ]{}[appex:redsnil0]{}) If ${\mathcal{R}}$ is commutative and reduced (has no nontrivial nilpotent elements), then the groups $SNil_{0}({\mathcal{R}})$ and $Nil_{0}({\mathcal{R}})$ coincide.
The nilpotent case of $(1)$ is straightforward, and is Exercise \[exer:nilpotentstavanishexer\] below. Both $(2)$ and the invertible case of $(1)$ are nontrivial to prove. Part $(1)$ uses localization and K-theoretic techniques for localization; in the non-commutative case, this requires some deep K-theoretic results of Neeman and Ranicki about non-commutative localization of rings. Part $(2)$ uses work of Nenashev on presentations for $K_{1}$ of exact categories. We will not go into more detail about the structure of these proofs, but instead refer the reader to [@BoSc1].
[(Ap. ]{}[appex:nilstvanish]{})\[exer:nilpotentstavanishexer\] If $A$ is a nilpotent matrix over ${\mathcal{R}}$ then ${\textnormal{St}}_{\mathfrak{N}}(A)$ vanishes.
There are rings for which $SNil_{0}({\mathcal{R}})$ does not vanish. For example, the ring $\mathbb{Q}[t^{2},t^{3},z,z^{-1}]$ is commutative and reduced, and has nontrivial $NK_{1}$ (see Example (1) in \[exa:nk1example\]).\
By part (2) of the above theorem, it follows that the $\mathfrak{N}$-stabilizers can be nontrivial, and can change depending on the matrix. There is a conjectured analogous version of part $(2)$ in the non-commutative case, which is more technical to state (see [@BoSc1 Conjecture 5.20]). In general, we do not have a complete understanding of the groups ${\textnormal{St}}_{\mathfrak{N}}(A)$, and the following problem was posed in [@BoSc1 Problem 5.21]:
\[prob:elementarystabl\] Given a square matrix $A$ over ${\mathcal{R}}$, give a satisfactory description of the elementary stabilizer ${\textnormal{St}}_{\mathfrak{N}}(A)$. In particular, determine when ${\textnormal{St}}_{\mathfrak{N}}(A)$ is trivial.
The SE and SSE relations in the context of endomorphisms
--------------------------------------------------------
Using the results above, we can now give another view on what the relations SE-${\mathcal{R}}$ and SSE-${\mathcal{R}}$ mean in the context of endomorphisms, and how they fit in with the similarity relation over a ring. Given square matrices $A,B$ over ${\mathcal{R}}$, we say
1. $B$ is a zero extension of $A$ if there exists some matrix $C$ over ${\mathcal{R}}$ such that $B = \begin{pmatrix} A & C & \\ 0 & 0 \end{pmatrix}$ or $B = \begin{pmatrix} A & 0 \\ C & 0 \end{pmatrix}$.
2. $B$ is a nilpotent extension of $A$ if there exists some matrix $C$ over ${\mathcal{R}}$ and some nilpotent matrix $N$ over ${\mathcal{R}}$ such that $B = \begin{pmatrix} A & C & \\ 0 & N \end{pmatrix}$ or $B = \begin{pmatrix} A & 0 \\ C & N \end{pmatrix}$.
Zero and nilpotent extensions fit nicely into the context of the category $\textbf{Nil}{\mathcal{R}}$; see [(Ap. ]{}[apprem:extsequences]{}).
\[thm:endorelations\] Let ${\mathcal{R}}$ be a ring.
1. SSE-${\mathcal{R}}$ is the equivalence relation on square matrices over ${\mathcal{R}}$ generated by:
1. Similarity
2. Zero extensions
2. SE-${\mathcal{R}}$ is the equivalence relation on square matrices over ${\mathcal{R}}$ generated by:
1. Similarity
2. Nilpotent extensions
Part $(1)$ is Proposition \[prop:MallerShub\] from Lecture 2.\
For part $(2)$, one direction is easy. If $A$ and $B$ are similar, they are certainly SE-${\mathcal{R}}$. To see why $A$ is shift equivalent to $\begin{pmatrix} A & B \\ 0 & N \end{pmatrix}$ for any $B$ and $N$ nilpotent, note there exists $l$ such that $$\begin{pmatrix} A & B \\ 0 & N \end{pmatrix}^{l} = \begin{pmatrix} A^{l} & C \\ 0 & 0 \end{pmatrix}$$ for some $C$. Then $\begin{pmatrix} A & B \\ 0 & N \end{pmatrix}$ and $A$ are shift equivalent with lag $l$ using $$R = \begin{pmatrix} I \\ 0 \end{pmatrix}, \hspace{.1in} S = \begin{pmatrix} A^{l} & C \end{pmatrix}.$$ For the other direction, suppose $A$ and $B$ are SE-${\mathcal{R}}$. Then the classes $[A]_{sse}, [B]_{sse}$ lie in the same fiber of the map $\pi$, so by Theorem \[thm:ssesefibers\] above, there exists a nilpotent matrix $N$ over ${\mathcal{R}}$ such that $A \oplus N$ is SSE-${\mathcal{R}}$ to $B$. Then $A \oplus N$ and $B$ are connected by a chain of similarities and extensions by zero. Since $A$ and $A \oplus N$ are related by an extension by a nilpotent, the result follows.\
The proof that $A$ is shift equivalent over ${\mathcal{R}}$ to $\begin{pmatrix} A & 0 \\ B & N \end{pmatrix}$ for any $B$ and nilpotent $N$ is analogous.
Appendix 6
----------
This appendix contains some remarks and solutions for exercises for Lecture 6.
\[apprem:projectivetofree\] For a ring ${\mathcal{R}}$, shift equivalence and strong shift equivalence may be defined in the context of finitely generated projective ${\mathcal{R}}$-modules as follows. If $f \colon P \to P, g \colon Q \to Q$ are endomorphisms of finitely generated projective ${\mathcal{R}}$-modules, then:
1. $f$ and $g$ are strong shift equivalent (over $\textbf{Proj}{\mathcal{R}}$) if there exists module homomorphisms $r \colon P \to Q, s \colon Q \to P$ such that $f=sr, g = rs$.
2. $f$ and $g$ are shift equivalent (over $\textbf{Proj}{\mathcal{R}}$) if there exists module homomorphisms $r \colon P \to Q, s \colon Q \to P$ and $l \ge 1$ such that $f^{l}=sr, g^{l} = rs$.
Suppose now $f \colon P \to P$ is an endomorphism of a finitely generated projective ${\mathcal{R}}$-module. There exists a finitely generated projective ${\mathcal{R}}$-module $Q$ such that $P \oplus Q$ is free, and $f$ is strong shift equivalent over $\textbf{Proj}{\mathcal{R}}$ to $f \oplus 0 \colon P \oplus Q \to P \oplus Q$ using $r \colon P \to P \oplus Q$ given by $r(x) = (x,0)$ and $s \colon P \oplus Q \to P$ given by $s(x,y) = f(x)$. It follows that, when considering strong shift equivalence and shift equivalence over $\textbf{Proj}{\mathcal{R}}$, we may without loss of generality work with free modules.
\[apprem:endtopolyeq\] For an example of the relationship between endomorphisms of ${\mathcal{R}}$-modules and certain classes of modules over the polynomial ring ${\mathcal{R}}[t]$ worked out more formally, see Theorem 2 in [@Grayson1977] and the discussion on page 441 there.
\[apprem:endo0intable\] Given a commutative ring ${\mathcal{R}}$ and an endomorphism $f \colon P \to $ of a finitely generated projective ${\mathcal{R}}$-module where ${\mathcal{R}}$ is commutative, one may add the polynomial $\det(I-tf)$ as an additional entry to the chart. The data $\det(I-tf)$ corresponds to, on the endomorphism side, the class of $[f]$ in the endomorphism class group $End_{0}({\mathcal{R}})$ (see [(Ap. ]{}[rem:ktheoryendomorphisms]{})).
\[apprem:elstnotation\] The presentation here of the elementary stabilizers differs from the one given in [@BoSc1]. Roughly speaking, here we use $Nil_{0}({\mathcal{R}})$ and the endomorphism side, whereas in [@BoSc1] the notation and definitions are in terms of $NK_{1}({\mathcal{R}})$ and the polynomial matrix side. More precisely, in [@BoSc1] the elementary stabilizer of a polynomial matrix $I-tA$ is defined to be $$E(A,{\mathcal{R}}) = \{U \in GL({\mathcal{R}}[t]) \mid U\textnormal{Orb}_{El({\mathcal{R}}[t])}(I-tA) \subset \textnormal{Orb}_{El({\mathcal{R}}[t])}(I-tA)\}$$ where $\textnormal{Orb}_{El({\mathcal{R}}[t])}(I-tA)$ denotes the set of matrices over ${\mathcal{R}}[t]$ which are elementary equivalent over ${\mathcal{R}}[t]$ to $I-tA$. There it is observed that $E(A,{\mathcal{R}})$ is a subgroup of $NK_{1}({\mathcal{R}})$. Given $A$ over ${\mathcal{R}}$, the map $$\begin{gathered}
E(A,{\mathcal{R}}) \to {\textnormal{St}}_{\mathfrak{N}}(A)\\
[I-tN] \mapsto [N]
\end{gathered}$$ defines a group isomorphism between $E(A,{\mathcal{R}})$ and ${\textnormal{St}}_{\mathfrak{N}}(A)$.
\[appex:redsnil0\] If ${\mathcal{R}}$ is commutative and reduced (has no nontrivial nilpotent elements), then the groups $SNil_{0}({\mathcal{R}})$ and $Nil_{0}({\mathcal{R}})$ coincide.
This is essentially Exercise \[appexer:reducednk1sk1\], just in the nilpotent endomorphism setting: if ${\mathcal{R}}$ is commutative and reduced, then $\det(I-tN)=1$ for any nilpotent matrix $N$ over ${\mathcal{R}}$.
\[appex:nilstvanish\] If $A$ is a nilpotent matrix over ${\mathcal{R}}$ then ${\textnormal{St}}_{\mathfrak{N}}(A)$ vanishes.
If $A$ is a nilpotent matrix over ${\mathcal{R}}$ and $[N] \in {\textnormal{St}}_{\mathfrak{N}}(A)$, then $[A \oplus N]_{sse} = [A]_{sse}$. Since both $A$ and $N$ are nilpotent, $A \oplus N$ is nilpotent, and by Theorem \[thm:sseeleeq\] this implies $[A \oplus N] = [A]$ in the group $Nil_{0}({\mathcal{R}})$. Thus $[N]=0$ in $Nil_{0}({\mathcal{R}})$.
\[apprem:extsequences\] The notion of zero and nilpotent extension can also be defined in terms of endomorphisms. Recall from [(Ap. ]{}[rem:ktheoryendomorphisms]{}) the category $\textbf{End}{\mathcal{R}}$ whose objects are pairs $(P,f)$ where $f \colon P \to P$ is an endomorphism of a finitely generated projective ${\mathcal{R}}$-module and a morphism from $(P,f)$ to $(Q,g)$ is an ${\mathcal{R}}$-module endomorphism $h \colon P \to Q$ such that $hf = gh$. Given $(P,f),(Q,g)$ in $\textbf{End}{\mathcal{R}}$, we say
1. $(Q,g)$ is a zero extension of $(P,f)$ if there exists some ${\mathcal{R}}$-module $P_{1}$ such that $Q = P \oplus P_{1}$ and either of the following happen:
1. There exists an ${\mathcal{R}}$-module homomorphism $h \colon P_{1} \to P$ such that $g = \begin{pmatrix} f & h & \\ 0 & 0 \end{pmatrix}$
2. There exists an ${\mathcal{R}}$-module homomorphism $h \colon P \to P_{1}$ such that $g = \begin{pmatrix} f & 0 & \\ h & 0 \end{pmatrix}$.
2. $(Q,g)$ is a nilpotent extension of $(P,f)$ if there exists some ${\mathcal{R}}$-module $P_{1}$ such that $Q = P \oplus P_{1}$ and either of the following happen:
1. There exists an ${\mathcal{R}}$-module homomorphism $h \colon P_{1} \to P$ and a nilpotent endomorphism $j \colon P_{1} \to P_{1}$ such that $g = \begin{pmatrix} f & h & \\ 0 & j \end{pmatrix}$
2. There exists an ${\mathcal{R}}$-module homomorphism $h \colon P \to P_{1}$ and a nilpotent endomorphism $j \colon P_{1} \to P_{1}$ such that $g = \begin{pmatrix} f & 0 & \\ h & j\end{pmatrix}$.
Recall in $\textbf{End}{\mathcal{R}}$ we say a sequence $(P_{1},f_{1}) \to (P_{2},f_{2}) \to (P_{3},f_{3})$ is exact if the corresponding sequence of ${\mathcal{R}}$-modules $P_{1} \to P_{2} \to P_{3}$ is exact. Zero extensions and nilpotent extensions have a nice interpretation in terms of certain exact sequences in the endomorphism category. Note that in $\textbf{End}{\mathcal{R}}$ the pair $(P,0)$ means the zero endomorphism of the ${\mathcal{R}}$-module $P$, and $(0,0)$ means the zero endomorphism of the zero ${\mathcal{R}}$-module. Since $(0,0)$ serves as a zero object in $\textbf{End}{\mathcal{R}}$, we can consider short exact sequences in $\textbf{End}{\mathcal{R}}$, by which we mean an exact sequence of the form $$(0,0) \to (P_{1},f_{1}) \to (P_{2},f_{2}) \to (P_{3},f_{3}) \to (0,0).$$ Given this, the following shows that zero extensions and nilpotent extensions are given (up to isomorphism) by certain short exact sequences in $\textbf{End}{\mathcal{R}}$.
Let ${\mathcal{R}}$ be a ring, and suppose $$(0,0) \to (P_{1},f_{1}) \stackrel{\alpha_{1}}\longrightarrow (P_{2},f_{2}) \stackrel{\alpha_{2}}\longrightarrow (P_{3},f_{3}) \to (0,0)$$ is a short exact sequence in $\textbf{End}{\mathcal{R}}$.
1. If $f_{1}=0$ then $(P_{2},f_{2})$ is isomorphic to a zero extension of $(P_{3},f_{3})$.
2. If $f_{3}=0$, then $(P_{2},f_{2})$ is isomorphic to a zero extension of $(P_{1},f_{1})$.
3. If $f_{1}$ is nilpotent, then $(P_{2},f_{2})$ is isomorphic to a nilpotent extension of $(P_{3},f_{3})$.
4. If $f_{3}$ is nilpotent, then $(P_{2},f_{2})$ is isomorphic to a nilpotent extension of $(P_{1},f_{1})$.
We will prove 4; the other are analogous. Since the sequence is exact there is a splitting map $\alpha_{1}^{\prime} \colon P_{2} \to P_{1}$ such that $\alpha_{1}^{\prime} \alpha_{1} = id$ on $P_{1}$, and an ${\mathcal{R}}$-module isomorphism $\beta \colon P_{2} \to P_{1} \oplus P_{3}$ given by $\beta(x) = (\alpha_{1}^{\prime}(x),\alpha_{2}(x))$ so that the following diagram commutes $$\xymatrix{
0 \ar[r] & P_{1} \ar[r]^{\alpha_{1}} \ar[d]_{id} & P_{2} \ar[d]_{\beta} \ar[r]^{\alpha_{2}} & P_{3} \ar[r] \ar[d]^{id} & 0\\
0 \ar[r] & P_{1} \ar[r]^{i} & P_{1} \oplus P_{3} \ar[r]^{q} & P_{3} \ar[r] & 0\\
}$$ where $i \colon P_{1} \to P_{1} \oplus P_{3}$ by $i(x) = (x,0)$ and $q \colon P_{1} \oplus P_{3} \to P_{3}$ by $q(x,y) = y$. Define $g = \beta f_{2} \beta^{-1}$, so $g \colon P_{1} \oplus P_{3} \to P_{1} \oplus P_{3}$. We may write $g = \begin{pmatrix} g_{1} & g_{2} \\ g_{2}^{\prime} & g_{3} \end{pmatrix}$ where $(P_{1},g_{1}),(P_{3},g_{3})$ are in $\textbf{End}{\mathcal{R}}$, and $g_{2} \colon P_{3} \to P_{1}, g_{2}^{\prime} \colon P_{1} \to P_{3}$. For any $x \in P_{1}$ we have $$g(x,0) = \beta f_{2} \beta^{-1}(x,0) = \beta f_{2} \alpha_{1}(x) = \beta \alpha_{1} f_{1}(x) = (f_{1}(x),0)$$ so $g_{2}^{\prime}(x) = 0$ and $g_{1}(x) = f_{1}(2)$. Since $x$ was arbitrary, it follows that $g_{2}^{\prime}=0$ and $g_{1} = f_{1}$. Likewise, one can check that $g_{3} = f_{3}$. Altogether $f_{2}$ is isomorphic to $g = \begin{pmatrix} f_{1} & g_{2} \\ 0 & f_{3} \end{pmatrix}$, and since $f_{3}$ is nilpotent, this is a nilpotent extension of $(P_{1},f_{1})$.
Automorphisms of SFTs {#sectionAut}
=====================
We turn now to discussing automorphisms of shifts of finite type. In general, an automorphism of a dynamical system is simply a self-conjugacy of the given system. The collection of all automorphisms of a given system forms a group, the size of which can vary greatly depending on the system in question. It turns out that a nontrivial mixing shift of finite type possesses a very rich group of automorphisms.\
It’s maybe unsurprising that, even in the context of the classification problem for shifts of finite type (Problem \[prob:classificationprob\] in Lecture 1), the study of automorphisms plays an important role. Partly, this role is indirect: various tools and ideas which were originally introduced to study automorphism groups of shifts of finite type (e.g. sign-gyration, introduced later in this lecture) in fact turned out to be important tools for the conjugacy problem. For example, the dimension representation plays a role in constructing counterexamples to Williams’ conjecture in the reducible case (see [@S21]). Some of this we will discuss in Lecture 8.\
The goal of this lecture is only to give a brief tour through some of main ideas in the study of automorphism groups for shifts of finite type. At the end of the lecture we mention some newer developments, as well as a small collection of problems and conjectures that have guided some of the direction for studying the automorphism groups.\
We will continue to use the following notation. For a matrix $A$ over $\mathbb{Z}_{+}$, we let $(X_{A},\sigma_{A})$ denote the edge shift of finite type (as defined in Section \[sec:edgesfts\] of Lecture 1) corresponding to the graph associated to $A$ (i.e. the graph $\Gamma_{A}$ as defined in Lecture 1). Since any shift of finite type is topologically conjugate to an edge shift $(X_{A},\sigma_{A})$ for some $\mathbb{Z}_{+}$-matrix $A$ (see e.g. [@LindMarcus1995 Theorem 2.3.2]), and automorphism groups of topologically conjugate systems are isomorphic, we will only consider edge shifts $(X_{A},\sigma_{A})$. The fundamental case is when $A$ is primitive with the topological entropy of the shift satisfying $h_{top}(\sigma_{A}) > 0$; with this in mind we make the following standing assumption.\
**Standing Assumption:** Throughout this lecture, unless otherwise noted, when considering an SFT $(X_{A},\sigma_{A})$ we assume $A$ is primitive with $\lambda_{A} > 1$, where $\lambda_{A}$ denotes the Perron-Frobenius eigenvalue of $A$.\
Since $h_{top}(\sigma_{A}) = \log \lambda_{A}$, where $h_{top}(\sigma_{A})$ is the topological entropy of the shift $\sigma_{A}$, the assumption on $\lambda_{A}$ is equivalent to the system $(X_{A},\sigma_{A})$ having positive entropy.\
Now let us say more precisely what we mean by an automorphism. We begin with a general definition, and specialize to shifts of finite type later. Recall by a topological dynamical system $(X,f)$ we mean a self-homeomorphism $f$ of a compact metric space $X$.
Let $(X,f)$ be a topological dynamical system. An automorphism of $(X,f)$ is a homeomorphism $\alpha \colon X \to X$ such that $\alpha f = f \alpha$. The collection of automorphisms of $(X,f)$ forms a group under composition, which we call the group of automorphisms of $(X,f)$, and we denote this group by ${\textnormal{Aut}}(f)$.
In other words, an automorphism of $(X,f)$ is simply a self-conjugacy of the system $(X,f)$, and the automorphism group is the group of all self-conjugacies of $(X,f)$.\
It is straightforward to check that if two systems $(X,f)$ and $(Y,g)$ are topologically conjugate then their automorphism groups ${\textnormal{Aut}}(f)$ and ${\textnormal{Aut}}(g)$ are isomorphic.
Let $X$ be a Cantor set and $f \colon X \to X$ be the identity map, i.e. $f(x) = x$ for all $x \in X$. Then ${\textnormal{Aut}}(f) = \textnormal{Homeo}(X)$ is the group of all homeomorphisms of the Cantor set.
Recall from Section \[subsec:symbolicdynamics\] a subshift is a system $(X,\sigma)$ which is a subsystem of some full shift $(\mathcal{A}^{\mathbb{Z}},\sigma)$.
For a subshift $(X,\sigma)$, the shift $\sigma$ is itself is always an automorphism of $(X,\sigma)$, i.e. $\sigma \in {\textnormal{Aut}}(\sigma)$. Whenever $(X,\sigma)$ has an aperiodic point, $\sigma$ is clearly infinite order in the group ${\textnormal{Aut}}(\sigma)$.
\[exa:0blockcodeauto\] Let $(X_{3},\sigma_{3})$ denote the full shift on the symbol set $\{0,1,2\}$ and define an automorphism $\alpha \in {\textnormal{Aut}}(\sigma_{3})$ using the block code $$\alpha_{0} \colon x \mapsto x + 1 \textnormal{ mod } 3, \hspace{.03in} x \in \{0,1,2\}.$$ Thus for example, $\alpha$ acts like the following: $$\begin{gathered}
\ldots 01020102011{\overset{\bullet}}{0}202220102110 \ldots \\
\downarrow \hspace{.03in} \alpha \\
\ldots 12101210122{\overset{\bullet}}{1}010001210221 \ldots\end{gathered}$$ This automorphism is order $3$, i.e. $\alpha^{3}=\textnormal{id}$.
As we’ll see later, automorphism groups of shifts of finite type contain a large supply of nontrivial automorphisms. Here is an interesting example of a subshift whose only automorphisms are powers of the shift.
\[ex:sturmiansubshift\] Let $\alpha$ be an irrational, and consider the rotation map $R_{\alpha} \colon [0,1) \to [0,1)$ given by $R_{\alpha}(x) = x + \alpha \textnormal{ mod } 1$. Consider the indicator map $I_{\alpha} \colon [0,1) \to \{0,1\}$ given by $I_{\alpha}(z)=0$ if $z \in [0,1-\alpha)$ and $I_{\alpha}(z)=1$ if $z \in [1-\alpha,1)$. Now we can define a subshift $(X_{\alpha},\sigma_{X_{\alpha}})$ of the full shift on two symbols $(\{0,1\}^{\mathbb{Z}},\sigma)$ to be the orbit closure of locations of orbits of points under the map $R_{\alpha}$, i.e. we let $$X_{\alpha} = \overline{\{I_{\alpha}(R_{\alpha}^{k}(z)) \mid k \in \mathbb{Z}, z \in [0,1)\}}.$$ The subshift $(X_{\alpha},\sigma_{X_{\alpha}})$ is known as a Sturmian subshift, and it is a folklore result (see [@Olli2013] or [@DDMP2016] for a proof) that ${\textnormal{Aut}}(\sigma_{X_{\alpha}}) = \langle \sigma_{X_{\alpha}} \rangle$, so as a group ${\textnormal{Aut}}(\sigma_{X_{\alpha}})$ is isomorphic to $\mathbb{Z}$.
The subshift in the last example has zero topological entropy, and the structure of its automorphism group is very easy to understand (as a group it’s just $\mathbb{Z})$. In many cases, the automorphism groups of subshifts with such “low-complexity” dynamics (of which Example \[ex:sturmiansubshift\] is an example) have more constrained automorphism groups, in contrast to the automorphism groups of shifts of finite type (see [(Ap. ]{}[apprem:lowcomplexity]{}) for a brief discussion of this, and for what we mean here by low-complexity).\
By the Curtis-Hedlund-Lyndon Theorem (Theorem \[thm:chlthm\]), any automorphism of a subshift $(X,\sigma)$ is induced by a block code. This leads immediately to the following observation:
If $(X,\sigma)$ is a subshift, then ${\textnormal{Aut}}(\sigma)$ is a countable group.
Thus for a shift of finite type $(X_{A},\sigma_{A})$, ${\textnormal{Aut}}(\sigma_{A})$ is always a countable group. Under our assumptions that $(X_{A},\sigma_{A})$ is mixing with positive entropy, ${\textnormal{Aut}}(\sigma_{A})$ is also always infinite.
It turns out that ${\textnormal{Aut}}(\sigma_{A})$ possesses a rich algebraic structure. Example \[exa:0blockcodeauto\] above was induced by a block code of range $0$, but for arbitrarily large $R \in \mathbb{N}$ there are automorphisms which can only be induced by block codes of range $R$ or greater (indeed, given an SFT $(X_{A},\sigma_{A})$ and a non-negative number $R$, there are only finitely many automorphisms in ${\textnormal{Aut}}(\sigma_{A})$ having range $\le R$). To give an indication that ${\textnormal{Aut}}(\sigma_{A})$ is quite large, consider the following results regarding different types of subgroups that can arise in ${\textnormal{Aut}}(\sigma_{A})$.
\[thm:largesubgroups\] Let $(X_{A},\sigma_{A})$ be a shift of finite type where $A$ is a primitive matrix with $\lambda_{A} > 1$.
1. (Boyle-Lind-Rudolph in [@BLR88]) The group ${\textnormal{Aut}}(\sigma_{A})$ contains isomorphic copies of each of the following groups:
1. Any finite group.
2. $\bigoplus\limits_{i=1}^{\infty}\mathbb{Z}$.
3. The free group on two generators $\mathbb{F}_{2}$.
2. (Kim-Roush in [@S27]) For any $n \ge 2$, let $(X_{n},\sigma_{n})$ denote the full shift on $n$ symbols. Then ${\textnormal{Aut}}(\sigma_{n})$ is isomorphic to a subgroup of ${\textnormal{Aut}}(\sigma_{A})$.
3. (Kim-Roush in [@S27]) Any countable, locally finite, residually finite group embeds into ${\textnormal{Aut}}(\sigma_{A})$.
In particular, by part $(1)$, ${\textnormal{Aut}}(\sigma_{A})$ is never amenable. By part $(2)$, for full shifts, the isomorphism types of groups that can appear as subgroups of ${\textnormal{Aut}}(\sigma_{n})$ is independent of $n$.\
Recall a group $G$ is residually finite if the intersection of all its subgroups of finite index is trivial. A finitely presented group $G$ is said to have solvable word problem if there is an algorithm to determine whether a word made from generators is the identity in the group.
[(Ap. ]{}[appex:denppresfin]{})\[exer:denseppresfinite\] If $(X,\sigma)$ is a subshift whose periodic points are dense in $X$, then ${\textnormal{Aut}}(\sigma)$ is residually finite.
\[prop:resfinsolwp\] Let $(X_{A},\sigma_{A})$ be a shift of finite type where $A$ is a primitive matrix with $\lambda_{A} > 1$. Then both of the following hold:
1. The group ${\textnormal{Aut}}(\sigma_{A})$ is residually finite.
2. The group ${\textnormal{Aut}}(\sigma_{A})$ contains no finitely generated group with unsolvable word problem.
Such an SFT has a dense set of periodic points (see [@LindMarcus1995 Sec. 6.1]), so (1) follows from Exercise \[exer:denseppresfinite\]. For (2), see [@BLR88 Prop. 2.8].
Since a subgroup of a residually finite group must be residually finite, both parts of the previous proposition give some necessary conditions for a group to embed as a subgroup of ${\textnormal{Aut}}(\sigma_{A})$. For example, it follows that the additive group of rationals $\mathbb{Q}$ cannot embed into ${\textnormal{Aut}}(\sigma_{A})$, since $\mathbb{Q}$ under addition is not residually finite (however, the additive group $\mathbb{Q}$ can embed into the automorphism group of a certain minimal subshift - see [@BLR88 Example 3.9]). Still, we do not have a good understanding of what types of countable groups can be isomorphic to a subgroup of ${\textnormal{Aut}}(\sigma_{A})$.\
An important tool for constructing automorphisms in ${\textnormal{Aut}}(\sigma_{A})$ is the use of “markers”. We’ll forego describing marker methods here, instead referring the reader to [@BLR88 Sec. 2]; but we note that, for example, all three parts of Theorem \[thm:largesubgroups\] make use of markers. We’ll see another perspective on marker automorphisms when discussing simple automorphisms below.
Simple Automorphisms
--------------------
In [@Nasu88], Nasu introduced a class of automorphisms known as *simple automorphisms*, which we’ll define shortly. Automorphisms built from compositions of these simple automorphisms encompasses the collection of automorphisms defined using marker methods (see [@BoyleNasu's] for a presentation of this), and give rise to an important subgroup of ${\textnormal{Aut}}(\sigma_{A})$.\
Let $A$ be a square matrix over $\mathbb{Z}_{+}$, and let $\Gamma_{A}$ be its associated directed graph. A *simple graph symmetry*[^3] of $\Gamma_{A}$ is a graph automorphism of $\Gamma_{A}$ which fixes all vertices. A simple graph symmetry of $\Gamma_{A}$ gives a 0-block code and hence a corresponding automorphism in ${\textnormal{Aut}}(\sigma_{A})$. Given $\alpha \in {\textnormal{Aut}}(\sigma_{A})$, we call $\alpha$ a *simple graph automorphism* if it is induced by a simple graph symmetry of $\Gamma_{A}$, and we call $\alpha \in {\textnormal{Aut}}(\sigma_{A})$ a *simple automorphism* if it is of the form $$\alpha = \Psi^{-1} \gamma \Psi$$ where $\Psi \colon (X_{A},\sigma_{A}) \to (X_{B},\sigma_{B})$ is a conjugacy to some shift of finite type $(X_{B},\sigma_{B})$ and $\gamma \in {\textnormal{Aut}}(\sigma_{B})$ is a simple graph automorphism in ${\textnormal{Aut}}(\sigma_{B})$.\
Let $A = \begin{pmatrix} 2 & 2 \\ 1 & 1 \end{pmatrix}$ and label the edges of $\Gamma_{A}$ by $a,\cdots,f$. The graph automorphism of $\Gamma_{A}$ drawn below defined by permuting the edges $c$ and $d$ is a simple graph symmetry of $\Gamma_{A}$, and the corresponding simple graph automorphism in ${\textnormal{Aut}}(\sigma_{A})$ is given by the block code of range 0 which swaps the letters $c$ and $d$ and leaves all other letters fixed. $$\label{fig:simplegraphsymmetry1}
\begin{gathered}
\xymatrix{
*+[F-:<2pt>]{1}
\ar@(u,ul)_a \ar@(d,dl)^b
\ar@/^/[rr]^{c}
\ar@/^2pc/[rr]^{d} & &
*+[F-:<2pt>]{2}
\ar@/^/[ll]^{f}
\ar@(ru,rd)^e
}
\end{gathered}$$
We define ${\textnormal{Simp}(\sigma_{A})}$ to be the subgroup of ${\textnormal{Aut}}(\sigma_{A})$ generated by simple automorphisms. It is immediate to check that ${\textnormal{Simp}(\sigma_{A})}$ is a normal subgroup of ${\textnormal{Aut}}(\sigma_{A})$.\
There is a conjugacy from the full 3-shift $(X_{3},\sigma_{3})$ on symbols $\{0,1,2\}$ to the edge shift of finite type $(X_{A},\sigma_{A})$ presented by the graph given in Figure \[fig:simplegraphsymmetry1\] on symbol set $\{a,b,c,d,e,f\}$. Here the matrix $A$ is given by $A = \begin{pmatrix} 2 & 2 \\ 1 & 1 \end{pmatrix}$, and a conjugacy $$\Psi \colon (X_{3},\sigma_{3}) \to (X_{A},\sigma_{A})$$ is given by the block code: $$\begin{array}{ccccc}
00 \mapsto a & \qquad & 10 \mapsto b & \qquad & 20 \mapsto f\\
01 \mapsto a & \qquad & 11 \mapsto b & \qquad & 21 \mapsto f\\
02 \mapsto d & \qquad & 12 \mapsto c & \qquad & 22 \mapsto e\\
\end{array}$$ with inverse given by $$\begin{array}{ccc}
a \mapsto 0 & \qquad & d \mapsto 0\\
b \mapsto 1 & \qquad & c \mapsto 1\\
e \mapsto 2 & \qquad & f \mapsto 2\\
\end{array}$$ Let $\gamma$ denote the simple automorphism in ${\textnormal{Aut}}(\sigma_{A})$ induced by the simple graph symmetry of $\Gamma_{A}$ shown in Figure \[fig:simplegraphsymmetry1\], which permutes the edges $c$ and $d$, and let $\beta = \Psi^{-1}\gamma\Psi$. Then $\beta \in {\textnormal{Simp}(\sigma_{3})}$, and acts for example like $$\begin{gathered}
\ldots 112002{\overset{\bullet}}{0}2120011 \ldots\\
\Psi \Big\downarrow\\
\ldots bcfadf{\overset{\bullet}}{d}fcfaab \ldots\\
\gamma \Big\downarrow\\
\ldots bdfacf{\overset{\bullet}}{c}fdfaab \ldots\\
\Psi^{-1} \Big\downarrow\\
\ldots 102012{\overset{\bullet}}{1}202001 \ldots\\\end{gathered}$$
Notice that $\beta$ essentially scans a string of $0,1,2$’s, and swaps $12$ with $02$.
${\textnormal{Simp}(\sigma_{A})}$ is an important subgroup of ${\textnormal{Aut}}(\sigma_{A})$, and we’ll come back to it later.
The center of ${\textnormal{Aut}}(\sigma_{A})$
----------------------------------------------
Understanding the structure of ${\textnormal{Aut}}(\sigma_{A})$ as a group is not easy. One useful result is the following, proved by Ryan in ’72/’74.
\[thm:ryanstheorem\] If $A$ is irreducible (in particular, if $A$ is primitive) then the center of ${\textnormal{Aut}}(\sigma_{A})$ is generated by $\sigma_{A}$.
Ryan’s Theorem essentially says the center of ${\textnormal{Aut}}(\sigma_{A})$ is as small as it could possibly be. In fact, for $A$ irreducible, every normal amenable subgroup of ${\textnormal{Aut}}(\sigma_{A})$ is contained in the subgroup generated by $\sigma_{A}$; see [(Ap. ]{}[apprem:amenradical]{}).\
In [@Kopra2018], Kopra proved a finitary version of Ryan’s Theorem: namely, for any nontrivial irreducible shift of finite type, there exists a subgroup generated by two elements whose centralizer is generated by the shift map. Prior to this, Salo in [@Salo2019] had proved there is a finitely generated subgroup (needing more than two generators) of the automorphism group of the full shift on four symbols whose centralizer is generated by the shift map.\
Ryan’s Theorem can be used to distinguish, up to isomorphism, automorphism groups of certain subshifts of finite type. The idea is to use Ryan’s Theorem in conjunction with the set of possible roots of the shift. For a subshift $(X,\sigma)$, define the root set of $\sigma$ to be $\textnormal{root}(\sigma) = \{k \in \mathbb{N} \mid \textnormal{ there exists } \alpha \in {\textnormal{Aut}}(\sigma) \textnormal{ such that } \alpha^{k}=\sigma \}$. The following exercise demonstrates this technique.
[(Ap. ]{}[appexer:ryanthmdist]{})\[exer:ryanthmdist\]
1. Show that if $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ are irreducible shifts of finite type such that ${\textnormal{Aut}}(\sigma_{A})$ and ${\textnormal{Aut}}(\sigma_{B})$ are isomorphic, then $\textnormal{root}(\sigma_{A}) = \textnormal{root}(\sigma_{B})$.
2. Let $(X_{2},\sigma_{2}), (X_{4},\sigma_{4})$ denote the full shift on 2 symbols and on 4 symbols, respectively. Show that $\textnormal{root}(\sigma_{2}) \ne \textnormal{root}(\sigma_{4})$. Use part (1) to conclude that ${\textnormal{Aut}}(\sigma_{2})$ and ${\textnormal{Aut}}(\sigma_{4})$ are not isomorphic as groups.
The exercise above can be generalized to some other values of $m$ and $n$; one can find this written down in [@HKS] (also see [@BLR88 Ex. 4.2] for an example where the method is used to distinguish automorphism groups in the non-full shift case). For a full shift $(X_{n},\sigma_{n})$, it turns out that $k \in \textnormal{root}(\sigma_{n})$ if and only if $n$ has a $k$th root in $\mathbb{N}$ (see [@Lind84 Theorem 8]).\
Currently, the technique of using Ryan’s Theorem in conjunction with $\textnormal{root}(\sigma_{A})$ is the only method known to us which can show two explicit nontrivial mixing shifts of finite type have non-isomorphic automorphism groups. We do not at the moment know how to distinguish automorphism groups with identical root sets; in particular, despite being introduced by Hedlund in the 60’s, we still do not know whether ${\textnormal{Aut}}(\sigma_{2})$ and ${\textnormal{Aut}}(\sigma_{3})$ are isomorphic (see Problem \[prob:fullshiftautisoproblem\] in Section \[subsec:openproblemsaut\]).
Representations of ${\textnormal{Aut}}(\sigma_{A})$
---------------------------------------------------
So how can we study ${\textnormal{Aut}}(\sigma_{A})$? One way is to try to find good representations of it. There are two main classes of representations that we know of:
1. Periodic point representations, and representations derived from these.
2. The dimension representation.
The first, the periodic point representations (and ones derived from them), are quite natural to consider. They also lead to the sign and gyration maps, which are also quite natural (once defined). The second, the dimension representation, is essentially a linear representation, and is based on the dimension group associated to the shift of finite type in question.
We start with the second one, the dimension representation.
Dimension Representation {#subsec:dimrep}
------------------------
We briefly recall the definition, introduced in Section \[sec:SEZdirectlimits\] in Lecture 2, of the dimension group associated to a $\mathbb{Z}_{+}$-matrix. Given an $r \times r$ matrix $A$ over $\mathbb{Z}_{+}$ the eventual range subspace of $A$ is $ER(A) = \mathbb{Q}^{r}A^{r}$ (we will have matrices act on row vectors throughout), and the dimension group associated to $A$ is $$G_{A} = \{ x \in ER(A) \mid x A^{k} \in \mathbb{Z}^{r} \cap ER(A) \textnormal{ for some }k \ge 0\}.$$ Recall also the group $G_{A}$ comes equipped with an automorphism (of abelian groups) $\delta_{A} \colon G_{A} \to G_{A}$ (the automorphism $\delta_{A}$ was denoted by $\hat{A}$ in Lecture 2, but we’ll use the notation $\delta_{A}$). The automorphism $\delta_{A}$ of $G_{A}$ makes $G_{A}$ into a $\mathbb{Z}[t,t^{-1}]$-module by having $t$ act by $\delta_{A}^{-1}$, but we will usually just refer to the pair $(G_{A},\delta_{A})$ to indicate we are considering both $G_{A}$ and $\delta_{A}$ together. Then by an automorphism of $(G_{A},\delta_{A})$ we mean a group automorphism $\Psi \colon G_{A} \to G_{A}$ which satisfies $\Psi \delta_{A} = \delta_{A} \Psi$; in other words, an automorphism of the pair is equivalent to an automorphism of $G_{A}$ as a $\mathbb{Z}[t,t^{-1}]$-module. Let ${\textnormal{Aut}}(G_{A})$ denote the group of automorphisms of the pair $(G_{A},\delta_{A})$.\
The group $G_{A}$ is isomorphic, as an abelian group, to the direct limit $\varinjlim \{\mathbb{Z}^{r}, x \mapsto xA\}$.\
When $A$ is over $\mathbb{Z}_{+}$ (which is the case for a matrix presenting an edge shift of finite type), $G_{A}$ has a positive cone $G_{A}^{+} = \{v \in G_{A} \mid vA^{k} \in \mathbb{Z}^{r}_{+} \textnormal{ for some } k\}$ making $G_{A}$ into an ordered abelian group. The automorphism $\delta_{A}$ maps $G_{A}^{+}$ into $G_{A}^{+}$, and when we want to keep track of the order structure we refer to the triple $(G_{A},G_{A}^{+},\delta_{A})$. An automorphism of the triple $(G_{A},G_{A}^{+},\delta_{A})$ then means an automorphism of $(G_{A},\delta_{A})$ which preserves $G_{A}^{+}$.\
[(Ap. ]{}[appexer:dimgroupfullshiftcomp]{})\[exer:dimgroupfullshiftcomp\] When $A = (n)$ (the case of the full-shift on $n$ symbols), the triple $(G_{n},G_{n}^{+},\delta_{n})$ is isomorphic to the triple $(\mathbb{Z}[\frac{1}{n}],\mathbb{Z}_{+}[\frac{1}{n}],m_{n})$, where $m_{n}$ is the automorphism of $\mathbb{Z}[\frac{1}{n}]$ defined by $m_{n}(x) = x \cdot n$.
The following exercise shows that for a mixing shift of finite type $(X_{A},\sigma_{A})$, the group of automorphisms of $(G_{A},G_{A}^{+},\delta_{A})$ is index two in ${\textnormal{Aut}}(G_{A},\delta_{A})$.
[(Ap. ]{}[appexer:indextwoorderautos]{})\[exer:indextwoorderautos\] Let $A$ be a primitive matrix and suppose $\Psi$ is an automorphism of $(G_{A},\delta_{A})$. By considering $G_{A}$ as a subgroup of $ER(A)$, show that $\Psi$ extends to a linear automorphism $\tilde{\Psi} \colon ER(A) \to ER(A)$ which multiplies the Perron eigenvector of $A$ by some quantity $\lambda_{\Psi}$. Show that $\Psi$ is also an automorphism of the ordered abelian group $(G_{A},G_{A}^{+},\delta_{A})$ if and only if $\lambda_{\Psi}$ is positive.
Krieger gave a definition of a triple $(D_{A},D_{A}^{+},d_{A})$ which is isomorphic to the triple $(G_{A},G_{A}^{+},\delta_{A})$ using only topological/dynamical data intrinsic to the system $(X_{A},\sigma_{A})$ [(Ap. ]{}[rem:kriegerconstruction]{}).\
A topological conjugacy between shifts of finite type $\Psi \colon (X_{A},\sigma_{A}) \to (X_{B},\sigma_{B})$ induces an isomorphism $\Psi_{*} (G_{A},G_{A}^{+},\delta_{A}) \stackrel{\cong}\longrightarrow (G_{B},G_{B}^{+},\delta_{B})$. This is easiest to see using Krieger’s intrinsic definition of $(G_{A},G_{A}^{+},\delta_{A})$ (see [(Ap. ]{}[rem:kriegerconstruction]{})). One can also see this in terms of the conjugacy/strong shift equivalence framework developed in Lecture 2, as follows. Given a conjugacy $\alpha \colon (X_{A},\sigma_{A}) \to (X_{B},\sigma_{B})$, from Lecture 2 we know that corresponding to $\alpha$ is some strong shift equivalence from $A$ to $B$ $$A = R_{1}S_{1}, A_{2} = S_{1}R_{1}, \ldots A_{n} = R_{n}S_{n}, B = S_{n}R_{n}.$$ Then we can define an isomorphism from $(G_{A},G_{A}^{+},\delta_{A})$ to $(G_{B},G_{B}^{+},\delta_{B})$ by $$\pi(\alpha) = \prod_{i=1}^{n}R_{i}.$$ A priori, it is not clear that $\pi(\alpha)$ is actually well-defined, since the strong shift equivalence we choose to associate to $\alpha$ may not be unique. However, it turns out that $\pi(\alpha)$ is indeed well-defined; this will be a consequence of material in Lecture 8. Since an automorphism of $(X_{A},\sigma_{A})$ is just a self-conjugacy of $(X_{A},\sigma_{A})$, it follows that any $\alpha \in {\textnormal{Aut}}(\sigma_{A})$ induces an isomorphism $\alpha_{*} \colon (G_{A},G_{A}^{+},\delta_{A}) \stackrel{\cong}\longrightarrow (G_{A},G_{A}^{+},\delta_{A})$, and there is a well-defined homomorphism $$\pi_{A} \colon {\textnormal{Aut}}(\sigma_{A}) \to {\textnormal{Aut}}(G_{A},G_{A}^{+},\delta_{A}).$$ The homomorphism $\pi_{A}$ is known as the *dimension representation* of ${\textnormal{Aut}}(\sigma_{A})$.
The automorphism $\sigma_{A} \in {\textnormal{Aut}}(\sigma_{A})$ corresponds to the strong shift equivalence $$A = (A)(I), \hspace{.03in} A = (I)(A).$$ In particular, we have for any shift of finite type $(X_{A},\sigma_{A})$ $$\pi_{A}(\sigma_{A}) = \delta_{A} \in {\textnormal{Aut}}(G_{A},G_{A}^{+},\delta_{A}).$$
When $A = (3)$, $ER(A) = \mathbb{Q}$, and as mentioned above, the dimension triple is isomorphic to $(\mathbb{Z}[\frac{1}{3}],\mathbb{Z}_{+}[\frac{1}{3}],m_{3})$ where $m_{3}(x) = 3x$. Thus ${\textnormal{Aut}}(G_{3},G_{3}^{+},\delta_{3}) \cong \mathbb{Z}$, where $\mathbb{Z}$ is generated by $\delta_{3}$. The dimension representation then looks like $$\pi_{3} \colon {\textnormal{Aut}}(\sigma_{3}) \to {\textnormal{Aut}}(\mathbb{Z}[\frac{1}{3}],\mathbb{Z}_{+}[\frac{1}{3}],\delta_{3}) \cong \mathbb{Z} = \langle \delta_{3} \rangle$$ $$\pi_{3} \colon \sigma_{3} \mapsto \delta_{3}.$$
More generally, the following proposition describes how the dimension representation behaves for full shifts. Given $n \in \mathbb{N}$, let $\omega(n)$ denote the number of distinct prime divisors of $n$.
\[prop:fullshiftdimautos\] Given $n \ge 2$, there is an isomorphism ${\textnormal{Aut}}(G_{n},G^{+}_{n},\delta_{n}) \cong \mathbb{Z}^{\omega(n)}$ and the map $\pi_{n} \colon {\textnormal{Aut}}(\sigma_{n}) \to {\textnormal{Aut}}(G_{n},G^{+}_{n},\delta_{n})$ is surjective.
From Exercise \[exer:dimgroupfullshiftcomp\] we know $(G_{n},G_{n}^{+},\delta_{n}) \cong (\mathbb{Z}[\frac{1}{n}],\mathbb{Z}_{+}[\frac{1}{n}],\delta_{n})$. The result follows since the group ${\textnormal{Aut}}(\mathbb{Z}[\frac{1}{n}],\mathbb{Z}_{+}[\frac{1}{n}],\delta_{n})$ is free abelian with basis given by the maps $\delta_{p_{i}} \colon x \mapsto x\cdot p_{i}$ where $p_{i}$ is a prime dividing $n$. For the surjectivity part of $\pi_{n}$, see [@BLR88].
In general, the dimension representation may not be surjective (see [@S19]), and the following question is still open:\
\[prob:dimreprange\] Given a mixing shift of finite type $(X_{A},\sigma_{A})$, what is the image of the dimension representation $\pi_{A} \colon {\textnormal{Aut}}(\sigma_{A}) \to {\textnormal{Aut}}(G_{A},G_{A}^{+},\delta_{A})$?
Problem \[prob:dimreprange\] is of relevance for the classification problem (see [(Ap. ]{}[apprem:dimreprangeclassification]{})).\
In [@BLR88 Theorem 6.8] it is shown that if the non-zero eigenvalues of $A$ are simple, and no ratio of distinct eigenvalues is a root of unity, then for all sufficiently large $m$ the dimension representation $\pi_{A}^{(m)} \colon {\textnormal{Aut}}(\sigma_{A}^{m}) \to {\textnormal{Aut}}(G_{A^{m}},G_{A^{m}}^{+},\delta_{A^{m}})$ is onto. Long [@Long2013] showed the “elementary” construction method of [@BLR88 Theorem 6.8] is not in general sufficient to reveal the full image of the dimension representation.\
An automorphism $\alpha \in {\textnormal{Aut}}(\sigma_{A})$ is called *inert* if $\alpha$ lies in the kernel of $\pi_{A}$, and we denote the subgroup of inerts by $${\textnormal{Inert}(\sigma_{A})} = \ker \pi_{A}.$$
The subgroup ${\textnormal{Inert}(\sigma_{A})}$ is, roughly speaking, the heart of ${\textnormal{Aut}}(\sigma_{A})$, and in general, we do not know how to distinguish the subgroup of inert automorphisms among different shifts of finite type. The following exercise shows that constructions using marker methods or simple automorphisms always lie in ${\textnormal{Inert}(\sigma_{A})}$.
[(Ap. ]{}[appexer:simpininert]{})\[exer:simpininerts\] For any shift of finite type $(X_{A},\sigma_{A})$, we have ${\textnormal{Simp}(\sigma_{A})} \subset {\textnormal{Inert}(\sigma_{A})}$. (Hint: Use [(Ap. ]{}[rem:kriegerconstruction]{}))
As evidence that ${\textnormal{Inert}(\sigma_{A})}$ contains much of the complicated algebraic structure of ${\textnormal{Aut}}(\sigma_{A})$, consider the case of a full shift over a prime number of symbols, i.e. $A = (p)$ for some prime $p$. In this case, ${\textnormal{Aut}}(G_{p},G_{p}^{+},\delta_{p}) \cong \mathbb{Z}$ is generated by $\delta_{p}$, and the map $$\pi_{p} \colon {\textnormal{Aut}}(\sigma_{p}) \to {\textnormal{Aut}}(G_{p},G_{p}^{+},\delta_{p})$$ is a split surjection, with a splitting map being given by $\delta_{p} \mapsto \sigma_{p}$. This shows ${\textnormal{Aut}}(\sigma_{p})$ is isomorphic to a semi-direct product of ${\textnormal{Inert}(\sigma_{p})}$ and $\mathbb{Z}$. Since $\sigma_{p}$ lies in the center of ${\textnormal{Aut}}(\sigma_{p})$, in fact this semi-direct product is isomorphic to a direct product, and we have $${\textnormal{Aut}}(\sigma_{p}) \cong {\textnormal{Inert}(\sigma_{p})} \times \mathbb{Z}.$$
Periodic point representation
-----------------------------
For an SFT $(X_{A},\sigma_{A})$ and $k \in \mathbb{N}$ we let $P_{k}$ denote the $\sigma_{A}$-periodic points of least period $k$, and $Q_{k}$ the set of $\sigma_{A}$-orbits of length $k$ (both $P_{k}$ and $Q_{k}$ depend on $\sigma_{A}$ of course - we suppress this in the notation since it’s usually clear from context). For a shift of finite type, the set $P_{k}$ is always finite, and we have $$|P_{k}| = k|Q_{k}|.$$
Let $\alpha \in {\textnormal{Aut}}(\sigma_{A})$ and let $k \in \mathbb{N}$. Since $\alpha$ is a bijection which commutes with $\sigma_{A}$, $\alpha$ maps $P_{k}$ to itself and thus induces a permutation of $P_{k}$ which we’ll denote by $$\rho_{k}(\alpha) \in {\textnormal{Sym}}(P_{k})$$ where ${\textnormal{Sym}}(P)$ of a set $P$ denotes the group of permutations of $P$ (we use the convention that if $P = \emptyset$ then ${\textnormal{Sym}}(P)$ is the group containing only one element).
It is straightforward to check that this assignment $\alpha \mapsto \rho_{k}(\alpha)$ defines a homomorphism $$\rho_{k} \colon {\textnormal{Aut}}(\sigma_{A}) \to {\textnormal{Sym}}(P_{k}).$$
The automorphism $\alpha$ must also respect $\sigma_{A}$-orbits, and it follows that $\alpha$ induces a permutation of the set $Q_{k}$ which we denote $$\xi_{k}(\alpha) \in {\textnormal{Sym}}(Q_{k}).$$ Thus, we also get a homomorphism $$\xi \colon {\textnormal{Aut}}(\sigma_{A}) \to {\textnormal{Sym}}(Q_{k}).$$
These homomorphisms assemble into homomorphisms $$\begin{gathered}
\rho \colon {\textnormal{Aut}}(\sigma_{A}) \to \prod_{k=1}^{\infty}{\textnormal{Sym}}(P_{k})\\
\rho(\alpha) = (\rho_{1}(\alpha),\rho_{2}(\alpha),\ldots).
\end{gathered}$$ and $$\begin{gathered}
\xi \colon {\textnormal{Aut}}(\sigma_{A}) \to \prod_{k=1}^{\infty}{\textnormal{Sym}}(Q_{k})\\
\xi(\alpha) = (\xi_{1}(\alpha),\xi_{2}(\alpha),\ldots).
\end{gathered}$$ The map $\rho$ is called the *periodic point representation* of ${\textnormal{Aut}}(\sigma_{A})$, and $\xi$ is called the *orbit representation*.\
When $A$ is irreducible, the map $\rho$ is injective (this follows from the fact that for irreducible $A$, periodic points are dense in $(X_{A},\sigma_{A})$ - see [@LindMarcus1995 Sec. 6.1]). Clearly $\xi$ can not be injective since $\sigma_{A} \in \xi$. However, it turns out $\sigma_{A}$ generates the whole kernel of $\xi$, from a theorem of Boyle-Krieger.
\[thm:injximap\] If $(X_{A},\sigma_{A})$ is an irreducible shift of finite type, then $\ker \xi = \langle \sigma_{A} \rangle$.
Fix $k \in \mathbb{N}$ and $\alpha \in {\textnormal{Aut}}(\sigma_{A})$. The periodic point representation $\rho_{k}(\alpha)$ is obtained by restricting $\alpha$ to the finite subsystem $P_{k}$ of $(X_{A},\sigma_{A})$, and $\rho_{k}(\alpha)$ lies in the automorphism group ${\textnormal{Aut}}(\sigma_{A}|_{P_{k}})$ of this finite system. It was observed in [@MR887501] that the automorphism group ${\textnormal{Aut}}(\sigma_{A}|_{P_{k}})$ is isomorphic to the semidirect product $\left(\mathbb{Z}/k\mathbb{Z}\right)^{Q_{k}} \rtimes {\textnormal{Sym}}(Q_{k})$ [(Ap. ]{}[apprem:semidirectaut]{}), and this leads to considering possible abelian factors of these automorphism groups ${\textnormal{Aut}}(\sigma_{A}|_{P_{k}})$. This motivates the following gyration maps, which were introduced by Boyle and Krieger in [@MR887501].
\[def:sg\] Fix $k \in \mathbb{N}$. We define the $k$th gyration map $g_{k} \colon {\textnormal{Aut}}(\sigma_{A}) \to \mathbb{Z}/k\mathbb{Z}$ as follows. Let $\alpha \in {\textnormal{Aut}}(\sigma_{A})$, let $Q_{k} = \{O_{1},\ldots,O_{I(k)}\}$ denote the set of orbits in $Q_{k}$, and choose, for each $1 \le i \le k$, some representative point $x_{i} \in O_{i}$. Then $\alpha(x_{i}) \in O_{\xi_{k}(\alpha)(i)}$, so there exists some $r(\alpha,i) \in \mathbb{Z} / k\mathbb{Z}$ such that $\alpha(x_{i}) = \sigma_{n}^{r(\alpha,i)}(x_{\xi_{k}(\alpha)(i)})$. Now define $$g_{k} = \sum_{i=1}^{I(k)} r(\alpha,i) \in \mathbb{Z} / k\mathbb{Z}.$$ Boyle and Krieger showed this map is independent of the choices of $x_{i}$’s, and is a homomorphism, so we get homomorphisms $$g_{k} \colon {\textnormal{Aut}}(\sigma_{n}) \to \mathbb{Z} / k\mathbb{Z}.$$ Now we can define the *gyration representation* by $$\begin{gathered}
g \colon {\textnormal{Aut}}(\sigma_{n}) \to \prod_{k=1}^{\infty}\mathbb{Z} / k\mathbb{Z}\\
g(\alpha) = (g_{1}(\alpha),g_{2}(\alpha),\ldots).
\end{gathered}$$
Given $k$, consider ${\textnormal{sign}}\xi_{k} \colon {\textnormal{Aut}}(\sigma_{A}|_{P_{k}}) \to \mathbb{Z}/2\mathbb{Z}$, the map $\xi_{k}$ composed with the sign map to $\mathbb{Z}/2\mathbb{Z}$. The gyration map $g_{k}$, together with ${\textnormal{sign}}\xi_{k}$, determines the abelianization of ${\textnormal{Aut}}(\sigma_{A}|_{P_{k}})$: any other map from ${\textnormal{Aut}}(\sigma_{A}|_{P_{k}})$ to an abelian group factors through the map $$\begin{gathered}
g_{k} \times {\textnormal{sign}}\xi_{k} \colon {\textnormal{Aut}}(\sigma_{A}|_{P_{k}}) \to \mathbb{Z}/k\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}\end{gathered}$$ (see [(Ap. ]{}[apprem:abelsigngyr]{})).
Inerts and the sign gyration compatibility condition {#subsec:sgcc}
----------------------------------------------------
A priori, it would seem the dimension representation and the periodic point representation need not have any relationship. Remarkably, this turns out not to be the case, and there is in fact a connection between them: for inert automorphisms (recall inert automorphisms are precisely the kernel of the dimension representation), there are certain conditions which relate the periodic orbit representation and the periodic point representation of the automorphism. This is formalized in the following way.
Say $\alpha \in {\textnormal{Aut}}(\sigma_{A})$ *satisfies SGCC (sign-gyration compatibility condition)* if the following holds: for every positive odd integer $m$ and every non-negative integer $i$, if $n=m2^{i}$, then
$$\begin{aligned}
{2} \\
g_n(\alpha ) &=0 \quad \ \ && \text{if } \quad \prod_{j=0}^{i-1}{\textnormal{sign}}\xi_{m2^{j}}(\alpha) = 1\\
g_n(\alpha ) &=\frac n2 && \text{if } \quad \prod_{j=0}^{i-1}{\textnormal{sign}}\xi_{m2^{j}}(\alpha) = -1 \ .\end{aligned}$$
The empty product we take to have the value 1.
Thus for $\alpha \in {\textnormal{Aut}}(\sigma_{A})$ satisfying SGCC, $g(\alpha)$ and ${\textnormal{sign}}\xi(\alpha)$ determine each other.\
An important step is to rephrase the SGCC condition in terms of certain homomorphisms, which we describe now. Consider now the ${\textnormal{sign}}$ homomorphisms as taking values in the group $\mathbb{Z}/2\mathbb{Z}$ (so if $\tau$ is an odd permutation, ${\textnormal{sign}}(\tau)=1 \in \mathbb{Z}/2\mathbb{Z}$). Define for $n \ge 2$ the SGCC homomorphism $$\begin{gathered}
SGCC_{n} \colon {\textnormal{Aut}}(\sigma_{A}) \to \mathbb{Z}/n\mathbb{Z}\\
SGCC_{n}(\alpha) = g_{n}(\alpha) + \left( \frac{n}{2}\right)\sum_{j>0}{\textnormal{sign}}\xi_{n/2^{j}}(\alpha)
\end{gathered}$$ where we define ${\textnormal{sign}}\xi_{n/2^{j}}(\alpha) = 0$ if $n/2^{j}$ is not an integer. The following is immediate to check, but very useful.
Let $(X_{A},\sigma_{A})$ be a mixing shift of finite type, and $\alpha \in {\textnormal{Aut}}(\sigma_{A})$. Then $\alpha$ satisfies SGCC if and only if for all $n \ge 2$, $SGCC_{n}(\alpha)=0$.
So which automorphisms satisfy SGCC? Amazingly enough, any inert automorphism does. This fact was the culmination of results obtained over several years (see [(Ap. ]{}[apprem:sgccremark]{})), and was finally proved by Kim and Roush in [@S23], using an important cocycle lemma of Wagoner. A more complete picture was subsequently given by Kim-Roush-Wagoner in [@S19]; we’ll describe this briefly here. The appropriate setting for a deeper understanding is Wagoner’s CW complexes, which are the subject of the next lecture.\
Suppose $A=RS, B=SR$ is a strong shift equivalence over $\mathbb{Z}_{+}$, and let $\phi_{R,S} \colon (X_{A},\sigma_{A}) \to (X_{B},\sigma_{B})$ be a conjugacy induced by this SSE. In [@S19], Kim-Rough-Wagoner showed that, using certain lexicographical orderings on each set of periodic points, one can compute $SGCC_{m}$ values, with respect to this choice of ordering on periodic points, analogous to how the $SGCC_{m}$ homomorphisms are defined for automorphisms. Moreover, they showed these values can be computed in terms of a (complicated) formula defined only using terms from the matrices $R,S$. In fact, this formula makes sense even if we start with a strong shift equivalence $A=RS, B=SR$ over $\mathbb{Z}$, and Kim-Roush-Wagoner showed that these formulas can be used to define homomorphisms $sgcc_{m} \colon {\textnormal{Aut}}(G_{A},\delta_{A}) \to \mathbb{Z}/m\mathbb{Z}$. Note that the domain of this homomorphism is ${\textnormal{Aut}}(G_{A},\delta_{A})$, i.e. automorphisms of the pair $(G_{A},\delta_{A})$ which don’t necessarily preserve the positive cone $G_{A}^{+}$. Altogether, Kim-Roush-Wagoner proved the following.
\[thm:sgccfactorization\] Let $(X_{A},\sigma_{A})$ be a mixing shift of finite type. For every $m \ge 2$ there exists a homomorphism $sgcc_{m} \colon {\textnormal{Aut}}(G_{A},\delta_{A}) \to \mathbb{Z}/m\mathbb{Z}$ such that the following diagram commutes $$\xymatrix{
{\textnormal{Aut}}(\sigma_{A}) \ar[r]^{\pi_{A}} \ar[dr]_{SGCC_{m}}& {\textnormal{Aut}}(G_{A},\delta_{A}) \ar[d]^{sgcc_{m}} \\
& \mathbb{Z}/m\mathbb{Z} \\
}$$ In particular, if $\alpha \in {\textnormal{Inert}(\sigma_{(})}\sigma_{A})$, then $SGCC_{m}(\alpha)=0$.
An explicit formula for $sgcc_{2}$ can be found in [@S19 Prop. 2.14], with a general formula for $sgcc_{m}$ described in [@S19 2.31].\
As shown in [@S23] and [@S19], that SGCC vanishes on any inert automorphism can be used to rule out certain actions on finite subsystems of the shift system. For example, the following was shown in [@S19] (based on a suggestion by Ulf Fiebig). Consider an automorphism $\alpha$ of the period 6 points of the full 2 shift $(X_{2},\sigma_{2})$ which acts by the shift on one of the orbits, and the identity on the remaining orbits. It is immediate to compute that $SGCC_{6}(\alpha) = 1 \in \mathbb{Z}/6\mathbb{Z}$. However ${\textnormal{Aut}}(G_{2},\delta_{2}) \cong \mathbb{Z}$ is generated by $\delta_{2}$, the image of the shift $\sigma_{2}$ under the dimension representation $\pi_{2}$, and $sgcc_{6}(\delta_{2}) = 3 \in \mathbb{Z}/6\mathbb{Z}$; by Theorem \[thm:sgccfactorization\], this implies the image of $SGCC_{6}$ in $\mathbb{Z}/6\mathbb{Z}$ must be the subgroup $\{0,3\} \subset \mathbb{Z}/6\mathbb{Z}$, which does not contain $1$. Thus $\alpha$ can not be the restriction of an automorphism in ${\textnormal{Aut}}(\sigma_{2})$. This (along with an additional example given in [@S19]) resolved a long standing open problem about lifting automorphisms from finite subsystems (see Problem \[prob:liftproblem1\] in Section \[subsec:openproblemsaut\]).
Actions on finite subsystems {#subsec:actionsonfinitesubs}
----------------------------
The SGCC conditions give necessary conditions for the action of an inert automorphism on finite subsystems of the shift system. A natural question is whether one can determine precisely what possible actions can be realized: that is, what are sufficient conditions for an automorphism of a finite subsystem to be the restriction of an inert automorphism? [(Ap. ]{}[apprem:actionsfinitesubsrem]{}) In [@MR1125880], Boyle and Fiebig characterized the possible actions of finite order inert automorphisms on finite subsystems of the shift. Then, in [@S14; @KRWForumI; @KRWForumII], Kim-Roush-Wagoner settled this question completely, by showing that the SGCC condition is also sufficient for lifting an automorphism of a finite subsystem to an automorphism of the shift. Together with the Boyle-Fiebig classification in [@MR1125880], this is used in [@S14; @KRWForumI; @KRWForumII] to resolve (in the negative) a long standing problem regarding finite order generation of the inert subgroup ${\textnormal{Inert}(\sigma_{A})}$; see Section \[subsec:openproblemsaut\].
Notable problems regarding ${\textnormal{Aut}}(\sigma_{A})$ {#subsec:openproblemsaut}
-----------------------------------------------------------
There have been a number of questions and conjectures that have been influential in the study of ${\textnormal{Aut}}(\sigma_{A})$, and we’ll describe a few of them here. This is by by no means intended to be an exhaustive list; instead, we simply highlight some problems that have been important (both historically, and still), as well as some problems that demonstrate the state of our ignorance regarding the group ${\textnormal{Aut}}(\sigma_{A})$. Some of these have been resolved in some cases, while some are open in all cases.\
Given a group $G$, let $\textnormal{Fin}(G)$ denote the (normal) subgroup of $G$ generated by elements of finite order.\
Recall for any shift of finite type $(X_{A},\sigma_{A})$, we have containments of subgroups ${\textnormal{Simp}(\sigma_{A})} \subset \textnormal{Fin}({\textnormal{Inert}(\sigma_{A})}) \subset {\textnormal{Inert}(\sigma_{A})}$. One general problem[^4] is the following:
When is it true that ${\textnormal{Inert}(\sigma_{A})} = \textnormal{Fin}({\textnormal{Inert}(\sigma_{A})})$?
The FOG problem is an outgrowth of a conjecture, originally posed by F. Rhodes to Hedlund in a correspondence, asking whether ${\textnormal{Aut}}(\sigma_{2})$ is generated by $\sigma_{2}$ and elements of finite order.\
Kim, Roush and Wagoner in [@KRWForumI; @KRWForumII] showed there exists a shift of finite type $(X_{B},\sigma_{B})$ such that the containment $\textnormal{Fin}({\textnormal{Inert}(\sigma_{B})}) \subset {\textnormal{Inert}(\sigma_{B})}$ is proper, showing the answer to FOG is ‘not always’ (see the discussion in Section \[subsec:actionsonfinitesubs\]). Prior to this, in [@Wagoner90eventual] Wagoner considered a stronger form of FOG, asking whether it was always true that ${\textnormal{Simp}(\sigma_{A})} = {\textnormal{Inert}(\sigma_{A})}$; this was sometimes referred to as the Simple Finite Order Generation Conjecture (SFOG). Kim and Roush in [@S25] showed (prior to their example showing FOG does not always hold) that SFOG does not always hold, giving an example of a shift of finite type $(X_{A},\sigma_{A})$ such that the containment ${\textnormal{Simp}(\sigma_{A})} \subset {\textnormal{Inert}(\sigma_{A})}$ is proper.\
Expanding on FOG, we have the following more general problem:
Given a shift of finite type $(X_{A},\sigma_{A})$, determine the index of the following subgroup containments:
1. \[item:partone\] ${\textnormal{Simp}(\sigma_{A})} \subset {\textnormal{Inert}(\sigma_{A})}$.
2. \[item:parttwo\] $\textnormal{Fin}({\textnormal{Inert}(\sigma_{A})}) \subset {\textnormal{Inert}(\sigma_{A})}$.
In particular, in each case, must the index be finite?
When ${\textnormal{Aut}}(G_{A})$ is torsion-free, every element of finite order in ${\textnormal{Aut}}(\sigma_{A})$ lies in ${\textnormal{Inert}(\sigma_{A})}$. In this case, the FOG problem is equivalent to determining whether the answer to Part (\[item:parttwo\]) of the Index Problem is one. In general, it is not known whether, for each part of the Index Problem, the index is finite or infinite. As noted earlier, in [@KRWForumI] an example is given of a mixing shift of finite type $(X_{A},\sigma_{A})$ for which the index of $\textnormal{Fin}({\textnormal{Inert}(\sigma_{A})})$ in ${\textnormal{Inert}(\sigma_{A})}$ is strictly greater than one. This relies on being able to construct an inert automorphism in ${\textnormal{Aut}}(\sigma_{A})$ which can not be a product of finite order automorphisms; this is carried out using the difficult constructions of Kim-Roush-Wagoner in [@KRWForumI; @KRWForumII], in which the polynomial matrix methods (introduced in Lecture 3) play an invaluable role (we do not know how to do such constructions without the polynomial matrix framework).\
However, whether FOG or even SFOG might hold in the case of a full shift $(X_{n},\sigma_{n})$ is still unknown.\
Finite order generation of the inerts for general mixing shifts of finite type is known to hold in the “eventual” setting; see [(Ap. ]{}[apprem:evensettingfog]{}).\
Williams in [@Williams73] asked whether any involution of a pair of fixed points of a shift of finite type can be extended to an automorphism of the whole shift of finite type. More generally, this grew into the following problem (stated in [@BLR88 Question 7.1]) about lifting actions on a finite collection of periodic points of the shift:
\[prob:liftproblem1\] Given a shift of finite type $(X_{A},\sigma_{A})$ and an automorphism $\phi$ of a finite subsystem $F$ of $(X_{A},\sigma_{A})$, does there exist $\tilde{\phi} \in {\textnormal{Aut}}(\sigma_{A})$ such that $\tilde{\phi}|_{F} = \phi$?
The answer to LIFT is also ‘not always’: Kim and Roush showed in [@S23], based on an example of Fiebig, that there exists an automorphism of the set of periodic six points in the full 2-shift which does extend to an automorphism of the full 2-shift.\
Roughly speaking, the LIFT problem involves two parts: determining the action of inert automorphisms on finite subsystems, and determining the range of the dimension representation. The first part has been resolved by Kim-Roush-Wagoner in [@KRWForumI; @KRWForumII]; see Section \[subsec:actionsonfinitesubs\]. The second part, to determine the range of the dimension representation, is still open in general (this was also stated in Problem \[prob:dimreprange\] in Section \[subsec:dimrep\]):
\[prob:dimreprange2\] Given a mixing shift of finite type $(X_{A},\sigma_{A})$, what is the image of the dimension representation $\pi_{A} \colon {\textnormal{Aut}}(\sigma_{A}) \to {\textnormal{Aut}}(G_{A},G_{A}^{+},\delta_{A})$? Is the image always finitely generated?
In [@S19], Kim and Roush showed there exists a shift of finite type for which the dimension representation is not surjective.\
In [@BLR88 Example 6.9], an example of a primitive matrix $A$ such that ${\textnormal{Aut}}(G_{A},G_{A}^{+},\delta_{A})$ is not finitely generated is given. This does not resolve the second part of Problem \[prob:dimreprange2\] though, since the range of the dimension representation is not known.\
Another question concerns the isomorphism type of the groups ${\textnormal{Aut}}(\sigma_{A})$. It is straightforward to check that conjugate shifts of finite type have isomorphic automorphism groups, and that ${\textnormal{Aut}}(\sigma_{A}) = {\textnormal{Aut}}(\sigma_{A}^{-1})$ always holds (note there exists shifts of finite type $(X_{A},\sigma_{A})$ which are not conjugate to their inverse; see for example Proposition \[transposeexerciseproof\] in Lecture 2). In [@BLR88 Question 4.1] the following was asked:
If ${\textnormal{Aut}}(\sigma_{A})$ and ${\textnormal{Aut}}(\sigma_{B})$ are isomorphic, must $(X_{A},\sigma_{A})$ be conjugate to either $(X_{B},\sigma_{B})$ or $(X_{B},\sigma_{B}^{-1})$?
A particular case of this which has been of interest is:
\[prob:fullshiftautisoproblem\] For which $m,n$ are the groups ${\textnormal{Aut}}(\sigma_{m})$ and ${\textnormal{Aut}}(\sigma_{n})$ isomorphic?
See Section \[subsec:stabilizedgroup\] for some results related to Problem \[prob:fullshiftautisoproblem\].
The stabilized automorphism group {#subsec:stabilizedgroup}
---------------------------------
Recently a new approach to the Aut-Isomorphism Problem, and the study of ${\textnormal{Aut}}(\sigma_{A})$ in general, has been undertaken in [@HKS]. The idea is to consider a certain stabilization of the automorphism group, using the observation that for all $k,m \ge 1$, ${\textnormal{Aut}}(\sigma_{A}^{k})$ is naturally a subgroup of ${\textnormal{Aut}}(\sigma_{A}^{km})$. Define the *stabilized automorphism group* of $(X_{A},\sigma_{A})$ to be $${\textnormal{Aut}^{(\infty)}(\sigma_{A})} = \bigcup_{k=1}^{\infty}{\textnormal{Aut}}(\sigma_{A}^{k})$$ where the union is taken in the group of all homeomorphisms of $X_{A}$. This is again a countable group. Similar to the definition of ${\textnormal{Aut}^{(\infty)}(\sigma_{A})}$, one defines a stabilized group of automorphisms of the dimension group by $${\textnormal{Aut}}^{(\infty)}(G_{A}) = \bigcup_{k=1}^{\infty} {\textnormal{Aut}}(G_{A},G_{A}^{+},\delta_{A}^{k}).$$ The group ${\textnormal{Aut}}^{(\infty)}(G_{A})$ is precisely the union of the centralizers of $\delta_{A}$ in the group ${\textnormal{Aut}}(G_{A},G_{A}^{+})$ of all order-preserving group automorphisms of $G_{A}$. Recall for a group $G$ we let $G_{\textnormal{ab}}$ denote the abelianization of $G$. In [@HKS], the following is proved.
\[thm:stabilizedtheorem\] Let $(X_{A},\sigma_{A})$ be a mixing shift of finite type. The dimension representation $$\pi_{A} \colon {\textnormal{Aut}}(\sigma_{A}) \to {\textnormal{Aut}}(G_{A})$$ extends to a stabilized dimension representation $$\pi_{A}^{(\infty)} \colon {\textnormal{Aut}^{(\infty)}(\sigma_{A})} \to {\textnormal{Aut}}^{(\infty)}(G_{A})$$ and the composition $${\textnormal{Aut}^{(\infty)}(\sigma_{A})} \stackrel{\pi_{A}^{(\infty)}}\longrightarrow {\textnormal{Aut}}^{(\infty)}(G_{A})\stackrel{\textnormal{ab}}\longrightarrow {\textnormal{Aut}}^{(\infty)}(G_{A})_{\textnormal{ab}}$$ is isomorphic to the abelianization of the stabilized automorphism group ${\textnormal{Aut}^{(\infty)}(\sigma_{A})}$. In particular, if ${\textnormal{Aut}}^{(\infty)}(G_{A})$ is abelian, then the commutator of ${\textnormal{Aut}^{(\infty)}(\sigma_{A})}$ coincides with the subgroup of stabilized inert automorphisms $${{\textnormal{Inert}}^{(\infty)}(\sigma_{A})} = \ker \pi_{A}^{(\infty)} = \bigcup_{k=1}^{\infty}\textnormal{Inert}(\sigma_{A}^{k}).$$
For example, in the case of a full shift $A=(n)$, it follows that ${\textnormal{Aut}^{(\infty)}(\sigma_{n})}_{ab}$ is isomorphic to $\mathbb{Z}^{\omega(n)}$, where $\omega(n)$ denotes the number of distinct prime divisors of $n$. As a corollary of this, if $\omega(m) \ne \omega(n)$, then ${\textnormal{Aut}^{(\infty)}(\sigma_{m})}$ and ${\textnormal{Aut}^{(\infty)}(\sigma_{n})}$ are not isomorphic. The corresponding result for non-stabilized automorphism groups is not known. For example, while the previously stated corollary implies ${\textnormal{Aut}^{(\infty)}(\sigma_{2})}$ and ${\textnormal{Aut}^{(\infty)}(\sigma_{6})}$ are not isomorphic, it is not currently known whether ${\textnormal{Aut}}(\sigma_{2})$ and ${\textnormal{Aut}}(\sigma_{6})$ are isomorphic or not.\
For a mixing shift of finite type, the classical automorphism group ${\textnormal{Aut}}(\sigma_{A})$ is always residually finite. It turns out that in the stabilized case, ${\textnormal{Aut}^{(\infty)}(\sigma_{A})}$ is never residually finite [@HKS Prop. 4.3]. In fact, in stark contrast, the following was proved in [@HKS]:
For any $n \ge 2$, the group of stabilized inert automorphisms ${{\textnormal{Inert}}^{(\infty)}(\sigma_{n})}$ is simple.
It would be very interesting to know whether the same result holds for all mixing shifts of finite type.\
Finally, we make a few comments about the connection between the stabilized setting for automorphism groups described above and algebraic K-theory. In fact, the idea of the groups ${\textnormal{Aut}^{(\infty)}(\sigma_{A})}$ is partly motivated by algebraic K-theory, where the technique of stabilization proves to be fundamental. Recall as outlined in Lecture 5, as a starting point for algebraic K-theory, given a ring ${\mathcal{R}}$, one can consider the stabilized general linear group $$GL({\mathcal{R}}) = \varinjlim GL_{n}({\mathcal{R}})$$ where $GL_{n}({\mathcal{R}}) \hookrightarrow GL_{n+1}({\mathcal{R}})$ via $A \mapsto \begin{pmatrix} A & 0 \\ 0 & I \end{pmatrix}$. Inside each $GL_{n}({\mathcal{R}})$ lies the subgroup $El_{n}({\mathcal{R}})$ generated by elementary matrices, and one likewise defines the stabilized group of elementary matrices by $$El({\mathcal{R}}) = \varinjlim El_{n}({\mathcal{R}}).$$ Whitehead showed (see Lecture 5) that, upon stabilizing, the explicitly defined subgroup $El({\mathcal{R}})$ coincides with the commutator of $GL({\mathcal{R}})$. From this viewpoint, one may interpret Theorem \[thm:stabilizedtheorem\] as a Whitehead-type result for shifts of finite type. In particular, in the case of a full shift $(X_{n},\sigma_{n})$ (or more generally a shift of finite type $(X_{A},\sigma_{A})$ where ${\textnormal{Aut}}^{(\infty)}(G_{A})$ is abelian), after stabilizing, the commutator subgroup of ${\textnormal{Aut}^{(\infty)}(\sigma_{A})}$ coincides with the subgroup ${{\textnormal{Inert}}^{(\infty)}(\sigma_{A})}$[^5].
Mapping class groups of subshifts
---------------------------------
Recall from Section \[subsec:backgroundstuff\] that two homeomorphisms are flow equivalent if there is a homeomorphism of their mapping tori which takes orbits to orbits and preserve the direction of the suspension flow. For a subshift $(X,\sigma)$, an analog of the automorphism group in the setting of flow equivalence is given by the mapping class group $\mathcal{M}(\sigma)$, which is defined to be the group of isotopy classes of self-flow equivalences of the subshift $(X,\sigma)$.\
In [@BCmcg] a study of the mapping class group for shifts of finite type was undertaken. There it was shown that, for a nontrivial irreducible shift of finite type $(X_{A},\sigma_{A})$, the mapping class group $\mathcal{M}(\sigma_{A})$ is not residually finite. While the periodic point representations do not exist for $\mathcal{M}(\sigma_{A})$, a vestige of the dimension representation survives in the form of the Bowen-Franks representation of $\mathcal{M}(\sigma_{A})$. It was also shown that ${\textnormal{Aut}}(\sigma_{A}) / \langle \sigma_{A} \rangle$ embeds into $\mathcal{M}(\sigma_{A})$, and there is an analog of block codes, known as flow codes.\
See also [@SYmcg] for a study of the mapping class group in the case of a minimal subshift.
Appendix 7
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This appendix contains some proofs, remarks, and solutions of various exercises through Lecture 7.\
\[apprem:lowcomplexity\] Recall from Section \[subsec:ctsshiftcommmaps\] that for a subshift $(X,\sigma)$, we let $\mathcal{W}_{n}(X)$ denote the set of $X$-words of length $n$. We define the complexity function (of $X$) $P_{X} \colon \mathbb{N} \to \mathbb{N}$ by $P_{X}(n) = |\mathcal{W}_{n}(X)|$. Thus $P_{X}(n)$ simply counts the number of $X$-words of length $n$. For a shift of finite type $(Y,\sigma)$ with positive entropy, the function $P_{Y}(n)$ grows exponentially in $n$; for example, for the full shift $(X_{m},\sigma_{m})$ on $m$ symbols, $P_{X_{m}}(n) = m^{n}$. For a subshift $(X_{\alpha},\sigma_{\alpha})$ of the form given in Example \[ex:sturmiansubshift\], the complexity satisfies $P_{X_{\alpha}}(n) = n+1$ (such subshifts are called Sturmian subshifts). This is the slowest possible growth of complexity function for an infinite subshift: a theorem of Morse and Hedlund [@MH1938] from 1938 shows that for an infinite subshift $(X,\sigma)$, we must have $P_{X}(n) \ge n+1$.\
There has been a great deal of interest in studying the automorphism groups of subshifts with slow-growing complexity functions. Numerous results show that such low complexity subshifts often have much more tame automorphism groups, in comparison to subshifts possessing complexity functions of exponential growth (e.g. shifts of finite type). We won’t attempt to survey these results, but refer the reader to [@DDMP2016; @HP1989; @Coven1971; @Olli2013; @SaloTorma2015; @CyrKra2016; @CyrKra2015; @CyrKra2016strexp].
\[appex:denppresfin\] If $(X,\sigma)$ is a subshift whose periodic points are dense in $X$, then ${\textnormal{Aut}}(\sigma)$ is residually finite.
Given $n \in \mathbb{N}$, let $P_{n}(X)$ denote the set of points of least period $n$ in $X$. Since $X$ is a subshift, $|P_{n}(X)|<\infty$ for every $n$. If $\alpha \in {\textnormal{Aut}}(\sigma)$, then since $\alpha$ commutes with $\sigma$, for any $n$ the set $P_{n}(X)$ is invariant under $\alpha$. It follows there are homomorphisms $$\begin{gathered}
\rho_{n} \colon {\textnormal{Aut}}(\sigma) \to {\textnormal{Sym}}(P_{n}(X))\\
\rho_{n} \colon \alpha \mapsto \alpha|_{P_{n}(X)}
\end{gathered}$$ where ${\textnormal{Sym}}(P_{n}(X))$ denotes the group of permutations of the set $P_{n}(X)$. Now suppose $\alpha \in {\textnormal{Aut}}(\sigma)$ and $\rho_{n}(\alpha)=\textnormal{id}$ for all $n$. Then $\alpha$ fixes every periodic point in $X$; since the periodic points are dense in $X$ (by assumption) and $\alpha$ is a homeomorphism, $\alpha$ must be the identity. This shows ${\textnormal{Aut}}(\sigma)$ is residually finite.
\[apprem:amenradical\] Any discrete group $G$ possesses a maximal normal amenable subgroup $\textnormal{Rad}(G)$ known as the *amenable radical* of $G$. By Ryan’s Theorem, the center of ${\textnormal{Aut}}(\sigma_{A})$ is the subgroup generated by $\sigma_{A}$, and hence is contained in $\textnormal{Rad}(G)$. In [@FST19] it was shown by Frisch, Schlank and Tamuz that, in the case of a full shift, $\textnormal{Rad}({\textnormal{Aut}}(\sigma_{n}))$ is precisely the center of ${\textnormal{Aut}}(\sigma_{n})$, i.e. the subgroup generated by $\sigma_{n}$. In [@Yang2018] Yang extended this result, proving that for any irreducible shift of finite type $(X_{A},\sigma_{A})$, $\textnormal{Rad}({\textnormal{Aut}}(\sigma_{A}))$ also coincides with the center of ${\textnormal{Aut}}(\sigma_{A})$ (in fact, Yang also proves the result for any irreducible sofic shift as well).
\[appexer:ryanthmdist\]
1. Show that if $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ are irreducible shifts of finite type such that ${\textnormal{Aut}}(\sigma_{A})$ and ${\textnormal{Aut}}(\sigma_{B})$ are isomorphic, then $\textnormal{root}(\sigma_{A}) = \textnormal{root}(\sigma_{B})$.
2. Let $(X_{2},\sigma_{2}), (X_{4},\sigma_{4})$ denote the full shift on 2 symbols and on 4 symbols, respectively. Show that $\textnormal{root}(\sigma_{2}) \ne \textnormal{root}(\sigma_{4})$. Use part (1) to conclude that ${\textnormal{Aut}}(\sigma_{2})$ and ${\textnormal{Aut}}(\sigma_{4})$ are not isomorphic as groups.
For part (1), suppose $\Psi \colon {\textnormal{Aut}}(\sigma_{A}) \to {\textnormal{Aut}}(\sigma_{B})$ is an isomorphism and $k \in \textnormal{root}(\sigma_{A})$. By Ryan’s Theorem, $\Psi(\sigma_{A}) = \sigma_{B}$ or $\Psi(\sigma_{A})=\sigma_{B}^{-1}$. Choose $\alpha \in {\textnormal{Aut}}(\sigma_{A})$ such that $\alpha^{k} = \sigma_{A}$. If $\Psi(\sigma_{A}) = \sigma_{B}$, then we have $(\Psi(\alpha))^{k} = \Psi(\alpha^{k}) = \Psi(\sigma_{A}) = \sigma_{B}$ so $k \in \textnormal{root}(\sigma_{B})$. If $\Psi(\sigma_{A}) = \sigma_{B}^{-1}$, then we have $(\Psi(\alpha^{-1})^{k} = \Psi(\alpha^{-k}) = \Psi(\sigma_{A}^{-1}) = \sigma_{B}$ so again $k \in \textnormal{root}(\sigma_{B})$. Thus $\textnormal{root}(\sigma_{A}) \subset \textnormal{root}(\sigma_{B})$. The proof that $\textnormal{root}(\sigma_{B}) \subset \textnormal{root}(\sigma_{A})$ is analogous.\
For part (2), choose a topological conjugacy $F \colon (X_{4},\sigma_{4}) \to (X_{2},\sigma_{2}^{2})$. If we let $s = F^{-1}\sigma_{2}F \in {\textnormal{Aut}}(\sigma_{4})$, then $s \in {\textnormal{Aut}}(\sigma_{4})$ and $s^{2} = \sigma_{4}$, so $2 \in \textnormal{root}(\sigma_{4})$. We claim $2 \not \in \textnormal{root}(\sigma_{2})$. To see this, suppose toward a contradiction that $\beta \in {\textnormal{Aut}}(\sigma_{2})$ satisfies $\beta^{2} = \sigma_{2}$. There are precisely two points $x,y$ of least period 2 in $(X_{2},\sigma_{2})$, so $\beta^{2}$ must act by the identity on the points $x,y$. But $\sigma_{2}(x)=y$, a contradiction.
\[appexer:dimgroupfullshiftcomp\] When $A = (n)$ (the case of the full-shift on $n$ symbols), the triple $(G_{n},G_{n}^{+},\delta_{n})$ is isomorphic to the triple $(\mathbb{Z}[\frac{1}{n}],\mathbb{Z}_{+}[\frac{1}{n}],m_{n})$, where $m_{n}$ is the automorphism of $\mathbb{Z}[\frac{1}{n}]$ defined by $m_{n}(x) = x \cdot n$.
The eventual range of $A$ is $\mathbb{Q}$. Given $\frac{p}{q} \in \mathbb{Q}$, $2^{k}\frac{p}{q} \in \mathbb{Z}_{+}$ if and only if $p \in \mathbb{Z}_{+}$ and $q$ is a power of $2$.
\[appexer:indextwoorderautos\] Let $A$ be a primitive matrix and suppose $\Psi$ is an automorphism of $(G_{A},\delta_{A})$. By considering $G_{A}$ as a subgroup of $ER(A)$, show that $\Psi$ extends to a linear automorphism $\tilde{\Psi} \colon ER(A) \to ER(A)$ which multiplies a Perron eigenvector of $A$ by some nonzero real number $\lambda_{\Psi}$. Show that $\Psi$ is also an automorphism of the ordered abelian group $(G_{A},G_{A}^{+},\delta_{A})$ if and only if $\lambda_{\Psi}$ is positive.
That $\Psi$ extends to a linear automorphism $\tilde{\Psi}$ of $ER(A)$ is immediate: given $v \in ER(A)$, write $v = \frac{1}{q}w$ where $w$ is integral, and define $\tilde{\Psi}(v) = \frac{1}{q}\Psi(w)$. The linear map $\Psi$ commutes with $\delta_{A}$ on $G_{A}$, so $\tilde{\Psi}$ commutes with $\delta_{A}$ as a linear automorphism of $ER(A)$. Since $A$ is primitive, a Perron eigenvector $v_{\lambda_{A}}$ for $\lambda_{A}$ spans a one-dimensional eigenspace for $\delta_{A}$, which hence must be preserved by $\tilde{\Psi}$. Thus $v_{\lambda_{A}}$ is also an eigenvector for $\tilde{\Psi}$, and has some corresponding eigenvalue $\lambda_{\Psi}$.\
For the second part, we’ll use the following proposition (a proof of which we include at the end).
\[mixingdimgroupprop\] Suppose $A$ is an $N\times N$ primitive matrix over ${\mathbb R}$. Let the spectral radius be $\lambda$ and let $v$ be a positive eigenvector, $vA = \lambda v$. Given $x$ in ${\mathbb R}^N$, let $c_x$ be the real number such that $x= c_xv + u_x$, with $u_x$ a vector in the $A$-invariant subspace complementary to $<v>$. Suppose $x$ is not the zero vector. Then $xA^n$ is nonnegative for large $n$ iff $c_x>0$.
To finish the exercise, suppose $0 \ne w \in G_{A}^{+}$, and write $w = c_{w}v_{\lambda_{A}}+u_{w}$ as in the proposition. Since $w \in G_{A}^{+}$, $c_{w} > 0$. Then $\tilde{\Psi}(w) = c_{w}\lambda_{\Psi}v_{\lambda_{A}} + \tilde{\Psi}(u_{w})$. Since $\lambda_{\Psi} > 0$, $c_{w}\lambda_{\Psi} > 0$, so the proposition implies $\Psi(w) \in G_{A}^{+}$ as desired.\
*Proof of Proposition \[mixingdimgroupprop\]*. The Perron Theorem tells us the positive eigenvector and complementary invariant subspace exist, with $\varlimsup_n ||u_xA^n||^{1/n} < \lambda $. Consequently, for large $n$, $xA^n$ is a positive vector if $c_x>0$ and $xA^n$ is a negative vector if $c_x < 0$. Given $c_x=0 $ and $x\neq 0$, no vector $w=u_xA^n $ can be nonnegative or nonpositive, because this would imply $\lim_n ||xA^n||^{1/n}= \lim_n ||wA^n||^{1/n} = \lambda $, a contradiction.
\[rem:kriegerconstruction\] Consider an edge shift of finite type $(X_{A},\sigma_{A})$. Here is an outline of Krieger’s construction of an ordered abelian group which is isomorphic to $(G_{A},G_{A}^{+},\delta_{A})$; our presentation follows the one given in [@LindMarcus1995 Sec. 7.5]. Recall we are assuming that $A$ is a $k \times k$ irreducible matrix.\
By an *$m$-ray* we mean a subset of $X_{A}$ given by $$R(x,m) = \{y \in X_{A} \mid y_{(-\infty,m]}=x_{(-\infty,m]}\}$$ for some $x \in X_{A}, m \in \mathbb{Z}$. An *$m$-beam* is a (possibly empty) finite union of $m$-rays. By a *ray* we mean an $m$-ray for some $m \in \mathbb{Z}$; likewise, by a *beam* we mean an $m$-beam for some $m$. It is easy to check that if $U$ is an $m$-beam for some $m$, and $n \ge m$, then $U$ is also an $n$-beam. Given an $m$-beam $$U = \bigcup_{i=1}^{j}R(x^{(i)},m),$$ define $v_{U,m} \in \mathbb{Z}^{k}$ to be the vector whose $J$th component is given by $$\#\{x^{(i)} \in U \mid \textnormal{ the edge corresponding to }x_{m}^{(i)} \textnormal{ ends at state } J\}.$$ We define two beams $U$ and $V$ to be equivalent if there exists $m$ such that $v_{U,m} = v_{V,m}$, and let $[U]$ denote the equivalence class of a beam $U$. We will make the collection of equivalence classes of beams into a semi-group as follows. Since $A$ is an irreducible matrix and $0 < h_{top}(\sigma_{A}) = \log \lambda_{A}$, given two beams $U,V$, we may find beams $U^{\prime}, V^{\prime}$ such that $$[U]=[U^{\prime}], \hspace{.23in} [V] = [V^{\prime}], \hspace{.23in} U^{\prime} \cap V^{\prime} = \emptyset,$$ and we let $D_{A}^{+}$ denote the abelian monoid defined by the operation $$[U] + [V] = [U^{\prime} \cup V^{\prime}]$$ where the class of the empty set serves as the identity for $D_{A}^{+}$. Now let $D_{A}$ denote the group completion of $D_{A}^{+}$; thus elements of $D_{A}$ are formal differences $[U]-[V]$. Then $D_{A}$ is an ordered abelian group with positive cone $D_{A}^{+}$. The map $d_{A} \colon D_{A} \to D_{A}$ induced by $$d_{A}([U]) = [\sigma_{A}(U)]$$ is a group automorphism of $D_{A}$ which preserves $D_{A}^{+}$, and the triple $(D_{A},D_{A}^{+},d_{A})$ is Krieger’s dimension triple for the SFT $(X_{A},\sigma_{A})$.\
The connection between Krieger’s triple $(D_{A},D_{A}^{+},d_{A})$ and the ordered abelian group triple $(G_{A},G_{A}^{+},\delta_{A})$ is given by the following proposition.
\[prop:kridimgroupiso\] There is a semi-group homomorphism $\theta \colon D_{A}^{+} \to G_{A}^{+}$ induced by the map $$\theta([U]) = \delta_{A}^{-k-n}(v_{U,n}A^{k}), \hspace{.29in} U \textnormal{ an } n\textnormal{-beam}.$$ The map $\theta$ satisfies $\theta(D_{A}^{+}) = G_{A}^{+}$, and induces an isomorphism $\theta \colon D_{A} \to G_{A}$ such that $\theta \circ d_{A} = \delta_{A} \circ \theta$. Thus $\theta$ induces an isomorphism of triples $$\theta \colon (D_{A},D_{A}^{+},d_{A}) \to (G_{A},G_{A}^{+},\delta_{A}).$$
\[apprem:dimreprangeclassification\] In [@S17], Kim and Roush describe how the problem of classifying general (i.e. not necessarily irreducible) shifts of finite type up to topological conjugacy can be broken into two parts: classifying mixing shifts of finite type up to conjugacy, and determining the range of the dimension representation in the mixing shift of finite type case. That the dimension representation need not always be surjective was also instrumental in the Kim-Roush argument in [@S21] that shift equivalence over $\mathbb{Z}_{+}$ need not imply strong shift equivalence over $\mathbb{Z}_{+}$ in the reducible setting.
\[appexer:simpininert\] For any shift of finite type $(X_{A},\sigma_{A})$, we have ${\textnormal{Simp}(\sigma_{A})} \subset {\textnormal{Inert}(\sigma_{A})}$.
This is easiest seen using Krieger’s presentation [(Ap. ]{}[rem:kriegerconstruction]{}). First suppose $\alpha \in {\textnormal{Simp}(\sigma_{A})}$ is induced by a simple graph symmetry of $\Gamma_{A}$. If $U$ is an $m$-beam in $X_{A}$, then $\alpha(U)$ is an $m$-beam, and $v_{\alpha(U),m} = v_{U,m}$. It follows that $[U] = [\alpha(U)]$, so $\alpha$ acts by the identity on the group $D_{A}$, and hence on $G_{A}$.\
Now suppose $\beta = \Psi^{-1}\alpha\Psi$ where $\Psi \colon (X_{A},\sigma_{A}) \to (X_{B},\sigma_{B})$ is a topological conjugacy and $\alpha \in {\textnormal{Simp}(\sigma_{B})}$ is induced by a simple graph symmetry of $\Gamma_{B}$. If $U$ is an $m$-beam in $X_{A}$, then by the previous part $\alpha \Psi([U]) = \alpha ([\Psi(U)]) = [\Psi(U)] = \Psi([U])$, so $$\beta([U]) = \Psi^{-1}\alpha\Psi([U]) =\Psi^{-1}\Psi([U]) = [U].$$ Thus $\beta$ acts by the identity on $G_{A}$. Since ${\textnormal{Simp}(\sigma_{A})}$ is generated by automorphisms in the form of $\beta$, this finishes the proof.
\[apprem:semidirectaut\] Let us write ${\textnormal{Aut}}(P_{k},\sigma_{A})$ for ${\textnormal{Aut}}(\sigma_{A}|_{P_{k}})$. For each orbit $q \in Q_{k}$ choose a point $x_{q} \in q$. There is a surjective homomorphism $${\textnormal{Aut}}(P_{k},\sigma_{A}) \stackrel{\pi}\longrightarrow {\textnormal{Sym}}(Q_{k})$$ since any $\alpha \in {\textnormal{Aut}}(P_{k},\sigma_{A})$ must preserve $\sigma_{A}$-orbits, and the map $\pi$ is split by the map $i \colon {\textnormal{Sym}}(Q_{k}) \to {\textnormal{Aut}}(P_{k},\sigma_{A})$ defined by, for $\tau \in {\textnormal{Sym}}(Q_{k})$, setting $$i(\tau)(\sigma_{A}^{i}(x_{q})) = \sigma_{A}^{i}x_{\tau(q)}, \qquad 0 \le i \le k-1.$$ The kernel of $\pi$ is isomorphic to $\left(\mathbb{Z}/k \mathbb{Z}\right)^{Q_{k}}$ with an isomorphism given by $$\begin{gathered}
\left(\mathbb{Z}/k \mathbb{Z}\right)^{Q_{k}} \to \ker \pi\\
g \mapsto \alpha_{g}, \qquad \alpha_{g}(\sigma_{A}^{i}x_{q}) = \sigma_{A}^{i+g(q)}x_{q}, \qquad 0 \le i \le k-1
\end{gathered}$$ and it follows ${\textnormal{Aut}}(P_{k},\sigma_{A})$ is isomorphic to the semidirect product $\left(\mathbb{Z}/k\mathbb{Z}\right)^{Q_{k}} \rtimes {\textnormal{Sym}}(Q_{k})$. The action of ${\textnormal{Sym}}(Q_{k})$ on $\left(\mathbb{Z}/k\mathbb{Z}\right)^{Q_{k}}$ is determined as follows. Let $g \in \left(\mathbb{Z}/k\mathbb{Z}\right)^{Q_{k}}$, so $g \colon Q_{k} \to \mathbb{Z}/k\mathbb{Z}$. Then $\alpha_{g} \in \ker \pi$, and given some $i(\tau)$ for some $\tau \in {\textnormal{Sym}}(Q_{k})$, $$i(\tau)^{-1}\alpha_{g}i(\tau) = \alpha_{g \circ \tau}.$$
\[apprem:abelsigngyr\] For a group $G$, let $G_{ab}$ denote the abelianization. Using the notation from \[apprem:semidirectaut\], we have an isomorphism $\Phi \colon {\textnormal{Aut}}(\sigma_{A}|_{P_{k}}) \to \left(\mathbb{Z}/k\mathbb{Z}\right)^{Q_{k}} \rtimes {\textnormal{Sym}}(Q_{k})$. The abelianization of $\left(\mathbb{Z}/k\mathbb{Z}\right)^{Q_{k}} \rtimes {\textnormal{Sym}}(Q_{k})$ is isomorphic to ${\textnormal{Sym}}(Q_{k})_{ab} \times (\left(\mathbb{Z}/k\mathbb{Z}\right)^{Q_{k}})_{{\textnormal{Sym}}(Q_{k})}$, where $(\left(\mathbb{Z}/k\mathbb{Z}\right)^{Q_{k}})_{{\textnormal{Sym}}(Q_{k})}$ is the quotient of $\left(\mathbb{Z}/k\mathbb{Z}\right)^{Q_{k}}$ by the subgroup generated by elements of the form $\tau^{-1}g\tau-g$, $\tau \in {\textnormal{Sym}}(Q_{k})_{ab}, g \in \left(\mathbb{Z}/k\mathbb{Z}\right)^{Q_{k}}_{ab}$. Now the abelianization of ${\textnormal{Sym}}(Q_{k})$ is given by $\textnormal{sign} \colon {\textnormal{Sym}}(Q_{k}) \to \mathbb{Z}/2$, and the map $$\begin{gathered}
\left(\mathbb{Z}/k\mathbb{Z}\right)^{Q_{k}} \to \mathbb{Z}/k\mathbb{Z}\\
g \mapsto \sum_{q \in Q_{k}}g(q)
\end{gathered}$$ maps elements of the form $\tau^{-1}g\tau-g$ to 0, and induces an isomorphism $$(\left(\mathbb{Z}/k\mathbb{Z}\right)^{Q_{k}})_{{\textnormal{Sym}}(Q_{k})} \stackrel{\cong}\longrightarrow \mathbb{Z}/k\mathbb{Z}.$$
\[apprem:sgccremark\] SGCC, and the question of which automorphisms satisfy SGCC, has a history spanning a number of years. The SGCC condition was introduced by Boyle and Krieger in [@MR887501], where it was also proved that, in the case of many SFT’s, it holds for any inert automorphism which is a product of involutions. This was followed up by a number of more general results, summarized in the following theorem.
\[thm:sgcc\] Let $(X_{A},\sigma_{A})$ be a shift of finite type. An automorphism $\alpha \in {\textnormal{Aut}}(\sigma_{A})$ satisfies SGCC if:
1. (Boyle-Krieger in [@MR887501]) $\alpha$ is inert and a product of involutions (not for all SFT’s, but many, including the full shifts).
2. (Nasu in [@Nasu88]) $\alpha$ is a simple automorphism.
3. (Fiebig in [@FiebigThesis]) $\alpha$ is inert and finite order.
4. (Kim-Roush in [@S23], with a key ingredient by Wagoner) $\alpha$ is inert.
\[apprem:actionsfinitesubsrem\] Williams first asked (around 1975) whether any permutation of fixed points of a shift of finite type could be lifted to an automorphism. Williams was motivated in part by the classification problem: he was studying an example of two shifts of finite type which were shift equivalent, one of which clearly had an involution of fixed points, while it was not obvious whether the other did. It is interesting to note that, many years later, the automorphism groups proved instrumental in addressing the classification problem.
\[apprem:evensettingfog\] In [@Wagoner90eventual] Wagoner proved that the inert automorphisms are generated by simple automorphisms in the “eventual” setting: namely, given a primitive matrix $A$ and inert automorphism $\alpha \in {\textnormal{Inert}(\sigma_{A})}$, there exists some $m \ge 1$ such that, upon considering $\alpha \in \textnormal{Aut}(\sigma_{A}^{m})$, $\alpha$ lies in $\textnormal{Simp}(\sigma_{A}^{m})$. In [@MR964880] Boyle gave an alternative proof of this, and also gave a stronger form of the result.
Wagoner’s strong shift equivalence complex, and applications {#sectionWagoner}
============================================================
In the late 80’s, Wagoner introduced certain CW complexes as a tool to study strong shift equivalence. These CW complexes provide an algebraic topological/combinatorial framework for studying strong shift equivalence, and have played a key role in a number of important results in the study of shifts of finite type. Among these, one of the most significant was the construction of a counterexample to Williams’ Conjecture in the primitive case, which was found by Kim and Roush in [@S11][^6]. Wagoner independently developed another framework for finding counterexamples, and in [@Wagoner2000] gave a different proof, using matrices generated from Kim and Roush’s method in [@S11], of the existence of a counterexample to Williams’ Conjecture. Both the Kim and Roush strategy, and Wagoner’s strategy, take place in the setting of Wagoner’s strong shift equivalence complexes.\
The goal in this last lecture is to give a brief introduction to these complexes. After defining and discussing them, we’ll give a short introduction into how the Kim-Roush and Wagoner strategies for producing counterexamples work. This will be very much an overview, and we will not go into details.\
In summary, our aim here is not to describe the construction of counterexamples to Williams’ Conjecture in any detail, but instead to give an overview of how Wagoner’s spaces are built, how the counterexample strategies make use of them, and where they leave the state of the classification problem.\
Wagoner’s SSE complexes
-----------------------
Suppose we have matrices $A, B$ over $\mathbb{Z}_{+}$, and a strong shift equivalence from $A$ to $B$ $$A = A_{0} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.02in} $R_{1},S_{1}$}}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.02in} $R_{1},S_{1}$}}$}{O}{c}{F}{T}{S}\mkern1mu}}} \end{tiny} A_{1} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.02in} $R_{2},S_{2}$}}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.02in} $R_{2},S_{2}$}}$}{O}{c}{F}{T}{S}\mkern1mu}}} \end{tiny} \hspace{.05in}\cdots \begin{tiny} \hspace{.03in} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.02in}$R_{n-1},S_{n-1}$}}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.02in}$R_{n-1},S_{n-1}$}}$}{O}{c}{F}{T}{S}\mkern1mu}}} \hspace{.03in} \end{tiny} A_{n-1} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.02in} $R_{n},S_{n}$}}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.02in} $R_{n},S_{n}$}}$}{O}{c}{F}{T}{S}\mkern1mu}}} \end{tiny} A_{n} = B$$ where for each $i \ge 1$, $A_{i-1} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.02in} $R_{i},S_{i}$}}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.02in} $R_{i},S_{i}$}}$}{O}{c}{F}{T}{S}\mkern1mu}}} \end{tiny} A_{i}$ indicates an elementary strong shift equivalence $$A_{i-1} = R_{i}S_{i}, \qquad A_{i} = S_{i}R_{i}.$$ We can visualize this as a path (at the moment we use the term path informally; it will be made precise later)
where each arrow in this picture represents an elementary strong shift equivalence. From Williams’ Theorem (Theorem \[rfwtheorem\]), there is a conjugacy $C \colon (X_{A},\sigma_{A}) \to (X_{B},\sigma_{B})$ given by $$C = \prod_{i=1}^{n}c(R_{i},S_{i})$$ where for each $i$, $c(R_{i},S_{i}) \colon (X_{A_{i-1}},\sigma_{A_{i-1}}) \to (X_{A_{i}},\sigma_{A_{i}})$ is a conjugacy induced by the ESSE $A_{i-1} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.02in} $R_{i},S_{i}$}}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.02in} $R_{i},S_{i}$}}$}{O}{c}{F}{T}{S}\mkern1mu}}} \end{tiny} A_{i}$.
Now suppose, with the matrices $A, B$ over $\mathbb{Z}_{+}$, we have two SSE’s from $A$ to $B$. We then have two paths of ESSE’s from $A$ to $B$\
and a pair of conjugacies corresponding to each path $$C_{1} \colon (X_{A},\sigma_{A}) \to (X_{B},\sigma_{B})$$ $$C_{2} \colon (X_{A},\sigma_{A}) \to (X_{B},\sigma_{B})$$ and one may ask: when do two such paths induce the same conjugacy? Can we determine this from the matrix entries in the paths themselves? Alternatively, is there a space in which we can actually consider these as paths, in which two paths are homotopic if and only if they give rise to the same conjugacy? Wagoner’s complexes are a way to do this, and one of the key insights in Wagoner’s complexes is determining the correct relations on matrices to accomplish this. These relations are known as the Triangle Identities. Since the Triangle Identities lead directly to the definition of Wagoner’s Complexes [(Ap. ]{}[apprem:wagonermarkovspace]{}), we’ll define both simultaneously.
Let ${\mathcal{R}}$ be a semiring. We define a CW-complex $SSE({\mathcal{R}})$ as follows:
1. The 0-cells of $SSE({\mathcal{R}})$ are square matrices over ${\mathcal{R}}$.
2. An edge $(R,S)$ from vertex $A$ to vertex $B$ corresponds to an elementary strong shift equivalence over ${\mathcal{R}}$ from $A$ to $B$:\
where $A=RS, B = SR$.
3. 2-cells are given by triangles\
which satisfy the *Triangle Identities*:\
$$\label{eqn:triangleidentities}
R_{1}R_{2} = R_{3}, \hspace{.23in} R_{2}S_{3} = S_{1}, \hspace{.23in} S_{3}R_{1} = S_{2}.$$
The definition of $SSE({\mathcal{R}})$ makes sense for any semiring. For this lecture however, we will consider the case where ${\mathcal{R}}$ may be one of:
1. $ZO = \{0,1\}$
2. $\mathbb{Z}_{+}$
3. $\mathbb{Z}$.
Wagoner also defines $n$-cells in $SSE({\mathcal{R}})$ for $n \ge 3$ in [@Wagoner90triangle], but we won’t need these in this lecture.\
Note that edges have orientations in $SSE({\mathcal{R}})$. Recall also that, for an edge from $A$ to $B$ given by a SSE $(R,S)$, we may choose an elementary conjugacy $c(R,S)$ (see \[rem:crsconj\]), and this choice of $c(R,S)$ does not only depend on $R$ and $S$ but also on some choice of simple automorphisms. By Williams’ Decomposition Theorem (Theorem \[rfwtheorem\]; see also [(Ap. ]{}[decomp]{})), if $C \colon (X_{A},\sigma_{A}) \to (X_{B},\sigma_{B})$ is a topological conjugacy, then there is a strong shift equivalence $$A = A_{0} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.02in} $R_{1},S_{1}$}}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.02in} $R_{1},S_{1}$}}$}{O}{c}{F}{T}{S}\mkern1mu}}} \end{tiny} A_{1} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.02in} $R_{2},S_{2}$}}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.02in} $R_{2},S_{2}$}}$}{O}{c}{F}{T}{S}\mkern1mu}}} \end{tiny} \hspace{.05in}\cdots \begin{tiny} \hspace{.03in} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.02in}$R_{n-1},S_{n-1}$}}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.02in}$R_{n-1},S_{n-1}$}}$}{O}{c}{F}{T}{S}\mkern1mu}}} \hspace{.03in} \end{tiny} A_{n-1} \begin{tiny} {\mathrel{\ThisStyle{ \setbox0=\hbox{$\SavedStyle\mkern3mu_{\text{\hspace{.02in} $R_{n},S_{n}$}}\mkern3mu$} \mkern1mu\stackengine{1\LMpt}{ \stretchto{\scaleto{\SavedStyle\mkern-1mu\sim\mkern-1mu}{.28\wd0}}{1\ht0} }{$\SavedStyle_{\text{\hspace{.02in} $R_{n},S_{n}$}}$}{O}{c}{F}{T}{S}\mkern1mu}}} \end{tiny} A_{n} = B$$ such that $$C = \prod_{i=1}^{n}c(R_{i},S_{i})^{s(i)}$$ with each $c(R_{i},S_{i})$ an elementary conjugacy corresponding to the ESSE given by $R_{i}, S_{i}$, and $s(i) = 1$ if $A_{i-1} = R_{i}S_{i}, A_{i} = S_{i}R_{i}$, while $s(i) = -1$ if $A_{i} = R_{i}S_{i}, A_{i-1} = R_{i}S_{i}$. This presentation $C$ of the conjugacy gives us a path in $SSE(\mathbb{Z}_{+})$
Note that some arrows are drawn in reverse, as needed so that the conjugacy $C$ matches the conjugacy given by following the path. Likewise, given a path $\gamma$ in $SSE({\mathcal{R}})$ between $A$ and $B$ $$\gamma = \prod_{i=1}^{m}\left(R_{i},S_{i}\right)^{s(i)}$$ there is a corresponding conjugacy $$\tilde{\gamma} = \prod_{i=1}^{m}c(R_{i},S_{i})^{s(i)} \colon (X_{A},\sigma_{A}) \to (X_{B},\sigma_{B}).$$
In particular, vertices of $SSE(\mathbb{Z}_{+})$ correspond to specific presentations of shifts of finite type (edge shift construction), and edges to specific conjugacies (elementary conjugacy coming from an elementary strong shift equivalence). Note that any path between two vertices in these complexes is homotopic to a path following a sequence of edges.\
Recall from Lecture 1 that a matrix $A$ is degenerate if it has a zero row or zero column; otherwise, it is nondegenerate. Following Wagoner, we only allow nondegenerate matrices as vertices. It is at times important to work with the larger space $SSE_{deg}({\mathcal{R}})$ which allows degenerate vertices; see for example [@BW04]. It turns out that the inclusion $SSE(\mathbb{Z}_{+}) \to SSE_{deg}(\mathbb{Z}_{+})$ induces an isomorphism on $\pi_{0}$ [@BW04] and also an isomorphism on $\pi_{1}$ [@Epperlein2019] for each path-component.
Homotopy groups for Wagoner’s complexes and ${\textnormal{Aut}}(\sigma_{A})$
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For a semiring ${\mathcal{R}}$ and square matrix $A$ over ${\mathcal{R}}$, we let $SSE({\mathcal{R}})_{A}$ denote the path-component of $SSE({\mathcal{R}})$ containing the vertex $A$. From Williams’ Theorem, the vertices $A,B$ in $SSE(\mathbb{Z}_{+})$ are in the same path-component if and only if the edge shifts $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ are topologically conjugate.\
From the perspective of homotopy theory, the Triangle Identities dictate basic moves for paths in $SSE({\mathcal{R}})$ to be homotopic. So why the Triangle Identities? The following result of Wagoner explains their importance. In the statement of the theorem, given $A,B$ and two conjugacies $\phi_{1}, \phi_{2} \colon (X_{A},\sigma_{A}) \to (X_{B},\sigma_{B})$, we say $\phi_{1} \sim_{simp} \phi_{2}$ if there exists simple automorphisms $\gamma_{1} \in {\textnormal{Simp}(\sigma_{A})}, \gamma_{2} \in {\textnormal{Simp}(\sigma_{B})}$ such that $\gamma_{2} \phi_{1} \gamma_{1} = \phi_{2}$ ($\sim_{simp}$ defines an equivalence relation on the set of conjugacies between $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$).
\[thm:wagonerhomotopyofpaths\] For the spaces $SSE(ZO), SSE(\mathbb{Z}_{+})$ defined above, both of the following hold:
1. Given vertices $A,B$ in $SSE(ZO)$, two paths in $SSE(ZO)$ from $A$ to $B$ are homotopic in $SSE(ZO)$ if and only if they induce the same conjugacy from $(X_{A},\sigma_{A})$ to $(X_{B},\sigma_{B})$.
2. Given vertices $A,B$ in $SSE(\mathbb{Z}_{+})$, two paths in $SSE(\mathbb{Z}_{+})$ from $A$ to $B$ are homotopic in $SSE(\mathbb{Z}_{+})$ if and only if they induce the same conjugacy from $(X_{A},\sigma_{A})$ to $(X_{B},\sigma_{B})$ modulo the relation $\sim_{simp}$.
Item $(2)$ in the above is perhaps expected; recall the construction given in Section \[subsec:sseandclassification\] of Lecture 1 for associating conjugacies with SSE’s over $\mathbb{Z}_{+}$ requires a choice of labels for certain edges. This choice is where ambiguity up to conjugating by simple automorphisms may arise.\
Theorem \[thm:wagonerhomotopyofpaths\] gives the first two parts of the following theorem of Wagoner. For a space $X$ with point $x \in X$, let $\pi_{k}(X,x)$ denote the $k$th homotopy group based at $x$.
\[thm:wagonerreptheorem\] Let $A$ be a square matrix over $ZO$. Then:
1. ${\textnormal{Aut}}(\sigma_{A}) \cong \pi_{1}(SSE(ZO),A)$.
2. ${\textnormal{Aut}}(\sigma_{A}) / {\textnormal{Simp}(\sigma_{A})} \cong \pi_{1}(SSE(\mathbb{Z}_{+}),A)$.
3. ${\textnormal{Aut}}(G_{A},\delta_{A}) \cong \pi_{1}(SSE(\mathbb{Z}),A)$.
It is immediate from the definition of the SSE spaces that the set $\pi_{0}(SSE(\mathbb{Z}_{+}))$ may be identified with the set of conjugacy classes of shifts of finite type. Moreover, $\pi_{0}(SSE(\mathbb{Z}))$ may be identified with the set of strong shift equivalence classes of matrices over $\mathbb{Z}$.\
Upon using the identifications above, the composition map $$\pi_{1}(SSE(ZO),A) \to \pi_{1}(SSE(\mathbb{Z}_{+}),A) \to \pi_{1}(SSE(\mathbb{Z}),A)$$ induced by the natural inclusions $SSE(ZO) \hookrightarrow SSE(\mathbb{Z}_{+}) \hookrightarrow SSE(\mathbb{Z})$ is isomorphic to the dimension representation factoring as $${\textnormal{Aut}}(\sigma_{A}) \to {\textnormal{Aut}}(\sigma_{A})/{\textnormal{Simp}(\sigma_{A})} \to {\textnormal{Aut}}(G_{A}),$$ i.e. the diagram $$\xymatrix{
\pi_{1}(SSE(ZO),A) \ar[d]^{\cong} \ar[r] & \pi_{1}(SSE(\mathbb{Z}_{+}),A) \ar[d]^{\cong} \ar[r] & \pi_{1}(SSE(\mathbb{Z}),A) \ar[d]^{\cong}\\
{\textnormal{Aut}}(\sigma_{A}) \ar[r] & {\textnormal{Aut}}(\sigma_{A})/{\textnormal{Simp}(\sigma_{A})} \ar[r] & {\textnormal{Aut}}(G_{A},\delta_{A})
}$$ commutes.
Wagoner also proves that $\pi_{k}(SSE(ZO),A) = 0$ for $k \ge 2$. This implies $SSE(ZO)_{A}$ is a model for the classifying space of ${\textnormal{Aut}}(\sigma_{A})$, i.e. $SSE(ZO)_{A}$ is homotopy equivalent to $B{\textnormal{Aut}}(\sigma_{A})$ [(Ap. ]{}[apprem:classifyingspacerem]{}). Thus, for example, we have $${\textnormal{Aut}}(\sigma_{A})_{ab} \cong H_{1}({\textnormal{Aut}}(\sigma_{A}),\mathbb{Z}) \cong H_{1}(SSE(ZO)_{A},\mathbb{Z}).$$ It is worth remarking that, at the moment, we do not know what the abelianization ${\textnormal{Aut}}(\sigma_{A})_{ab}$ is for any positive entropy shift of finite type $(X_{A},\sigma_{A})$ (however it is at least known, from [@BLR88 Theorem 7.8], that ${\textnormal{Aut}}(\sigma_{A})_{ab}$ is not finitely generated).\
Wagoner also introduced complexes $SE({\mathcal{R}})$ defined analogously to $SSE({\mathcal{R}})$ (see [(Ap. ]{}[apprem:secomplexdef]{}) for a definition). Since an ESSE over ${\mathcal{R}}$ also gives an SE over ${\mathcal{R}}$, there is a continuous inclusion map $i_{{\mathcal{R}}} \colon SSE({\mathcal{R}}) \to SE({\mathcal{R}})$. Wagoner proved in [@MR1012941] that, in the case ${\mathcal{R}}$ is a principal ideal domain, this map $i_{{\mathcal{R}}}$ is a homotopy equivalence, and that $\pi_{n}(SSE({\mathcal{R}}),A) = \pi_{n}(SE({\mathcal{R}}),A) = 0$ for all $n \ge 2$ and any $A$. The map $i_{{\mathcal{R}}}$ cannot be a homotopy equivalence for a general ring ${\mathcal{R}}$ [(Ap. ]{}[apprem:nothomeqingeneral]{}).\
Wagoner’s complexes, and the results of Theorem \[thm:wagonerreptheorem\], have recently been generalized to a groupoid setting in [@Epperlein2019]. This setting simplifies some of the proofs and extends Wagoner’s construction to shifts of finite type carrying a free action by a finite group, as well as more general shifts of finite type over arbitrary finitely generated groups.
Counterexamples to Williams’ Conjecture
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A counterexample to Williams’ Conjecture in the primitive case was given by Kim and Roush in [@S11]. In [@Wagoner2000], Wagoner also verified the counterexamples using a different framework. Both methods for detecting the counterexamples take place in the setting of Wagoner’s SSE complexes, and build on a great deal of work by many authors. We outline the techniques here; one may also see Wagoner’s survey article [@Wagoner99] for an exposition regarding the counterexamples.\
Since our goal is only to give a brief introduction to how these counterexamples arise, we won’t actually list the explicit matrices involved; they can be found in [@S11] or [@Wagoner2000]). Instead, we focus on the strategy used to prove that they in fact *are* counterexamples.\
To start, both strategies roughly follow the same initial idea. As mentioned in Section \[subsec:comparingseandsse\], to find a counterexample to Williams’ Conjecture it is sufficient to find a pair of primitive matrices which are connected by a path in $SSE(\mathbb{Z})$, and show they cannot be connected by a path through $SSE(\mathbb{Z}_{+})$. We can formalize this approach in terms of homotopy theory (this is an important viewpoint, although not necessary to understand the Kim-Roush counterexample, as we will see). Consider $SSE(\mathbb{Z}_{+})$ as a subcomplex of $SSE(\mathbb{Z})$, and, upon fixing a base point $A$ in $SSE(\mathbb{Z}_{+})$, consider the long exact sequence in homotopy groups based at $A$ for the pair $\left(SSE(\mathbb{Z}),SSE(\mathbb{Z}_{+})\right)$:
$$\cdots \pi_{1}(SSE(\mathbb{Z}),A) \to \pi_{1}(SSE(\mathbb{Z}),SSE(\mathbb{Z}_{+}),A) \to \pi_{0}(SSE(\mathbb{Z}_{+}),A) \to \pi_{0}(SSE(\mathbb{Z}),A).$$
Here $\pi_{1}(SSE(\mathbb{Z}),SSE(\mathbb{Z}_{+}),A)$ denotes the set of homotopy classes of paths with base point in $SSE(\mathbb{Z}_{+})$ and end point equal to $A$, and the map $\pi_{1}(SSE(\mathbb{Z}),SSE(\mathbb{Z}_{+}),A) \to \pi_{0}(SSE(\mathbb{Z}_{+}),A)$ is defined by sending the homotopy class of a path $\gamma$ to the component containing $\gamma(0)$ (details regarding this sequence can be found in [@HatcherBook Ch. 4, Thm. 4.3]). The last three terms in this sequence are not actually groups, but just pointed sets. Still, exactness makes sense, by defining the kernel to be the pre-image of the base point. The base point in $\pi_{1}(SSE(\mathbb{Z}),SSE(\mathbb{Z}_{+}),A)$ is given by the homotopy class of a path which lies entirely in $SSE(\mathbb{Z}_{+})$. In particular, the set $\pi_{1}(SSE(\mathbb{Z}),SSE(\mathbb{Z}_{+}),A)$ has only one element if and only if every path beginning in $SSE(\mathbb{Z}_{+})$ and ending at $A$ is homotopic to a path lying entirely in $SSE(\mathbb{Z}_{+})$.\
In this setup, the goal is to now find a function
$$F \colon \pi_{1}\left(SSE(\mathbb{Z}),SSE(\mathbb{Z}_{+}),A\right) \to G$$
to some group $G$; for computability, we would like $G$ abelian. Then to find a counterexample, it would be enough to find matrices $A$ and $B$ and a path $\gamma$ in $SSE(\mathbb{Z})$ from $A$ to $B$ such that $F(\gamma) \ne 0$, while $F(\beta) = 0$ for any $\beta \in \pi_{1}(SSE(\mathbb{Z}),A)$. Note that from Theorem \[thm:wagonerreptheorem\] we know $\pi_{1}(SSE(\mathbb{Z}),A) \cong {\textnormal{Aut}}(G_{A},\delta_{A})$, so in light of the long exact sequence in homotopy written above, being able to compute generators for ${\textnormal{Aut}}(G_{A},\delta_{A})$ plays an important role here.\
Put another way, we want to find some abelian group $G$, a primitive matrix $A$, and some function $F$ from edges in $SSE(\mathbb{Z})_{A}$ to $G$ which satisfies all of the following: $$\begin{gathered}
\label{eqn:FandG1}
F(\alpha \star \beta) = F(\alpha) + F(\beta)\\
\label{eqn:FandG2}
\textnormal{If } \gamma_{1} \textnormal{ and } \gamma_{2} \textnormal{ are homotopic paths, then } F(\gamma_{1}) = F(\gamma_{2})\\
\label{eqn:FandG3}
F(\gamma) = 0 \textnormal{ if } \gamma \textnormal{ lies in } SSE(\mathbb{Z}_{+})\\
\label{eqn:Fnonvanishing}
F(\gamma_{A,B}) \ne 0 \textnormal{ for some path } \gamma_{A,B} \textnormal{ between some pair of matrices } A,B\end{gathered}$$ where $\alpha \star \beta$ denotes concatenation of paths.\
Kim and Roush, and independently Wagoner, found functions $F_{m}$ each satisfying $\eqref{eqn:FandG1}, \eqref{eqn:FandG2}, \eqref{eqn:FandG3}$ for $G = \mathbb{Z}/m$ for paths contained in any component of a matrix $A$ satisfying $tr(A^{k}) = 0$ for all $1 \le k \le m$. Finally, for $m=2$, Kim and Roush found a pair of matrices $A,B$ and a path $\gamma_{A,B}$ satisfying .
Kim-Roush relative sign-gyration method: {#subsec:kimroushrelsg}
-----------------------------------------
Let $(X_{A},\sigma_{A})$ be a mixing shift of finite type, and recall from Section \[subsec:sgcc\] the sign-gyration-compatability-condition homomorphisms $$\begin{gathered}
SGCC_{m} \colon {\textnormal{Aut}}(\sigma_{A}) \to \mathbb{Z}/m\mathbb{Z}\\
SGCC_{m} = g_{m} + \left(\frac{m}{2}\right)\sum_{j>0}{\textnormal{sign}}\xi_{m/2^{j}}.
\end{gathered}$$
Given $\alpha \in {\textnormal{Aut}}(\sigma_{A})$, for any $m$, $SGCC_{m}(\alpha)$ is defined in terms of the action of $\alpha$ on the periodic points up to level $m$.\
The idea behind the Kim and Roush technique is to define, for each $m$, a relative sign-gyration-compatibility-condition map $$sgc_{m} \colon \pi_{1}(SSE(\mathbb{Z}),SSE(\mathbb{Z}_{+}),A) \to \mathbb{Z}/m\mathbb{Z}.$$
To start, suppose $A \stackrel{(R,S)}\longrightarrow B$ is an edge in $SSE(\mathbb{Z}_{+})$ given by a strong shift equivalence $A=RS, B=SR$ over $\mathbb{Z}_{+}$. Associated to this (by Theorem \[rfwtheorem\]) is an elementary conjugacy $$c(R,S) \colon (X_{A},\sigma_{A}) \to (X_{B},\sigma_{B}).$$ Recall this conjugacy $c(R,S)$ is not determined by $(R,S)$, but is only defined up to composition with simple automorphisms in the domain and range. Given $m$, choose some orderings on the set of orbits whose lengths divide $m$, and a distinguished point in each such orbit, for each of $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$; in [@S19], these choices are made using certain lexicographic rules on the set of periodic points. The conjugacy $c(R,S)$ induces a bijection between the respective periodic point sets for $\sigma_{A}$ and $\sigma_{B}$, and we may define, with respect to the choices of orderings and distinguished points in each orbit, the sign and gyration maps, and hence define $SGCC_{m}(c(R,S)) \in \mathbb{Z}/m\mathbb{Z}$. If $RS \ne SR$, the value $SGCC_{m}(c(R,S))$ may depend on the choices of orderings and distinguished points.\
In [@S19], Kim-Roush-Wagoner showed that for such a conjugacy $c(R,S)$, there is a formula $sgcc_{m}(R,S)$ for $SGCC_{m}(R,S)$ in terms of the entries from the matrices $R,S$. This was used to prove Theorem \[thm:sgccfactorization\], that $SGCC$ factors through the dimension representation. We note that this formula for $sgcc_{m}(R,S)$ in general depends on the choice of orderings on the periodic points. Furthermore, the formulas defined in [@S19] are very complicated for large $m$. In [@S11], Kim and Roush defined $sgc_{m}$, a slightly different version [(Ap. ]{}[apprem:sgcvssgcc]{}) of $sgcc_{m}$, that also computes $SGCC_{m}$ in terms of entries from $R$ and $S$; for $m=2$, it takes the form $$sgc_{2}(R,S) = \sum_{\underset{k > l}{i<j}}R_{ik}S_{ki}R_{jl}S_{lj} + \sum_{\underset{k \ge l}{i<j}}R_{ik}S_{kj}R_{jl}S_{li} + \sum_{i,j}\frac{1}{2}R_{ij}(R_{ij}-1)S_{ji}^{2}.$$
In other words, for an elementary strong shift equivalence $A=RS, B=SR$, we have $SGCC_{m}(R,S) = sgc_{m}(R,S)$. The formula given above for $sgc_{2}$ uses orderings on the fixed points and period two points defined by certain lexicographic rules given in [@S19]. For the counterexamples to Williams’ Conjecture, only $sgc_{2}$ is needed.\
We can extend $SGCC_{m}$ from elementary conjugacies $c(R,S)$ to paths in $SSE(\mathbb{Z}_{+})$: given a path $$\gamma = \prod_{i=1}^{J}\left(R_{i},S_{i}\right)^{s(i)}$$ define $$SGCC_{m}(\gamma) = \sum_{i}^{J}s(i)SGCC_{m}(R_{i},S_{i}).$$
Note from the above we also know that $$SGCC_{m}(\gamma) = \sum_{i}^{J}s(i)sgc_{m}(R_{i},S_{i}).$$
Now suppose we have a basic triangle in $SSE(\mathbb{Z}_{+})$ with edges $(R_{1},S_{1}), (R_{2},S_{2}), (R_{3},S_{3})$. If $c(R_{3},S_{3}) = c(R_{1},S_{1})c(R_{2},S_{2})$ then using the fact that $SGCC_{m}$ is defined in terms of dynamical data coming from the corresponding conjugacies, a calculation [@S19 Prop. 2.9] shows that $$SGCC_{m}(R_{1},S_{1}) + SGCC_{m}(R_{2},S_{2}) = SGCC_{m}(R_{3},S_{3}).$$ But by Theorem \[thm:wagonerhomotopyofpaths\], up to conjugating by simple automorphisms, we do have $c(R_{3},S_{3}) = c(R_{1},S_{1})c(R_{2},S_{2})$; since $SGCC_{m}$ vanishes on simple automorphisms (Theorem \[thm:sgccfactorization\]), this gives an addition formula for $SGCC_{m}$ over triangles in $SSE(\mathbb{Z}_{+})$.\
Now suppose we have an elementary strong shift equivalence $A=RS,B=SR$ over $\mathbb{Z}$ (so not necessarily in $\mathbb{Z}_{+}$). The $sgc_{m}$ formulas still make sense, so we can define $sgc_{m}(\gamma)$ for any path $\gamma$ in $SSE(\mathbb{Z})$. If $sgc_{m}$ also satisfies an addition formula for triangles in $SSE(\mathbb{Z})$, then $sgc_{m}$ will give us an extension of $SGCC_{m}$ to $SSE(\mathbb{Z})$. This turns out to be the case, and is a consequence of the following Cocycle Lemma.
\[lemma:cocyclelemma\] If the edges $(R_{1},S_{1}), (R_{2},S_{2}), (R_{3},S_{3})$ form a basic triangle in $SSE(\mathbb{Z})$, then $$sgc_{m}(R_{1},S_{1}) + sgc_{m}(R_{2},S_{2}) = sgc_{m}(R_{3},S_{3}).$$
The Cocycle Lemma was first proved in [@S19] in the case when the triangle contains a vertex which is strong shift equivalent over $\mathbb{Z}$ to a nonnegative primitive matrix. The version above, which does not require a primitivity assumption, was given in [@S11], with a much shorter proof suggested by Mike Boyle.\
Putting all of the above together, for a matrix $A$, the map $$sgc_{m} \colon \pi_{1}(SSE(\mathbb{Z}),SSE(\mathbb{Z}_{+}),A) \to \mathbb{Z}/m\mathbb{Z}$$ satisfies and .\
Now suppose that $A$ satisfies $tr(A^{k})=0$ for all $1 \le k \le m$ and $(R,S)$ is an edge in $SSE(\mathbb{Z}_{+})$ from $A$ to $B$. Then both $(X_{A},\sigma_{A})$ and $(X_{B},\sigma_{B})$ have no points of period $k$ for any $1 \le k \le m$, and the dynamically defined $SGCC_{m}(R,S)$ must vanish; since $sgc_{m} = SGCC_{m}$ on edges in $SSE(\mathbb{Z}_{+})$, this implies $sgc_{m}(R,S) = 0$. It follows that on path-components of matrices $A$ with $tr(A^{k})=0$ for all $1 \le k \le m$, the map $sgc_{m}$ also satisfies .\
Finally, using $m=2$, in [@S11] Kim and Roush found two primitive matrices $A,B$ and a path $\gamma$ in $SSE(\mathbb{Z})$ from $A$ to $B$ such that all of the following hold:
1. $tr(A)=tr(A^{2})=0$.
2. $sgc_{2}(\alpha) = 0$ for any $\alpha \in \pi_{1}(SSE(\mathbb{Z}),A)$.
3. $sgc_{2}(\gamma) \ne 0$.
It follows these matrices $A$ and $B$ are strong shift equivalent over $\mathbb{Z}$, but not strong shift equivalent over $\mathbb{Z}_{+}$. The matrices $A$ and $B$ given in [@S11] are $7 \times 7$.
Wagoner’s $K_{2}$-valued obstruction map: {#subsec:wagonerk2obs}
------------------------------------------
Wagoner, influenced by ideas from pseudo-isotopy theory, constructed a map $F$ satisfying the three conditions \[eqn:FandG1\] – \[eqn:FandG3\] landing in the $K$-theory group $K_{2}(\mathbb{Z}[t]/(t^{m+1}))$. In [@Wagoner2000] Wagoner then used this framework to detect counterexamples with matrices found using the technique given by Kim and Roush in [@S11]. The Kim-Roush relative-sign-gyration-compatability method of the previous section enjoys the fact that it is motivated by dynamical data relating directly to the shift systems, being based on ideas from sign-gyration. Wagoner’s method is not as easily connected to the dynamics, but offers some alternative benefits, namely:
1. Landing in $K_{2}$, it connects directly with algebraic K-theory.
2. It operates within the polynomial matrix framework.
3. It is perhaps suggestive of more general strategies for studying the refinement of strong shift equivalence over a ring by strong shift equivalence over the ordered part of a ring, i.e. part (3) in the picture in Lecture 6 describing Williams’ Problem.
So how does Wagoner’s construction work? We recall two facts about the group $K_{2}({\mathcal{R}})$ from Section \[subsec:k2ofaring\]:
1. $K_{2}({\mathcal{R}})$ is an abelian group.
2. An expression of the form $\prod_{i=1}^{k}E_{i} =1$, where $E_{i}$ are elementary matrices over ${\mathcal{R}}$, can be used to construct an element of $K_{2}({\mathcal{R}})$.
For $m \ge 1$, let $SSE_{2m}(\mathbb{Z}_{+})$ denote the subcomplex of $SSE(\mathbb{Z}_{+})$ consisting of path-components which have a vertex $A$ such that $tr(A^{k})=0$ for all $1 \le k \le 2m$. Wagoner’s construction proceeds as follows:
1. Consider an edge in $SSE(\mathbb{Z})$ from $A$ to $B$. As shown in Lecture 3, this gives matrices $E_{1},F_{1}$ in $El(\mathbb{Z}[t])$ over $\mathbb{Z}[t]$ such that $$E_{1}(I-tA)F_{1} = I-tB.$$
2. Suppose the matrix $A$ satisfies $tr(A^{k}) = 0$ for all $1 \le k \le m$. In [@Wagoner2000 Prop. 4.9] it is shown there exist matrices $E_{2},F_{2}$ in $El(\mathbb{Z}[t])$ and $A^{\prime}$ over $\mathbb{Z}[t]$ such that $E_{2}(I-tA)F_{2} = I-t^{m+1}A^{\prime}$. Doing the same for $B$ yields matrices $E_{3},F_{3}$ in $El(\mathbb{Z}[t])$ and some $B^{\prime}$ over $\mathbb{Z}[t]$ such that $$E_{2}(I-tA)F_{2} = I-t^{m+1}A^{\prime}$$ $$E_{3}(I-tB)F_{3} = I-t^{m+1}B^{\prime}.$$
3. Combining steps $(1)$ and $(2)$ we have matrices $X,Y$ in $El(\mathbb{Z}[t])$ such that $$X(I-t^{m+1}A^{\prime})Y = I-t^{m+1}B^{\prime}.$$ Passing to $\mathbb{Z}[t] / (t^{m+1})$, we get $$XY = I.$$ We can now use this expression to produce an element of $K_{2}(\mathbb{Z}[t]/(t^{m+1}))$.
Wagoner shows this assignment defined above is additive with respect to concatenation of paths given by two subsequent edges, so one can extend it to arbitrary paths. Thus, given an edge $\gamma$ in $SSE(\mathbb{Z})$, applying the above gives an element $F(\gamma) \in K_{2}(\mathbb{Z}[t]/(t^{m+1}))$. Then, given a path $\gamma$ between two vertices $A$ and $B$ in $SSE_{2m}(\mathbb{Z})$, Wagoner shows:
1. The element $F(\gamma)$ in $K_{2}(\mathbb{Z}[t]/(t^{m+1}))$ produced by the above construction is independent of the choices of elementary matrices made in the construction.
2. If $A,B$ are nonnegative and $\gamma^{\prime}$ is another path in $SSE(\mathbb{Z})$ from $A$ to $B$ such that $\gamma$ and $\gamma^{\prime}$ are homotopic (with endpoints fixed), then $F(\gamma) = F(\gamma^{\prime})$ in $K_{2}(\mathbb{Z}[t]/(t^{m+1}))$.
3. If the path $\gamma$ lies entirely in $SSE_{2m}(\mathbb{Z}_{+})$, then the corresponding element $F(\gamma)$ in $K_{2}(\mathbb{Z}[t]/(t^{m+1}))$ vanishes.
Altogether this defines a function $$\Phi_{2m} \colon \pi_{1}(SSE(\mathbb{Z}),SSE_{2m}(\mathbb{Z}_{+}),A) \to K_{2}(\mathbb{Z}[t]/(t^{m+1}))$$ satisfying the properties \[eqn:FandG1\] – \[eqn:FandG3\] for $A$ in $SSE_{2m}(\mathbb{Z}_{+})$.\
Let $K_{2}(\mathbb{Z}[t]/(t^{m+1}),(t))$ denote the kernel of the split surjection $K_{2}(\mathbb{Z}[t]/(t^{m+1})) \to K_{2}(\mathbb{Z})$ induced by the ring map $\mathbb{Z}[t]/(t^{m+1}) \to \mathbb{Z}$ induced by $t \to 0$. Wagoner proved that the maps $\Phi_{2m}$ defined above actually lands in $K_{2}(\mathbb{Z}[t]/(t^{m+1}),(t))$. This is a significant fact, since van der Kallen proved in [@vdK] that $K_{2}(\mathbb{Z}[t]/(t^{2}),(t)) \cong \mathbb{Z}/2$. This calculation by van der Kallen was used by Wagoner to explicitly compute [@Wagoner2000 Eq. 1.21] $\Phi_{2}$, and to detect some explicit counterexamples in [@Wagoner2000].
Some remarks and open problems {#WagonerSubSecRemarks}
------------------------------
At the $m=2$ level, each method outlined above gives a map $$sgc_{2} \colon \pi_{1}(SSE(\mathbb{Z}),SSE_{2}(\mathbb{Z}_{+}),A) \to \mathbb{Z}/2\mathbb{Z}$$ $$\Phi_{2} \colon \pi_{1}(SSE(\mathbb{Z}),SSE_{2}(\mathbb{Z}_{+}),A) \to K_{2}(\mathbb{Z}[t]/(t^{2}),(t)) \cong \mathbb{Z}/2\mathbb{Z}.$$ While these were developed independently, remarkably, it was shown by Kim and Roush in the Appendix of [@Wagoner2000] that $\Phi_{2} = sgc_{2}$. Wagoner explicitly poses the problem in [@Wagoner99 Number 6] to determine, for larger $m$, the relationship between $\Phi_{2m}$ and $sgc_{m}$.\
Finally, let us note that both the Kim-Roush method and Wagoner’s method rely on the non-existence of periodic points at certain low levels. In Wagoner’s case, without vanishing trace conditions, step $(2)$ above can not be carried out. Moreover, step $(3)$ also relies on the vanishing trace conditions. As a result, Wagoner’s construction is *only* defined in the case of shifts of finite type lacking periodic points of certain low order levels. For the Kim-Roush technique, the non-existence of low-order periodic points comes in when one wants to conclude that the assignment from edges to some element of $\mathbb{Z}/m$ vanishes along any path through $SSE(\mathbb{Z}_{+})$: for an edge in $SSE(\mathbb{Z}_{+})$, the assignment coincides with the relative sign-gyration numbers associated to a conjugacy, which, in the absence of any periodic points of the given levels, must vanish.\
In light of this, neither method is able to produce more than a finite index refinement of the strong shift equivalence class of a given primitive matrix $A$ over $\mathbb{Z}_{+}$, since $(X_{A},\sigma_{A})$ will, above some level $k$ depending on $A$, eventually contain periodic points at all levels larger than $k$.
To finish, we highlight two open problems (Problem \[prob:finiterefinementproblem1\] below was mentioned informally in the discussion following Conjecture \[conj:williams\] in Lecture 1):\
If $A$ is shift equivalent over $\mathbb{Z}_{+}$ to the $1 \times 1$ matrix $(n)$, must $A$ be strong shift equivalent over $\mathbb{Z}_{+}$ to $(n)$? In other words, does Williams’ Conjecture hold in the case of full shifts?
\[prob:finiterefinementproblem1\] For a primitive matrix $A$, is the refinement of the SE-$\mathbb{Z}_{+}$-equivalence class of $A$ by SSE-$\mathbb{Z}_{+}$ finite?
Finally, we think the complexes $SSE(ZO), SSE(\mathbb{Z}_{+})$ and $SSE(\mathbb{Z})$ probably have much more to offer, and obtaining a deeper understanding of them would be valuable for studying both strong shift equivalence and the conjugacy problem for shifts of finite type.
Appendix 8
----------
This appendix contains some proofs, remarks, and solutions of various exercises throughout Lecture 8.\
\[apprem:wagonermarkovspace\] Prior to considering the strong shift equivalence spaces $SSE({\mathcal{R}})$, Wagoner also introduced a related ‘space of Markov partitions’ for a shift of finite type; we won’t describe these here, and instead refer the reader to [@Wagoner90triangle; @MR1748178; @Epperlein2019].
\[apprem:classifyingspacerem\] For a discrete group $G$, a classifying space is a path-connected space $BG$ such that $\pi_{1}(BG) \cong G$ and $\pi_{k}(BG) = 0$ for all $k \ge 2$. The space $BG$ has the property that $H_{k}(G,\mathbb{Z})$, the integral group homology of the group $G$, is isomorphic to $H_{k}(BG,\mathbb{Z})$, the integral singular homology of the space $BG$. See [@WeibelHomAlgBook 6.10.4] for details.
\[apprem:secomplexdef\] For a semiring ${\mathcal{R}}$, the shift equivalence space $SE({\mathcal{R}})$ is the CW complex defined as follows.
1. The 0-cells of $SE({\mathcal{R}})$ are square matrices over ${\mathcal{R}}$.
2. An edge from vertex $A$ to vertex $B$ corresponds to a shift equivalence over ${\mathcal{R}}$ from $A$ to $B$, i.e. matrices $R,S$ over ${\mathcal{R}}$ and $k \ge 1$ such that $$A^{k}=RS, \qquad B^{k} = SR, \qquad AR = RB, \qquad SA = BS.$$
3. 2-cells are given by triangles
such that $$R_{1}R_{2} = R_{3}.$$
Higher cells are defined in the same way as for the SSE spaces. It is immediate from the definition that $\pi_{0}(SE({\mathcal{R}}))$ is in bijective correspondence with the set of shift equivalence classes of matrices over ${\mathcal{R}}$.
\[apprem:nothomeqingeneral\] We’ll show here that $i_{{\mathcal{R}}}$ cannot in general be a homotopy equivalence. The map $i_{{\mathcal{R}}}$ induces a map of sets $i_{{\mathcal{R}},*} \colon \pi_{0}(SSE({\mathcal{R}})) \to \pi_{0}(SE({\mathcal{R}}))$. We can identify $\pi_{0}(SSE({\mathcal{R}}))$ with the set of SSE-classes of matrices over ${\mathcal{R}}$ and $\pi_{0}(SE({\mathcal{R}})$ with the set of SE-classes of matrices over ${\mathcal{R}}$, and upon making these identifications, the map $i_{{\mathcal{R}},*}$ agrees with the map $\pi$ given in . Theorem \[thm:ssesefibers\] from Lecture 6 gives a description of the fibers of this map in terms of some K-theoretic data. In particular, from Corollary \[cor:nk1vanishandsse\] we know that this map $i_{{\mathcal{R}},*} \colon \pi_{0}(SSE({\mathcal{R}})) \to \pi_{0}(SE({\mathcal{R}}))$ is not always an injection. Thus Wagoner’s result that $i_{{\mathcal{R}}}$ is a homotopy equivalence when ${\mathcal{R}}$ is a principal ideal domain can not hold in the case $NK_{1}({\mathcal{R}}) \ne 0$; as we see, it need not even induce an injection on the level of $\pi_{0}$.
\[apprem:sgcvssgcc\] As pointed out in [@S11 Section 8], the maps $sgc_{m}$ and $sgcc_{m}$ are not the same in general. However, they do yield the same value on path-components containing a primitive matrix whose trace is zero. The definition for $sgcc_{m}$ requires a component with a matrix which is shift equivalent to a primitive matrix, whereas the map $sgc_{m}$ does not. See [@S11 Section 8] for more details regarding the difference between $sgc_{m}$ and $sgcc_{m}$.
[^1]: At the beginning of the book *Algebraic K-theory and Its Applications* [@RosenbergBook], the author Jonathan Rosenberg writes “Algebraic K-theory is the branch of algebra dealing with linear algebra over a ring”.
[^2]: To be convinced of the difficulties in determining whether $SK_{1}$ vanishes, see the introduction of Oliver’s very thorough book [@OliverBook].
[^3]: We use the term graph symmetry instead of graph automorphism to avoid confusion between automorphisms of graphs and automorphisms of subshifts.
[^4]: What we call the Finite Order Generation Problem here was historically posed as a conjecture. Here we opted instead for the word ’problem’, since this conjecture is known to be false in general.
[^5]: In fact, something stronger is true: the commutator of ${\textnormal{Aut}^{(\infty)}(\sigma_{A})}$ coincides with the stabilized group of simple automorphisms; see [@HKS].
[^6]: Earlier counterexamples to Williams’ Conjecture in the reducible case were found by Kim and Roush - see [@S21].
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---
abstract: 'The detection of gravitational wave usually requires to match the measurement data with a large number of templates, which is computationally very expensive. Compressed sensing methods allow one to match the data with a small number of templates and interpolate the rest. However, the interpolation process is still computationally expensive. In this article, we designed a novel method that only requires to match the data with a few templates, yet without needing any interpolation process. The algorithm worked well for signals with relatively high SNRs. It also showed promise for low SNRs signals.'
author:
- Yan Wang
title: Fast detection and automatic parameter estimation of a gravitational wave signal with a novel method
---
*Introduction*–While gravitational wave (GW) signals contain invaluable physical information, extracting this information from the noisy data is quite challenging. Most of the time, GW signals are weaker than the instrumental noise at any instant, but they are predictable and long lived [@Sathyaprakash09]. This gives a way to build up signal-to-noise ratio (SNR) over time by tracking the signals coherently with matched filtering [@Jaranowski12]. However, this requires the templates to be exactly the same as the true signal to recover the optimal SNR, or at least resemble the true signal sufficiently in order not to lose much SNR [@Owen96]. Since the template waveforms depend on several parameters, one needs to match the data with a huge number of templates in the high dimensional parameter space. Therefore, a normal grid-based search is usually computationally extremely expensive, or even prohibitive. The reduction of the computational cost lies in the center of the modern GW data analysis.
There are several categories of algorithms, successfully reducing the computational cost, such as reduced bases (RB) [@Field11], singular value decomposition (SVD) [@Cannon10] and principal component analysis (PCA) [@Heng09]. These methods make use of the fact that each template is strongly correlated with the templates in its neighbourhood in the parameter space. Therefore, its SNR can be effectively interpolated from the SNRs of the templates in its neighbourhood. In other words, the likelihood surface on the grid of the template bank has special properties (sparsity), which allows the compressed sensing [@Candes06] algorithms to apply. Instead of using all the templates in the bank, one only needs to calculate the SNRs of a few so-called basis templates (which are different from the original templates), and then interpolate the SNRs of all the other templates in the bank. It is extremely fast to perform matched filtering on that few basis templates comparing to the original bank of templates. However, the interpolation (or sometimes referred to as the reconstruction) process is still computationally expensive.
We wish to design a novel method, which requires to perform matched filtering on a few templates, and in the meantime does not require any interpolation stage (or can automatically reconstruct the parameters of the GW signal). However, this method currently requires a relatively high SNR of the signal. The detailed description of the method and the preliminary simulation results are shown in the following.
*GW data analysis routine*–First of all, we briefly review the convention and notations of the GW data analysis. Usually, the measurement data can be expressed as $s = A h_* + n$, where $n$ is the noise, $A$ is the amplitude of the signal, $h_*$ is the normalized signal in the measurement, which satisfies $\langle h_*|h_* \rangle=1$. The inner product of two time series $a(t)$ and $b(t)$ is defined as follows $$\begin{aligned}
\langle a|b\rangle=\int_{-\infty}^{\infty}\frac{\tilde{a}^*(f)\tilde{b}(f)}{S_n(f)}df,\end{aligned}$$ where $\tilde{a}(f),\tilde{b}(f)$ are the Fourier transforms of $a(t)$ and $b(t)$. $S_n(f)$ is the so-called two-sided noise power spectral density (PSD), usually defined as $\textrm{E}[\tilde{n}^*(f')\tilde{n}(f)]=S_n(f)\delta(f-f')$.
The GW data analysis problem that we want to solve is formulated as follows. For a set of normalized candidate templates $h_i=h(\Theta_i)$ (we choose the template index $i=1,\dots,2^N$ for convenience) characterized by parameters $\Theta_i$, we want to determine which one is present in the measurement, hence obtaining the parameters $\Theta_*$ of the signal. Notice that $\Theta$ denotes a set of waveform parameters. For clarity, we require the templates to be nearly independent $ \langle h_i|h_j \rangle \,\ll 1,\, (i\neq j)$. This is not generally true for a whole template bank. However, one can easily divide the entire template bank into a group of smaller template banks, within which the templates are nearly independent.
We assume that the true signal $h_*$ belongs to the template family, $*\in \{1,2,\dots,2^N\}$. The inner product between the measurement data and a template is denoted as $$\begin{aligned}
x_i &\equiv& \langle s|h_i \rangle \nonumber \\
&=& A \langle h_*|h_i \rangle + \langle n|h_i \rangle,\end{aligned}$$ thus the expectation and the variance are $$\begin{aligned}
\textrm{E}(x_i) &=& A \delta_{*,i} \\
\textrm{Var}(x_i) &=& \textrm{E}[\langle h_i|n \rangle \langle n|h_i \rangle] \nonumber \\
&=& \langle h_i | h_i \rangle = 1.\end{aligned}$$ By identifying the largest inner product $x_*$, we can detect the signal $h_*$ and estimate its parameters $\Theta_*$. When the inner product $x_*$ is much larger than its standard deviation $\sqrt{\textrm{Var}(x_*)}=1$, the significance is high. The above shows a normal search strategy, which requires to perform $2^N$ inner products.
*The novel method*–In the following, we will describe a novel search algorithm. First, we express the waveform indices $i$ in binary, hence each index is an $N$-digit binary number (e.g. $001011011\dots$). Then, we define $N$ sets $\mathcal{P}_k$ ($k=1,2,\dots,N$) such that $\mathcal{P}_k$ consists of all the indices $i$ whose $k$-th digit is $1$. A new template family is defined based on these sets $$\begin{aligned}
H_k = \sum_{i\in \mathcal{P}_k} h_i.\end{aligned}$$ The inner products of these new templates with the measurement data are $$\begin{aligned}
X_k &\equiv& \langle s|H_k \rangle \nonumber \\
&=& \sum_{i\in \mathcal{P}_k} \langle s|h_i \rangle.\end{aligned}$$ The expectation of $X_k$ is $$\begin{aligned}
\textrm{E}(X_k)=\left\{
\begin{aligned}
A,\;\;\;\;\; *\in \mathcal{P}_k\\
0,\;\;\;\;\; *\notin \mathcal{P}_k
\end{aligned}
\right.\end{aligned}$$ The variance can be calculated as follows $$\begin{aligned}
\textrm{Var}(X_k)&=& \textrm{E}[\langle n | \sum_{i\in \mathcal{P}_k} h_i \rangle^2] \nonumber \\
&=& \sum_{i,j\in \mathcal{P}_k} \langle h_i | h_j \rangle.\end{aligned}$$ Since the templates $h_i$ are nearly independent, we have $$\begin{aligned}
\textrm{Var}(X_k)&=& \sum_{i\in \mathcal{P}_k} \langle h_i | h_i \rangle \nonumber \\
&=& 2^{N-1}.\end{aligned}$$ Suppose $*\in \mathcal{P}_a$ and $*\notin \mathcal{P}_b$, then $$\begin{aligned}
\textrm{E}(X_a-X_b)&=& A \\
\textrm{Var}(X_a-X_b) &=& \textrm{E}[\langle n | \sum_{i\in \mathcal{P}_a} h_i - \sum_{j\in \mathcal{P}_b} h_j \rangle ] \nonumber \\
&=& \sum_{i\in \{\mathcal{P}_a\cup \mathcal{P}_b - \mathcal{P}_a \cap \mathcal{P}_b\}} \langle h_i | h_i \rangle \nonumber \\
&=& 2^{N-1}.\end{aligned}$$ When the expectation $A$ is much larger than the standard deviation $2^{(N-1)/2}$, we can set some threshold $\mathcal{T}$ between $A$ and $2^{(N-1)/2}$. Based on this threshold, a binary number can be obtained as follows: if $X_k>\mathcal{T}$, the $k$-th bit of this binary number is $1$, otherwise its $k$-th digit is set as $0$. This binary number can be converted to a decimal number $i_0$. The method identifies the waveform $h_{i_0}$ with parameters $\Theta_{i_0}$ to be most probably present in the data. In this new approach, we have used $N$ templates instead of $2^N$ templates to detect the signal and estimate its parameters. The computational cost is thus reduced from $\mathcal{C}\cdot 2^N$ to $\mathcal{C}\cdot N$. Notice that, if each inner product of the data and a template provides one bit of information (above or below a certain threshold), $N$ is the minimum required number of templates to distinguish $2^N$ sets of candidate parameters.
*Simulation*–To exemplify the performance of the novel method, we consider the following chirp waveform family $$\begin{aligned}
h(t;f,\dot{f}) = \mathcal{A}\cos (2\pi f t+ \pi \dot{f} t^2),\end{aligned}$$ where $\mathcal{A}$ is the normalization constant, $f$ and $\dot{f}$ are the two intrinsic parameters to be estimated. We have simulated $100$ seconds measurement data at $1\,$kHz with different SNRs. The parameters of the true signal are $f_*=100\,$Hz and $\dot{f}_*=0.2\,$Hz/s. We have considered $2^6$ candidate waveforms with the parameter mesh grid $$f=\{70,80,90,100,110,120,130,140\}\,\textrm{Hz},$$ $$\dot{f}=\{-0.3,-0.2,-0.1,0,0.1,0.2,0.3,0.4\}\,\textrm{Hz/s}.$$
The threshold is simply chosen as $\mathcal{T} = c\cdot\max(X_k)$, where we have tried several values of the coefficient $c$. The SNR varies from $8$ to $50$ with a uniform spacing $3$. For each combination of SNR and the threshold, we carried out a Monte Carlo simulation with $1000$ different noise realizations. If the algorithm identifies the true signal and its true parameters, the detection is successful. The success rate is called the detection rate. Fig. \[fig:detectionRate\] shows the detection rate at different SNRs and thresholds, where the color bar indicates the value of the coefficient $c$. The best performance is realized by setting the coefficient $c$ around $0.5$. For signals with SNR higher than $30$, the detection rate of the algorithm is above $99\%$. Thus, the algorithm with the least number of new templates works efficiently at relatively hight SNRs. However, at low SNRs, the detection rate is low. We will see whether we could improve the detection rate by slightly increase the computational cost.
![ \[fig:detectionRate\] Detection rate at different SNRs and thresholds. The color bar indicates the value of the coefficient $c$. The algorithm achieves the optimal performance, when $c$ is around $0.5$. The detection rate is above $99\%$, when SNR is above $30$.](dRate.png){width="50.00000%"}
*Features of the algorithm*–For the set of $2^N$ independent templates $h_i$, if $2^N$ is smaller than the number of samples in the observation data, $x_i=\langle s|h_i\rangle$ are also independent. To characterize the performance of the algorithms, we want to examine to what extent can the noise mimic a signal. Since the signal part of $x_i$ only contributes a DC bias to its probability distribution, we can ignore the DC part and only consider the random part of $x_i$, which is $\langle n|h_i \rangle$. It can be shown without much effort that the probability density function of the maximum of these $2^N$ random variables $x_i$ is the following $$\begin{aligned}
p_{\max}(x) = \frac{2^N}{\sqrt{2^{2^{N+1}-1}\pi}} \left[ 1 + \textrm{erf}\left( \frac{x}{\sqrt{2}} \right) \right]^{2^N-1}e^{-\frac{x^2}{2}}, \nonumber \\\end{aligned}$$ where the error function $\textrm{erf}(x)$ is defined as $\textrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-x^2}dx$. Similarly, the probability density function of the maximum of the $N$ random variables $X_k$ turns out to be the following $$\begin{aligned}
p_{\max}(X) = \frac{N}{\sqrt{2^{3N-2}\pi}} \left[ 1 + \textrm{erf}\left( \frac{X}{\sqrt{2^N}} \right) \right]^{N-1}e^{-\frac{X^2}{2^N}}. \nonumber \\\end{aligned}$$ For the case we considered, we have $N=6$. The probability density functions and cumulative distribution functions of the random part of $x_i$ and $X_k$ are shown in Fig. \[fig:PDF\], which tells us how large SNRs could be mimicked by pure noise. As expected, in case of $X_k$, the noise could mimic larger SNRs. This can also be seen from the larger standard deviation of $X_k$. In fact, this is the reason for the drop in the detection rate at low SNRs in Fig. \[fig:detectionRate\].
![ \[fig:PDF\] The probability density functions and cumulative distribution functions of the random part of $x_i$ and $X_k$, which are $\langle n|h_i\rangle$ and $\langle n|H_k\rangle$.](PDF.png){width="50.00000%"}
Next, let us examine the role of the threshold $\mathcal{T}=\frac{1}{2}\max(X_k)$. In the previous simulations, we have six inner products $X_k,\;(k=1,\dots,6)$, each corresponding to an SNR achieved by $H_i$. Since the detection criteria only depends on the ratio between the inner products, it is convenient to look at their pie charts. In Fig. \[fig:SNRpie\], we show the pie charts for different SNRs, where the color bar represents the indices of the inner products. Take Fig. \[fig:SNRpie\] (a) for instance. The inner products $X_1,\,X_3,\,X_4$ contribute most part of the summation $\sum_{k=1}^6 X_k$, while $X_2,\,X_5,\,X_6$ are much smaller. According to the criteria we designed before, only $X_1,\,X_3,\,X_4$ are above the threshold. Therefore, we obtain the index $101100_2=44$ of the template, which most resembles the signal in the data. Similarly, Fig. \[fig:SNRpie\] (b)-(e) all successfully identify the correct template in case of different SNRs. Fig. \[fig:SNRpie\] (f) shows a failure case. According to the previous criteria, this pie chart gives a wrong index $101001_2=41$. In fact, even if one bit of the binary is wrongly determined, we end up with a completely different template (and its corresponding parameters). This is also a main reason why the detection rate at low SNRs drops so quickly.
\
\
*Improve the performance of the algorithm*–Now we discuss a simple and straightforward way to improve the performance of the algorithm by slightly increasing the computational cost. Let us look at the failure case in Fig. \[fig:SNRpie\] (f) again. The largest inner product is $X_1$, which contributes 30 percent of the entire SNR pie. The threshold, which was set to half of the largest inner product, turns out to be $15$ percent. Therefore, among the six inner products, $X_1,\,X_3$ are significantly above the threshold, $X_2,\,X_5$ are significantly below, while $X_4,\,X_6$ are close to the threshold. In the end, the binary bits corresponding to $X_4$ and $X_6$ (i.e. the 4th and 6th bits) were determined wrongly, which leads to a detection failure. However, the binary bits corresponding to $X_1,\,X_2,\,X_3$ and $X_5$ are correctly determined, and we are confident about that in the blind search. In fact, we are not so confident about the bits corresponding to $X_4$ and $X_6$, since they are just slightly above or below the threshold. If we leave these two binary bits undetermined, we end up with a binary number $101\textrm{y}0\textrm{y}_2$, where we have used $\textrm{y}$ to denote undetermined bits. It implies that the true signal might match one of the four templates $101000_2=40$, $101100_2=44$, $101001_2=41$ and $101101_2=45$. By simply calculating the inner products of the data and these four templates, we will know which one matches the true signal.
Hence, we can modify the algorithm according to the above procedure. In the beginning, we calculate $X_k,\,(k=1,\dots,6)$ and the threshold $\mathcal{T}= c\cdot\max(X_k)$. Then, we identify two $X_k$, which are closest to the threshold $\mathcal{T}$, and leave two binary bits corresponding to these two $X_k$ undetermined. We determine other binary bits in the same way as before. A binary number with two unknown bits is thus constructed. It corresponds to four original templates $h_i$. In the end, we calculate the inner product between the data and these four templates, and detect the signal. Following this procedure, we carried out a similar simulation as before. The detection rate is plotted in Fig. \[fig:detectionRate2\] with different combinations of $c$ values and SNRs. Comparing with Fig. \[fig:detectionRate\], the modified algorithm has significantly improved the performance. The detection rate is increased at all SNRs. We also observe that $c=0.5$ is still the optimal choice. For the curve $c=0.5$, the detection rate is 100% above SNR 30 and 96% at SNR=20. This strategy can be easily generalized by assigning a probability to each binary bit according to $X_k$, hence obtaining the probability of each $h_i$ present in the data. However, this is out of the scope of the current article. We will discuss it in the future work.
![ \[fig:detectionRate2\] Detection rate at different SNRs and thresholds. The color bar indicates the value of the coefficient $c$. ](dRate_mod2.png){width="50.00000%"}
*Conclusion and future work*–We have designed a novel algorithm for GW data analysis. Instead of using $2^N$ normal waveform templates, this new algorithm uses only $N$ combinations of the original waveforms as the new templates. By calculating the inner products between these $N$ new templates with the data and comparing these inner products with some threshold, we can construct a binary number with $N$ bits. From this binary number, we can determine which normal template in the original template bank best matches the signal in the data, without any reconstruction process. Therefore, this new algorithm can greatly reduce the computational cost in certain circumstances. However, it requires relatively high SNRs. We have discussed a simple and straightforward way to improve the performance of the algorithm. By leaving two most unconfident binary bits undetermined and calculating four additional inner products, we can significantly improve the performance of the algorithm at low SNRs. The detection rate of the modified algorithm is $100\%$ for 1000 different noise realizations for each SNR larger than 25. For SNR lower than 25, further improvements are demanded. We reserve that for future work.
One possible way to improve the algorithm is to construct additional $H_k,\,(k=N+1,\dots)$ for auxiliary use, such as to determine unconfident binary bits, to suppress the noise in $X_k$, etc. One can also set more sophisticated thresholds. We have used a threshold only depending on the relative values between the inner products $X_k$ for simplicity. A threshold also depending on the absolute values of the inner products would help, since the probability distribution of the random part of $X_k$ depends only on the absolute SNRs.
We have only carried out simulations for a bank of nearly independent templates. In the future, we will do a simulation for an entire template bank. The correlation between templates need also to be studied, since it could be used to reduce the noise in the detection statistic.
Y. W. was partially supported by DFG Grant No. SFB/TR 7 Gravitational Wave Astronomy and DLR (Deutsches Zentrum für Luft- und Raumfahrt). Y. W. also would like to thank the German Research Foundation for funding the Cluster of Excellence QUEST-Center for Quantum Engineering and Space-Time Research.
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abstract: |
0.5cm
We extend the one pion exchange model at quark level to include the short distance contributions coming from $\eta$, $\sigma$, $\rho$ and $\omega$ exchange. This formalism is applied to discuss the possible molecular states of $D\bar{D}^{*}/\bar{D}D^{*}$, $B\bar{B}^{*}/\bar{B}B^{*}$, $DD^{*}$, $BB^{*}$, the pseudoscalar-vector systems with $C=B=1$ and $C=-B=1$ respectively. The “$\delta$ function” term contribution and the S-D mixing effects have been taken into account. We find the conclusions reached after including the heavier mesons exchange are qualitatively the same as those in the one pion exchange model. The previous suggestion that $1^{++}$ $B\bar{B}^{*}/\bar{B}B^{*}$ molecule should exist, is confirmed in the one boson exchange model, whereas $DD^{*}$ bound state should not exist. The $D\bar{D}^{*}/\bar{D}D^{*}$ system can accomodate a $1^{++}$ molecule close to the threshold, the mixing between the molecule and the conventional charmonium has to be considered to identify this state with X(3872). For the $BB^{*}$ system, the pseudoscalar-vector systems with $C=B=1$ and $C=-B=1$, near threshold molecular states may exist. These bound states should be rather narrow, isospin is violated and the $I=0$ component is dominant. Experimental search channels for these states are suggested.
author:
- 'Gui-Jun Ding$^{a}$'
- 'Jia-Feng Liu$^{a}$'
- 'Mu-Lin Yan$^{a,b}$'
title: '[**Dynamics of Hadronic Molecule in One-Boson Exchange Approach and Possible Heavy Flavor Molecules** ]{}'
---
Introduction\[sec:introduction\]
================================
Since 1970s it is widely believed that Quantum Chromodynamics should accomodate a richer spectrum than just $q\bar{q}$ and $qqq$ resonances, many possible nonconventional structures are suggested, e.g. glueballs ($gg$, $ggg$,...), hybrid mesons ($q\bar{q}g$) and multiquark states($qq\bar{q}\bar{q}$, $qqqq\bar{q}$, $qqqqqq$, $qqq\bar{q}\bar{q}\bar{q}$). Unfortunately, so far there is still no uncontroversial evidence for nonconventional states experimentally except the hadronic molecules. The deuteron is a well-known example of hadronic molecule, and the approximate $10^5$ known nuclear levels are all hadronic molecule. In the past few years, many new states have been reported, a striking feature is that some of them are close to the thresholds of certain two hadrons, which inspires the possible interpretation of hadronic molecule.
Hadronic molecule is an old idea, about thirty years ago the possible hadronic molecules consisting of two charm mesons are suggested[@Voloshin:1976ap], and $\psi(4040)$ was proposed to be a P wave $D^{*}\bar{D}^{*}$ molecule [@De; @Rujula:1976qd]. Since in general molecule is weakly bound, the separation between the two hadrons in the molecule should be large. We can picture the two hadrons as interacting via a meson exchange potential [@Sakurai:1960ju]. At large distance, one pion exchange is dominant. Guided by the binding of deuteron, Tornqvist performed a systematic study of possible deuteronlike two-meson bound states [@Tornqvist:1991ks; @Tornqvist:1993ng]. The role of pion exchange in forming hadronic molecules was studied by Ericson and Karl [@Ericson:1993wy]. Recently Close et al. [@Thomas:2008ja] performed a pedagogic analysis of the overall sign, in addition they included the contribution of the “$\delta$ function” term which gives a $\delta$ function in the effective potential when no regularization is used. In these original work, only long distance one pion exchange has been considered, and the short distance contributions are neglected. In Ref. [@swanson] Swanson assumed that the short distance dynamics is governed by the one gluon exchange induced constituent quark interchange mechanism, which results in state mixing.
In the model of the nucleon-nucleon interaction, the long range part of the nucleon-nucleon force is quantitatively accounted for by the one $\pi$ exchange. However, the short and intermediate range interactions are governed by more complex dynamics. Combining the well-established one $\pi$ exchange with the exchange of heavier bosons (e.g. scalar and vector mesons) to describe the behavior at short distance has been proved to be a very successful approach [@Nagels:1975fb; @Nagels:1977ze; @Machleidt:1987hj]. Physically, the scalar and vector meson exchange describes part of multiple pion exchange effects. For the two $\pi$ exchange, if they interact and correlate in a P wave state, such a exchange can be modeled by $\rho$ exchange. If the two correlated $\pi$ pair is in a S-wave state, Durso et al. showed that one can approximate them by the exchange of a scalar $\sigma$ meson [@Durso:1980vn]. Similarly, the correlated 3 $\pi$ exchange can be approximated by the exchange of one $\omega$ meson.
Inspired by the nucleon-nucleon interaction, we shall represent the short distance interactions by the heavier bosons $\eta$, $\sigma$, $\rho$ and $\omega$ exchange instead of the quark interchange. The effective potential between two hadrons is obtained by summing over the interactions between light quarks or antiquarks as in the original work [@Tornqvist:1991ks; @Tornqvist:1993ng; @Thomas:2008ja]. It is well-known that one pion exchange between two light quarks results in two terms: the isospin dependent spin-spin interaction and tensor force. After taking into account the heavy bosons exchange, six additional terms appear including the spin-isospin independent central term, only isospin dependent term, isospin independent spin-spin interaction and tensor force, both isospin dependent and independent spin-orbit interactions. Consequently the situation becomes more complex than the only one pion exchange model. In our model, both the “$\delta$ function” term and the S-D mixing effects would be considered, which have be shown to play an important role in the binding [@Tornqvist:1991ks; @Tornqvist:1993ng; @Thomas:2008ja]. In this work, we first give a good description of the deuteron in our model, which is an unambiguous hadronic molecule, then apply this formalism to the heavy flavor pseudoscalar-vector (PV) systems. Thus the predictions for the possible heavy flavor PV molecules are base on a solid and reliable foundation. This is a greater advantage over other approaches dealing with the dynamics of hadronic molecule, such as one boson exchange in the effective field theory [@Ding:2008gr; @Liu:2008fh] and residual strong force with pairwise interactions [@Wong:2003xk; @Ding:2008mp] etc.
The paper is organized as follows. In section II, the formalism of the one boson exchange model is presented, the effective potentials from pseudoscalar, scalar and vector meson exchange are given explicitly. In section III, we give the meson parameters involved in our model and the boson-quark couplings which are extracted from the boson-nucleon couplings. The formalism is applied to the deuteron in section IV, the $D\bar{D}^{*}/\bar{D}D^{*}$ system and the molecular interpretation of X(3872) are investigated in section V. We further apply the one boson exchange approach to other heavy flavor PV systems in section VI, and possible molecular states are discussed. Section VII is our conclusions and discussions section. The expressions for the matrix elements of the spin relevant operators are analytically given in the Appendix.
The formalism of one-boson exchange model
=========================================
The construction of one-boson exchange interaction is constrained by the symmetry principle. To the leading order in the boson fields and their derivative, the effective interaction Lagrangian describing the coupling between the constituent quarks and the exchange boson fields is as follows [@Nagels:1975fb; @Nagels:1977ze; @Machleidt:1987hj] $$\begin{aligned}
\nonumber \rm
{Pseudoscalar:}&&~~~~~\mathcal{L}_p=-g_{pqq}\bar{\psi}(x)i\gamma_5\psi(x)\varphi(x)\\
\nonumber \rm
{Scalar:}&&~~~~~\mathcal{L}_s=-g_{sqq}\bar{\psi}(x)\psi(x)\phi(x)\\
\label{1}\rm{Vector:}&&~~~~~\mathcal{L}_v=-g_{vqq}\bar{\psi}(x)\gamma_{\mu}\psi(x)v^{\mu}(x)-\frac{f_{vqq}}{2m_{q}}\bar{\psi}(x)\sigma_{\mu\nu}\psi(x)\partial^{\mu}v^{\nu}(x)\end{aligned}$$ Here $m_q$ is the constituent quark mass, $\psi(x)$ is the constituent quark Dirac spinor field, $\varphi(x)$, $\phi(x)$ and $v^{\mu}(x)$ are the isospin-singlet pseudoscalar, scalar and vector boson fields respectively. In this work we take $m_q\equiv
m_u=m_d\simeq313$ MeV, since we concentrate on constituent up and down quarks. If the isovector bosons are involved, the couplings enter in the form $\bm{\tau\cdot\varphi}$, $\bm{\tau\cdot\phi}$ and $\bm{\tau\cdot v^{\mu}}$ respectively, where $\bm{\tau}$ is the well-known Pauli matrices. For the pseudoscalar, another interaction term is allowed $\mathcal{L}'_p=\frac{f_{pqq}}{m_{p}}\bar{\psi}(x)\gamma^{\mu}\gamma_5\psi(x)\partial_{\mu}\varphi(x)$ , where $m_p$ is the exchange pseudoscalar mass, this Lagrangian has been used by Tornqvist [@Tornqvist:1991ks; @Tornqvist:1993ng] and Close [@Thomas:2008ja]. By partial integration and using the equation of motion, one can easily show that $\mathcal{L}_p$ and $\mathcal{L}'_p$ are equivalent provided the coupling constants are related by $$\label{2}\frac{f_{pqq}}{m_p}=\frac{g_{pqq}}{2m _q}$$
From the above effective interactions, the effective potential between two quarks in momentum space can be calculated straightforwardly following the standard procedure. To the leading order in $\mathbf{q^2}/m^2_q$, where $\mathbf{q}$ is the momentum transfer, the potentials are
1. [Pseudoscalar boson exchange]{} $$\begin{aligned}
\nonumber V_p(\mathbf{q})&=&-\frac{g^2_{pqq}}{4m^2_q}\frac{(\bm{\sigma}_i\cdot\mathbf{q})(\bm{\sigma}_j\cdot\mathbf{q})}{\mathbf{q}^2+\mu^2_p}\\
\label{3}&&=-\frac{g^2_{pqq}}{12m^2_q}\left[\frac{\mathbf{q}^2}{\mathbf{q}^2+\mu^2_p}\;\bm{\sigma}_i\cdot\bm{\sigma}_j+\frac{\mathbf{q}^2S_{ij}(\hat{\mathbf{q}})}{\mathbf{q^2}+\mu^2_p}\right]\end{aligned}$$ where $S_{ij}(\hat{\mathbf{q}})=3(\bm{\sigma}_i\cdot\hat{\bm{q}})(\bm{\sigma}_j\cdot\hat{\bm{q}})-\bm{\sigma}_i\cdot\bm{\sigma}_j$, we have used $\mu^2_p=m^2_p-q^2_0$ instead of $m^2_p$ to approximately account for the recoil effect [@Tornqvist:1991ks; @Tornqvist:1993ng; @Thomas:2008ja].
2. [Scalar boson exchange ]{} $$\begin{aligned}
\label{4}V_s(\mathbf{q})=-\frac{g^2_{sqq}}{\mathbf{q}^2+\mu^2_s}\big(1+\frac{\mathbf{q}^2}{8m^2_q}\big)-\frac{g^2_{sqq}}{2m^2_q}\frac{i\mathbf{S}_{ij}\cdot(\mathbf{p}\times\mathbf{q})}{\mathbf{q}^2+\mu^2_s}\end{aligned}$$ where $\mathbf{S}_{ij}\equiv\frac{1}{2}(\bm{\sigma}_i+\bm{\sigma}_j)$, $\mu^2_s=m^2_s-q^2_0$ with $m_s$ the exchange scalar meson mass, and $\mathbf{p}$ denotes the total momentum.
3. [Vector boson exchange]{} $$\begin{aligned}
\nonumber V_v(\mathbf{q})&=&\frac{g^2_{vqq}}{\mathbf{q}^2+\mu^2_v}-\frac{g^2_{vqq}+4g_{vqq}f_{vqq}}{8m^2_q}\frac{\mathbf{q}^2}{\mathbf{q}^2+\mu^2_v}+\frac{(g_{vqq}+f_{vqq})^2}{12m^2_q}\frac{\mathbf{q}^2S_{ij}(\hat{\mathbf{q}})-2\mathbf{q}^2(\bm{\sigma}_i\cdot\bm{\sigma}_j)}{\mathbf{q}^2+\mu^2_v}\\
\label{5}&&-\frac{3g^2_{vqq}+4g_{vqq}f_{vqq}}{2m^2_q}\;\frac{i\mathbf{S}_{ij}\cdot(\mathbf{p}\times\mathbf{q})}{\mathbf{q}^2+\mu^2_v}\end{aligned}$$ where $\mu^2_v=m^2_v-q^2_0$ approximately reflects the recoil effect with $m_v$ the exchange vector meson mass.
The effective potential in configuration space is obtained by Fourier transforming the momentum space potential. $$\label{6}V_i(\mathbf{r})=\frac{1}{(2\pi)^3}\int
d^3\mathbf{q}\;e^{i\mathbf{q}\cdot\mathbf{r}}V_i(\mathbf{q})$$ where $i=p$, $s$ and $v$ respectively. However, the resulting potentials are singular, which contains delta function, so the potentials have to be regularized. Considering the internal structure of the involved hadrons, one usually introduces form factor at each vertex. Here the form factor is taken as $$\label{7}F(q)=\frac{\Lambda^2-m^2}{\Lambda^2-q^2}=\frac{\Lambda^2-m^2}{X^2+\mathbf{q}^2}$$ where $\Lambda$ is the so-called regularization parameter, $m$ and $q$ are the mass and the four momentum of the exchanged boson respectively with $X^2=\Lambda^2-q^2_0$. The form factor suppresses the contribution of high momentum, i.e. small distance. The presence of such a form factor is dictated by the extended structure of the hadrons. The parameter $\Lambda$, which governs the range of suppression, can be directly related to the hadron size that is approximately proportional to $1/\Lambda$. However, since the question of hadron size is still very much open, the value of $\Lambda$ is poorly known phenomenologically, and it is dependent on the model and application. In the nucleon-nucleon interaction, the $\Lambda$ in the range 0.8-1.5 GeV has been used to fit the data. For the present application to heavy mesons, which have a much smaller size than nucleon, we would expect a larger regularization parameter $\Lambda$. We can straightforwardly obtain the effective potentials between two quarks in configuration space. For convenience, the following dimensionless functions are introduced. $$\begin{aligned}
\nonumber H_0(\Lambda,m_{ex},\mu,r)&=&\frac{1}{\mu r}\big(e^{-\mu
r}-e^{-Xr}\big)-\frac{\Lambda^2-m^2_{ex}}{2\mu X}\,e^{-Xr}\\
\nonumber H_1(\Lambda,m_{ex},\mu,r)&=&-\frac{1}{\mu r}\big(e^{-\mu
r}-e^{-Xr}\big)+\frac{X(\Lambda^2-m^2_{ex})}{2\mu^3}\,e^{-Xr}\\
\nonumber H_2(\Lambda,m_{ex},\mu,r)&=&\big(1+\frac{1}{\mu
r}\big)\frac{1}{\mu^2r^2}e^{-\mu
r}-\big(1+\frac{1}{Xr}\big)\frac{X}{\mu}\frac{1}{\mu^2r^2}e^{-Xr}-\frac{\Lambda^2-m^2_{ex}}{2\mu^2}\frac{e^{-Xr}}{\mu
r}\\
\nonumber H_3(\Lambda,m_{ex},\mu,r)&=&\big(1+\frac{3}{\mu
r}+\frac{3}{\mu^2r^2}\big)\frac{1}{\mu r}e^{-\mu
r}-\big(1+\frac{3}{Xr}+\frac{3}{X^2r^2}\big)\frac{X^2}{\mu^2}\frac{e^{-Xr}}{\mu
r}-\frac{\Lambda^2-m^2_{ex}}{2\mu^2}\big(1+Xr\big)\frac{e^{-Xr}}{\mu r}\\
\nonumber
G_1(\Lambda,m_{ex},\tilde{\mu},r)&=&\frac{1}{\tilde{\mu}r}\;\big[\cos(\tilde{\mu
r})-e^{-Xr}\big]+\frac{X(\Lambda^2-m^2_{ex})}{2\tilde{\mu}^3}e^{-Xr}\\
\nonumber G_3(\Lambda,m_{ex},\tilde{\mu},r)&=&-\big[\cos(\tilde{\mu
r})-\frac{3\sin{(\tilde{\mu}r)}}{\tilde{u}r}-\frac{3\cos(\tilde{\mu}r)}{\tilde{\mu}^2r^2}\big]\frac{1}{\tilde{\mu}r}-\big(1+\frac{3}{Xr}+\frac{3}{X^2r^2}\big)\frac{X^2}{\tilde{\mu}^2}\frac{e^{-Xr}}{\tilde{\mu}r}\\
\label{8}&&-\frac{\Lambda^2-m^2_{ex}}{2\tilde{\mu}^2}\big(1+Xr\big)\frac{e^{-Xr}}{\tilde{\mu}r}\end{aligned}$$ Then the effective potentials between two quarks from one-boson exchange are
1. [Pseudoscalar boson exchange]{} $$\label{9}V_p(\mathbf{r})=\left\{\begin{array}{cc}
\frac{g^2_{pqq}}{4\pi}\frac{\mu^3_p}{12m^2_q}\big[-H_1(\Lambda,m_{p},\mu_p,r)\,\bm{\sigma}_i\cdot\bm{\sigma}_j+H_3(\Lambda,m_{p},\mu_p,r)S_{ij}(\hat{\mathbf{r}})\big],&\mu^2_p>0\\
\frac{g^2_{pqq}}{4\pi}\frac{\tilde{\mu}^3_p}{12m^2_q}\big[-G_1(\Lambda,m_{p},\tilde{\mu}_p,r)\,\bm{\sigma}_i\cdot\bm{\sigma}_j+G_3(\Lambda,m_{p},\tilde{\mu}_p,r)S_{ij}(\hat{\mathbf{r}})\big],&~~\mu^2_p=-\tilde{\mu}^2_p<0
\end{array}\right.$$ with $S_{ij}(\hat{\mathbf{r}})=3(\bm{\sigma}_i\cdot\hat{\mathbf{r}})(\bm{\sigma}_j\cdot\hat{\mathbf{r}})-\bm{\sigma}_i\cdot\bm{\sigma}_j$
2. [Scalar boson exchange]{} $$\label{10}V_s(\mathbf{r})=-\mu_s\frac{g^2_{sqq}}{4\pi}\left[H_0(\Lambda,m_s,\mu_s,r)+\frac{\mu^2_s}{8m^2_q}H_1(\Lambda,m_s,\mu_s,r)+\frac{\mu^2_s}{2m^2_q}H_2(\Lambda,m_s,\mu_s,r)\mathbf{L}\cdot\mathbf{S}_{ij}\right]$$ Here $\mathbf{L}=\mathbf{r}\times\mathbf{p}$ is the angular momentum operator.
3. [Vector boson exchange]{} $$\begin{aligned}
\nonumber
V_v(\mathbf{r})&=&\frac{\mu_v}{4\pi}\bigg\{g^2_{vqq}H_0(\Lambda,m_v,\mu_v,r)-\frac{(g^2_{vqq}+4g_{vqq}f_{vqq})\mu^2_v}{8m^2_q}H_1(\Lambda,m_v,\mu_v,r)\\
\nonumber&&-(g_{vqq}+f_{vqq})^2\frac{\mu^2_v}{12m^2_q}\Big[H_3(\Lambda,m_v,\mu_v,r)S_{ij}(\hat{\mathbf{r}})+2H_1(\Lambda,m_v,\mu_v,r)(\bm{\sigma_i\cdot\bm{\sigma}_j})\Big]\\
\label{11}&&-(3g^2_{vqq}+4g_{vqq}f_{vqq})\frac{\mu^2_{v}}{2m^2_q}H_2(\Lambda,m_v,\mu_v,r)\mathbf{L}\cdot\mathbf{S}_{ij}\bigg\}\end{aligned}$$ For $I=1$ isovector boson exchange, the above potential should be multiplied by the operator $\bm{\tau}_i\cdot\bm{\tau}_j$ in the isospin space. We have included the contribution of the “$\delta$ function” term in the above potentials, which gives the delta function when no regularization is used, since this contribution turns out to be important [@Thomas:2008ja]. The effective potential between two hadrons are obtained by summing the interactions between light quarks or antiquarks via one boson exchange.
Meson parameters and coupling constants
=======================================
As the well-known nuclear-nuclear interaction in the one boson exchange model, we shall take into account the contributions from pseudoscalar mesons $\pi$ and $\eta$ exchange, that from scalar meson $\sigma$ exchange, and those from vector mesons $\rho$ and $\omega$ exchange. The basic input parameters are the boson masses and the effective coupling constants between the exchanged bosons and the constituent quarks. The meson masses with their quantum numbers are taken from the compilation of the Particle Data Group [@pdg]. For the constituent quark-meson coupling constants, one may derive suitable estimates from the phenomenologically known $\pi
NN$, $\eta NN$, $\sigma NN$, $\rho NN$ and $\omega NN$ coupling constants using the Goldberger-Treiman relation. Riska and Brown have demonstrated that the nucleon resonance transition couplings to $\pi$, $\rho$ and $\omega$ can be derived in the single-quark operator approximation [@Riska:2000gd], which are in good agreement with the experimental data. Along the same way, we can straightforwardly derive the following relations between the boson-quark couplings and the boson-nucleon couplings, $$\begin{aligned}
\nonumber&&g_{\pi qq}=\frac{3}{5} \frac{m_q}{m_N}\,g_{\pi
NN},~~~~g_{\eta qq}=\frac{m_q}{m_{N}}\,g_{\eta NN}\\
\nonumber&&g_{\rho qq}=g_{\rho NN},~~~~~~~~~~~~f_{\rho
qq}=\frac{3}{5}\frac{m_q}{m_{N}}f_{\rho
NN}-(1-\frac{3}{5}\frac{m_q}{m_N})g_{\rho NN}\\
\nonumber&&g_{\omega qq}=\frac{1}{3}\,g_{\omega
NN},~~~~~~~~f_{\omega qq}=\frac{m_q}{m_N}f_{\omega
NN}-(\frac{1}{3}-\frac{m_q}{m_N})g_{\omega NN}\\
\label{12}&&g_{\sigma qq}=\frac{1}{3}\,g_{\sigma NN}\end{aligned}$$ where $m_N$ is the nucleon mass. In the present work, the constituent up(down) quark mass $m_{u(d)}$ is taken to be usual value $m_{u(d)}\simeq313$ MeV, which is about one third of the nucleon mass. The effective boson-nucleon coupling constants are taken from the well-known Bonn model [@Machleidt:1987hj], and a typical set of parameters is shown in Table \[parameter\]. The uncertainty of the effective couplings will be taken into account later, all the coupling constants except $g_{\pi NN}$ would be reduced by a factor of two, since the experimental value for $g_{\pi
NN}$ has been determined accurately from pion-nucleon and nucleon-nucleon scatterings. In the following, we shall explore the possible molecular states consisting of a pair heavy flavor pseudoscalar and vector mesons, their masses are taken from Particle Data Group [@pdg]: ${m_{D^{0}}=1864.84}$ MeV, ${
m_{D^{\pm}}=1869.62}$ MeV, ${m_{D^{*0}}=2006.97}$ MeV, ${m_{D^{*\pm}}=2010.27}$ MeV, ${m_{B^{0}}=5279.53}$ MeV, ${
m_{B^{\pm}}=5279.15}$ MeV and ${m_{B^{*}}=5325.1}$ MeV.
Boson $I^{G}(J^P)$ Mass (MeV) $g^2/4\pi$ $f^2/4\pi$
------------- ----------------- --------------- --------------- ---------------
$\pi^{\pm}$ $1^-(0^-)$ 139.57 14.9
$\pi^0$ $1^-(0^-)$ 134.98 14.9
$\eta$ $0^+(0^-)$ 547.85 3.0
$\sigma$ $0^+(0^+)$ 600.0 7.78
$\rho$ $1^+(1^-)$ 775.49 0.95 35.35
$\omega$ $0^-(1^-)$ 782.65 20.0 0.0
: \[parameter\]Spin, parity, isospin, G-parity, the masses of the exchange bosons, and the meson-nucleon coupling constants in the model.
Deuteron from one boson exchange model
======================================
Deuteron is a uncontroversial proton-neutron bound state with $J=1$ and $I=0$. It has been established that the long distance one pion exchange is the main binding mechanism, and the tensor force plays a crucial role, which results in the ${\rm ^3S_1}$ and ${\rm ^3D_1}$ states mixing. Tornqvist and Close only considered the pion exchange contribution in Refs. [@Tornqvist:1993ng; @Thomas:2008ja], however, the scalar meson $\sigma$ exchange and the vector mesons $\rho$, $\omega$ exchange turn out to be important in providing the short distance repulsion and the intermediate range attraction, consequently, we shall take into account the contributions from the heavier boson exchange in the following. The effective potential becomes $$\begin{aligned}
\nonumber V^{d}(\mathbf{r})&=&V^d_{\pi}(\mathbf{r})+V^d_{\eta}(\mathbf{r})+V^d_{\sigma}(\mathbf{r})+V^d_{\rho}(\mathbf{r})+V^d_{\omega}(\mathbf{r})\\
\nonumber&\equiv&
V^{d}_C(r)+V^d_S(r)(\bm{\sigma}_1\cdot\bm{\sigma}_2)+V^d_I(r)(\bm{\tau}_1\cdot\bm{\tau}_2)+V^d_T(r)S_{12}(\mathbf{\hat{r}})+
V^d_{SI}(r)(\bm{\sigma}_1\cdot\bm{\sigma}_2)(\bm{\tau}_1\cdot\bm{\tau}_2)\\
\label{13}&&+V^d_{TI}(r)S_{12}(\mathbf{\hat{r}})(\bm{\tau}_1\cdot\bm{\tau}_2)+V^d_{LS}(r)(\mathbf{L}\cdot\mathbf{S})+V^d_{LSI}(r)(\mathbf{L}\cdot\mathbf{S})(\bm{\tau}_1\cdot\bm{\tau}_2)\end{aligned}$$ where $\mathbf{S}=\frac{1}{2}(\bm{\sigma}_1+\bm{\sigma}_2)$ is the total spin, and $\mathbf{L}$ is the relative angular momentum operator. In the isospin symmetry limit, the components $V^{d}_C(r)$, $V^d_S(r)$ etc are given by $$\begin{aligned}
\nonumber V^d_C(r)&=&-\frac{g^2_{\sigma
NN}}{4\pi}\,m_{\sigma}\Big[H_0(\Lambda,m_{\sigma},m_{\sigma},r)+\frac{m^2_{\sigma}}{8m^2_N}H_1(\Lambda,m_{\sigma},m_{\sigma},r)\Big]+\frac{g^2_{\omega
NN}}{4\pi}\,m_{\omega}H_0(\Lambda,m_{\omega},m_{\omega},r)\\
\nonumber&&-\frac{g^2_{\omega NN}+4g_{\omega NN}f_{\omega
NN}}{4\pi}\frac{m^3_{\omega}}{8m^2_N}H_1(\Lambda,m_{\omega},m_{\omega},r)\\
\nonumber V^d_S(r)&=&-\frac{g^2_{\eta
NN}}{4\pi}\frac{m^3_{\eta}}{12m^2_N}H_1(\Lambda,m_{\eta},m_{\eta},r)-\frac{(g_{\omega
NN}+f_{\omega
NN})^2}{4\pi}\frac{m^3_{\omega}}{6m^2_{N}}H_1(\Lambda,m_{\omega},m_{\omega},r)\\
\nonumber V^d_I(r)&=&\frac{g^2_{\rho
NN}}{4\pi}m_{\rho}H_0(\Lambda,m_{\rho},m_{\rho},r)-\frac{g^2_{\rho
NN}+4g_{\rho NN}f_{\rho
NN}}{4\pi}\frac{m^3_{\rho}}{8m^2_{N}}H_1(\Lambda,m_{\rho},m_{\rho},r)\\
\nonumber V^{d}_T(r)&=&\frac{g^2_{\eta
NN}}{4\pi}\frac{m^3_{\eta}}{12m^2_{N}}H_3(\Lambda,m_{\eta},m_{\eta},r)-\frac{(g_{\omega
NN}+f_{\omega
NN})^2}{4\pi}\frac{m^3_{\omega}}{12m^2_{N}}H_3(\Lambda,m_{\omega},m_{\omega},r)\\
\nonumber V^{d}_{SI}(r)&=&-\frac{g^2_{\pi
NN}}{4\pi}\frac{m^3_{\pi}}{12m^2_{N}}H_1(\Lambda,m_{\pi},m_{\pi},r)-\frac{(g_{\rho
NN}+f_{\rho
NN})^2}{4\pi}\frac{m^3_{\rho}}{6m^2_{N}}H_1(\Lambda,m_{\rho},m_{\rho},r)\\
\nonumber V^{d}_{TI}(r)&=&\frac{g^2_{\pi
NN}}{4\pi}\frac{m^3_{\pi}}{12m^2_{N}}H_3(\Lambda,m_{\pi},m_{\pi},r)-\frac{(g_{\rho
NN}+f_{\rho
NN})^2}{4\pi}\frac{m^3_{\rho}}{12m^2_{N}}H_3(\Lambda,m_{\rho},m_{\rho},r)\\
\nonumber V^{d}_{LS}(r)&=&-\frac{g^2_{\sigma
NN}}{4\pi}\frac{m^3_{\sigma}}{2m^2_{N}}H_2(\Lambda,m_{\sigma},m_{\sigma},r)-\frac{3g^2_{\omega
NN}+4g_{\omega NN}f_{\omega
NN}}{4\pi}\frac{m^3_{\omega}}{2m^2_{N}}H_2(\Lambda,m_{\omega},m_{\omega},r)\\
\label{14} V^{d}_{LSI}(r)&=&-\frac{3g^2_{\rho NN}+4g_{\rho
NN}f_{\rho
NN}}{4\pi}\frac{m^3_{\rho}}{2m^2_{N}}H_2(\Lambda,m_{\rho},m_{\rho},r)\end{aligned}$$ In the basis of ${\rm ^3S_1}$ and ${\rm ^3D_1}$ states, the deuteron potential can be written in the matrix form as $$\begin{aligned}
\nonumber
V^d&=&\big[V^d_C(r)+V^d_S(r)-3V^d_I(r)-3V^d_{SI}(r)\big]\left(\begin{array}{cc}1&0\\
0&1\end{array}\right)+\big[9V^d_{LSI}(r)-3V^d_{LS}(r)\big]\left(\begin{array}{cc}0&0\\
0&1\end{array}\right)\\
\label{15}&&+\big[V^d_{T}(r)-3V^d_{TI}(r)\big]\left(\begin{array}{cc}0&\sqrt{8}\\
\sqrt{8}&-2\end{array}\right)\end{aligned}$$ Taking into account the centrifugal barrier from D wave and solving the corresponding two channel Schr$\ddot{\rm o}$dinger equation numerically via the Fortran77 package FESSDE2.2 [@fessde], which can fastly and accurately solve the eigenvalue problem for systems of coupled Schr$\ddot{\rm o}$dinger equations, we find the binding energy $\varepsilon_d\simeq 2.25$ MeV for the cutoff parameter $\Lambda=808$ MeV, and the corresponding wavefunction is presented in Fig. \[deuteron\_wavefunction\]. If we reduce half of the effective coupling constants except $g_{\pi NN}$, the binding energy is found to be about 2.28 MeV with $\Lambda=970$ MeV. From the wavefunction one can calculate the static properties of deuteron such as the root of mean square radius, the D wave probability, the magnetic moment and the quadrupole moment, which are in agreement with experimental data. We would like to note that the small binding energy of deuteron is a cancellation result of different contributions of opposite signs. The detailed results are listed in Table \[deuteron-static-properties\], it is obvious the results are sensitive to the regularization parameter $\Lambda$, and the same conclusion has been drawn in the one pion exchange model [@Tornqvist:1993ng; @Thomas:2008ja]. The binding energy variation with respect to $\Lambda$ is shown in Fig. \[deuteron\_binding\_energy\_variation\], the dependence is less sensitive than the one pion exchange model. It is obvious that the binding energy variation with $\Lambda$ is dependent on the coupling constants. For the coupling constants listed in Table \[parameter\], the binding energy no longer monotonically increases with $\Lambda$ in contrast with the one pion exchange model. To understand this peculiar behavior, we plot the three components of the deuteron effective potential in Eq.(\[15\]) in Fig. \[deuteron\_effective\_potential\]. We can see that both $V_{11}(\Lambda,r)$ and $V_{22}(\Lambda,r)$ potentials are repulsive, and they increase with $\Lambda$ at short distance. However, at intermediate distance the relation $|V_{12}(\Lambda=1.2\rm {GeV},r)|<|V_{12}(\Lambda=0.8\rm
{GeV},r)|<|V_{12}(\Lambda=0.9\rm {GeV},r)|<|V_{12}(\Lambda=1.6\rm
{GeV},r)|$ is satisfied, the $V_{12}(\Lambda,r)$ doesn’t monotonically increases with $\Lambda$. Therefore the non-monotonous behavior in Fig. \[deuteron\_binding\_energy\_variation\]a mainly comes from the non-monotonous dependence of $V_{12}(\Lambda,r)$ potential on $\Lambda$, which is a cancellation result of various contributions. As has been discussed above, the heavy flavor system should admit a larger $\Lambda$ than the deuteron. Therefore the above values of $\Lambda$ with which the smaller deuteron binding energy is reproduced, would be assumed to be the lower bound in the following.
[|cccccc|]{}$\Lambda({\rm MeV})$&$~~~{\rm \varepsilon_d}(\rm MeV)$&$~~~{\rm
r}_{\rm rms}({\rm fm})$&$~~~{\rm P_D:P_S}(\%)$& $~~~~~{\rm
\mu_d}(\mu_N)$ & ${\rm Q_d(fm^2)}$\
808&2.25 &3.85 & 5.66:94.34& 0.85&0.27\
900&5.33&2.77& 7.44:92.56 &0.84&0.20\
1000&4.96&2.87 &7.37:92.63& 0.84&0.21\
\
$\Lambda({\rm MeV})$&$~~~{\rm \varepsilon_d}(\rm MeV)$&$~~~{\rm
r}_{\rm rms}({\rm fm})$&$~~~{\rm P_D:P_S}$& $~~~~~{\rm
\mu_d}(\mu_N)$ & ${\rm Q_d(fm^2)}$\
970 &2.28 &3.84 & 6.52:93.48&0.84&0.28\
1100& 5.65 & 2.70& 8.92:91.08&0.83&0.20\
1200&8.89 &2.28 & 10.26:89.74 & 0.82 &0.16\
![The deuteron ${\rm ^3S_1}$ and ${\rm ^3D_1}$ wavefunction with binding energy $\varepsilon_d\simeq 2.25$ MeV and $\Lambda\simeq 808$ MeV. []{data-label="deuteron_wavefunction"}](deuteron_wavefunction.EPS)
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![\[deuteron\_binding\_energy\_variation\]The deuteron binding energy variation with respect to the regularization parameter $\Lambda$. Fig. \[deuteron\_binding\_energy\_variation\]a corresponds to the coupling constants shown in Table \[parameter\], and Fig. \[deuteron\_binding\_energy\_variation\]b for the couplings reduced by half.](deuteron_binding_energy_variation1.EPS "fig:") ![\[deuteron\_binding\_energy\_variation\]The deuteron binding energy variation with respect to the regularization parameter $\Lambda$. Fig. \[deuteron\_binding\_energy\_variation\]a corresponds to the coupling constants shown in Table \[parameter\], and Fig. \[deuteron\_binding\_energy\_variation\]b for the couplings reduced by half.](deuteron_binding_energy_variation2.eps "fig:")
(a) (b)
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------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[deuteron\_effective\_potential\] The three components of the deuteron effective potential in Eq.(\[13\]), Fig. \[deuteron\_effective\_potential\] a, Fig. \[deuteron\_effective\_potential\] b and Fig. \[deuteron\_effective\_potential\] c respectively illustrate the $V_{11}(\Lambda,r)$, $V_{12}(\Lambda,r)$ and $V_{22}(\Lambda,r)$ components. The solid line, dashed, dotted and dash-dotted lines correspond to $\Lambda=0.8$ GeV, 0.9 GeV, 1.2 GeV and 1.6 GeV respectively.](V11.eps "fig:") ![\[deuteron\_effective\_potential\] The three components of the deuteron effective potential in Eq.(\[13\]), Fig. \[deuteron\_effective\_potential\] a, Fig. \[deuteron\_effective\_potential\] b and Fig. \[deuteron\_effective\_potential\] c respectively illustrate the $V_{11}(\Lambda,r)$, $V_{12}(\Lambda,r)$ and $V_{22}(\Lambda,r)$ components. The solid line, dashed, dotted and dash-dotted lines correspond to $\Lambda=0.8$ GeV, 0.9 GeV, 1.2 GeV and 1.6 GeV respectively.](V12.eps "fig:") ![\[deuteron\_effective\_potential\] The three components of the deuteron effective potential in Eq.(\[13\]), Fig. \[deuteron\_effective\_potential\] a, Fig. \[deuteron\_effective\_potential\] b and Fig. \[deuteron\_effective\_potential\] c respectively illustrate the $V_{11}(\Lambda,r)$, $V_{12}(\Lambda,r)$ and $V_{22}(\Lambda,r)$ components. The solid line, dashed, dotted and dash-dotted lines correspond to $\Lambda=0.8$ GeV, 0.9 GeV, 1.2 GeV and 1.6 GeV respectively.](V22.eps "fig:")
(a) (b) (c)
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Possible ${D\bar{D}^{*}/\bar{D}D^{*}}$ hadronic molecule and X(3872)
=====================================================================
The narrow charmoniumlike state X(3872) was discovered by the Belle collaboration in the decay $B^{+}\rightarrow K^{+}+X(3872)$ followed by $X(3872)\rightarrow J/\psi\pi^{+}\pi^{-}$ with a statistical significance of 10.3$\sigma$ [@Choi:2003ue]. The existence of X(3872) has been confirmed by CDF [@Acosta:2003zx], D0 [@Abazov:2004kp] and Babar collaboration [@Aubert:2004ns]. the CDF collaboration measured the X(3872) mass to be ${\rm
(3871.61\pm0.16(stat)\pm0.19(sys.))}$ MeV. Its quantum number is strongly preferred to be $1^{++}$ [@Abulencia:2006ma]. In the one pion exchange model, Tornqvist suggested that X(3872) is a $1^{++}$ $D\bar{D}^{*}/\bar{D}D^{*}$ molecule and isospin is strongly broken [@Tornqvist:2004qy]. Recently Close et al. re-analyzed X(3872) in the same model, the critical overall sign is corrected and the contribution of the “$\delta$ function” term is included [@Thomas:2008ja]. Swanson have taken into account both the long rang pion exchange and short range contribution arising from constituent quark interchange [@swanson]. Recently Zhu et al. dynamically studied the binding of X(3872) in the heavy quark effective theory [@Liu:2008fh]. In this section, we will investigate the $1^{++}$ $D\bar{D}^{*}/\bar{D}D^{*}$ system from the one boson exchange model at quark level, where the short range interactions are represented by the heavier bosons $\eta$, $\sigma$, $\rho$ and $\omega$ exchange instead of the quark interchange.
There is only a sign difference $(-1)^{G}$ between the quark-quark interaction and quark-antiquark interaction, and the magnitudes are the same, where $G$ is the $G$-parity of the exchanged meson. The diagrams contributing to the $D\bar{D}^{*}$ and $\bar{D}D^{*}$ interactions are displayed in Fig. \[interaction diagram\]. Because of the parity conservation, $D\bar{D}^{*}$ can only scatter into $D^{*}\bar{D}$ via the pseudoscalar $\pi$ and $\eta$ exchange, and $D\bar{D}^{*}$ scatters into $D\bar{D}^{*}$ with the scalar $\sigma$ exchange, whereas the vector mesons $\rho$ and $\omega$ exchange contribute to both processes. The effective potential for the $1^{++}$ $D\bar{D}^{*}/\bar{D}D^{*}$ system is $$\begin{aligned}
\nonumber V^{X}(\mathbf{r})&=&-V^{X}_{\pi}(\mathbf{r})+V^{X}_{\eta}(\mathbf{r})+V^{X}_{\sigma}(\mathbf{r})+V^{X}_{\rho}(\mathbf{r})-V^{X}_{\omega}(\mathbf{r})\\
\nonumber&\equiv&
V^{X}_C(r)+V^X_S(r)(\bm{\sigma}_i\cdot\bm{\sigma}_j)+V^X_I(r)(\bm{\tau}_i\cdot\bm{\tau}_j)+V^X_T(r)S_{ij}(\mathbf{\hat{r}})+
V^X_{SI}(\mu_k,r)(\bm{\sigma}_i\cdot\bm{\sigma}_j)(\bm{\tau}_i\cdot\bm{\tau}_j)\\
\label{16}&&+V^X_{TI}(\mu_k,r)S_{ij}(\mathbf{\hat{r}})(\bm{\tau}_i\cdot\bm{\tau}_j)+V^X_{LS}(r)(\mathbf{L}\cdot\mathbf{S}_{ij})+V^X_{LSI}(r)(\mathbf{L}\cdot\mathbf{S}_{ij})(\bm{\tau}_i\cdot\bm{\tau}_j)\end{aligned}$$ with $i$ and $j$ is the index of light quark or antiquark, $\mu_k(k=1,2,3,4)$ takes four different values due to the mass difference within the $D$, $D^{*}$ and $\pi$ isospin multiplets. The eight components $V^{X}_C(r)$, $V^X_S(r)$ etc are given by $$\begin{aligned}
\nonumber V^{X}_C({r})&=&-\frac{g^2_{\sigma
qq}}{4\pi}m_{\sigma}\Big[H_0(\Lambda,m_{\sigma},m_{\sigma},r)+\frac{m^2_{\sigma}}{8m^2_q}H_1(\Lambda,m_{\sigma},m_{\sigma},r)\Big]-\frac{g^2_{\omega
qq}}{4\pi}m_{\omega}H_0(\Lambda,m_{\omega},m_{\omega},r)\\
\nonumber&&+\frac{g^2_{\omega qq}+4g_{\omega qq}f_{\omega
qq}}{4\pi}\frac{m^3_{\omega}}{8m^2_q}H_1(\Lambda,m_{\omega},m_{\omega},r)\\
\nonumber V^{X}_S({r})&=&-\frac{g^2_{\eta
qq}}{4\pi}\frac{\mu^3_5}{12m^2_q}H_1(\Lambda,m_{\eta},\mu_5,r)+\frac{(g_{\omega
qq}+f_{\omega
qq})^2}{4\pi}\frac{\mu^3_7}{6m^2_q}H_1(\Lambda,m_{\omega},\mu_7,r)\\
\nonumber V^{X}_I({r})&=&\frac{g^2_{\rho
qq}}{4\pi}m_{\rho}H_0(\Lambda,m_{\rho},m_{\rho},r)-\frac{g^2_{\rho
qq}+4g_{\rho qq}f_{\rho qq}}{4\pi}\frac{m^3_{\rho}}{8m^2_q}H_1(\Lambda,m_{\rho},m_{\rho},r)\\
\nonumber V^{X}_{T}({r})&=&\frac{g^2_{\eta
qq}}{4\pi}\frac{\mu^3_5}{12m^2_q}H_3(\Lambda,m_{\eta},\mu_5,r)+\frac{(g_{\omega
qq}+f_{\omega
qq})^2}{4\pi}\frac{\mu^3_7}{12m^2_q}H_3(\Lambda,m_{\omega},\mu_7,r)\\
\nonumber
V^{X}_{SI}(\mu,{r})&=&\left\{\begin{array}{cc}\frac{g^2_{\pi
qq}}{4\pi}\frac{\mu^3}{12m^2_{q}}H_1(\Lambda,m_{\pi^{\pm,0}},\mu,r)-\frac{(g_{\rho
qq}+f_{\rho
qq})^2}{4\pi}\frac{\mu^3_6}{6m^2_q}H_1(\Lambda,m_{\rho},\mu_6,r),&\mu^2>0\\
\frac{g^2_{\pi
qq}}{4\pi}\frac{\tilde{\mu}^3}{12m^2_{q}}G_1(\Lambda,m_{\pi^{\pm,0}},\tilde{\mu},r)-\frac{(g_{\rho
qq}+f_{\rho
qq})^2}{4\pi}\frac{\mu^3_6}{6m^2_q}H_1(\Lambda,m_{\rho},\mu_6,r),&~~~~~~\mu^2=-\tilde{\mu}^2<0
\end{array}\right.\\
\nonumber V^{X}_{TI}(\mu,r)&=&\left\{\begin{array}{cc}
-\frac{g^2_{\pi
qq}}{4\pi}\frac{\mu^3}{12m^2_q}H_3(\Lambda,m_{\pi^{\pm,0}},\mu,r)-\frac{(g_{\rho
qq}+f_{\rho
qq})^2}{4\pi}\frac{\mu^3_6}{12m^2_q}H_3(\Lambda,m_{\rho},\mu_6,r),&\mu^2>0\\
-\frac{g^2_{\pi
qq}}{4\pi}\frac{\tilde{\mu}^3}{12m^2_q}G_3(\Lambda,m_{\pi^{\pm,0}},\tilde{\mu},r)-\frac{(g_{\rho
qq}+f_{\rho
qq})^2}{4\pi}\frac{\mu^3_6}{12m^2_q}H_3(\Lambda,m_{\rho},\mu_6,r),&~~~~~~\mu^2=-\tilde{\mu}^2<0
\end{array}
\right.\\
\nonumber V^{X}_{LS}(r)&=&-\frac{g^2_{\sigma
qq}}{4\pi}\frac{m^3_{\sigma}}{2m^2_q}H_2(\Lambda,m_{\sigma},m_{\sigma},r)+\frac{3g^2_{\omega
qq}+4g_{\omega qq}f_{\omega
qq}}{4\pi}\frac{m^3_{\omega}}{2m^2_q}H_2(\Lambda,m_{\omega},m_{\omega},r)\\
\label{17} V^{X}_{LSI}(r)&=&-\frac{3g^2_{\rho qq}+4g_{\rho
qq}f_{\rho
qq}}{4\pi}\frac{m^3_{\rho}}{2m^2_q}H_2(\Lambda,m_{\rho},m_{\rho},r)\end{aligned}$$ where $$\begin{aligned}
\nonumber
\mu^2_1&=&m^2_{\pi^0}-(m_{D^{*0}}-m_{D^{0}})^2,~~~~~\mu^2_2=m^2_{\pi^{\pm}}-(m_{D^{*0}}-m_{D^{\pm}})^2\\
\nonumber
\mu^2_3&=&m^2_{\pi^{\pm}}-(m_{D^{*\pm}}-m_{D^{0}})^2,~~~~~\mu^2_4=m^2_{\pi^0}-(m_{D^{*\pm}}-m_{D^{\pm}})^2\\
\nonumber\mu^2_5&=&m^2_{\eta}-(m_{D^{*0}}-m_{D^{0}})^2,~~~~~~~\mu^2_6=m^2_{\rho}-(m_{D^{*0}}-m_{D^{0}})^2\\
\label{18}\mu^2_5&=&m^2_{\omega}-(m_{D^{*0}}-m_{D^{0}})^2\end{aligned}$$
![$D\bar{D}^{*}$ and $\bar{D}D^{*}$ interaction in one boson exchange model at quark level, where the thick line represents heavy quark or antiquark, and the thin line denotes light quark or antiquark.[]{data-label="interaction diagram"}](scattering.eps)
These $\mu^2$ parameters approximately represent the recoil effect due to different values of $m_{D}$ and $m_{D^{*}}$ as in Refs. [@Tornqvist:1993ng; @Thomas:2008ja]. For the $\eta$, $\sigma$, $\rho$ and $\omega$ exchange processes, the mass difference of $m_{D^{0}}$ and $m_{D^{\pm}}$ as well as $m_{D^{*0}}$ and $m_{D^{*\pm}}$ are neglected, since they are much smaller comparing with $m_{\eta}$, $m_{\rho}$ and $m_{\omega}$. X(3872) is very close to the $D^{0}\bar{D}^{*0}$ threshold, however, it is about 8.3 MeV below the $D^{+}D^{*-}$ threshold. Hence, isospin symmetry is drastically broken [@Close:2003sg; @Tornqvist:2004qy]. For the $J^{PC}=1^{++}$ $D\bar{D}^{*}/\bar{D}D^{*}$ system, they can be in S wave or D wave similar to the deuteron, then the wavefunction of this system is written as $$\begin{aligned}
\nonumber |X(3872)\rangle&=&\frac{u_1(r)}{r}\frac{1}{\sqrt{2}}|(D^{0}\bar{D}^{*0}+\bar{D}^{0}D^{*0})_S\rangle+\frac{u_2(r)}{r}\frac{1}{\sqrt{2}}|(D^{0}\bar{D}^{*0}+\bar{D}^{0}D^{*0})_D\rangle\\
\label{19}&&+\frac{u_3(r)}{r}\frac{1}{\sqrt{2}}|(D^{+}D^{*-}+D^{-}D^{*+})_S\rangle+\frac{u_4(r)}{r}\frac{1}{\sqrt{2}}|(D^{+}D^{*-}+D^{-}D^{*+})_D\rangle\end{aligned}$$ where the subscript $S$ and $D$ denote the system in $S$ wave and $D$ wave respectively. $u_1(r)$, $u_2(r)$, $u_3(r)$ and $u_4(r)$ are the spatial wavefunctions. There are four channels coupled with each other as has been shown above, and we might as well choose the basis to be $|1\rangle\equiv\frac{1}{\sqrt{2}}|(D^{0}\bar{D}^{*0}+\bar{D}^{0}D^{*0})_S\rangle$, $|2\rangle\equiv\frac{1}{\sqrt{2}}|(D^{0}\bar{D}^{*0}+\bar{D}^{0}D^{*0})_D\rangle$, $|3\rangle\equiv\frac{1}{\sqrt{2}}|(D^{+}D^{*-}+D^{-}D^{*+})_S\rangle$ and $|4\rangle\equiv\frac{1}{\sqrt{2}}|(D^{+}D^{*-}+D^{-}D^{*+})_D\rangle$. Using the analytical formula for the matrix elements presented in the appendix, the effective potential for $1^{++}$ $D\bar{D}^{*}/\bar{D}D^{*}$ can be written in the matrix form as $$\begin{aligned}
\nonumber
V^{X}(r)&=&\big[V^{X}_C(r)+V^{X}_S(r)\big]\left(\begin{array}{cccc}
1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&1
\end{array}\right)+\big[V^{X}_I(r)+V^{X}_{SI}(\mu_k,r)\big]\left(\begin{array}{cccc}
-1&0&-2&0\\
0&-1&0&-2\\
-2&0&-1&0\\
0&-2&0&-1
\end{array}\right)\\
\nonumber&&+V^{X}_{T}(r)\left(\begin{array}{cccc}
0&-\sqrt{2}&0&0\\
-\sqrt{2}&1&0&0\\
0&0&0&-\sqrt{2}\\
0&0&-\sqrt{2}&1
\end{array}\right)+V^{X}_{TI}(\mu_k,r)\left(\begin{array}{cccc}
0&\sqrt{2}&0&2\sqrt{2}\\
\sqrt{2}&-1&2\sqrt{2}&-2\\
0&2\sqrt{2}&0&\sqrt{2}\\
2\sqrt{2}&-2&\sqrt{2}&-1
\end{array}\right)\\
\label{20}&&+V^{X}_{LS}(r)\left(\begin{array}{cccc}
0&0&0&0\\
0&-3/2&0&0\\
0&0&0&0\\
0&0&0&-3/2
\end{array}\right)+V^{X}_{LSI}(r)\left(\begin{array}{cccc}
0&0&0&0\\
0&3/2&0&3\\
0&0&0&0\\
0&3&0&3/2
\end{array}\right)\end{aligned}$$ In the above equation, the value of $\mu^2_k$ is $\mu^2_1$ for the up-left $2\times2$ matrix elements, and it is equal to $\mu^2_4$ for the down-right $2\times2$ matrix elements. There is ambiguity in choosing $\mu^2_k$ value for the processes $D^{0}\bar{D}^{*0}\rightarrow D^{*+}D^-$ or $D^{+}D^{*-}\rightarrow
D^{*0}\bar{D}^{0}$, accordingly $\mu^2_k$ can take the value $\mu^2_2$ or $\mu^2_3$ for the off-diagonal $2\times2$ matrix elements, the numerical results for both choices would be given in the following. The different $\mu_k$ values is due to the isospin symmetry breaking from the mass difference within the $D$, $D^{*}$ and $\pi$ isospin multiplets. Taking into account the centrifugal barrier from D wave and solving the four channel coupled Schr$\ddot{\rm o}$dinger equation using the package FESSDE2.2, the numerical results are listed in Table \[x3872\]. It is remarkable that the $1^{++}$ $D\bar{D}^{*}/\bar{D}D^{*}$ system could accomodate a molecular state with mass about 3871.6 MeV for $\Lambda=808$ MeV, it is very close to the central value of X(3872) mass 3871.61 MeV. The corresponding wavefunction is shown in Fig. \[x3872 wavefunction\], it is obvious that the $D^{0}\bar{D}^{*0}+\bar{D}^{0}D^{*0}$ component dominates over the $D^{+}D^{*-}+D^{-}D^{*+}$ component. Since the spatial wavefunctions $u_1(r)$ and $u_3(r)$ have the same sign, the same is true for $u_2(r)$ and $u_4(r)$, thus the $I=0$ component in this state is predominant, it would be a isospin singlet in the isospin symmetry limit. From the results in Table \[x3872\], we notice that the predictions about the static properties for the two $\mu^2$ choices are very similar to each other, and the difference is small. The isospin symmetry is strongly broken especially for the states near the threshold. The uncertainties induced by the effective coupling constants are considered, we reduce half of the couplings except $g_{\pi NN}$, and the numerical results are given in Table \[x3872\] as well. For both choices of the coupling constants, the binding energy and other static properties are sensitive to the regularization parameter $\Lambda$, and the bound state mass dependence on $\Lambda$ is displayed in Fig. \[x3872 binding energy variation\]. It is obvious that the bound state mass decreases monotonically with the regularization parameter $\Lambda$ as in the one pion exchange model. In short summary, the predictions are qualitatively the same as those in the one pion exchange model, even after we have included the contributions from $\eta$, $\sigma$, $\rho$ and $\omega$ exchange. Since unexpectedly large branch ratio of $X(3872)\rightarrow\psi(2S)\gamma$ recently was reported [@Fulsom:2008rn], we have to take into account the mixing between the $1^{++}$ $D\bar{D}^{*}/\bar{D}D^{*}$ molecule and the conventional charmonium state in order to identify this state with X(3872). This is outside the range of the present work.
[|c|cccc|]{}
$\mu^2$ &$\Lambda({\rm MeV})$&$~~~{\rm M}(\rm MeV)$&$~~~{\rm r}_{\rm
rms}({\rm fm})$&$~~~{\rm
P^{00}_S:P^{00}_D:P^{+-}_S:P^{+-}_D}(\%)$\
& 808& 3871.6 & 7.02 & 90.76:0.56:8.11:0.56\
& 840& 3870.4 &2.84 & 78.23:1.08:19.59:1.11\
$\mu^2_3$ &850&3869.8 & 2.45& 75.26:1.21:22.29:1.23\
&900& 3865.9 & 1.61 & 65.17:1.89:31.06:1.88\
& 1000& 3849.2 & 1.08 & 53.06:4.72:37.65:4.57\
& 808& 3871.7& 11.34 & 94.40:0.38:4.86:0.36\
& 840 & 3870.7& 3.19& 80.74:0.99:17.26:1.01\
$\mu^2_2$ & 850 & 3870.2 & 2.68 & 77.44:1.14:20.28:1.15\
& 900& 3866.4&1.66 & 66.23:1.85:30.09:1.83\
& 1000& 3849.9 & 1.09& 53.35:4.69:37.43:4.53\
\
$\mu^2$& $\Lambda({\rm MeV})$&$~~~{\rm M}(\rm MeV)$&$~~~{\rm r}_{\rm
rms}({\rm fm})$&$~~~{\rm P^{00}_S:P^{00}_D:P^{+-}_S:P^{+-}_D}(\%)$\
&970& 3869.1& 2.13& 70.65:1.65:26.02:1.69\
$\mu^2_3$ &1100& 3860.1& 1.25&57.24:2.98:36.83:2.95\
&1200& 3848.2 &1.00 & 51.80:4.40:39.46:4.33\
&970&3869.5 &2.28 & 72.56:1.57:24.28:1.60\
$\mu^2_2$ &1100&3860.8& 1.27 &57.76:2.94:36.38:2.92\
&1200& 3849.0& 1.01 & 52.04:4.38:39.29:4.30\
![\[x3872 wavefunction\] The four components spatial wavefunctions of the $1^{++}$ $D\bar{D}^{*}/\bar{D}D^{*}$ system with $\Lambda=808$ MeV. ](x3872_wavefunction.EPS)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[x3872 binding energy variation\]The variation of the $1^{++}$ $D\bar{D}^{*}/\bar{D}D^{*}$ bound state mass with respect to $\Lambda$. (a) corresponds to the coupling constants shown in Table \[parameter\], and (b) for the couplings reduced by half.](x3872_binding_energy1.EPS "fig:") ![\[x3872 binding energy variation\]The variation of the $1^{++}$ $D\bar{D}^{*}/\bar{D}D^{*}$ bound state mass with respect to $\Lambda$. (a) corresponds to the coupling constants shown in Table \[parameter\], and (b) for the couplings reduced by half.](x3872_binding_energy2.eps "fig:")
(a) (b)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Possible molecular states of other heavy flavor PV systems
==========================================================
$B\bar{B}^{*}/\bar{B}B^{*}$ system
----------------------------------
For the $1^{++}$ $B\bar{B}^{*}/\bar{B}B^{*}$ system, the kinetic energy is greatly reduced due to the heavier mass of $B$ meson, and the interaction potential has features similar to those of the $D\bar{D}^{*}/\bar{D}D^{*}$ system except that the former is deeper than the latter. Therefore molecular states should be more easily formed. Following the same procedure as the $D\bar{D}^{*}/\bar{D}D^{*}$ case, the numerical results are shown in Table \[bottom analog of x3872\], where the $\mu^2$ ambiguity is considered. For the same value of $\Lambda$, the $B\bar{B}^{*}/\bar{B}B^{*}$ system is really more strongly bound than the $D\bar{D}^{*}/\bar{D}D^{*}$ system, its binding energy is a few tens of MeV, and the same was predicted in the one pion exchange model [@Tornqvist:1993ng; @Thomas:2008ja] and in the models [@Liu:2008fh; @liu-chiral]. It is obvious that the predictions about the static properties for the two $\mu^2$ choices are approximately the same. We notice that the isospin symmetry breaking is less stronger than the charm system, this is because the mass difference of $B^{0}$ and $B^{+}$ is smaller than that of $D^{0}$ and $D^{+}$ as well as $D^{*0}$ and $D^{*+}$. It is notable that there may be two molecular states for appropriate values of $\Lambda$. The corresponding wavefunctions for $\Lambda=1000$ MeV and $\mu^2=\mu^2_{bb1}\equiv
m^2_{\pi^{\pm}}-(m_{B^{*}}-m_{B^{0}})^2$ are displayed in Fig. \[wavefunction of bottom analog\], the first state is tightly bound, whereas the second is loosely bound. We notice that the first state is almost an isospin singlet, and the $I=0$ component is dominant for the second state. This state can no longer be produced through $B$ meson decay because of its large mass, and we have to resort to hadron collider. We can search for this state at Tevatron via $p\bar{p}\rightarrow\pi^{+}\pi^{-}\Upsilon(1S)$, and LHC is more promising.
$DD^{*}$ system with $C=2$
--------------------------
The interaction potentials arise from the one boson exchange between two antiquarks instead of a quark and antiquark pair, hence both the $\pi$ exchange and $\omega$ exchange potentials have overall opposite sign relative to the $D\bar{D}^{*}/\bar{D}D^{*}$ case. In this case we have four coupled channels $(D^{+}D^{*0})_S$, $(D^{+}D^{*0})_D$, $(D^{0}D^{*+})_S$ and $(D^{0}D^{*+})_D$. The numerical results are given in Table \[static properties of C=2 system\]. For $\Lambda=808$ MeV or $\Lambda=970$ MeV, we find no bound state. A bound state with mass about 3873.9 MeV appears for $\Lambda=1600$ MeV (about 3873.1 MeV for $\Lambda=1900$ MeV if the couplings except $g_{\pi NN}$ are reduced half), and the corresponding wavefunction is shown in Fig. \[wavefunction of C=2\]. We notice that the wavefunctions of $^3S_1$ $D^{+}D^{*0}$ and $D^{0}D^{*+}$ have opposite signs, the same is true for the $^3D_1$ $D^{+}D^{*0}$ and $D^{0}D^{*+}$ wavefunctions, therefore this state would be a isospin singlet in the isospin symmetry limit. We notice that the $^3D_1$ probability are much larger than $^3S_1$ probability for the state with $\Lambda=1800$ MeV, although there is centrifugal barrier for the D wave state. Thus the S-D mixing effect induced by the tensor force is especially crucial for this state. In short, the bound state of the $DD^{*}$ system appears only for the regularization parameter $\Lambda$ as large as 1600 MeV or 1900 MeV, which is beyond the range of 0.8 to 1.5 GeV favored by the nucleon-nucleon interaction. Moreover, the parameters that allow X(3872) to emerge as a $D\bar{D}^{*}/\bar{D}D^{*}$ molecule exclude the $DD^{*}$ bound state, as can be seen from the results in section V. Consequently we tend to conclude that the $DD^{*}$ molecular state may not exist.
$BB^{*}$ system with $B=2$
--------------------------
The situation is very similar to the $DD^{*}$ system except the different mass of $D$ mesons and $B$ mesons, we list the numerical results in Table \[static properties of B=2 system\]. We find a marginally bound state with mass 10603.9 MeV for $\Lambda=808$ MeV, which is very close to the $BB^{*}$ threshold. Its binding energy is much smaller than that of the $1^{++}$ $B\bar{B}^{*}/\bar{B}B^{*}$ system, however, the binding energy is less sensitive to $\Lambda$ than the latter case. Fig. \[wavefunction of B=2\] displays the wavefunction of the bound state solution with mass 10602.3 MeV and $\Lambda=900$ MeV. It is obvious that the $B^{+}B^{*0}$ and $B^{0}B^{*+}$ wavefunctions have the opposite sign, then the $I=0$ component is dominant in this state. If the couplings except $g_{\pi
NN}$ are reduced by half, a weakly bound state with mass about 10601.5 MeV is found as well assuming $\Lambda=970$ MeV. These indicates a weakly bound $BB^{*}$ should exist, This is consistent with the results of Manohar and Wise form heavy quark effective theory [@manohar]. For a loosely bound molecule, the leading source of decay is dissociation, to a good approximation the dissociation will proceed via the free space decay of the constituent mesons. The spin-parity forbids its decay into $BB$, therefore the $BB^{*}$ molecule is a very narrow state and it mainly decays into $BB\gamma$.
Pseudoscalar-vector system with $C=B=1$
---------------------------------------
This system could have the same quantum as $B_c$ meson or its antiparticle, and it is different from all the systems discussed above, eight channels instead of four channels are coupled with each other under the one boson exchange interaction, i.e. $(D^{+}B^{*0})_S$, $(D^{+}B^{*0})_D$, $(D^{0}B^{*+})_S$, $(D^{0}B^{*+})_D$, $(D^{*+}B^{0})_S$, $(D^{*+}B^{0})_D$, $(D^{*0}B^{+})_S$ and $(D^{*0}B^{+})_D$. We can investigate the possible bound states along the same line, although it is somewhat lengthy and tedious. There is ambiguity in choosing the $\mu^2$ value as well, for the $DB^{*}\rightarrow D^{*}B$ scattering process, we could take $\mu^2=m^2_{ex}-(m_{D^{*}}-m_{D})^2$ or $\mu^2=m^2_{ex}-(m_{B^{*}}-m_{B})^2$, where $m_{ex}$ is the mass of the exchanged boson. Specifically for $D^{+}B^{*0}\rightarrow
D^{*+}B^{0}$ via $\pi$ exchange, we can choose $\mu^2=m^2_{\pi^0}-(m_{D^{*+}}-m_{D^{+}})^2$ or $\mu^2=m^2_{\pi^0}-(m_{B^{*0}}-m_{B^{0}})^2$. This ambiguity has been taken into account in our analysis. The numerical results are given in Table \[static properties of C=B=1 system\], For $\Lambda=808$ MeV, we find no bound state. With the choice $\mu^2=m^2_{ex}-(m_{D^{*}}-m_{D})^2$, a bound state with mass 7189.7 MeV is found for $\Lambda=850$ MeV, However, this solution disappears if one chooses $\mu^2=m^2_{ex}-(m_{B^{*}}-m_{B})^2$. Only when $\Lambda$ is around 880 MeV, the bound state solutions can be found for both $\mu^2$ choices. The difference of the static properties for the two $\mu^2$ choices is relatively larger than that of the above systems considered, this is because of the larger difference between $m_{D^{*}}-m_{D}\simeq140$ MeV and $m_{B^{*}}-m_{B}\simeq45$ MeV. We notice that the $D^{0}B^{*+}$ component has the largest probability in the states, since the threshold of $D^{0}B^{*+}$ is lower than that of $D^{+}B^{*0}$, $D^{*+}B^{0}$ and $D^{*0}B^{+}$. The wavefunction of the state with mass about 7185.9 MeV and $\Lambda=900$ MeV is shown in Fig. \[wavefunction of C=B=1\], it is obvious all the eight components of the spatial wavefunction have the same sign, consequently this state would be isospin singlet in the isospin symmetry limit. Similar pattern of bound state solutions is predicted if the coupling constants except $g_{\pi NN}$ are reduced by half. This state is difficult to be produced, since both $c$ and $\bar{b}$ have to be produced simultaneously. The direct production of this state at hadron collider such as LHC and Tevatron is most promising, and the indirect production via top quark decay is a possible alternative. Once produced, it should be very stable, $DB\pi$ and $DB\gamma$ are the main decay channels.
Pseudoscalar-vector system with $C=-B=1$
----------------------------------------
The effective interaction potentials are induced by one boson exchange between two antiquarks, therefore both the $\pi$ exchange and $\omega$ exchange contributions give opposite sign between the $C=B=1$ system and the $C=-B=1$ system, nevertheless the overall signs of $\eta$, $\sigma$ and $\rho$ exchange potentials remain. We have eight coupled channels as well, $(D^{+}B^{*-})_S$, $(D^{+}B^{*-})_D$, $(D^{0}\bar{B}^{*0})_S$, $(D^{0}\bar{B}^{*0})_D$, $(D^{*+}B^{-})_S$, $(D^{*+}B^{-})_D$, $(D^{*0}\bar{B}^{0})_S$ and $(D^{*0}\bar{B}^{0})_D$ are involved. The numerical results are given in Table \[static properties of C=-B=1 system\]. It is remarkable that the $\mu^2$ and $\Lambda$ dependence of the bound state solutions is similar to the $C=B=1$ case. With the same $\mu^2$ and $\Lambda$ values, the predictions for the static properties of the two systems are not drastically different from each other. Concretely for $\Lambda=900$ MeV and $\mu^2=m^2_{ex}-(m_{D^{*}}-m_{D})^2$, we find a bound state with mass 7187.6 MeV for the $C=-B=1$ system, and the mass of $C=B=1$ bound state is 7185.9 MeV, the difference is about 1.7 MeV. The corresponding wavefunction with $\Lambda=900$ MeV is plotted in Fig. \[wavefunction of C=-B=1\], which can be roughly obtained by reversing the overall sign of the third, fourth, seventh and eighth components of the $C=B=1$ system wavefunction in Fig. \[wavefunction of C=B=1\]. To understand the similarity of the predictions for the $C=B=1$ and $C=-B=1$ system, we turn to the one $\pi$ exchange model, the effective potential comprises a spin-spin potential proportional to $(\bm{\sigma}_i\cdot\bm{\sigma}_j)(\bm{\tau}_i\cdot\bm{\tau}_j)$ and a tensor potential proportional to $S_{ij}(\hat{\mathbf{r}})(\bm{\tau}_i\cdot\bm{\tau}_j)$, where the isospin matrix $\bm{\tau}_i$ and the spin matrix $\bm{\sigma}_i$ only act on the light quarks. In the basis of the eight channels listed above, these two operators can be written as $8\times 8$ matrices $$\begin{aligned}
\nonumber&&
(\bm{\sigma}_i\cdot\bm{\sigma}_j)(\bm{\tau}_i\cdot\bm{\tau}_j)\longrightarrow\left(\begin{array}{cccccccc}
0&0&0&0&-1&0&2&0\\
0&0&0&0&0&-1&0&2\\
0&0&0&0&2&0&-1&0\\
0&0&0&0&0&2&0&-1\\
-1&0&2&0&0&0&0&0\\
0&-1&0&2&0&0&0&0\\
2&0&-1&0&0&0&0&0\\
0&2&0&-1&0&0&0&0
\end{array}\right)\\
\label{21}&&
S_{ij}(\hat{\mathbf{r}})(\bm{\tau}_i\cdot\bm{\tau}_j)\longrightarrow\left(\begin{array}{cccccccc}
0&0&0&0&0&\sqrt{2}&0&-2\sqrt{2}\\
0&0&0&0&\sqrt{2}&-1&-2\sqrt{2}&2\\
0&0&0&0&0&-2\sqrt{2}&0&\sqrt{2}\\
0&0&0&0&-2\sqrt{2}&2&\sqrt{2}&-1\\
0&\sqrt{2}&0&-2\sqrt{2}&0&0&0&0\\
\sqrt{2}&-1&-2\sqrt{2}&2&0&0&0&0\\
0&-2\sqrt{2}&0&\sqrt{2}&0&0&0&0\\
-2\sqrt{2}&2&\sqrt{2}&-1&0&0&0&0
\end{array}\right)\end{aligned}$$ For the $C=B=1$ pseudoscalar-vector system, the corresponding matrix representations are obtained by replacing 2 and $2\sqrt{2}$ with -2 and $-2\sqrt{2}$ respectively in Eq.(\[21\]). It is obvious both operators contribute to only the off-diagonal $4\times4$ matrix elements. As a result, the eigenvalues of the corresponding Schr$\ddot{\rm o}$dinger equation for the $C=B=1$ and $C=-B=1$ cases are exactly the same, if the small mass difference within the isospin multiplets is neglected, and the eingen-wavefunction of one system can be obtained from another by reversing the overall sign of the third, fourth, seventh and eighth components. Therefore the heavy bosons $\eta$, $\sigma$, $\rho$ and $\omega$ exchange contributes to effective potential, and the pion exchange contribution is still dominant. In short summary, even after including shorter distance contributions from $\eta$, $\sigma$, $\rho$ and $\omega$ exchange, the results obtained are qualitatively the same as those in the one $\pi$ exchange model. The same conclusion has been reached for all the system consider above.
Conclusions and discussions
===========================
Motivated by the nucleon-nucleon interaction, we have represented the short range interaction by heavier mesons $\eta$, $\sigma$, $\rho$ and $\omega$ exchange. The effective potentials between two hadrons are obtained by summing the interactions between light quarks or antiquarks via one boson exchange. The potential becomes more complicated than that in the one pion exchange model, and there are six additional terms which are proportional to $\mathbf{1}$, $\bm{\tau}_i\cdot\bm{\tau}_j$, $\bm{\sigma}_i\cdot\bm{\sigma}_j$, $S_{ij}(\hat{\mathbf{r}})$, $\mathbf{L}\cdot\mathbf{S}_{ij}$ and $(\mathbf{L}\cdot\mathbf{S}_{ij})(\bm{\tau}_i\cdot\bm{\tau}_j)$ respectively.
We first apply the one boson exchange formalism to the deuteron, then generalize to $D\bar{D}^{*}/\bar{D}D^{*}$, $B\bar{B}^{*}/\bar{B}B^{*}$, $DD^{*}$, $BB^{*}$, PV systems with $C=B=1$ and $C=-B=1$. S-D mixing effects has been taken into account, and the uncertainties from the regularization parameter $\Lambda$ and effective coupling constants are considered. We find the conclusions reached are qualitatively the same as those in the one pion exchange model. This implies that the long range $\pi$ exchange effects dominate the physics of a weakly bound hadronic molecule, and we can safely use one pion exchange model to qualitatively discuss the binding of molecule candidates. Since the predictions for the binding energy and other static properties are sensitive to the regularization parameter $\Lambda$ and the effective couplings, we are not able to predict the binding energies very precisely. If the potential is so strong that binding energy is large enough, we would be quite confident that such bound state must exist. However, the exact binding energy will depend on the details of the regularization and the effective couplings involved. Our results indicate that the $1^{++}$ $B\bar{B}^{*}/\bar{B}B^{*}$ molecule should exist, whereas $DD^{*}$ bound state doesn’t exist. For $\Lambda=808$ MeV (970 MeV), the binding energy, D wave probability and other static properties of deuteron are produced, meanwhile near threshold $1^{++}$ $D\bar{D}^{*}/\bar{D}D^{*}$ molecule is predicted. To identify this state with X(3872), the mixing between this $D\bar{D}^{*}/\bar{D}D^{*}$ molecule and the conventional charmonium state should be further considered to be consistent with the recent experimental data on $X(3872)\rightarrow\psi(2S)\gamma$ [@Fulsom:2008rn]. For the $BB^{*}$ system, the PV systems with $C=B=1$ and $C=-B=1$, near threshold molecular states may exist. Similar to the $1^{++}$ $D\bar{D}^{*}/\bar{D}D^{*}$ molecule, these states should be rather stable, isospin is drastically broken, and the $I=0$ component is dominant. Direct production of the above doubly heavy states at Tevatron and LHC is the most promising way. We can search for the $1^{++}$ $B\bar{B}^{*}/\bar{B}B^{*}$ molecule via $p\bar{p}\rightarrow\pi^{+}\pi^{-}\Upsilon(1S)$ at Tevatron. The $BB^{*}$ bound state mainly decays into $BB\gamma$ if it really exists. The dominant decay channels of the heavy flavor PV bound state with $C=B=1$ are $DB\pi$ and $DB\gamma$, and the possible heavy flavor PV bound state with $C=-B=1$ mainly decays into $D\bar{B}\pi$ and $D\bar{B}\gamma$.
In our model, the involved parameters include the effective quark-boson couplings, the masses of the exchanged bosons and the hadrons inside the molecule. Therefore this model is quite general, it can be widely used to dynamically study the possible molecular candidates. We will further apply the one boson exchange model to baryon-antibaryon system etc, and compare the predictions with the recent experimental observations [@progress].
We acknowledge Prof. Dao-Neng Gao for stimulating discussions. This work is supported by the China Postdoctoral Science foundation (20070420735). Jia-Feng Liu is supported in part by the National Natural Science Foundation of China under Grant No.10775124.
The matrix elements of the spin relevant operators\[appendix\]
==============================================================
For initial state consisting of two mesons $A$ and $B$, with relative angular momentum $L$, total spin $S$ and total angular momentum $J$, its wavefunction is written as $$\begin{aligned}
\nonumber&&|(AB)LS,JM_J\rangle=\sum_{M_L,M_S} \langle
LM_L;SM_S|JM_J\rangle\; |LM_L\rangle|SM_S\rangle\\
\label{a1}&&=\sum_{S_{13},S_{24}}\hat{S}_A\hat{S}_B\hat{S}_{13}\hat{S}_{24}\left\{\begin{array}{ccc}
1/2&1/2&S_A\\
1/2&1/2&S_B\\
S_{13}&S_{24}&S
\end{array}\right\}|L(S_{13}S_{24})S,JM_J\rangle\end{aligned}$$ where $\hat{S}=\sqrt{2S+1}$. For the convenience of calculating the matrix elements of the spin-orbit operator $\mathbf{L}\cdot\mathbf{S}_{24}$, we can recouple the state as $$\begin{aligned}
\nonumber|(AB)LS,JM_J\rangle=&&\sum_{S_{13},S_{24},J_{LS}}(-1)^{L+S+J}\hat{S}_A\hat{S}_B\hat{S}_{13}\hat{S}_{24}\hat{S}\hat{J}_{LS}\left\{\begin{array}{ccc}
L&S_{24}&J_{LS}\\
S_{13}&J&S
\end{array}\right\}\left\{\begin{array}{ccc}
1/2&1/2&S_A\\
1/2&1/2&S_B\\
S_{13}&S_{24}&S
\end{array}\right\}\\
\label{a2}&&\times|(LS_{24})J_{LS}S_{13},JM_{J}\rangle\end{aligned}$$ In the same way, we can recouple the the final state $|(A'B')L'S',J'M'_J\rangle$ via the Wigner 6-j and 9-j coefficients. In the following, we shall present the matrix elements of four light quark operators involved in the work, which is helpful to calculating the matrix representation of the effective interactions.
1. [The unit operator $\mathbf{1}$ ]{}\
Using Eq.(\[a1\]), it is obvious that $$\begin{aligned}
\nonumber&&\label{a3}\langle(A'B')L'S',J'M'_J|\mathbf{1}|(AB)LS,JM_J\rangle=\delta_{LL'}\delta_{SS'}\delta_{JJ'}\delta_{M_JM'_J}\sum_{S_{13},S_{24}}\hat{S}_A
\hat{S}'_{A}\hat{S}_{B}\hat{S}'_{B}\hat{S}^2_{13}\hat{S}^2_{24}\\
\label{a3}&&\times\left\{\begin{array}{ccc}
1/2&1/2&S_A\\
1/2&1/2&S_B\\
S_{13}&S_{24}&S
\end{array}\right\}
\left\{\begin{array}{ccc}
1/2&1/2&S'_A\\
1/2&1/2&S'_B\\
S_{13}&S_{24}&S
\end{array}\right\}=\delta_{LL'}\delta_{SS'}\delta_{S_AS'_{A}}\delta_{S_BS'_{B}}\delta_{JJ'}\delta_{M_JM'_J}\end{aligned}$$
2. [The spin-spin operator $\bm{\sigma}_2\cdot\bm{\sigma}_4$]{} $$\begin{aligned}
\nonumber&&\langle(A'B')L'S',J'M'_J|\bm{\sigma}_2\cdot\bm{\sigma}_4|(AB)LS,JM_J\rangle=\delta_{LL'}\delta_{SS'}\delta_{JJ'}\delta_{M_JM'_J}\sum_{S_{13},S_{24}}\hat{S}_A\hat{S}'_{A}
\hat{S}_B\hat{S}'_{B}\hat{S}^2_{13}\hat{S}^2_{24}\\
\label{a4}&&\times[2S_{24}(S_{24}+1)-3]\left\{\begin{array}{ccc}
1/2&1/2&S_A\\
1/2&1/2&S_B\\
S_{13}&S_{24}&S
\end{array}\right\}
\left\{\begin{array}{ccc}
1/2&1/2&S'_A\\
1/2&1/2&S'_B\\
S_{13}&S_{24}&S
\end{array}\right\}\end{aligned}$$ where the spin operators $\bm{\sigma}_2$ and $\bm{\sigma}_4$ only act on the light quarks and antiquarks
3. [The spin-orbit operator $\mathbf{L}\cdot\mathbf{S}_{24}$]{} $$\begin{aligned}
\nonumber&&\langle(A'B')L'S',J'M'_J|\mathbf{L}\cdot\mathbf{S}_{24}|(AB)LS,JM_J\rangle=\delta_{LL'}\delta_{JJ'}\delta_{M_JM'_J}\sum_{S_{13},S_{24},J_{LS}}(-1)^{S+S'+2L+2J}\hat{S}_A
\hat{S}'_A\\
\nonumber&&\times\hat{S}_B\hat{S}'_B\hat{S}\hat{S}'\hat{S}^2_{13}\hat{S}^2_{24}\hat{J}^2_{LS}\frac{1}{2}[J_{LS}(J_{LS}+1)-L(L+1)-S_{24}(S_{24}+1)]\left\{\begin{array}{ccc}
L&S_{24}&J_{LS}\\
S_{13}&J&S
\end{array}\right\}
\\
\label{a5}&&\times\left\{\begin{array}{ccc}
L&S_{24}&J_{LS}\\
S_{13}&J&S'
\end{array}\right\}\left\{\begin{array}{ccc}
1/2&1/2&S_A\\
1/2&1/2&S_B\\
S_{13}&S_{24}&S
\end{array}\right\}
\left\{\begin{array}{ccc}
1/2&1/2&S'_A\\
1/2&1/2&S'_B\\
S_{13}&S_{24}&S
\end{array}\right\}\end{aligned}$$ where $\mathbf{S}_{24}=\frac{1}{2}(\bm{\sigma}_2+\bm{\sigma}_4)$, $\mathbf{L}$ is the relative spatial angular momentum. The matrix elements of $\mathbf{L}\cdot\mathbf{S}_{24}$ can be calculated by the Wigner-Echart theorem [@angular], and the same result has been obtained.
4. [The tensor operator $S_{24}(\hat{\mathbf{r}})\equiv3(\bm{\sigma}_2\cdot\hat{\bf r})(\bm{\sigma}_4\cdot\hat{\bf
r})-\bm{\sigma}_2\cdot\bm{\sigma}_4$]{}\
It can be checked that the tensor operator $S_{24}(\hat{r})$ is proportional to the scalar product of two rank-2 tensor operators $Y_{2m}$ and $S^{(2)}_m$ with $m=0,\pm1,\pm2$, where $Y_{2m}$ is the spherical harmonic function of degree 2, and the five components of $S^{(2)}_m$ are $$\begin{aligned}
\nonumber&&S^{(2)}_2=\frac{1}{2}S_{2+}S_{4+},~~~S^{(2)}_1=-\frac{1}{2}(S_{20}S_{4+}+S_{2+}S_{40}),~~~S^{(2)}_0=-\frac{\sqrt{6}}{12}(S_{2-}S_{4+}-4S_{20}S_{40}+S_{2+}S_{4-})\\
\label{a6}&&S^{(2)}_{-1}=\frac{1}{2}(S_{2-}S_{40}+S_{20}S_{4-}),~~~S^{(2)}_{-2}=\frac{1}{2}S_{2-}S_{4-}\end{aligned}$$ Here $S_{2+}=\frac{1}{2}(\sigma_{2x}+i\sigma_{2y})$, $S_{20}=\frac{1}{2}\sigma_{20}$ and $S_{2-}=\frac{1}{2}(\sigma_{2x}-i\sigma_{2y})$. The same convention applies to $S_{4,\pm}$ and $S_{40}$, the spin operators $\bm{\sigma}_2$ and $\bm{\sigma}_4$ only act on the light quark and antiquarks. Using the Wigner-Echart theorem, the matrix element of this tensor operator can be obtained, although it is somewhat lengthy. $$\begin{aligned}
\nonumber&&\langle(A'B')L'S',J'M'_J|S_{24}(\hat{\mathbf{r}})|(AB)LS,JM_J\rangle=\delta_{JJ'}\delta_{M_JM'_J}\frac{2}{3}\sqrt{30}\sum_{S_{13},S_{24}}\delta_{S_{24},1}(-1)^{J+L+L'+2S'+S_{13}+S_{24}}\\
\nonumber&&\times\hat{S}_A\hat{S}'_A\hat{S}_B\hat{S}'_B\hat{S}\hat{S}'\hat{L}\hat{L}'\hat{S}^2_{13}\hat{S}^4_{24}\left\{\begin{array}{ccc}
L'&S'&J\\
S&L&2
\end{array}\right\}\left\{\begin{array}{ccc}
S_{24}&S'&S_{13}\\
S&S_{24}&2
\end{array}\right\}\left(\begin{array}{ccc}
L'&2&L\\
0&0&0
\end{array}\right)\left\{\begin{array}{ccc}
1/2&1/2&S_A\\
1/2&1/2&S_B\\
S_{13}&S_{24}&S
\end{array}\right\}\\
\label{a7}&&\times\left\{\begin{array}{ccc}
1/2&1/2&S'_A\\
1/2&1/2&S'_B\\
S_{13}&S_{24}&S'
\end{array}\right\}\end{aligned}$$ The above expression is apparently different from the results in Ref. [@Thomas:2008ja], However, the numerical results of all the matrix elements are the same.
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---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[wavefunction of bottom analog\] The spatial wavefunctions of $1^{++}$ $B\bar{B}^{*}/\bar{B}B^{*}$ molecule with $\Lambda=1000$ MeV. There are two bound states with mass 10457.6 MeV and 10600.4 MeV respectively, Fig. \[wavefunction of bottom analog\]a is for the first state, and the Fig. \[wavefunction of bottom analog\]b for the second.](bottom_analog_wavefunction_fir.eps "fig:") ![\[wavefunction of bottom analog\] The spatial wavefunctions of $1^{++}$ $B\bar{B}^{*}/\bar{B}B^{*}$ molecule with $\Lambda=1000$ MeV. There are two bound states with mass 10457.6 MeV and 10600.4 MeV respectively, Fig. \[wavefunction of bottom analog\]a is for the first state, and the Fig. \[wavefunction of bottom analog\]b for the second.](bottom_analog_wavefunction_sec.eps "fig:")
(a) (b)
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[wavefunction of C=2\] The wavefunction for the $DD^{*}$ system assuming $\Lambda=1600$ MeV and $\mu^2=\mu^2_{cc1}\equiv m^2_{\pi^{0}}-(m_{D^{*+}}-m_{D^{+}})^2$, its mass approximately is 3873.9 MeV.](C2_wavefunction.eps)
![\[wavefunction of B=2\] The spatial wavefunction for the $BB^{*}$ system assuming $\Lambda=900$ MeV and $\mu^2=\mu^2_{bb1}\equiv m^2_{\pi^{0}}-(m_{B^{*}}-m_{B^{+}})^2$, its mass is 10602.3 MeV.](B2_wavefunction.eps)
![\[wavefunction of C=B=1\] The wavefunction of the $C=B=1$ pseudoscalar-vector system with $\Lambda=900$ MeV and $\mu^2=m^2_{ex}-(m_{D^{*}}-m_{D})^2$, the mass of this state is about 7185.9 MeV.](C=B=1_wavefunction.EPS)
![\[wavefunction of C=-B=1\] The wavefunction of the $C=-B=1$ pseudoscalar-vector system with $\Lambda=900$ MeV and $\mu^2=m^2_{ex}-(m_{D^{*}}-m_{D})^2$, its mass is about 7187.6 MeV.](C=-B=1_wavefunction.eps)
[|c|cccc|]{}
\
$\mu^2$ & $\Lambda({\rm MeV})$&$~~~{\rm M}(\rm MeV)$&$~~~{\rm
r}_{\rm
rms}({\rm fm})$&$~~~{\rm P^{00}_S:P^{00}_D:P^{+-}_S:P^{+-}_D}(\%)$\
& 808 & 10565.3 & 0.60 & 47.70:2.05:48.20:2.05\
$\mu^2_{b\bar{b}1}$ & 900& 10543.5& 0.59 & 44.11:5.69:44.50:5.70\
& 1000& 10457.6 & 0.52 & 27.52:22.41:27.64:22.44\
& &10600.4 & 1.73& 37.10:9.43:44.26:9.22\
&808 & 10565.3 & 0.60 & 47.70:2.05:48.20:2.05\
$\mu^2_{b\bar{b}2}$ &900& 10543.5 & 0.59 & 44.11:5.69:44.49:5.70\
& 1000& 10457.6 & 0.52 & 27.52:22.41:27.64:22.44\
& &10600.4 & 1.72 & 37.10:9.43:44.25:9.22\
\
$\mu^2$& $\Lambda({\rm MeV})$&$~~~{\rm M}(\rm MeV)$&$~~~{\rm r}_{\rm
rms}({\rm fm})$&$~~~{\rm
P^{00}_S:P^{00}_D:P^{+-}_S:P^{+-}_D}(\%)$\
& 970 & 10544.8 & 0.55 & 45.24:4.60:45.55:4.61\
$\mu^2_{b\bar{b}1}$ &1100 &10503.9 & 0.51 & 40.12:9.78:40.30:9.80\
& 1200 & 10443.1 & 0.46 & 33.66:16.27:33.77:16.29\
& & 10601.9 & 1.91 & 38.82:8.03:45.26:7.89\
& 970 & 10544.8 & 0.55 & 45.24:4.60:45.55:4.61\
$\mu^2_{b\bar{b}2}$& 1100 & 10503.9 & 0.51 & 40.12:9.78:40.30:9.80\
&1200 & 10443.1 & 0.46 & 33.66:16.27:33.77:16.29\
& &10601.9 & 1.91& 38.83:8.04:45.25:7.89\
[|c|cccc|]{}
\
$\mu^2$ & $\Lambda({\rm MeV})$&$~~~{\rm M}(\rm MeV)$&$~~~{\rm
r}_{\rm rms}({\rm fm})$&$~~~{\rm
P^{+0}_S:P^{+0}_D:P^{0+}_S:P^{0+}_D}(\%)$\
&1600& 3873.9 & 3.15 & 34.61:4.03:56.82:4.54\
$\mu^2_{cc1}$ &1700& 3865.1 & 1.30 & 20.29:28.37:22.63:28.71\
&1800& 3770.9 & 0.58 & 0.19:49.78:0.19:49.84\
& & 3872.8 & 2.48 & 41.47:1.38:55.54:1.63\
& 1600& 3873.9 & 3.16 & 34.54:4.04:56.87:4.55\
$\mu^2_{cc2}$ & 1700& 3865.1 & 1.30 & 20.31:28.35:22.65:28.70\
& 1800& 3771.0 & 0.58 & 0.19:49.78:0.19:49.84\
& & 3872.8 & 2.49 & 41.41:1.38:55.58:1.63\
\
$\mu^2$ &$\Lambda({\rm MeV})$&$~~~{\rm M}(\rm MeV)$&$~~~{\rm r}_{\rm
rms}({\rm fm})$&$~~~{\rm
P^{+0}_S:P^{+0}_D:P^{0+}_S:P^{0+}_D}(\%)$\
& 1900& 3873.1 & 2.53 & 36.05:5.90:51.55:6.50\
$\mu^2_{cc1}$ &2000& 3870.1 & 1.82 & 38.09:8.08:45.32:8.52\
& 2100& 3865.3 & 1.44 & 37.67:10.21:41.57:10.55\
& 2200& 3858.4 & 1.19 & 36.31:12.42:38.60:12.68\
&1900& 3873.1 & 2.54 & 36.01:5.91:51.58:6.51\
$\mu^2_{cc2}$ &2000& 3870.1 & 1.82 & 38.07:8.08:45.32:8.53\
&2100& 3865.4 & 1.44 & 37.66:10.22:41.57:10.55\
& 2200& 3858.4 & 1.19 & 36.30:12.42:38.59:12.68\
[|c|cccc|]{}
\
$\mu^2$ & $\Lambda({\rm MeV})$&$~~~{\rm M}(\rm MeV)$&$~~~{\rm
r}_{\rm rms}({\rm fm})$&$~~~{\rm
P^{+0}_S:P^{+0}_D:P^{0+}_S:P^{0+}_D}(\%)$\
&808& 10603.9 & 4.09 & 59.37:6.74:28.49:5.40\
$\mu^2_{bb1}$ &900& 10602.3 & 2.23 & 43.52:10.84:35.63:10.02\
&1000& 10598.8 & 1.61 & 37.22:14.37:34.48:13.93\
& 1100& 10592.2 & 1.27 & 31.33:19.34:30.24:19.09\
& 808& 10603.9 & 4.09 & 59.38:6.74:28.48:5.40\
$\mu^2_{bb2}$ & 900&10602.3 & 2.23 & 43.52:10.84:35.63:10.01\
&1000& 10598.8 & 1.61 & 37.22:14.37:34.48:13.93\
& 1100& 10592.2 & 1.27 & 31.33:19.34:30.24:19.09\
\
$\mu^2$ &$\Lambda({\rm MeV})$&$~~~{\rm M}(\rm MeV)$&$~~~{\rm r}_{\rm
rms}({\rm fm})$&$~~~{\rm
P^{+0}_S:P^{+0}_D:P^{0+}_S:P^{0+}_D}(\%)$\
& 970& 10601.5 & 1.99 & 39.93:13.08:34.67:12.31\
$\mu^2_{bb1}$ &1000 & 10600.7 & 1.82 & 38.34:14.02:34.30:13.34\
&1100& 10596.6 & 1.44 & 34.37:16.80:32.47:16.36\
&1200& 10590.3 & 1.19 & 31.33:19.33:30.32:19.03\
& 970& 10601.5 & 1.99 & 39.94:13.08:34.67:12.31\
$\mu^2_{bb2}$ &1000& 10600.7 & 1.82 & 38.34:14.02:34.30:13.34\
&1100& 10596.6 & 1.44 & 34.37:16.80:32.47:16.36\
&1200& 10590.3 & 1.19 & 31.33:19.33:30.32:19.03\
[|c|cccc|]{}\
[$\mu^2$]{} & [$ \Lambda({\rm
MeV})$]{}&[${\rm M}(\rm MeV)$]{}&[${\rm r}_{\rm
rms}({\rm fm})$]{}&[$~~{\rm
P^{+0}_S(DB^{*}):P^{+0}_D(DB^{*}):P^{0+}_S(DB^{*}):P^{0+}_D(DB^{*}):P^{+0}_S(D^{*}B):P^{+0}_D(D^{*}B):P^{0+}_S(D^{*}B):P^{0+}_D(D^{*}B)}(\%)$]{}\
&850 & 7189.7 & 5.54 & 8.10:0.02:89.69:0.02:0.87:0.36:0.67:0.26\
[$\mu^2_{\bar{b}c1}$]{}&880 & 7187.9 & 2.05 & 21.88:0.07:72.40:0.08:2.09:0.96:1.73:0.80\
&900 & 7185.9 & 1.58 & 27.34:0.11:65.19:0.12:2.54:1.35:2.16:1.19\
& 1000 & 7157.2 & 0.86 & 36.55:0.01:46.27:0.01:1.39:7.32:1.21:7.26\
& 850 & no bounded & — & —\
[$\mu^2_{\bar{b}c2}$]{} & 880 & 7189.5 & 4.11 & 10.70:0.02:86.98:0.02:0.79:0.50:0.61:0.39\
&900 & 7188.1 & 2.18 & 20.59:0.04:75.09:0.04:1.34:0.98:1.10:0.83\
& 1000 & 7161.1 & 0.88 & 36.86:0.22:47.18:0.23:0.48:7.33:0.41:7.31\
\
[$\mu^2$]{} &[$\Lambda({\rm
MeV})$]{}&[${\rm M}(\rm MeV)$]{}&[${\rm r}_{\rm
rms}({\rm fm})$]{}&[$~~{\rm
P^{+0}_S(DB^{*}):P^{+0}_D(DB^{*}):P^{0+}_S(DB^{*}):P^{0+}_D(DB^{*}):
P^{+0}_S(D^{*}B):P^{+0}_D(D^{*}B):P^{0+}_S(D^{*}B):P^{0+}_D(D^{*}B)}(\%)$]{}\
& 970 & 7189.2 & 3.13 & 16.76:0.07:78.38:0.08:1.75:0.85:1.45:0.67\
[$\mu^2_{\bar{b}c1}$]{} & 1000 & 7187.6 & 1.88 & 25.51:0.13:66.52:0.15:2.78:1.35:2.42:1.15\
& 1100 & 7177.1 & 1.03 & 34.50:0.42:49.32:0.45:4.94:2.96:4.62:2.81\
& 1200 & 7156.3 & 0.78 & 34.62:0.72:41.51:0.75:5.85:5.51:5.64:5.42\
& 970 & no bounded & — & —\
[$\mu^2_{\bar{b}c2}$]{} &1000 & 7189.6 & 4.90 & 11.19:0.03:86.02:0.03:0.90:0.62:0.74:0.49\
& 1100 & 7181.9 & 1.20 & 34.04:0.20:54.01:0.21:3.26:2.67:3.08:2.53\
& 1200 & 7163.0 & 0.83 & 36.21:0.33:43.97:0.34:4.08:5.57:4.00:5.51\
[|c|cccc|]{}
\
[$\mu^2$ ]{}& [$\Lambda({\rm
MeV})$]{}&[${\rm M}(\rm MeV)$]{}&[${\rm r}_{\rm
rms}({\rm fm})$]{}&[$~~~{\rm
P^{+-}_S(DB^{*}):P^{+-}_D(DB^{*}):P^{00}_S(DB^{*}):P^{00}_D(DB^{*}):P^{+-}_S(D^{*}B):P^{+-}_D(D^{*}B):P^{00}_S(D^{*}B):P^{00}_D(D^{*}B)}(\%)$]{}\
& 880 & 7189.2 & 3.11 & 15.16:0.04:80.15:0.05:2.21:0.45:1.63:0.32\
[$\mu^2_{bc1}$]{} & 900 & 7187.6 & 1.85 & 23.82:0.08:68.07:0.09:3.83:0.61:3.02:0.48\
& 1000 & 7169.0 & 0.79 & 30.56:0.28:46.30:0.30:11.86:0.74:9.27:0.69\
& 1050 & 7148.4 & 0.51 & 15.57:0.02:53.99:0.04:22.84:0.10:7.41:0.03\
& & 7154.3 & 0.63 & 53.57:0.36:16.62:0.34:5.65:0.59:22.22:0.66\
& 880 & no bounded & — & —\
[$\mu^2_{bc2}$]{} & 900 & 7189.7 & 5.17 & 9.30:0.02:88.09:0.02:1.25:0.27:0.87:0.19\
& 1000& 7176.2 & 0.91 & 30.77:0.20:50.54:0.21:9.62:0.70:7.32:0.65\
& 1050 & 7154.6 & 0.52 & 25.22:0.00:45.64:0.01:17.96:0.04:11.14:0.00\
& & 7163.3 & 0.70 & 46.16:0.31:27.72:0.30:7.95:0.64:16.25:0.68\
\
[$\mu^2$]{} &[$\Lambda({\rm
MeV})$]{}&[${\rm M}(\rm MeV)$]{}&[${\rm r}_{\rm
rms}({\rm fm})$]{}&[$~~~{\rm
P^{+-}_S(DB^{*}):P^{+-}_D(DB^{*}):P^{00}_S(DB^{*}):P^{00}_D(DB^{*}):P^{+-}_S(D^{*}B):P^{+-}_D(D^{*}B):P^{00}_S(D^{*}B):P^{00}_D(D^{*}B)}(\%)$]{}\
& 970 & 7189.4 & 3.57 & 15.68:0.06:78.98:0.07:2.29:0.58:1.90:0.44\
[$\mu^2_{bc1}$]{} &1020& 7185.6 & 1.38 & 30.23:0.20:56.36:0.21:5.82:1.03:5.25:0.90\
& 1100 & 7173.2 & 0.82 & 33.18:0.42:43.08:0.45:10.53:1.24:9.93:1.17\
& 1200 & 7147.4 & 0.59 & 30.96:0.68:35.69:0.70:15.05:1.28:14.40:1.25\
& 970 & no bounded & — & —\
[$\mu^{2}_{bc2}$]{} & 1020 & 7189.2 & 3.03 & 18.64:0.07:75.25:0.07:2.60:0.63:2.24:0.51\
& 1100 & 7180.5 & 1.00 & 33.61:0.29:47.78:0.30:8.09:1.16:7.68:1.09\
& 1200 & 7158.2 & 0.64 & 32.34:0.58:37.41:0.60:13.43:1.28:13.09:1.26\
|
---
abstract: 'In the present paper we study normal transport surfaces in four-dimensional Euclidean space $\mathbb{E}^{4}$ which are the generalization of surface offsets in $\mathbb{E}^{3}$. We find some results of normal transport surfaces in $\mathbb{E}^{4}$ of evolute and parallel type. Further, we give some examples of these type of surfaces.'
author:
- 'K. Arslan, B. Bulca, B. K. Bayram & G. Öztürk'
date: December 2014
title: '**Normal Transport Surfaces in Euclidean 4-space $\mathbb{E}^{4} $** '
---
Introduction
============
[^1]The geometric modelling of free-form curves and surfaces is of central importance for sophisticated CAD/CAM systems. Apart from the pure construction of these curves and surfaces, the analysis of their quality is equally important in the design and manufacturing process. It is for example very important to test the convexity of a surface, to pinpoint inflection points, to visualize flat points and to visualize technical smoothness of surface [@HH1].
The 3D offsets or parallel surfaces are very widely used in many applications. these include tool path generation for 3N machining. However 3D offsets are particularly important and useful as pre-process modifications to CAD geometry. By defining an 3D offset means moving a surface of a 3D model by a constant “d” in a direction normal to the surface of the model. Offset techniques for surfaces has been extensively studied by Mechawa ([@Ma]) and Pham ([@P]). Generally offsets of 3D models are achieved by first offsetting all surfaces of the model and then trimming and extending these offsets to reconstruct a closed 3D model ([@Fa], [@Fo]).
Further, focal surfaces are known in the field of line congruences. Line congruences have been introduced in the field of visualization by Hagen and Pottmann (see, [@Hha]). Focal surfaces are also used as a surface interrogation tool to analyses the “quality” of the surface before further processing of the surface, for example in a NC-milling operation (see [HH1]{}).
The generalized focal surfaces are related to hedgehog diagrams. Instead of drawing surface normals proportional to a surface value, only the point on the surface normal proportional to the function is drawing. The loci of all these points is the generalized focal surface. This method was introduced by Hagen and Hahmann ([@HH1], [@HH2]) and is based on the concept of focal surface which are known from line geometry. The focal surfaces are the loci of all focal points of special congruence, the normal congruence. Recently the present authors considered parallel and focal surfaces and their curvature properties (see [@OA]).
The normal transport surface $\widetilde{M}$ of $M$ are generalization of offset surfaces to $4$-dimensional Euclidean space $\mathbb{E}^{4}$ [@Fr]. Observe that, evolute surfaces and parallel type surfaces in $\mathbb{E}^{4}$ are the special type normal transport surfaces [@Kr], [@Ce], [@Fr]. Parallel type surface are widely used in geometry and mathematical physics. We want to refer the reader to da Costa [@Co] for an application in quantum mechanics in curved spaces.
The paper organized as follows. In section $2,$ we briefly considered basic concepts of surfaces in Euclidean spaces. In section $3,$ we consider some known results about the surfaces with flat normal bundle. In the final section, we consider normal transport surfaces in $\mathbb{E}^{4}.$ Further we give some examples of evolute and parallel type surfaces in $\mathbb{E}^{4}$.
**Preliminaries**
=================
In the present section we recall definitions and results of [@Fr]. Let $M $ be a local surface in $\mathbb{E}^{n+2}$ given with the regular patch $x(u,v)$ : $(u,v)\in D\subset \mathbb{E}^{2}$. The tangent space $T_{p}(M)$ to $M$ at an arbitrary point $p=x(u,v)$ of $M$ is spanned by $\left \{
x_{u},x_{v}\right \} $. Further, the coefficients of the first fundamental form of $M$ are given by $$g_{_{11}}=\left \langle x_{u},x_{u}\right \rangle ,g_{_{12}}=\left \langle
x_{u},x_{v}\right \rangle ,g_{22}=\left \langle x_{v},x_{v}\right \rangle ,
\label{2.1}$$where $\left \langle ,\right \rangle $ is the Euclidean inner product. Let us denote by$$ds^{2}=\sum \limits_{i,j=1}^{2}g_{ij}du^{i}du^{j}. \label{2.2}$$
Let us choose $n$ linearly independent, orthogonal unit normal vectors $N_{\alpha },$ $\alpha =1,2,...,n$ spanning the normal space $T_{p}^{\perp }M$ at point $p=x(u,v).$ For each $p\in M$, consider the decomposition $T_{p}\mathbb{E}^{n+2}=T_{p}M\oplus T_{p}^{\perp }M,$ where $T_{p}^{\perp }M$ is the orthogonal component of $T_{p}M$ in $\mathbb{E}^{n+2}$. Let $\overset{\sim }{\nabla }$ be the Riemannian connection of $\mathbb{E}^{n+2}$ then the Gauss equation of the surface $M$ is given by $$x_{u^{i}u^{j}}=\widetilde{\nabla }_{x_{u^{i}}}x_{u^{j}}=\sum_{k=1}^{2}\Gamma
_{ij}^{k}x_{u^{k}}+\sum_{\alpha =1}^{n}\ c_{ij}^{\alpha }N_{\alpha },
\label{2.3}$$where $$c_{ij}^{\alpha }=\left\langle x_{u^{i}u^{j}},N_{\alpha }\right\rangle ;\text{
}c_{ij}^{\alpha }=c_{ji}^{\alpha }, \label{2.4}$$are the coefficients of the second fundamental form and$$\Gamma _{ij}^{k}=\sum_{l=1}^{2}g^{lk}\left( \frac{\partial g_{jl}}{\partial
u^{i}}+\frac{\partial g_{li}}{\partial u^{j}}-\frac{\partial g_{ij}}{\partial u^{l}}\right) , \label{2.5}$$are the Christoffel symbols corresponding to $x(u,v)$.
Further, the Weingarten equation of the surface $M$ is given by$$(N_{\alpha })_{u^{i}}=\widetilde{\nabla }_{x_{u^{i}}}N_{\alpha
}=-\sum_{k=1}^{2}\ c_{\alpha }^{ik}x_{u^{k}}+\sum_{\beta =1}^{n}\
T_{i}^{\alpha \beta }N_{\beta }, \label{2.6}$$where$$c_{\alpha }^{ik}=\sum_{j=1}^{2}\ c_{ij}^{\alpha }g^{jk};\text{ }c_{\alpha
}^{ik}=c_{\alpha }^{ki}, \label{2.7}$$are the Weingarten forms of M with respect to some unit normal vector $N_{\alpha }$ and $$T_{i}^{\alpha \beta }=\left \langle (N_{\alpha })_{u^{i}},N_{\beta }\right
\rangle ;T_{i}^{\alpha \beta }=-T_{i}^{\beta \alpha }\text{, }i=1,2,
\label{2.8}$$are the torsion coefficients with $\alpha ,\beta =1,...,n$ and $$\left( g^{ij}\right) _{i,j=1,2}=\frac{1}{g}\left[
\begin{array}{cc}
g_{22} & -g_{12} \\
-g_{21} & g_{11}\end{array}\right] ,\text{ }g=g_{11}g_{22}-g_{12}^{2}\text{.} \label{2.9}$$A simple calculation shows that$$c_{ij}^{\alpha }=\left \langle x_{u^{i}u^{j}},N_{\alpha }\right \rangle
=-\left \langle x_{u^{i}},\left( N_{\alpha }\right) _{u^{j}}\right \rangle .
\label{2.9*}$$
The Gaussian curvature of the surface $M$ is defined by$$K=\sum_{\alpha =1}^{n}K_{\alpha },\text{ \ \ }K_{\alpha }=\frac{c_{11}^{\alpha }c_{22}^{\alpha }-(c_{12}^{\alpha })^{2}}{g}. \label{2.10}$$where $K_{\alpha }$ is the $\alpha ^{th}$ Gaussian curvature of the surface $M$. The Gaussian curvature vanishes identically for so-called flat surface. Observe that $$\ K_{\alpha }=c_{\alpha }^{11}c_{\alpha }^{22}-(c_{\alpha }^{12})^{2}.
\label{2.11}$$
The mean curvature vector field $\overrightarrow{H}$ of the surface $M$ is defined by$$\overrightarrow{H}=\sum_{\alpha =1}^{n}H_{\alpha }N_{\alpha }, \label{2.12}$$where $$H_{\alpha }=\frac{1}{2}\sum \limits_{i,j=1}^{2}\text{\ }g^{ij}c_{ij}^{\alpha }=\frac{g_{22}c_{11}^{\alpha }+g_{11}c_{22}^{\alpha
}-2g_{12}c_{12}^{\alpha }}{2g}, \label{2.13}$$is the $\alpha ^{th}$** **mean curvature of the surface $M$ with respect to the unit normal vector $N_{\alpha }.$ The mean curvature $H$ of $M $ is defined by $H=\left \Vert \overrightarrow{H}\right \Vert .~$The mean curvature (vector) vanishes identically for so-called minimal surface. Observe that $$H_{\alpha }=\frac{c_{\alpha }^{11}+c_{\alpha }^{22}}{2}. \label{2.14}$$
The curvature tensor of the normal bundle $NM$ of the surface $M$ is defined by$$\begin{array}{l}
S_{ij}^{\alpha \beta }=\left( T_{i}^{\alpha \beta }\right)
_{u^{j}}-\left( T_{j}^{\alpha \beta }\right)
_{u^{i}}+\sum\limits_{\sigma =1}^{n}\left( T_{i}^{\alpha \sigma
}T_{j}^{\sigma \beta }-T_{j}^{\alpha \sigma
}T_{i}^{\sigma \beta }\right) , \\
=\sum\limits_{m,n=1}^{2}\left( c_{1m}^{\alpha }c_{n2}^{\beta
}-c_{2m}^{\alpha }c_{n1}^{\beta }\right) \text{\ }g^{mn};1\leq
\alpha ,\beta
\leq n.\end{array}
\label{2.15*}$$The equality $$S_{N}^{\alpha \beta }=\frac{1}{\sqrt{g}}S_{12}^{\alpha \beta }, \label{2.16}$$is called the normal sectional curvature with respect to the plane $\Pi
=span\left\{ x_{u},x_{v}\right\} $. For the case $n=2$ the scalar curvature of its normal bundle is defined as$$K_{N}=S_{N}^{12}=\frac{1}{\sqrt{g}}S_{12}^{12}. \label{2.17}$$which is also called normal curvature of the surface $M$ in $\mathbb{E}^{4}.$ Observe that$$K_{N}=\frac{1}{\sqrt{g}}\left( \left( T_{2}^{12}\right) _{u}-\left(
T_{1}^{12}\right) _{v}\right) . \label{2.18}$$
**Known Results**
=================
Let M be a local surface in $\mathbb{E}^{n+2}$ given with the surface patch $x(u,v)$ : $(u,v)\in D\subset \mathbb{E}^{2}.$ The mean curvature vector $\overrightarrow{H}$ is called parallel in the normal bundle if and only if $$\left( H_{\alpha }\right) _{u}^{\bot }=0,\left( H_{\alpha }\right)
_{v}^{\bot }=0, \label{4.1}$$or equivalently$$\left( H_{\alpha }\right) _{u^{i}}=\sum \limits_{\beta
=1}^{n}H_{\beta }T_{i}^{\alpha \beta }. \label{4.2}$$for all $i=1,2$, $\alpha =1,...,n$ with respect to an arbitrary orthonormal frame $N_{1},...N_{n}~$[@Fr].
[@Fr] The mean curvature vector $\overrightarrow{H}$ is called parallel in the normal bundle if and only if the squared mean curvature $\left \Vert
\overrightarrow{H}\right \Vert ^{2}$of $M$ is a constant function.
Suppose $H_{\alpha }\neq 0$ and the mean curvature vector $\overrightarrow{H}
$ is parallel in the normal bundle. Multiplying the first order differential equation (\[4.2\]) by $H_{\alpha }$ gives $$H_{\alpha }\left( H_{\alpha }\right) _{u^{i}}=\sum \limits_{\beta
=1}^{n}H_{\alpha }H_{\beta }T_{i}^{\alpha \beta }=0,$$for all $\alpha =1,...,n.$ Summing over $\alpha $ shows$$\begin{aligned}
\frac{1}{2}\frac{\partial }{\partial u^{i}}\left \Vert \overrightarrow{H}\right \Vert ^{2} &=&\sum \limits_{\alpha =1}^{n}H_{\alpha }\left(
H_{\alpha }\right) _{u^{i}} \\
&=&\sum \limits_{\alpha =1}^{n}\sum \limits_{\beta =1}^{n}H_{\alpha
}H_{\beta }T_{i}^{\alpha \beta }=0,\end{aligned}$$where the right hand side vanishes automatically due to the skew-symmetric of the torsion coefficients [@Fr]. Thus, one get $$\left \Vert \overrightarrow{H}\right \Vert ^{2}=\sum \limits_{\alpha
=1}^{n}H_{\alpha }^{2}=const.$$The converse statement of the theorem is trivial.
A local surface of $\mathbb{E}^{n+2}$ is said to have flat normal bundle if and only if the orthonormal frame $N_{1},...N_{n}$ of $M$ is of torsion free.
The existence of flat normal bundle of $M$ is equivalent to say that normal curvature $K_{N}$ of $M$ vanishes identically.
The following classification result due to Chen from [@Ch].
Let $M$ be an immersed surface in $\mathbb{E}^{n+2}$. If $\overrightarrow{H}\neq 0$ is parallel in the normal bundle then either $M$ is a minimal surface of a hypersphere of $\mathbb{E}^{n+2}$, or it has flat normal bundle.
**Normal Transport Surfaces in** $\mathbb{E}^{4}$
=================================================
Let $M$ and $\widetilde{M}$ be two smooth surfaces in Euclidean $4$-space $\mathbb{E}^{4}$ and let $\varphi :M\rightarrow \widetilde{M}$ be a diffeomorphism. Then the surface $\widetilde{M}$ enveloping family of normal $2$-planes to $M$ is the normal transport of $M$ in $\mathbb{E}^{4}$ [Fr]{}. Furthermore, let $\overrightarrow{x}$ be a position (radius) vector of $p\in M,$ and $\widetilde{x}$ be the position (radius) vector of the point $\varphi (p)\in \widetilde{M}.$ Then the mapping $\varphi :M\rightarrow
\widetilde{M}$ has the form$$\widetilde{x}=x+\overrightarrow{w},~~\overrightarrow{w}\in T_{p}^{\perp }M.$$where, $\overrightarrow{p\varphi (p)}=\overrightarrow{w}(p),~\overrightarrow{w}(p)\in T_{p}^{\perp }M$ is the normal vector to $M$. For the case$$\overrightarrow{w}(p)=\sum \limits_{i=1}^{2}f_{i}(u,v)N_{i}(u,v),$$the normal transport surface $\widetilde{M}$ of $M$ given by $$\widetilde{M}:\widetilde{x}(u,v)=x(u,v)+\sum\limits_{i=1}^{2}f_{i}(u,v)N_{i}(u,v), \label{5.1}$$where $f_{i}$ $(i=1,2)$ are offset functions and $N_{1},N_{2}\in
T_{p}^{\perp }M$ [@Fr].
The tangent space to $\widetilde{M}$ at an arbitrary point $p=\widetilde{x}(u,v)$ of $\widetilde{M}$ is spanned by$$\begin{array}{l}
\vspace{2mm}\widetilde{x}_{u}=x_{u}+f_{1}\left( N_{1}\right)
_{u}+f_{2}\left( N_{2}\right) _{u}+(f_{1})_{u}N_{1}+(f_{2})_{u}N_{2}, \\
\widetilde{x}_{v}=x_{v}+f_{1}\left( N_{1}\right) _{v}+f_{2}\left(
N_{2}\right) _{v}+(f_{1})_{v}N_{1}+(f_{2})_{v}N_{2}.\end{array}
\label{5.2}$$Further, using the Weingarten equation (\[2.6\]) we get$$\begin{array}{l}
\left( N_{1}\right) _{u}=-\left( c_{1}^{11}x_{u}+c_{1}^{12}x_{v}\right)
+T_{1}^{12}N_{2} \\
\left( N_{2}\right) _{u}=-\left( c_{2}^{11}x_{u}+c_{2}^{12}x_{v}\right)
-T_{1}^{12}N_{1} \\
\left( N_{1}\right) _{v}=-\left( c_{1}^{21}x_{u}+c_{1}^{22}x_{v}\right)
+T_{2}^{12}N_{2} \\
\left( N_{2}\right) _{v}=-\left( c_{2}^{21}x_{u}+c_{2}^{22}x_{v}\right)
-T_{2}^{12}N_{2}.\end{array}
\label{5.3}$$So, substituting (\[5.3\]) into (\[5.2\]) we get$$\begin{array}{l}
\widetilde{x}_{u}=\left( 1-f_{1}c_{1}^{11}-f_{2}c_{2}^{11}\right)
x_{u}-\left( f_{1}c_{1}^{12}+f_{2}c_{2}^{12}\right) x_{v} \\
\text{ \ \ \ \ \ }+\left( (f_{1})_{u}-f_{2}T_{1}^{12}\right) N_{1}+\left(
(f_{2})_{u}+f_{1}T_{1}^{12}\right) N_{2},\end{array}
\label{5.4}$$$$\begin{array}{l}
\widetilde{x}_{v}=-\left( f_{1}c_{1}^{21}+f_{2}c_{2}^{21}\right)
x_{u}+\left( 1-f_{1}c_{1}^{22}-f_{2}c_{2}^{22}\right) x_{v} \\
\text{ \ \ \ \ \ }+\left( (f_{1})_{v}-f_{2}T_{2}^{12}\right) N_{1}+\left(
(f_{2})_{v}+f_{1}T_{2}^{12}\right) N_{2}.\end{array}
\label{5.4*}$$
The normal transport surfaces in $3$-dimensional Euclidean space $\mathbb{E}^{3}$ have the parametrization $$\widetilde{M}:\widetilde{x}(u,v)=x(u,v)+F(u,v)~N(u,v),$$where $N(u,v)\in T_{p}^{\perp }M$ and $F$ is a real valued function in the parameter $(u,v)$. In fact, these surfaces are known as surface offsets in $\mathbb{E}^{3}~$and $F$ is its offset function [@H].
If the offset function depends on the principal curvatures $k_{1}$ and $k_{2} $ of $M$ then one can choose the variable offset function as;
1. $F=k_{1}k_{2},~$Gaussian curvature,
2. $F=\frac{1}{2}(k_{1}+k_{2}),$ mean curvature,
3. $F=k_{1}^{2}+k_{2}^{2},$ energy functional,
4. $F=\left \vert k_{1}\right \vert +\left \vert k_{2}\right \vert ,$ absolute functional,
5. $F=k_{i},1\leq i\leq 2,$ principal curvature,
6. $F=\frac{1}{k_{i}},$ focal points,
7. $F=const.,$ parallel surface.
The different offset functions listed above can now be used the interrogate and visualize of the surfaces (see [@OA]). Using different offset functions, one can construct a one-parameter family of various normal transport surfaces from a given surface of $4$-dimensional Euclidean space $\mathbb{E}^{4}$.
In the following definition we construct some special normal transport surfaces in $\mathbb{E}^{4}$ which are the generalization of some generalized focal surfaces give before$.$
i\) The normal transport surface $\widetilde{M}_{H}$ given with the parametrization $$\widetilde{M}_{H}:\widetilde{x}(u,v)=x(u,v)+H_{1}(u,v)~N_{1}(u,v)+H_{2}(u,v)~N_{2}(u,v), \label{5.8}$$is called normal transport surface of $H$-type, where $f_{\alpha
}(u,v)=H_{\alpha }$ $(\alpha =1,2)$ are the offset functions.
ii\) The normal transport surface $\widetilde{M}_{K}$ given with the parametrization $$\widetilde{M}_{K}:\widetilde{x}(u,v)=x(u,v)+K_{1}(u,v)~N_{1}(u,v)+K_{2}(u,v)~N_{2}(u,v), \label{5.9}$$is called normal transport surface of $K$-type, where $f_{\alpha
}(u,v)=K_{\alpha }$ $(\alpha =1,2)$ are the offset functions.
Parallel Surfaces in $\mathbb{E}^{4}$
-------------------------------------
The normal transport surface $\widetilde{M}$ of $M$ is called parallel surface of $M$ in $\mathbb{E}^{4}$ if the equality $$\left \langle \widetilde{x}_{u_{i}},N_{\alpha }\right \rangle =0,\text{ }1\leq i,\alpha \leq 2, \label{5.5}$$holds for all $N_{\alpha }\in T_{p}^{\perp }M$ [@Fr].
If the functions $f_{1}$ and $f_{2}$ are constant then it is easy to see that $\widetilde{M}$ is a parallel surface of $M$ and vice versa, at least if the surfaces are immersed in $\mathbb{E}^{3}.$ The parallelity of $\widetilde{M}$ in $\mathbb{E}^{4}$ depends on the normal curvature $K_{N}$ of $M$ [@Fr]. Parallel type surface are widely used in geometry and mathematical physics. We want to refer the reader to da Costa [@Co] for an application in quantum mechanics in curved spaces.
Let $\widetilde{M}$ be a parallel surface of $M$ in $\mathbb{E}^{4}$. Then by use of (\[5.4\]) and (\[5.4\*\]) with (\[5.5\]) one can get$$\begin{aligned}
0 &=&\left \langle \widetilde{x}_{u},N_{1}\right \rangle
=(f_{1})_{u}-f_{2}T_{1}^{12}, \notag \\
0 &=&\left \langle \widetilde{x}_{v},N_{1}\right \rangle
=(f_{1})_{v}-f_{2}T_{2}^{12}, \label{5.6} \\
0 &=&\left \langle \widetilde{x}_{u},N_{2}\right \rangle
=(f_{2})_{u}+f_{1}T_{1}^{12}, \notag \\
0 &=&\left \langle \widetilde{x}_{v},N_{2}\right \rangle
=(f_{2})_{v}+f_{1}T_{2}^{12}. \notag\end{aligned}$$Differentiating the first two equations and making use of the other equations shows us $$\begin{aligned}
(f_{1})_{uv}+f_{1}T_{2}^{12}T_{1}^{12}-f_{2}\left( T_{1}^{12}\right) _{v}
&=&0, \label{5.7} \\
(f_{1})_{vu}+f_{1}T_{1}^{12}T_{2}^{12}-f_{2}\left( T_{2}^{12}\right) _{u}
&=&0. \notag\end{aligned}$$Thus a computation of the left hand sides of (\[5.7\]) brings$$-f_{2}\left \{ \left( T_{1}^{12}\right) _{v}-\left( T_{2}^{12}\right)
_{u}\right \} =0.$$So, by the use of (\[2.18\]) we can conclude that the normal curvature $K_{N}$ of $M$ vanishes identically [@Fr]. Consequently, we obtain the following result of S. Fröhlich.
[@Fr] The normal transport surface $\widetilde{M}$ of $M$ is parallel if and only if $M$ has flat normal bundle.
We obtain the following result.
The normal transport surface $\widetilde{M}$ of $M$ is parallel if and only if the squared sum of the offset functions is constant, i.e., $\sum \limits_{i=1}^{2}f_{i}^{2}(u,v)=const.$
From the expressions in (\[5.6\]) we get$$\begin{array}{l}
(f_{1})_{u}f_{1}+(f_{2})_{u}f_{2}=0, \\
(f_{1})_{v}f_{1}+(f_{2})_{v}f_{2}=0.\end{array}
\label{5.7.*}$$which completes the proof.
We give the following example.
The normal transport surface $\widetilde{M}$ of $M$ is given with the patch$$\widetilde{X}(u,v)=X(u,v)+r\cos u~N_{1}(u,v)+r\sin u~N_{2}(u,v),$$is a parallel surface of $M$ in $\mathbb{E}^{4}.$
Let $M$ be a non-minimal local surface in $\mathbb{E}^{4}$ and $\widetilde{M}_{H}$ its normal transport surface. If $\widetilde{M}_{H}$ is a parallel surface of $M$ in $\mathbb{E}^{4}$ then by Theorem 3 $M$ has vanishing normal curvature. Furthermore, by the use of (\[5.7.\*\]) we get $$\begin{aligned}
\left( H_{1}\right) _{u}H_{1}+\left( H_{2}\right) _{u}H_{2} &=&0, \\
\left( H_{1}\right) _{v}H_{1}+\left( H_{2}\right) _{v}H_{2} &=&0.\end{aligned}$$Thus, $\left \Vert \overrightarrow{H}\right \Vert ^{2}=\sum
\limits_{\alpha =1}^{2}H_{\alpha }^{2}$ is a constant function. So, we conclude that the mean curvature vector $\overrightarrow{H}$ of $M$ is parallel in the normal bundle. Thus, we have proved the following result.
Let $M$ be a non-minimal local surface in $\mathbb{E}^{4}$. Then the normal transport surface $\widetilde{M}_{H}$ of $M$ in $\mathbb{E}^{4}$ is parallel if and only if the mean curvature vector $\overrightarrow{H}$ of $M$ is parallel in the normal bundle.
Let $M$ be a non-flat local surface in $\mathbb{E}^{4}$ and $\widetilde{M}_{K}$ its normal transport surface. If $\widetilde{M}_{K}$ is a parallel surface of $M$ in $\mathbb{E}^{4}$ then by Theorem 3 $\widetilde{M}_{K}$ has vanishing normal curvature. Furthermore, by the use of (\[5.7.\*\]) we get $$\begin{aligned}
\left( K_{1}\right) _{u}K_{1}+\left( K_{2}\right) _{u}K_{2} &=&0, \\
\left( K_{1}\right) _{v}K_{1}+\left( K_{2}\right) _{v}K_{2} &=&0.\end{aligned}$$Thus, we conclude that $K=\sum \limits_{\alpha =1}^{2}K_{\alpha
}^{2}$ is a constant function, i.e., $M$ has constant Gauss curvature. Thus, we have proved the following result.
Let $M$ be a non-flat local surface in $\mathbb{E}^{4}$. Then the normal transport surface $\widetilde{M}_{K}$ of $M$ in $\mathbb{E}^{4}$ is parallel if and only if the Gaussian curvature of $M$ is a non-zero constant.
Evolute Surfaces in $\mathbb{E}^{4}$
------------------------------------
The normal transport surface $\widetilde{M}$ of $M$ is called evolute surface of $M$ in $\mathbb{E}^{4}$ if the equality $$\left \langle \widetilde{x}_{u_{i}},x_{u_{j}}\right \rangle =0,\text{ }1\leq
i,j\leq 2, \label{6.1}$$holds for all $x_{u_{j}}\in T_{p}M$ .
Observe that, The tangent $2$-planes at a point $p\in M$ and at the corresponding point $\varphi (p)\in \widetilde{M}$ are mutually orthogonal, and the vector $\overrightarrow{p\varphi
(p)}=~\overrightarrow{w}(p)$, $\overrightarrow{w}(p)\in T_{p}^{\perp }M$ is the normal vector to $M$ [Ce]{}.
Let $\widetilde{M}$ be a evolute surface of $M$ in $\mathbb{E}^{4}$. Then by use of (\[5.4\]) with (\[6.1\]) we can get$$\begin{aligned}
0 &=&\left \langle \widetilde{x}_{u},x_{u}\right \rangle =\left(
1-f_{1}c_{1}^{11}-f_{2}c_{2}^{11}\right) g_{11}-\left(
f_{1}c_{1}^{12}+f_{2}c_{2}^{12}\right) g_{21}, \notag \\
0 &=&\left \langle \widetilde{x}_{u},x_{v}\right \rangle =\left(
1-f_{1}c_{1}^{11}-f_{2}c_{2}^{11}\right) g_{12}-\left(
f_{1}c_{1}^{12}+f_{2}c_{2}^{12}\right) g_{22}, \label{6.2} \\
0 &=&\left \langle \widetilde{x}_{v},x_{u}\right \rangle =-\left(
f_{1}c_{1}^{12}+f_{2}c_{2}^{12}\right) g_{11}+\left(
1-f_{1}c_{1}^{22}-f_{2}c_{2}^{22}\right) g_{21}, \notag \\
0 &=&\left \langle \widetilde{x}_{v},x_{v}\right \rangle =-\left(
f_{1}c_{1}^{12}+f_{2}c_{2}^{12}\right) g_{12}+\left(
1-f_{1}c_{1}^{22}-f_{2}c_{2}^{22}\right) g_{22}. \notag\end{aligned}$$From now on we assume that the surface patch $x(u,v)$ satisfies the metric condition $g_{12}=0.$ So the equations in (\[6.2\]) turn into$$\begin{aligned}
f_{1}c_{1}^{11}+f_{2}c_{2}^{11} &=&1, \notag \\
f_{1}c_{1}^{22}+f_{2}c_{2}^{22} &=&1, \label{6.3} \\
f_{1}c_{1}^{12}+f_{2}c_{2}^{12} &=&0. \notag\end{aligned}$$Consequently by the use of (\[6.3\]) with (\[2.14\]) we get $$f_{1}H_{1}+f_{2}H_{2}=1. \label{6.4}$$So, we obtain the following result.
Let $M$ be local surface in $\mathbb{E}^{4}$ with $g_{12}=0$. Then the normal transport surface $\widetilde{M}$ in $\mathbb{E}^{4}$ is evolute surface of $M~$if and only if the first and second mean curvatures $H_{1},~H_{2}~$satisfies the condition (\[6.4\]).
Let $M$ be local surface in $\mathbb{E}^{4}$ with $g_{12}=0$. Then the normal transport surface $\widetilde{M}_{H}$ in $\mathbb{E}^{4}$ is evolute surface of $M~$if and only if the mean curvature of $M$ is equal to one.
In [@Ce] M. A. Cheshkova gave the following results.
[@Ce] Let $M$ be local surface in $\mathbb{E}^{4}$. If the normal transport surface $\widetilde{M}$ in $\mathbb{E}^{4}$ is evolute surface of $M~$then $M$ has flat normal bundle.
[@Ce] The minimal surfaces have no evolutes.
Let $M$ is a translation surface $x(u,v)=\alpha (u)+\beta (v)$ in $\mathbb{E}^{4}$ , then the translation curves $\alpha (u)=\left( \alpha _{1}(u),\alpha
_{2}(u),0,0\right) $ and $\beta (v)=\left( 0,0,\beta _{1}(v),\beta
_{2}(v)\right) $ are plane curves of mutually orthogonal $2$-planes. The surface $\widetilde{M}$ is a translation surface, and its translation curves $\widetilde{\alpha }(u)$, $\widetilde{\beta }(u)$ are the evolutes of the curves $\alpha (u)$, $\beta (u)$. If $u,v,\kappa _{\alpha },\kappa _{\beta }$ and $\{t_{\alpha },n_{\alpha }\},\{t_{\beta },n_{\beta }\}$ are the arc length, the curvature, and the Frenet frame of the curves $\alpha (u)$ and $\beta (v)$, correspondingly, then $$\begin{aligned}
\widetilde{x}(u,v) &=&\alpha (u)+\frac{1}{\kappa _{\alpha }}n_{\alpha
}(u)+\beta (v)+\frac{1}{\kappa _{\beta }}n_{\beta }(v) \\
&=&\alpha (u)+\beta (v)+\frac{1}{\kappa _{\alpha }}n_{\alpha }(u)+\frac{1}{\kappa _{\beta }}n_{\beta }(v) \\
&=&x(u,v)+\frac{1}{\kappa _{\alpha }}n_{\alpha }(u)+\frac{1}{\kappa _{\beta }}n_{\beta }(v).\end{aligned}$$The tangent space to $\widetilde{M}$ at an arbitrary point $p=\widetilde{x}(u,v)$ of $\widetilde{M}$ is spanned by$$\begin{array}{l}
\vspace{2mm}\widetilde{x}_{u}=\left( \frac{1}{\kappa _{\alpha }}\right)
^{\prime }n_{\alpha }(u), \\
\widetilde{x}_{v}=\left( \frac{1}{\kappa _{\beta }}\right) ^{\prime
}n_{\beta }(v).\end{array}$$Consequently, the normal transport surface $\widetilde{M}$ of $M$ satisfies the equality $$\left \langle \widetilde{x}_{u_{i}},x_{u_{j}}\right \rangle =0.$$Hence, $\widetilde{M}$ is the evolute of $M$ [@Ce].
**An Application**
==================
Rotation surfaces were studied in [@Vr] by Vranceanu as surfaces in $\mathbb{E}^{4}$ which are defined by the following parametrization $$M:x(u,v)=(r(v)\cos v\cos u,r(v)\cos v\sin u,r(v)\sin v\cos u,r(v)\sin v\sin
u) \label{7.1}$$where $r(v)$ is a real valued non-zero function.
We have the following result.
Let $\widetilde{M}$ be a normal transport surface of the Vranceanu surface $M $ given with the parametrization (\[5.1\]). If $\ \widetilde{M}$ is an evolute surface of $M$ in $\mathbb{E}^{4}$ then $$\widetilde{M}:\widetilde{x}(u,v)=\lambda \mu e^{\mu v}(-\sin v\cos u,-\sin
v\sin u,\cos v\cos u,\cos v\sin u), \label{7.2}$$where $\lambda $ and $\mu $ are non zero constants.
Let $M$ be a Vranceanu surfaces given with the parametrization (\[7.1\]). We choose a moving frame $\{X_{u},X_{v},N_{1},N_{2}\}$ such that $X_{u},X_{v} $ are tangent to $M$ and $N_{1},N_{2}$ normal to $M$ as given the following (see, [@Yo1]): $$\begin{aligned}
x_{u} &=&r(-\cos v\sin u,\cos v\cos u,-\sin v\sin u,\sin v\cos u), \\
x_{v} &=&(B(v)\cos u,B(v)\sin u,C(v)\cos u,C(v)\sin u), \\
N_{1} &=&\frac{1}{A}(-C(v)\cos u,-C(v)\sin u,B(v)\cos u,B(v)\sin u), \\
N_{2} &=&(-\sin v\sin u,\sin v\cos u,\cos v\sin u,-\cos v\cos u),\end{aligned}$$where $$\begin{aligned}
A(v) &=&\sqrt{r^{2}(v)+(r^{\prime }(v))^{2}},\text{ }\,B(v)=r^{\prime
}(v)\cos v-r(v)\sin v, \\
C(v) &=&r^{\prime }(v)\sin v+r(v)\cos v.\end{aligned}$$
Suppose that $\widetilde{M}$ is the normal transport surface of the Vranceanu surface $M$ in $\mathbb{E}^{4}$ then we have $$\begin{aligned}
\left \langle \widetilde{x}_{u},x_{u}\right \rangle &=&r^{2}(v)-f_{1}\left(
\frac{r^{2}(v)}{\sqrt{r^{2}(v)+(r^{\prime }(v))^{2}}}\right) , \notag \\
\left \langle \widetilde{x}_{u},x_{v}\right \rangle &=&f_{2}r(v),
\label{7.3} \\
\left \langle \widetilde{x}_{v},x_{u}\right \rangle &=&f_{2}r(v), \notag \\
\left \langle \widetilde{x}_{v},x_{v}\right \rangle &=&r^{2}(v)+(r^{\prime
}(v))^{2}+f_{1}\left( \frac{r(v)r^{\prime \prime }(v)-2(r^{\prime
}(v))^{2}-r^{2}(v)}{\sqrt{r^{2}(v)+(r^{\prime }(v))^{2}}}\right) . \notag\end{aligned}$$
Furthermore, if $\widetilde{M}$ is an evolute surface of the Vranceanu surface $M$ in $\mathbb{E}^{4}$ then using (\[6.1\]) with (\[7.3\]) we obtain$$\begin{array}{l}
\vspace{2mm}f_{2}=0, \\
f_{1}=\sqrt{r^{2}(v)+(r^{\prime }(v))^{2}}.\end{array}
\label{7.4}$$Moreover, from the first and fourth equations of (\[7.3\]) one can get $$r(v)r^{\prime \prime }(v)-(r^{\prime }(v))^{2}=0$$which has a non-trivial solution $$r(v)=\lambda e^{\mu v} \label{7.5}$$
As a consequence of (\[7.4\]) with (\[7.5\]) we get the desired result.
The Vranceanu surface given with $r(v)=\lambda e^{\mu v}$ is a flat surface with vanishing normal curvature [@ABBKMO].
[99]{} K. Arslan, B.K. Bayram, B. Bulca, Y.H. Kim, C. Murathan and G. Öztürk, *Vranceanu surfaces with pointwise 1-type gauss map*, Indian J. Pura Appl. Math. 42(2011), 41-51.
B.-Y. Chen, *Surfaces with parallel mean curvature vector*, Bull. Amer. Math. Soc. 78(1972), 709–710.
M. A. Cheshkova, *Evolute surfaces in* $\mathbb{E}^{4},$ Mathematical Notes, Vol. 70(2001), 870–872.
R.C.T. da Costa, *Constraints in quantum mechanics*, Physical Review A, 25(1982), 2893–2900.
R. T. Farouki, *Exact Offset Procedures for Simple Solids*, Computer Aided Geometric Design, 2(1985), 257-279.
M. Forsyth. *Shelling and offsetting bodies*, Proceedings of the third ACM symposium on Solid modeling and applications, Salt Lake City, Utah, United States, 373-381, May 17-19, 1995.
S. Fröhlich, *Surfaces-in-Euclidean-Space,* www.scribd.com/doc, 2013.
S. Hahmann. * Visualization techniques for surface analysis*, in C. Bajaj (ed.): Advanced techniques, John Viley, 1999.
H. Hagen and S. Hahmann. *Generalized Focal Surfaces: A New Method for Surface Interrogation,* Proceeding, Visualization’92, Boston-1992, 70-16.
H. Hagen and S. Hahmann. *Visualization of curvature behavior of free-form curves and surfaces*, CAD 27(1995), 545-552.
H. Hagen, H. Pottmann. *A* *Divivier, Visualization Functions on Surface*, Journal of Visualization and Animation, 2(1991), 52-58.
L.N. Krivonosov. *Parallel and Normal Correspondence of two-dimensional Surfaces in the four-dimensional Euclidean Space* $\mathbb{E}^{4}$, Amer. Math. Soc. Transl. 92(1970), 139-150.
T. Maekawa. *An Overview of Offset Curves and Surfaces*, Computer Aided Design Vol. 31(1999), 165-173.
B. Özdemir and K. Arslan. *On generalized focal surfaces in* $\mathbb{E}^{3}$, Rev. Bull. Calcutta Math. Soc. 16 (2008), 23–32.
B. Pham. *Offset curves and surfaces: a brief survey*, Computer-Aided Design 24(1992), 223–229.
G. Vranceanu, *Surfaces de Rotation Dans* $\mathbb{E}^{4}.$ Rev. Roumaine Math. Pures Appl. 22(1977), 857-862.
D.W. Yoon, *Rotation Surfaces with Finite Type Gauss Map in* $\mathbb{E}^{4}$. Indian J. pura appl.Math. 32(2001), no.12, 1803-1808.
----------------------------------
Kadri Arslan & Betül Bulca
Department of Mathematics
Uludağ University
16059 Bursa, TURKEY
E-mails: arslan@uludag.edu.tr,
bbulca@uludag.edu.tr
----------------------------------
----------------------------------
Bengü Kiliç Bayram
Department of Mathematics
Balikesir University
Balikesir, TURKEY
E-mail: benguk@balikesir.edu.tr,
----------------------------------
--------------------------------
Günay Öztürk
Department of Mathematics
Kocaeli University
Kocaeli, TURKEY
E-mail: ogunay@kocaeli.edu.tr,
--------------------------------
[^1]: 2000 Mathematics Subject Classifications: 53A04, 53C42
Key words and phrases: Translation surface, parallel surface, evolute surface, focal surface
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---
abstract: 'We present an axiomatic approach to earthquake forecasting in terms of multi-component random fields on a lattice. This approach provides a method for constructing point estimates and confidence intervals for conditional probabilities of strong earthquakes under conditions on the levels of precursors. Also, it provides an approach for setting multilevel alarm system and hypothesis testing for binary alarms. We use a method of comparison for different earthquake forecasts in terms of the increase of Shannon information. ’Forecasting’ and ’prediction’ of earthquakes are equivalent in this approach.'
author:
- 'V.Gertsik [^1], M.Kelbert[^2], A.Krichevets[^3]'
title: 'Earthquake forecasting: Statistics and Information'
---
The methodology of selecting and processing of relevant information about the future occurrence of potentially damaging earthquakes has reached a reasonable level of maturity over the recent years. However, the problem as a whole still lacks a comprehensive and generally accepted solution. Further efforts for optimization of the methodology of forecasting would be productive and well-justified.
A comprehensive review of the modern earthquake forecasting state of knowledge and guidelines for utilization can be found in [\[]{}Jordan et all., 2011[\]]{}. Note that all methods of evaluating the probabilities of earthquakes are based on a combination of geophysical, geological and probabilistic models and considerations. Even the best and very detailed models used in practice are in fact only ’caricatures’ of immensely complicated real processes.
A mathematical toolkit for earthquake forecasting is well presented in the paper [\[]{}Harte and Vere-Jones, 2005[\]]{}. This work is based on the modeling of earthquake sequences in terms of the marked point processes. However, the mathematical technique used is quite sophisticated and does not provide direct practical tools to investigate the relations of the structure of temporal-spatial random fields of precursors to the appearance of strong earthquakes.
The use of the multicomponent lattice models (instead of marked point processes) gives a different/novel way of investigating these relations in a more elementary way. Discretization of space and time allows us to separate the problem in question into two separate tasks. The first task is the selection of relevant precursors, i.e., observable and theoretically explained physical and geological facts which are casually related to a high probability of strong earthquakes. Particularly, this task involves the development of models of seismic events and computing probabilities of strong earthquakes in the framework of these models. Such probabilities are used as precursors in the second task.
The second task is the development of methodology of working with these precursors in order to extract the maximum information about the probabilities of strong earthquakes. This is the main topic of this paper.
Our approach allows us to obtain the following results:
$\bullet$ Estimates of probabilities of strong earthquakes for given values of precursors are calculated in terms of the frequencies of historic data.
$\bullet$ Confidence intervals are also constructed to provide reasonable bounds of precision for point estimates.
$\bullet$ Methods of predictions (i.e., binary alarm announcement [\[]{}Keilis-Borok, 1996[\]]{}, [\[]{}Keilis-Borok, Kossobokov, 1990[\]]{}, [\[]{}Holiday et all., 2007[\]]{}) and forecasting (i.e., calculating probabilities of earthquakes [\[]{}Jordan et all., 2011[\]]{}, [\[]{}Kagan and Jackson, 2000[\]]{}, [\[]{}Harte and Vere-Jones, 2005[\]]{}, [\[]{}WGNCEP[\]]{}) are equivalent in the following sense: the setting of some threshold for probability of earthquakes allows to update the alarm level. On the other hand, the knowledge of the alarm domain based on historical data allows us to evaluate the probabilities of earthquakes. In a sense, the prediction is equivalent to hypothesis testing as well, see Section 11.
$\bullet$ In our scheme we propose a scalar statistic which is the ratio of actual increment of information to the maximal possible increment of information. This statistic allows us to linearly order all possible forecasting algorithms. Nowadays the final judgement about the quality of earthquake forecasting algorithms is left to experts. This arrangement puts the problem outside the scope of natural sciences which are trying to avoid subjective judgements.
The foundation of our proposed scheme is the assumption that the seismic process is random and cannot be described by a purely deterministic model. Indeed, if the seismic process is deterministic then the inaccuracy of the forecast could be explained by the non-completeness of our knowledge about the seismic events and non-precision of the available information. This may explain, at least in principle, attacks from the authorities addressed to geophysicists who failed to predict a damaging earthquake. However, these attacks have no grounds if one accepts that the seismic process is random. At the end of the last century (February-April 1999) a group of leading seismologists organized a debate via the web to form a collective opinion of the scientific community on the topic: ’Is the reliable prediction of individual earthquakes a realistic scientic goal?’ (see http://www.nature.com/nature/debates/earthquake/).
Despite a considerable divergence in peripheral issues all experts taking part in the debate agreed on the following main principles:
$\bullet$ the deterministic prediction of an individual earthquake, within sufficiently narrow limits to allow a planned evacuation programme, is an unrealistic goal;
$\bullet$ forecasting of at least some forms of time-dependent seismic hazard can be justified on both physical and observational grounds.
The following facts form the basis of our agreement with this point of view.
The string-block Burridge-Knopov model, generally accepted as a mathematical tool to demonstrate the power-like Gutenberg-Richter relationship between the magnitude and the number of earthquakes, involves the generators of chaotic behaviour or dynamic stochasticity. In fact, the nonlinearity makes the seismic processes stochastic: a small change in the shift force may lead to completely different consequences. If the force is below the threshold of static friction the block is immovable, if the force exceeds this threshold it starts moving, producing an avalanche of unpredictable size.
This mechanism is widespread in the Earth. Suppose that the front propagation of the earthquake approaches a region of enhanced strength of the rocks. The earthquake magnitude depends on whether this region will be destroyed or remains intact. In the first case the front moves further on, in the second case the earthquake remains localized. So, if the strength of the rocks is below the threshold the first scenario prevails, if it is above the threshold the second scenario is adapted. The whole situation is usually labelled as a *butterfly effect*: infinitesimally small changes of strength and stress lead to macroscopic consequences which cannot be predicted because this infinitesimal change is below any precision of the measurement. For these reasons determinism of seismic processes looks more doubtful than stochasticity.
The only comment we would like to contribute to this discussion is that the forecasting algorithms based exclusively on the empirical data without consistent physical models could hardly be effective in practice (see Sections 12, 13 for more details).
In conclusion we discuss the problem of precursor selection and present a theorem by A. Krichevetz stating that using a learning sample for an arbitrary feature selection in pattern recognition is useless in principle.
Finally, note that our approach may be well-applicable for the space-time forecasting of different extremal events outside the scope of earthquake prediction.
Events and precursors on the lattice
====================================
In order to define explicitly estimates of probabilities of strong earthquakes we discretize the two-dimensional physical space and time, i.e., introduce a partition of three-dimensional space-time into rectangular cells with the space partition in the shape of squares and time partition in the shape of intervals. These cells should not intersect to avoid an ambiguity in computing the frequencies for each cell. In fact, the space cells should not be perfect squares because of the curvature of the earth’s surface, but this may be neglected if the region of forecasting is not too large.
So, we obtain a discrete set $\Omega_{K}$ with $N=I\times J\times K$ points which is defined as follows. Let us select a rectangular domain $A$ of the two-dimensional lattice with $I\times J$ points $x=(x_{i},y_{j});\, x_{i}=a\text{\texttimes}i;\, i=1,\ldots,I$ and $y_{j}=a\text{\texttimes}j;\, j=1,\ldots,J,$$a$ is the step of the lattice. A cell in $\Omega_{K}$ takes the shape of parallelepiped of height ${\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\textgreek{D}}\endgroup\else\textgreek{D}\fi}t$ with a square base. Clearly, any point in $\Omega_{K}$ has coordinates $(x_{i},y_{j},t_{k}),\, t_{k}=t+k\Delta t;\, k=0,\ldots,K.$ .
We say that *a seismic event* happens if an earthquake with magnitude greater than some pre-selected threshold $M_{0}$ is registered, and this earthquake is not foreshock or aftershock of another, more powerful earthquake (we put aside a technical problem of identification of foreshocks and aftershocks in the sequence of a seismic event). For any cell in our space-time grid we define *an indicator of an event*, i.e., a binary function $h(i,j,k)$. This function takes the value $1$ if at least one seismic event is registered in a given cell and $0$ otherwise. Suppose that for all points $(x_{i},y_{j},t_{k})$ the value of a vector precursor $\mathbf{\boldsymbol{f}}(i,j,k)=\left\{ f_{q}(i,j,k),\, q=1,...,Q\right\} $ is given. The components of the precursor $f_{q}(i,j,k),\, q=1,...,Q$ are the scalar statistics constructed on the base of our understanding of the phenomena that precede a seismic event.
Note that specifying an alarm domain as a circle with center at a lattice site and radius proportional to the maximal magnitude of the forecasted earthquakes leads to a contradiction. Indeed, suppose we announce an alarm for earthquakes with minimal magnitude $6$ in a domain $A_{6}$. Obviously, the same alarm should be announced in the domain $A_{7}$ as well. By the definition $A_{6}\subset A_{7}$ and we expect an earthquake with magnitude at least $7$ and do not expect an earthquake with magnitude at least $6$ in the domain $A_{7}\setminus A_{6}$. But this is absurd.
Mathematical assumptions
========================
A number of basic assumptions form the foundation of the mathematical tecnique of earthquake forecasting. In the framework of mathematical theory they can be treated as axioms but are, in fact, an idealization and simplification with respect to the description of the real phenomena. Below we summarize the basic assumptions which are routinely used in existing studies of seismicity and algorithms of earthquake forecasting even the authors do not always formulate them explicitly.
We accept the following assumptions or axioms of the mathematical theory:
\(i) The multicomponent random process $\left\{ h(i,j,k),\mathbf{f}(i,j,k)\right\} $, describing the joint evolution of the vector precursors and the indicator of seismic events, is stationary.
This assumption provides an opportunity to investigate the intrinsic relations between the precursors and the seismic events based on the historical data. In other words, the experience obtained by analysing the series of events in the past, is applicable to the future as the properties of the process do not depend on time.
In reality, this assumption holds only approximately and for a restricted time period. Indeed, plate tectonics destroys the stationarity for a number of reasons including some purely geometrical considerations. For instance, the movements of the plates leads to their collisions, their partial destruction and also changes their shapes. Nevertheless, the seismic process can be treated as quasi-stationary one for considerable periods of time. At the time when the system changes one quasi-stationary regime to another (say, nowadays, many researcher speak about the abrupt climate change) the reliability of any prediction including the forecast of seismic events is severely restricted.
\(ii) The multicomponent random process $\left\{ h(i,j,k),\mathbf{f}(i,j,k)\right\} $ is ergodic.
Any quantitative characteristic of seismicity more representative than a registration of an individual event is, in fact, the result of averaging over time. For instance, the Gutenberg-Richter law, applied to a given region relates the magnitude with the average number of earthquakes where the averaging is taken over a specific time interval. In order to associate with the time averaging a proper probabilistic characteristic of the process and make a forecast about the future one naturally needs the assumption of ergodicity. This exactly means that any averaging over time interval $[0,T]$ will converge to the stochastic average when $T\to\infty$. In view of ergodicity one can also construct unbiased and consistent estimates of conditional probabilities of strong earthquakes under conditions that the precursors take values in some intervals. Naturally, these estimates are the frequencies of observed earthquakes, i.e., ratios of the number of cells with seismic events and prescribed values of precursors to the total number of cells with the prescribed values of precursors. (Recall that an unbiased point estimate ${\hat{\theta}}$ of parameter $\theta$ satisfies the condition ${\bf E}{\hat{\theta}}=\theta$, and a consistent estimate converges to the true value $\theta$ when the sample size tends to infinity).
\(iii) Any statement about the value of the indicator of a seismic event $h(i,j,k)$ in the cell $(i,j,k)$ or its probability should be based on the values of the precursor ${\bf f}(i,j,k)$ only.
This assumption means that the precursor in the given cell accumulates all the relevant information about the past and the information about the local properties of the area that may be used for the forecast of the seismic event in this cell. In other words, the best possible precursor is used (which is not always the case in practice). As in the other cases, this assumption is only an approximation to reality, and the quality of a forecast depends on the quality of the selection and accumulation of relevant information in the precursors.
Below we present some corollaries and further specifications.
(iii-a) For any $k$ the random variables $h(i,j,k),\, i=1,...,I,\, j=1,\ldots,J$ are conditionally independent under the condition that the values of any measurable function $u(\mathbf{\boldsymbol{f}}(i,j,k))$ of the precursors ${\bf f}(i,j,k),\, i=1,\ldots,I,\, j=1,\ldots,J$ are fixed..
In practice this assumption means that the forecast for the time $t_{k}=t_{0}+k\Delta t$ cannot be affected by the values related to the future time intervals $(t_{k},t_{k}+\Delta t]$. In reality all of these events may be dependent, but our forecast does not use the information from the future after $t_{k}$.
(iii-b) The conditional distribution of the random variable *$h$* at a given cell depends on the values of the precursors at this cell *$\boldsymbol{\mathbf{f}}$* and is independent of all other variables.
(iii-c) The conditional probabilities $\mathrm{Pr}_{ij}\left\{ \mathit{h\mid u\mathrm{(}\boldsymbol{\mathbf{f}}\mathrm{)}}\right\} $ of the indicator of seismic events $h$ in the cell $(i,j,k)$, under condition $u\mathrm{(}\boldsymbol{\mathbf{f}}\mathrm{)}$ in this cell do not depend on the position of the cell in space (the time index $k$ related to this probablity may be dropped due to the stationarity of the process).
In other words, the rule for computing the conditional probability $\mathrm{Pr}_{ij}\left\{ \mathit{h\mid u\mathrm{(}\boldsymbol{\mathbf{f}}\mathrm{)}}\right\} $ based on the values of precursors is the same for all cells, and the space indices of probability $\mathrm{Pr}$ may be dropped. This condition is widely accepted in constructions of the forecasting algoritms but rarely formulated explicitly. However, the probability of a seismic event depends to a large extent on the local properties of the area. Hence, the quality of the forecasting depends on how adequately these properties are summarized in the precursors. This formalism properly describes the space inhomogenuity of the physical space because the stationary joint distribution of $\mathrm{Pr}_{ij}\left\{ \mathit{h\text{,}f\mathrm{\leq x}}\right\} $ for an arbitrary precursor $f$ depends, in general, on the position of the cell in the domain $A.$ Below we will use the distributions of precursors and indicators of seismic events in domain $A$ that do not depend on the spatial coordinates and have the following form
$$\mathrm{Pr}_{A}\left\{ \mathit{h\text{,}f\mathrm{\leq x}}\right\} =\frac{1}{\mathit{I\cdot J}}\sum_{\left(\mathit{i,j}\right)\in A}\mathbf{\mathrm{Pr}_{\mathrm{\mathit{ij}}}}\left\{ \mathit{h\text{,}f\mathrm{\leq x}}\right\} ,$$
$$\mathrm{\mathit{P}_{\mathit{A}}(\mathit{x})\equiv}\mathbf{\mathrm{Pr}_{\mathit{A}}}\left\{ \mathit{f\mathrm{\leq\mathit{x}}}\right\} =\frac{1}{\mathit{I\cdot J}}\sum_{\left(\mathit{i,j}\right)\in A}\mathbf{\mathrm{Pr}_{\mathrm{\mathit{ij}}}}\left\{ \mathit{f\leq\mathit{x}}\right\} ,$$
$$\mathrm{\mathit{p_{A}\equiv}}\mathrm{Pr}_{A}\left\{ \mathit{h\mathrm{=1}}\right\} =\frac{1}{\mathit{I\cdot J}}\sum_{\left(\mathit{i,j}\right)\in A}\mathbf{\mathrm{Pr}_{\mathrm{\mathit{ij}}}}\left\{ \mathit{h=\mathrm{1}}\right\} ,$$
(iii-d) Note that assumption (iii) implies that the conditional probabilities ${\bf Pr}\left(h\vert u({\bf f})\right)$ are computed via the probabilities $\mathit{\mathbf{\mathrm{P\text{r}_{\mathit{A}}}}}\left\{ \mathit{h\text{,}f\mathrm{\leq x}}\right\} $ only.
The properties listed above are sufficient to obtain the point estimates for the conditional probabilities of seismic events under conditions formulated in terms of the values of precursors. However, additional assumption are required for a testing of the forecasting algorithm:
\(iv) The random variables $\boldsymbol{\mathbf{f}}\mathrm{(}i,j,k),$ are conditionally independent under condition that $h(i,j,k)=1$.
Again, these conditions are not exactly true, however they may be treated as a reasonable approximation to reality. Indeed, if the threshold $M_{0}$ is sufficiently high than the strong earthquakes may be treated as rare events, and the cells where they are observed are far apart with a high probability. Any two events related to cells separated by the time intervals $\Delta t$ are asymptotically independent as $\Delta t\rightarrow\infty$ because the seismic process has decaying correlations (the mixing property in the language of random processes). The loss of dependence (or decaying memory) is related to the physical phemonema such as healing of the defects in the rocks, relaxation of strength due to viscosity, etc. As usual in physical theories, we accept an idealized model of the real phenomena applying this asymptotic property for large but finite intervals between localizations of seismic events.
The independence of strong earthquakes is not a new assumption, in the case of continuous space-time it is equivalent to the assumption that the locations of these events form a Poisson random field.. (Note that the distribution of strong earthquake should be homogeneous in space, because there is no information about the heterogeneity a priori .) The Poisson hypothesis is used in many papers, see, e.g. [\[]{}Harte and Vere-Jones , 2005[\]]{}. It is very natural for the analysis of the «tails» of the Gutenberg-Richter law for large magnitudes [\[]{}Pisarenko et al., 2008[\]]{}. Summing up, the development of the strict mathematical theory of earthquake forecasting does not require any additional assumption except those routinely accepted in the existing algorithms but usually not formulated explicitly.
The standard form for precursors
================================
The correct solution of the forecasting problem given the values of precursors $\mathbf{f}(i,j,k)=\big(f_{1}(i,j,k),\ldots,f_{Q}(i,j,k)\big)$ is provided by the estimate of conditional probability $\text{Pr\ensuremath{\left\{ \mathit{h\mid\mathbf{f}(i,j,k)}\right\} }}$ of the indicator of seismic event in the cell $(i,j,k)$. In practice this solution may be difficult to obtain because the number of events in catalog is not sufficient.
Indeed, the range of value of a scalar precursor is usually divided into a number $M$ of intervals, and only a few events are registered for any such interval. For a $Q$-dimensional precursor the number of $Q$-dimensional rectangles, covering the range, is already $M^{Q}$, and majority of them contains $0$ event. Only a small number of such rectangles contains one or more events, that is the precision of such an estimate of conditional probability is usually too low to have any practical value.
For this reason one constructs a new scalar precursor in the form of the scalar function of component of the vector precursor, and optimize its predictive power. This approach leads to additional complication as the units of measurement and the physical sense of different components of precursor are substantially different. In order to overcome this problem one uses some transformation to reduce all the components of the precursor to a standard form with the same sense and range of values.
Let us transform all the precursors $f_{q}(i,j,k),\, q=1,...,Q$ to variables with the values in $[0,1]$ providing estimates of conditional probabilities. So, after some transformation $F$ we obtain an estimate of $\mathrm{Pr}\left\{ h=\mathrm{1}\mid u({\bf f}(i,j,k))=1\right\} $, where $u$ is a characteristic function of some interval ${\bf B}$, i.e., the probability of event $h(i,j,k)=1$ under condition that this precursor takes the value ${\bf f}(i,j,k)\in{\bf B}$.
The transformation $F$ of a scalar precursor $f(i,j,k)$ is defined as follows. Fix an arbitrary small number $\varepsilon$. Let $L$ be a number of cells $(i,j,k)$ such that $h(i,j,k)=1$, and $Z_{l},\, l=1,\ldots,L,$ be the ordered statistics, i.e., the values $f(i,j,k)$ in these cells listed in non-decreasing order. Define a new sequence $z_{m},_{\,}m=0,...,M,$ from the ordered statistics $Z_{l}$ by the following recursion: $z_{0}=-\infty$, $z_{m}$ is defined as the first point in the sequence $Z_{l}$, such that $z_{m}-z_{m-1}\geq\varepsilon$. Next, construct the sequence $z_{m}^{*}=z_{m}+(z_{m+1}-z_{m})/2,$$\, m=1,\ldots,M-1$, and add the auxiliary elements $z_{0}^{*}=-\infty,\: z_{M}^{*}=\infty$. Define also a sequence $n_{m},\, m=1,\ldots,M$, where $n_{m}$ equals to the number of values in the sequence $Z_{l}$, such that $z_{m-1}^{*}\leq Z_{l}<z_{m}^{*}$. Finally, define the numbers $N_{m},\, m=1,...,M$ counting all cells such that $z_{m-1}^{*}\leq f(i,j,k)<z_{m}^{*},\, m=1,...M$. Observe that $\sum_{m=1}^{M}n_{m}=L,\:\sum_{m=1}^{M}N_{m}=N$, and use the ratios
$$\lambda=\frac{L}{N}\label{eq:7}$$
as estimate of unconditional probability of a seismic event in a given cell
$$p_{A}\equiv\mathrm{Pr_{\mathit{A}}}\left\{ \mathit{h\mathrm{(}i,j,k\mathrm{)}=\mathrm{1}}\right\} =\int_{-\infty}^{\infty}\mathrm{Pr}\left\{ \mathit{h=\mathrm{1\mid\mathit{x}}}\right\} dP_{A}(x).\label{eq:8}$$
The transformation $F$ is defined as follows
$$\begin{gathered}
g=Ff(i,j,k)=\frac{n_{m}}{N_{m}}\text{,}\:\text{if}\: z_{m-1}^{*}\leq f(i,j,k)<z_{m}^{*},\, m=1,\ldots,M.\label{eq:1}\end{gathered}$$
This definition implies that transformation $F$ replace the value of precursor for the frequency, i.e., the ratio of a number of cells containing a seismic event and the values of precursor from $[z_{m-1}^{*},z_{m}^{*})$ to the number of cells with the value of precursor in this range. These frequencies are the natural estimates of conditional probabilities $\mathrm{Pr}\left\{ \mathit{h=\mathrm{1\mid}z_{m-1}^{*}\leq f<z_{m}^{*}}\right\} ,\, m=1,\ldots,M$, computed with respect to stationary distribution $P_{A}\mathrm{(}x\mathrm{)}$:
$$\mathrm{Pr}\left\{ \mathit{h=\mathrm{1\mid}z_{m-1}^{*}\leq f<z_{m}^{*}}\right\} =\frac{\int_{z_{m-1}^{*}}^{z_{m}^{*}}\mathrm{Pr}\left\{ \mathit{h=\mathrm{1\mid\mathit{x}}}\right\} dP_{\Omega}(x)}{\int_{z_{m-1}^{*}}^{z_{m}^{*}}dP_{A}(x)}.\label{eq:2}$$
(This conditional probability can be written as $\mathrm{Pr}\left\{ \mathit{h=\mathrm{1\mid}u\mathrm{(}f)}\right\} ,$ where $u$ is the characteristic function of interval $[z_{m-1}^{*},z_{m}^{*})$). The function $g$ has a stepwise shape, and the length of the step in bounded from below by $\varepsilon$. It can be checked that there exist the limit $\tilde{g}=\underset{\varepsilon\rightarrow0}{\lim}\underset{K\rightarrow\infty}{\lim}g=\mathrm{Pr}\left\{ \mathit{h=\mathrm{1\mid}f}\right\} .$
The estimates of conditional probabilities in terms of the function $g$ are quite rough because typically the numbers $n_{m}$ are of the order 1. As a final result we will present below more sharp but less detailed estimates of conditional probabilities and confidence intervals for them.
Combinations of precursors
==========================
There are many ways to construct a single scalar precursor based on the vector precursor $(Ff_{q},\, q=1,\ldots,Q)$. Each such construction inevitably contains a number of parameters or degrees of freedom. These parameters (including the parameters used for construction of the precursors themselves) should be selected in a way to optimize the predictive power of the forecasting algorithm. The optimization procedure will be presented below, its goal is to adapt the parameters of precursors to a given catalog of earthquakes, that is to obtain the best possible retrospective forecast. However, this adaptation procedure creates a ghost information related with the specific features of the given catalog but not present in physical propertities of real seismicity. This ghost information will not be reproduced if the algorithm is applied to another catalog of earthquakes. It is necessary to increase the volume of the catalog and to reduce the number of free parameters to get rid of this ghost information.. Clearly, the first goal requires the considerable increase of the observation period and may be achieved in the remote future only. So, one concentrates on the reduction of number of degrees of freedom. The simplest ansatz including $Q-1$ parameters is the linear combination
$$f^{*}=Ff_{\text{1}}+\sum_{q=2}^{Q}c_{q-1}F\mathit{\mathit{f}_{q}}.\label{eq:3}$$
As a strictly monotonic function of precursor is a precursor itself the log-linear combination is an equally suitable choice
$$f^{*}=\ln\left(Ff_{\text{1}}\right)+\sum_{q=2}^{Q}c_{q-1}\ln\left(F\mathit{\mathit{f}_{q}}\right)\text{,}\label{eq:4}$$
Here $c_{q},\, q=1,...,Q-1$ are free parameters. The result of the procedure has the form $g=Ff^{*}.$
Alarm levels, point and interval estimations
=============================================
In view of (\[eq:1\]) the precursor $g$ is the set of estimates for probabilities $$\mathrm{Pr}\left\{ \mathit{h=\mathrm{1\mid}z_{l-1}^{*}\leq f(i,j,k)<z_{l}^{*}}\right\} ,\, l=1,...,L(f).$$ Its serious drawback is that typically $\left\{ z_{l-1}^{*}\leq f(i,j,k)<z_{l}^{*}\right\} $ correspond to single events, and therefore the precision of these estimates is very low (the confidence intervals discussed below may be taken as a convenient measure of precision). In order to increase the precision it is recommended to use the larger cells containing a larger number of events, that is a more coarse covering of the space where the precursor takes its values. In a sense, the precision of the estimation and the localization of the precursor values in its time-space region are related by a kind of uncertainty principle: the more precise estimate one wants to get the more coarse is the time-space range of their values and vice versa.
We adapt the following approach in order to achieve a reasonable compromise.
1\. For fixed thresholds $a_{s},\, s=1,...,S+1,\: a_{1}=1,\: a_{s}<a_{s+1},\: a_{S+1}=0,$ we define $\text{\ensuremath{\mathit{S}}}$ possible alarm levels $a_{s+1}\leq g(i,j,k)<a_{s}$ and subsets $\Omega_{s},\, s=1,...,S$, of the set $\Omega_{K}$ corresponding to alarm levels, i.e., $\Omega_{s}$ is a set of cells of $\Omega_{K}$, such that $a_{s+1}\leq g(i,j,k)<a_{s}$
There are different ways to choose the number $S$ of alarm levels and the thresholds $a_{s},\, s=2,...,S$. Say, fix $S=5$, and select $a_{s}=10^{-\alpha(s-1)}$. This is a natural choice of the alarm level because at $\alpha=1$ it corresponds to decimal places of the estimate of the conditional probability given by the precursor. The problem with $S=2$, i.e., two-level alarm, may be reduced to the hypothesis testing and discussed in more details below.
2\. Compute the point estimates $\theta_{\text{s}}$ of probabilities $\mathrm{Pr}\left\{ \mathit{h=\mathrm{1\mid}a_{s+\mathrm{1}}\leq g\mathrm{(}i,j,k\mathrm{)}<a_{s}}\right\} ,$ $s=1,...,S$, obtained via the distribution $P_{\Omega}(x)$ of precursor $g$ in the same way as in (\[eq:2\]). The property (iv) implies that for any domain $\Omega_{s}$ the binary random variables $h\mathrm{(}i,j,k\mathrm{)}$ are independent and identically distributed, i.e
$$\mathrm{Pr}\left\{ \mathit{h=\mathrm{1\mid}a_{s+1}\leq g(i,j,k)<a_{s}}\right\} \equiv p_{s},$$ and the unbiased estimate of $p_{s}$ takes the form
$$\theta_{s}=\frac{m_{s}}{n_{s}}\label{eq:9}$$
where $n_{s}$ stands for the number of cells in domain $\Omega_{s}$, and by $m_{s}$ we denote the number of cells in $\Omega_{s}$ containing seismic events.
3\. The random variable $m_{s}$ takes integer values between $0$ and $n_{s}$. The probabilities of these values are computed via the well-known Bernoulli formula $\mathrm{Pr}(m_{s}=k)=\binom{n_{s}}{k}\, p_{s}^{k}(1-p_{s})^{n_{s}-k}$. Let us specify the confidence interval covering the unknown parameter $p_{s}$ with the confidence level $\gamma$. In view of the integral Mouvre-Laplace theorem for $n_{s}$ large enough the statistics $\frac{(\theta_{s}-p_{s})\sqrt{n_{s}}}{\sqrt{p_{s}(1-p_{s})}}$ is approximately Gaussian N$(0,1)$ with zero mean and unit variance. Note that the values $n_{s}$ increase with time. Omitting straightforward calculations and replacing the parameter $p_{s}$ by its estimate $\theta_{s}$ we obtain that $\theta_{s}^{-}<\theta_{s}\text{<\ensuremath{\theta_{s}^{+}}},$ where $\theta_{s}^{-}=\theta_{s}-\frac{t_{\gamma}\sqrt{\theta_{s}(1-\theta_{s})}}{\sqrt{n_{s}}}$, $\theta_{s}^{+}=\theta_{s}+\frac{t_{\gamma}\sqrt{\theta_{s}(1-\theta_{s})}}{\sqrt{n_{s}}}$, and $t_{\gamma}$ is the solution of equation $\Phi(t_{\gamma})=\frac{\gamma}{2}$. Here $\Phi$ stands for the standard Gaussian distribution function.
4\. As a result of these considerations we introduce ’the precursor of alarms’ which indicates the alarm level: $\mathbf{R}(\mathbf{f}(i,j,k))=s(i,j,k)$. It will be used for calculations of point estimate and the confidence inteval in the form $\{\theta_{s(i,j,k)}^{-}<\theta_{s(i,j,k)}<\theta_{s(i,j,k)}^{+}\}$. This result will be use for prospective forecasting procedure.
The information gain and the precursor quality
==============================================
The construction of a ’combined’ precursor $\mathbf{R}$ involves parameters from formula (\[eq:3\]) or (\[eq:4\]) as well as parameters which appear in definition of each individual precursors $f_{q}$. It is natural to optimize the forecasting algorithm in such a way that the information gain related to the seismic events is maximal. In one-dimensional case the information gain as a measure of the forecast efficiency was first intoduced by Vere-Jones [\[]{}Vere-Jones, 1998[\]]{}. Here we exploit his ideas in the case of multidimensional space-time process.
Remind the notions of the entropy and information. Putting aside the mathematical subtlety (see [\[]{}Kelbert, Suhov, 2013[\]]{} for details) we follow below an intuitive approach of the book [\[]{}Prohorov, Rozanov, 1969[\]]{}. The information containing in a given text is, basically, the length of the shortest compression of this text without the loss of its content. The smallest length $S$ of the sequence of digits $0$ and $1$ (in a binary code) for counting $N$ different objects satisfies the relations $0\leq S-\log_{2}N\leq1$. So, the quantity $S\approx\log_{2}N$ characterizes the shortest length of coding the numbers of $N$ objects.
Consider an experiment that can produce one of $N$ non-intersecting events $\mathit{{\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\textcyr{\char192}}\endgroup\else\textcyr{\char192}\fi}}_{1},{\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\ldots}\endgroup\else\ldots\fi},\mathit{{\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\textcyr{\char192}}\endgroup\else\textcyr{\char192}\fi}_{N}}$ with probabilities $\mathit{q}_{1},{\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\ldots}\endgroup\else\ldots\fi},\mathit{q}_{N}$, respectively, $\mathit{q}_{1}+{\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\ldots}\endgroup\else\ldots\fi}+\mathit{q_{N}}=1$. A message informing about the outcomes of $n$ such independent identical experiments may look as a sequence $(A_{i_{1}},{\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\ldots}\endgroup\else\ldots\fi},A_{i_{n}})$, where $A_{i_{k}}$ is the outcome of the experiment $k$. But for long enough series of observations the frequency $n_{i}/n$ of event $\mathit{{\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\textcyr{\char192}}\endgroup\else\textcyr{\char192}\fi}}_{i}$ is very close to its probability $\mathit{q}_{i}$. It means that in our message $(A_{i_{1}},\text{\ensuremath{\ldots}},A_{i_{n}})$ the event $\mathit{{\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\textcyr{\char192}}\endgroup\else\textcyr{\char192}\fi}}_{i}$ appears $n_{i}$ times. The number of such messages is
$$N_{n}=\frac{n!}{n_{1}!...n_{N}!}.$$ By the Stirling formula the length of the shortest coding of these messages
$$S_{n}\approx\log_{2}N_{n}\approx-n\sum_{i=1}^{N}q_{i}\log_{2}q_{i}.$$ The quantity $S_{n}$ measures the uncertainty of the given experiment before its start, in our case we are looking for one of possible outcomes of $n$ independent trials. The specific measure of uncertainty for one trial
$$\frac{1}{n}S_{n}=\frac{1}{n}S_{n}(\mathit{q_{\mathrm{1}},{\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\ldots}\endgroup\else\ldots\fi},q_{N}})=-\sum_{i=1}^{N}q_{i}\log_{2}q_{i}$$ is known as Shannon’s entropy of distribution $\mathit{q\mathrm{_{1}},{\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\ldots}\endgroup\else\ldots\fi},q_{N}}$ (in physical literature it is also known as a measure of chaos or disorder). After one trial the uncertainty about the future outcomes decreases by the value $S=S_{n}-S_{n-1}$, this decrement equals to the *information gain* $I=S$, obtained as a result of single trial.
The quantity
$$S(h)=-p_{A}\log_{2}p_{A}-(1-p_{A})\log_{2}(1-p_{A})\label{eq:6}$$
is the (unconditional) entropy of distribution for indicator of seismic event $h$ in a space-time cell in the absence of any precursors. The conditional entropy $S(h\mid a_{s+1}\leq g<a_{s})$ under condition that in the cell $(i,j,k)$ the alarm level $s$ is set up equals
$$S(h\mid a_{s+1}\leq g<a_{s})=-p_{s}\log_{2}p_{s}-(1-p_{s})\log_{2}(1-p_{s})$$ The expected conditional entropy $S_{\boldsymbol{\mathbf{R}}}(h)$ of indicator of seismic events where the averaging in taken by the distribution of precursors $\mathrm{\mathbf{R}}$ takes the form
$$S_{\boldsymbol{\mathbf{R}}}(h)=-\sum_{s=1}^{S}\left[p_{s}\log_{2}p_{s}+(1-p_{s})\log_{2}(1-p_{s})\right]P_{A}(a_{s+1}\leq g<a_{s})\label{eq:5}$$
We conclude that the knowledge of the precursor values helps to reduce the uncertainty about the future experiment by $S(h)-S_{\mathrm{\mathbf{R}}}(h)$ which is precisely information $I(\mathrm{\mathbf{R}},h)$ obtained from the precursor. Taking into account (\[eq:6\]), (\[eq:5\]) and the fact that
$$p_{A}=\sum_{s=1}^{S}p_{s}P_{A}(a_{s+1}\leq g(i,j,k)<a_{s})$$ we specify the information gain as
$$\begin{gathered}
I(\mathrm{\mathbf{R}},h)=\sum_{s=1}^{S}\left[p_{s}\log_{2}\frac{p_{s}}{p_{A}}+(1-p_{s})\log_{2}\frac{1-p_{s}}{1-p_{A}}\right]P_{A}(a_{s+1}\leq g<a_{s}).\\\end{gathered}$$
By analogy with the one-dimensional case [\[]{}Kolmogorov, 1965[\]]{} the quantity $I(\mathrm{\mathbf{R}},h)$ may the called the mutual information about the random field $h$ that may be obtained from observations of random field $\mathbf{R}$. It is known that the information $I(\mathrm{\mathbf{R}},h)$ is non-negative and equals to $0$ if and only if the random fields $h$ and $\mathbf{R}$ are independent. This mutual information $I(\mathrm{\mathbf{R}},h)$ takes its maximal value $S(h)$ in an idealized case of absolutely exact forecast. The mutual information quantifies the information that the distributions of precursors contribute to that of the indicator of seismic event. For this reason it may be considered as an adequate scalar estimate for the quality of the forecast.
The quantity $I(\mathrm{\mathbf{R}},h)$ depends on the cell size, i.e., on the space discretization length $a$ and time interval $\Delta t$. We need a formal test to compare precursors defined for different size of the discretization cells. For this aim let us introduce the so-called ’efficiency’ of precursors as the ratio of information gains
$$r(\mathrm{\mathbf{R}},h)=\frac{I(\mathrm{\mathbf{R}},h)}{S(h)}.$$ This efficiency varies between $0$ and $1$ and serves as a natural estimate of information quality of precursors. It allows to compare different forecasting algorithms and select the best one.
A natural estimate of $S(h)$ based on (\[eq:7\]) and (\[eq:8\]) is defined as follows
$$\hat{S}(h)=-\lambda\log_{2}\lambda-(1-\lambda)\log_{2}(1-\lambda).\label{eq:12}$$
Taking into account (\[eq:9\]) and using an estimate of $P_{A}(a_{s+1}\leq g<a_{s})$ in the form of ratio $\tau_{s}=\frac{n_{s}}{N},$ we construct an estimate of $I(\mathrm{\mathbf{R}},h)$ as follows
$$\hat{I}(\mathrm{\mathbf{R}},h)=\sum_{s=1}^{S}\left[\theta_{s}\log_{2}\frac{\theta_{s}}{\lambda}+(1-\theta_{s})\log_{2}\frac{1-\theta_{s}}{1-\lambda}\right]\tau_{s},\label{eq:13}$$
$$\hat{r}(\mathrm{\mathbf{R}},h)=\frac{\hat{I}(\mathrm{\mathbf{R}},h)}{\hat{S}(h)}.\label{eq:14}$$
*The economical quality of forecast.* A natural economic measure for a quality of binary forecast is the economic risk or damage $r$ related to the earthquakes and the necessary protective measures. In mathematical statistics the risk is defined as the expectation of the loss function, in our case there are two types of losses: damage and expenses related to protection. For each cell of our grid the risk may be specified by the formula $$\begin{array}{l}
r=\alpha\mathrm{Pr}\text{\{}\mathit{h}(\mathit{i},\mathit{j},\mathit{k})=1,\eta(\mathit{i},\mathit{j},\mathit{k})=0\}+\beta\mathrm{Pr}\{\mathit{h}(\mathit{i},\mathit{j},\mathit{k})=0,\eta(\mathit{i},\mathit{j},\mathit{k})=1\}+\\
+\gamma\mathrm{Pr}\{\mathit{h}(\mathit{i},\mathit{j},\mathit{k})=1,\eta(\mathit{i},\mathit{j},\mathit{k})=1\},
\end{array}$$ here $\alpha$ stands for the average damage from a seismic event; $\beta$ stands for the average expenses for protection after a seismic alarm is announced; $\gamma$ stands for the average damage after the alarm, $\gamma=\alpha+\beta-\delta$, where $\delta$ is the damage prevented by the alarm. The coefficient in front of $\mathrm{Pr}\{\mathit{h}(\mathit{i},\mathit{j},\mathit{k})=0,\eta(\mathit{i},\mathit{j},\mathit{k})=0\}$, obviously, equals $0$, because in the absence both of a seismic event and an alarm there is no loss of any kind. Clearly, only the case when $\delta>\beta$ is economically justified, i.e., the gain from the prevention measures is positive. Obviously, $\delta$ should be less than $\alpha+\beta$, i.e., an earthquake cannot be profitable. Taking into account that $\alpha,\beta$ and $\gamma$ depend on the geographical position of the cell, we write the total risk as the summation over all cells in the region of a given forecast. In the simplest case of the absence of the spacial component, when a single cell represents a region of forecast, the expression for the risk is simplified as follows $r=\alpha\lambda\nu+\beta\tau+\gamma\lambda(1-\nu).$
However, the risk $r$, which is very useful for economical considerations and as a basis for an administrative decision, could hardly be used as a criteria for quality of seismic prediction. First of all, it cannot be computed in a consistent way because the coefficients $\alpha,\:\beta$ and $\gamma$ are not known in practice, and hence no effective way of its numerical evaluation is known. The computation of these coefficients is a difficult economic problem and goes far beyond of the competence of geophysicists. On the other hand, the readiness of the authority to commit resources to solving the problem depends on the quality of the geophysical forecast. This situation leads to a vicious cirle.
The second drawback of the economic risk as a criterion for the quality of prediction is related to the fact that it depends on many factors which have no relation to geophysics or earthquake prediction. These factors inlude the density of population, the number and size of industrial enterprises, infrastructure, etc. It also depends on subjective factors such as the williness of authorities to use resources for prevention of the damage from earthquakes. The natural sciences could hardly accept the criteria for the forecast quality which depend on the type of state organization, priorities of ruling parties, results of the recent elections, etc.
It seems reasonable to introduce a penalty related to the number of superfluous parameters in evaluating the quality of forecast pointing to the natural analogy with the Akaike test [\[]{}Akaike, 1974[\]]{} and similar methods in information theory. In our context the main parameter of importance is ${\hat{r}}({\bf R},t)$ and its limit as $t\to\infty$. This quantity does not involve the number of parameters directly. Probably, the rate of convergence depends on the number of parameters but this dependence is not studied yet.
The forecasting procedure
==========================
The number of time intervals, i.e., the number of observation $N$ used in the construction of estimates increases with the growth of observation time. So, the computation procedure requires constant innovations. On the other hand some computation time is required to ’adapt’ the model parameters to the updated information about seismic events via an iterative procedure. For these reasons we propose the following forecasting algorithm.
1\. Given initial parameter values at the moment $t_{K-1}=t_{0}+(K-2)\Delta t$ we optimize them to obtain the maximum of efficiency $\hat{r}(\mathrm{\mathbf{R}},h)$ of precursor in domain $\Omega_{K-1}.$ For this aim the Monte-Carlo methods is helpful: one perturbs the current values of parameters randomly and adapts the new values if the efficiency increases. The process continues before the value of efficiency stabilized, this may give a local maximum, so the precedure is repeated sufficient number of times. The choice of initial value on the first step of optimization procedure is somewhat arbitrary but a reasonable iteration procedure usually leads to consistent results. The opmization procedure takes the period of time $t_{K-1}<t\leq t_{K}$ .
2\. Next, we construct the forecast in the following way. At the moment $t_{K}$ the values of precursor $\hat{g}$ in each cell $(i,j,K+1)$ is computed with optimized parameters. Based on these parameter values the alarm levels, the point estimates and confidence intervals are computed in each cell as well as the values of efficiency of precursors.
3\. The estimates of stationary probabilities of seismic events in the cell $\bar{\theta}(i,j)$ are defined as follows:
$$\bar{\theta}(i,j)=\frac{1}{K}\sum_{k=1}^{K}\theta_{s(i,j,k)}.$$ they can be used for creation of the ofaps of seismic hazard in the region.
Retrospective and prospective informativities
=============================================
The efficiency of precursor which is achieved as a result of parameters optimization could be considered as *retrospective* as it is constructed by the precursors adaptation to the historical catalogs of seismic events. The prospective efficiency for the space-time domain $\Omega^{*}$ containing the cell in the ’future’ is based on the forecast. It is computed via formulas (\[eq:12\]), (\[eq:13\]), (\[eq:14\]) with the only difference that domain $\Omega_{\mathit{s}}^{*}$ consists from the cells where the forecasted alarm level is $s$. The efficiency of prospective forcast is smaller compared with the retrospective efficiency, however approaches this value with time. In principle, the prospective efficiency is an ultimate criteria of precursors quality and the retrospective efficiency could serve only for the preliminary selection of precursors and their adaptation to the past history of seismic events.
Testing of the forecasting algorithm
=====================================
The efficiency of precursor could be computed exactly only in an idealized case of infinite observation time. However, its estimate may be obtained based on the observation over a finite time interval. So, if an estimate produces a non-zero value not necessarily the real effects is present. It may be simply a random fluctuation even if the precursor provides no information about the future earthquake. For this reason we would like to check the hypothesis $H_{0}$ about the independence of a precursor and an event indicator with a reasonable level of confidence. In case the hypothesis is rejected one have additional assurance that the forecasting is real, not just a ghost.
So, consider the distributions
$$\mathit{P}_{A}(\mathit{x})=\frac{1}{\mathit{I\cdot J}}\sum_{\left(\mathit{i,j}\right)\in\Omega}\mathrm{P}\mathrm{r}_{ij}\left\{ \mathit{g\mathrm{(}i,j,k\mathrm{)\leq\mathit{x}}}\right\} ,$$ and $$\mathit{P_{A}^{\prime}}(\mathit{x})=\frac{1}{\mathit{I\cdot J}}\sum_{\left(\mathit{i,j}\right)\in A}\mathrm{P}\mathrm{r}_{ij}\left\{ \mathit{g\mathrm{(}i,j,k\mathrm{)\leq\mathit{x}}\mid}h\mathrm{(}i,j,k\mathrm{)=1}\right\}$$ The function $\mathit{P}_{A}(\mathit{P}_{A}^{-1}(y))=y$ of variable $y=\mathit{P}_{A}(x)$ provides an uniform distribution $F^{*}(y)=\mathrm{Pr}\big({\xi\leq y}\big)$ of some random variable $\xi$ on [\[]{}0,1[\]]{}. Next, consider a distribution function $G(y)=P_{A}^{\prime}(\mathit{P}_{A}^{-\mathrm{1}}(y))$ on [\[]{}0,1[\]]{}, and use a parametric representation for abcissa $\mathit{P}_{A}(\mathit{x})$ and ordinate $\mathit{P}_{A}^{\prime}(\mathit{x})$. If random fields $g$ and $h$ are independent the distribution functions $\mathit{P}_{A}(\mathit{x})=\mathit{P}_{A}^{\prime}(\mathit{x})$ and $G(y)$ are uniform. So, the hypothesis about the absence of forecasting, i.e., about the independence of $g$ and $h$, is equivalent to the hypothesis $H_{0}$ that the distribution $G(y)$ is uniform.
The empirical distribution $G_{L}(y)$ related to $G(y)$ is defined as follows. Denote by $u_{l},l=1,...L$ the values of the function $g(i,j,k)$ sorted in the non-decreasing order and beloning to the cells where $h\mathrm{(}i,j,k\mathrm{)=1.}$ Let $n_{l}$ be the numbers of cells such that $h\mathrm{(}i,j,k\mathrm{)=1,}\, g(i,j,k)=u_{l}.$ Denote by $m(u_{l})$ the numbers of cells from $\Omega$ such that $g(i,j,k)<u_{l}$, and define the empirical distribution $G_{L}(y)$ as a step-wise function with $G_{L}(0)=0$ and positive jumps of the size $\frac{n_{l}}{L}$ at points $y_{l}=\frac{m(u_{l})}{N}$.
The well-known methods of hypothesis testing requires that the function $G_{L}(y)$ has the same shape as for independent trials, i.e., random variables $u_{l},l=1,...L$ are independent in view of axiom (iv). Naturally, we accept the precursors such that the hypothesis $H_{0}$ is rejected with the reasonable level of confidence. (Remind, that the hypothesis is accepted if and only if its logical negation could be rejected based on the available observations. The fact that the hypothesis cannot be rejected does not mean at all that it should be accepted, it only means that the available observations don’t contradict this hypothesis. Say, the well-known fact that The Sun rise in the East does not contradict to our hypothesis, however it may not be considered as a ground for its acceptance.) For large values of $L$ the Kolmogorov statistics [\[]{}Kolmogorov, 1933a[\]]{} is helpful for this aim
$$D_{L}=\sup\mid G_{L}(y)-y\mid$$ with an asymptotic distribution
$$\underset{L\rightarrow\infty}{\lim}\mathrm{Pr\left\{ \mathit{\sqrt{L}D_{L}\leq z}\right\} =\sum_{\mathit{k}=-\infty}^{\infty}}\left(-1\right)^{k}\mathit{e^{-2k^{2}z^{2}},\, z>\mathrm{0},}$$ or Smirnov’s statistics [\[]{}Smirnov, 1939[\]]{}
$$D_{L}^{+}=\sup\left[G_{L}(y)-y\right],$$
$$D_{L}^{-}=-\inf\left[G_{L}(y)-y\right],$$ with asymptotic distribution
$$\underset{L\rightarrow\infty}{\lim}\mathrm{Pr\left\{ \mathit{\sqrt{L}D_{L}^{+}\leq z}\right\} =\underset{\mathit{L}\rightarrow\infty}{\lim}\mathrm{Pr\left\{ \mathit{\sqrt{L}D_{L}^{-}\leq z}\right\} =1-\mathit{e^{-2z^{2}},\, z>\mathrm{0.}}}}$$ The asymptotic expressions for these statistics can be used for $L>20$ ([\[]{}Bolshev, Smirnov, 1965[\]]{})..
The binary alarm and the hypothesis testing
===========================================
The prediction is the form of forecast when an alarm is announced in a given cell without a preliminary evaluation of probability of seismic event. In this case we can estimate the probabilities of events too. (If the alarm is announced in an arbitrary domain $\Omega$ we set up an alarm if at least arm.).
Let $M$ be the number of cells in $\Omega$ which are in the state of alarm, $M_{0}$ be the number of cells where the seismic event is present but no alarm was announced (the number of ’missed targets’). Denote by $\tau=\frac{M}{N}$ the share of the cells with alarm announced, $\lambda=\frac{L}{N}$ the share of the cells with seismic events, and $\nu=\frac{M_{0}}{M}$ the share of missed targets. Let a random variable $\eta(i,j,k),$ equal $1$ if an alarm is announced in the cell $(i,j,k)$, and $0$ otherwise. Obviously, the estimate of conditional probability $\mathrm{Pr\left\{ \mathit{h}(\mathit{i},\mathit{j},\mathit{k})=1\mid\eta(\mathit{i},\mathit{j},\mathit{k})=1\right\} }$ of the seismic event under the condition of alarm is $\frac{\lambda(1-\nu)}{\tau},$ and the estimate of conditional probability $\mathrm{Pr\left\{ \mathit{h}(\mathit{i},\mathit{j},\mathit{k})=1\mid\eta(\mathit{i},\mathit{j},\mathit{k})=0\right\} }$ of the seismic event under the condition of no alarm is $\frac{\lambda\nu}{1-\tau}.$
If the alarm is announced according to the procedure described in Section 5 the threshold $a_{1}$ specifying the acceptable domain of values for $g(i,j,k)$ should be treated as a free parameter and selected by maximizing the information efficiency $\hat{r}(\eta,h)$. The estimate of information increase for given values of $\tau$ and $\nu$ equals
$$\begin{gathered}
\hat{I}(\eta,h)=\lambda(1-\nu)\log_{2}\frac{1-\nu}{\tau}+\lambda\nu\log_{2}\frac{\nu}{1-\tau}+\\
+\left[\tau-\lambda(1-\nu)\right]\log_{2}\frac{\tau-\lambda(1-\nu)}{(1-\lambda)\tau}+\left(1-\tau-\lambda\nu\right)\log_{2}\frac{1-\tau-\lambda\nu}{(1-\lambda)(1-\tau)}.\end{gathered}$$
The value of $\eta(i,j,k)$ characterizes the results of checking two mutually exclusive simple hypothesis:
$H_{0}$: the distribution of $\tilde{g}\mathrm{(}i,j,k\mathrm{)}$ has the form $\mathrm{\mathit{P}_{\mathit{A}}^{0}(\mathit{x})\equiv}$ $P\mathrm{r}_{\mathit{A}}\left\{ \mathit{\tilde{g}\mathrm{(}i,j,k\mathrm{)\leq\mathit{x}\mid\mathit{h}\mathit{\mathrm{(}i\mathrm{,}j\mathrm{,}k}\mathrm{)=0}}}\right\} $, implying ’no seismic events’,\
or
$H_{1}:$ the distribution of $\tilde{g}\mathrm{(}i,j,k\mathrm{)}$ has the form $\mathrm{\mathit{P}_{\mathit{A}}^{1}(\mathit{x})\equiv}$ $P\mathrm{r}_{\mathit{A}}\left\{ \mathit{\tilde{g}\mathrm{(}i,j,k\mathrm{)}\mathrm{\leq\mathit{x}\mid\mathit{h}\mathit{\mathrm{(}i\mathrm{,}j\mathrm{,}k}\mathrm{)=1}}}\right\} $, implying the presence of seismic event.\
Statistics for checking of these hypothesis is the precursor $g\mathrm{(}i,j,k\mathrm{)}$, and the critical domain for $H_{0}$ has the form $\left\{ g\mathrm{(}i,j,k\mathrm{)}\geq a_{1}\right\} $. (If usual method of alarm announcement is used the relevant precursor plays the rôle of statistics and the critical domain is defined by the rule of the alarm announcement). The probability of first type error $$\alpha=\mathrm{Pr\left\{ \mathit{\eta}(\mathit{i},\mathit{j},\mathit{k})=1\mid\mathit{h}(\mathit{i},\mathit{j},\mathit{k})=0\right\} ,}$$ it is estimated as $\frac{\tau-\lambda(1-\nu)}{1-\lambda}$. The probability of second type error $$\beta=\mathrm{Pr\left\{ \mathit{\eta}(\mathit{i},\mathit{j},\mathit{k})=0\mid\mathit{h}(\mathit{i},\mathit{j},\mathit{k})=1\right\} ,}$$ it is estimated as $\nu.$ (Note that due to condition (iii) any test used for the checking these hypothesis should not depend on the coordinates of the cell).
The Neyman-Pearson theory allows to define the domain of all possible criteriaall possible criteria: in coordinates $(\alpha,\beta)$ it is a convex domain with a boundary $\Gamma$ which corresponds to the set of uniformly most powerful tests. This family may be defined in terms of the likelihood ratio $\Lambda(x)=\frac{\mathit{p}_{\mathit{A}}^{1}(\mathit{x})}{\mathit{p}_{\mathit{A}}^{0}(\mathit{x})}$ under condition that the distributions $\mathit{P}_{\mathit{A}}^{1}(\mathit{x})$ and $\mathit{P}_{\mathit{A}}^{0}(\mathit{x})$ has densities $\mathit{p}_{\mathit{A}}^{1}(\mathit{x})$ and $\mathit{p}_{\mathit{A}}^{0}(\mathit{x})$: $$\begin{cases}
\eta(i,j,k\mathrm{)}=1\:\mathrm{if\:}\Lambda(x)>\omega,\\
\eta(i,j,k\mathrm{)}=0\:\mathrm{if\:}\Lambda(x)<\omega
\end{cases}$$ where $\omega$ denotes the threshold. In the paper [\[]{}Gercsik, 2004[\]]{} we demonstrated that among all the tests with the images on the boundary $\Gamma$ there exists three different best tests. Here the term best may be understood in three different sense, i.e., maximizing the variational, correlational and informational efficiency. The most relevant criteria is the informational efficiency ${\hat{r}}(\eta,h)$.
The well-known Molchan’s error diagram [\[]{}Molchan, 1990[\]]{} where the probability of the first kind error is estimated by $\tau$ is constructed in the same way. However, it involve a comparison of two intersecting hypothesis:
$H_{0}$: the distribution $\tilde{g}\mathrm{(}i,j,k\mathrm{)}$ has the form $\mathrm{\mathit{P}_{\mathit{A}}^{0}(\mathit{x})\equiv P}\mathrm{r}_{\mathit{A}}\left\{ \mathit{\tilde{g}\mathrm{(}i,j,k\mathrm{)}\mathrm{\leq\mathit{x}}}\right\} $, i.e., the seismic event could either happen or not happen, and
$H_{1}:$ the distribution $\tilde{g}\mathrm{(}i,j,k\mathrm{)}$ has the form $\mathrm{\mathit{P}_{\mathit{A}}^{1}(\mathit{x})\equiv P}\mathrm{r}_{\mathit{A}}\left\{ \mathit{\tilde{g}\mathrm{(}i,j,k\mathrm{)}\leq\mathit{x}\mid\mathit{h}\mathit{\mathrm{(}i\mathrm{,}j\mathrm{,}k}\mathrm{)=1}}\right\} $, i.e., the seismic event will happen
.Note that the rejection of hypothesis $H_{0}$ leads to absurd results.
In the paper [\[]{}Molchan and Keilis-Borok, 2008[\]]{} the area of the alarm domain is defined in terms of non-homogeneous measure depending on the spacial coordinates, in terms of our paper it may be denoted as ${\bar{\theta}(i,j)}$. i.e., $\tau\backsim\sum_{i,j}\bar{\theta}(i,j)\eta(i,j,k\mathrm{)}$. This approach is used to eliminate the decrease of the share of alarmed sites $\tau$ with the extension of the domain when a purely safe and aseismic territory is included into consideration. It would be well-justified if the quantity $\tau$ could be accepted as an adequate criterion of the quality of forecast in its own right. On the other hand, it can be demonstrated that the information efficiency ${\hat{r}}(\eta,h)$ converges to a non-zero value $1-\nu$ when the number of cells with an alarm is fixed but the total number of cells tends to infinity. An inhomogeneous area of the territory under forecast which is proportional to $\bar{\theta}(i,j)$ does not enable us to calculate the informational efficiency. Moreover, it possesses a number of unnatural features from the point of view of evaluation of economical damage. A seismic event in the territory of low seismicity is more costly because no precautions are taken to prevent the damage of infrastructure. However, in this inhomogeneous area an alarm announced in an aseismic territory will have a smaller contribution than an alarm in a seismically active territory where the losses would be in fact smaller. We conclude that this approach ’hides’ the most costly events and does not provide a reasonable estimate of economic damage.
The choice of precursors
========================
We use the term ’empiric precursor of earthquake’ for any observable characterisric derived from the catalogi.e., phase transitions, cracks propagations, etc.) In the meteorological forecast the danger of using empirical precursors was highlighted by A. Kolmogorov in 1933 [\[]{}Kolmogorov, 1933[\]]{}. From that time the meteorological forecast relies on the physical precursors which are theoretically justified by the models of atmospheric dynamics. Below we will present A. Kolmogorov’s argument adapted to the case of seismic forecast. This demonstrates that the purely empirical precursors work well only for the given catalog from which they are derived. However, their eficiency deteriorates drastically when they are applied to any other independent catalog.
Consider a group of $k$ empirical precursors used for a forecast and and selected from a set of $n$ such groups. According to A. Kolmogorov’s remark the number $k$ is typically rather small. This is related to the fact that a number of strong earthquakes in catalog is unlikely to exceed a few dozen. As the values of precursors are random there exists a small probability $p$ that the efficiency of the forecast exceeds the given threshold ${\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\textcyr{\char209}}\endgroup\else\textcyr{\char209}\fi}$. Then the probability of event $\hat{r}(\mathrm{\mathbf{R}},h)\text{\ensuremath{\le}}\text{\textcyr{\char209}}$ equals $1{\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\textendash}\endgroup\else\textendash\fi}p$, and the probability of event $\hat{r}(\mathrm{\mathbf{R}},h)>\text{\textcyr{\char209}}$ for at least one collection of precursors equals $P=1-(1-p)^{n}$ and tends to $1$ as $n\rightarrow\infty$.. (According to Kolmogorov some arbitrariness of the assumption of independence is compensated by the large number of collections.)
Summing up, if the number of groups is large enough with probability close to 1 it is possible to find a group giving an effective retrospective forecast for a given catalog. In practice this is always the case as the number of empirical precursors could be increased indefinitely by variation of real parameters used in their construction. It is important to note that for such a group, which is highly eficient for the initial catalog, the probability that the eficiency is greater than $C$ is still equal to $p$ for any other catalog. In other words the larger the number of the groups of empirical precursors the less reliable forecast is. So, the collection of a large list of the empirical precursors is counter-productive.
Much more reliable are the physical precursors intrinsicly connected with the physical processes which preserve their values with the change of sample. The probability to find such a set of precursors by pure empirical choice is negligible because they are very rare in the immense collection of all possible precursors.
Image identification
====================
The possibility to use the pattern recognition formalism in seismic forecast is totally based on the acceptance of deterministic model of seismicity. It is necessary to assume that in principle there exits such a group of precursors which allows to determine with certainly whether a strong earthquake will happen or not. In this case one believes that all random errors are related to the incompleteness of this set of precursors.But if the seismicity is a random process then the image appears only after the earthquake and before it any set of values for precursors cannot guarantee the possible outcome and only the relevant probabilities may be a subject of scientific study. After the discovery of dynamic instability and generators of stochastic behavior of dynamical systems the deterministic model of seismicity is cast in doubts. Its potential acceptance requires substantial evidence which hardly exist at present.
In any case the results of pattern recognition procedure (i.e., a binary alarm) are useful if they are considered alongside with the results of statistical tests. They allows to calculate the estimate of probabilities of seismic events and informational efficiency.
However, the section of ’features’ for pattern recognition leads to the same difficulties as the selection of precursors: the ’features’ based on the observations only and not related to the physics of earthquakes are not helpful, and any hopes for ’perceptron education’ are not grounded. A successful supervised recognition is possible if the features has proved causal relation with pattern. This principle is illustrated by a simple but important theorem by A.N. Krichevets.
Let $A$ be a finite set, $B_{1},B_{2}\subset A,B_{1}\cap B_{2}=\emptyset$. We say that $B_{1}$ and $B_{2}$ are finite educational samples. Let $X\in A$, $X\notin B_{1}\cup B_{2}$ be a new object. Then among all classifications, i.e., subsets $(A_{1},A_{2})$ such that $B_{1}\subset A_{1}$, $B_{2}\subset A_{2}$, $A_{1}\cup A_{2}=A$, $A_{1}\cap A_{2}=\emptyset$ satisfying condition that either $B_{1}\cup X\subset A_{1}$ or $B_{2}\cup X\subset A_{2}$ exactly a half classifies $X$ as an object of sample $B_{1}$ and a half classifies $X$ as an object from $B_{2}$.
It is easy to define a one-to-one between classifications. Indeed, if $\left\{ A_{1},A_{2}\right\} ,$ $A_{1}\cup A_{2}=A,$ is a classification such that $B_{1}\subset A_{1},\, X\subset A_{1},\, B_{2}\subset A_{2},$ one maps it into the unique classification $\left\{ A_{1}^{\prime},A_{2}^{\prime}\right\} $ such that $B_{1}\subset A_{1}^{\prime},\, X\subset A_{2}^{\prime},\, B_{2}\subset A_{2}^{\prime},$ where $A_{1}^{\prime}=A_{1}\setminus X,\, A_{2}^{\prime}=A_{1}\cup X.$
A supervised pattern recognition is impossible. After the leaning procedure the probability to classify correctly a new object is the same as before leaning, i.e., $1/2$.
Demonstration of algorithm
==========================
A preliminary version of the forecast algorithm described above was used in the paper [\[]{}Ghertzik, 2008[\]]{} for California and the Sumatra-Andaman earthquake region. These computations serve as a demonstration of the efficiency of the method but their actual results should be taken with a pinch of salt because the selection of precursors does not appear well-justified from the modern point of view: the number of free parameters to be adapted in the precursor ‘stress indicator is too large.
****
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[^1]: Institute of Earthquake Prediction Theory and Mathematical Geophysics RAS, Moscow, RF, getrzik@ya.ru
[^2]: Dept. of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK. Institute of Earthquake Prediction Theory and Mathematical Geophysics RAS, m.kelbert@swansea.ac.uk
[^3]: Lomonosov MSU, Department of Psychology, Moscow, Russia, ankrich@mail.ru
|
---
abstract: |
The biggest overhead for the instantiation of a virtual machine in a cloud infrastructure is the time spent in transferring the image of the virtual machine into the physical node that executes it. This overhead becomes larger for requests composed of several virtual machines to be started concurrently, and the illusion of flexibility and elasticity usually associated with the cloud computing model may vanish. This poses a problem for both the resource providers and the software developers, since tackling those overheads is not a trivial issue.
In this work we implement and evaluate several improvements for virtual machine image distribution problem in a cloud infrastructure and propose a method based on BitTorrent and local caching of the virtual machine images that reduces the transfer time when large requests are made.
address: |
Instituto de F[í]{}sica de Cantabria — IFCA (CSIC—UC).\
Avda. los Castros s/n. 39005 Santander, Spain
author:
- '[Á]{}lvaro L[ó]{}pez Garc[í]{}a'
- 'Enol Fern[á]{}ndez del Castillo'
bibliography:
- 'references.bib'
title: Efficient image deployment in Cloud environments
---
Cloud Computing ,Image Deployment ,OpenStack ,Scheduling
Introduction {#sec:introduction}
============
As it is widely known, the Cloud Computing model is aimed on delivering resources (such as virtual machines, storage and network capacity) as an on demand service. The most accepted publication defining the Cloud from the United States National Institute of Standards and Technology (NIST), emphasizes the *rapid elasticity* as one of the essential characteristics of the Cloud Computing model: “capabilities can be elastically provisioned and released, (...), to scale rapidly outward and inward (...)” [@Mell2011]. Moreover, users and consumers consider them as the new key features that are more attractive [@Zhang2010; @Armbrust2010] when embracing the cloud.
If we take into consideration virtual machines delivered by an Infrastructure as a Service resource provider, these two outstanding features imply two different facts. On the one hand, **elasticity** is the ability to start and dispose one or several virtual machines (VMs) almost immediately. On the other hand, an **on demand** access implies that VMs are allocated whenever the user requires them, without prior advise and without human intervention from the Resource Provider (RP).
Any cloud must be able to deliver rapidly the requested machines to provide a satisfactory elastic and on-demand perception according to any Service Level Agreement SLA [@Aceto2012] established with the users or customers. With this fact in mind, the Cloud Management Frameworks (CMFs) —and as a consequence the resource providers operating a cloud— face a challenge when they are requested to provision a large number of resources, specially when running large infrastructures [@ChenZ2009].
These on-demand and elastic perceptions that a cloud should be able to deliver mostly depend on the time needed to serve the final service, so a rapid provisioning should be one of the objectives of any cloud provider. Besides the delays introduced by the Cloud Management Framework there are other factors contributing to this delay from the user standpoint. For instance, inter-datacenters transfers of large amounts of data [@Femminella2014] are a good example of contributors to the final delivery time. Any reduction in each of these factors will yield on a better reactivity of the cloud, leading to an increase of the ability to satisfy elastic requests on-demand.
Therefore, in order to deliver a rapid service, this spawning delay or penalty has to be decreased. It is the duty of the cloud provider to be able to provision efficiently the resources to the users, regardless of the size of the request, minimizing the costs of mapping the request into the underlying resources[@Manvi2014]. Hence, it is needed to study how current CMFs can minimize the start time of the virtual machines requested. Our main contribution in this paper is the proposal of an improvement of the current Coud Management Frameworks in two sides: firstly, the CMFs should implement more advanced and appropriated image transfer mechanisms; secondly, the cloud schedulers should be adapted so as to make use of local caches on the physical nodes. Moreover, in this paper:
- We will study how the deployment of images into the physical machines poses a problem to an Infrastructure as a Service (IaaS) resource provider and how it introduces a penalty towards the users.
- We will discuss several image transfer methods that alleviate this problem and review the related work.
- We implement and evaluate some of the described methods in an existing CMF.
- We propose an improvement of the scheduling algorithm to take profit of the VM images cached at the physical nodes.
The paper is structured as follows: In Section \[sec:problem\] we discuss and present the problem statement that. In Section \[sec:related\] the related research in the area is presented and discussed. Section \[sec:evaluation\] contains the evaluation of some of the methods described in the previous section. In Section \[sec:cache scheduler\] we propose a modification to the scheduling algorithms, and evaluate in combination with the studied image transfer methods. Finally, conclusions and future works are outlined in Section \[sec:conclusions\].
Problem statement {#sec:problem}
=================
Whenever a virtual machine is spawned its virtual image disk must be available at the physical node in advance. If the image is not available on that host, it needs to be transfered, therefore the spawning will be delayed until the transfer is finished. This problem is specially magnified if the request consist on more than a few virtual machines, as more data needs to be transfered over the network. As the underlying infrastructure increases its size the problem becomes also bigger, the number of requests need to be satisfied may become larger.
Regardless of the Cloud Management Framework (CMF) being used, the process of launching a VM in an IaaS cloud infrastructure comprises a set of common steps:
1. A VM image is created by somebody —e.g. by a system administrator or a seasoned user—, containing the desired software environment.
2. The image is uploaded to the cloud infrastructure image catalog or image repository. This image is normally stored as read-only, therefore, if further modifications (for example any user customization) need to be done on a given image, a new one must be created.
3. VMs based on this image are spawned into the physical machines.
4. The running VMs are customized on boot time to satisfy the user needs. This step is normally referred as contextualization and it is performed by the users.
The first two steps are normally performed once in the lifetime of a virtual machine image, meaning that once the image is created and is available in the catalog, then it is ready for being launched, so there is no need to recreate the image and upload it again. Therefore, assuming that the IaaS provider is able to satisfy the request (i.e. there are enough available resources to execute the requested VMs), whenever a user launches a VM, only the two former steps will introduce a delay in the boot time.
The last step, that is, the contextualization phase is made once the virtual instance has booted, and it is normally a user’s responsibility and beyond the Cloud Management Framework control [@Campos2013; @Li2012]. Hence, the field where a IaaS resource provider can take actions to reduce the boot time of a virtual machine is the spawning phase. This phase involves several management and preparation operations that will depend on the Cloud Management Framework being used. Generally, these operations will consist on one or several of the following steps:
Scheduling phase
: where the software selects the most suitable nodes to satisfy the user’s request.
Image transfer
: if the image data is not available in the selected physical machine, the CMF has to transfer it from the catalog into that host.
Image duplication
: once the image is available at the node. Some CMF duplicate the image before spawning the virtual machine. This way, the original image remains intact and it can be reused afterwards for another VM based on that same image.
Image preparation
: consisting in all the further image modifications prior to the virtual machine spawning, needed to satisfy the user’s request. For instance, this step can comprise the image resize, image format conversion, user-data injection into the image, file system checks, etc.
Taking as an example the OpenStack cloud testbed described in the experimental setup of Section \[sec:testbed\], Figure \[fig:bootchart-raw\] shows the boot sequence for an instance once the request is scheduled into a physical machine. In this request a 10GB image was launched with an additional local ephemeral disk of 80GB. This ephemeral empty space is created on the fly on the local disk of the physical machine, therefore it is not transferred over the network. In this initial setup, the images are stored in the catalog server and are transferred using HTTP when they are needed in the compute host.
![Chart of the boot process for one VM on an OpenStack cloud. The image used was 10GB large with an 80GB ephemeral disk.[]{data-label="fig:bootchart-raw"}](bootchart-crop.pdf){width="1\linewidth"}
As it can be seen, the OpenStack spawning process is broken down into several sub steps:
Resource claim
: The compute node checks if the requested resources are available, and claims them before spawning the instance.
Image download
: The image is fetched from the image catalog, and it is stored in the local disk.
Image duplication
: An exact replica image is created from the downloaded one.
Image resize
: The image is resized to fit into the size request by the user. Normally minimal images are stored in order to spare disk and save transfer times, therefore these images need to be resized into the correct final size.
Ephemeral disk creation
: An ephemeral virtual disk is created in the local disk. This virtual disk is created on the fly and it is normally located on the local machine disk, since it is a disposable space destroyed when the instance is terminated.
Data injection
: Any data specified by the user is injected into the image. This step needs to figure out the image layout and try to inject the data into the correct location. This is a prone to errors step since the image structure is unknown to the middleware and therefore it can fail. It could be avoided with the usage of contextualization, assuming that the images are properly configured.
Network configuration
: The virtual network is configured and set up in the physical node to ensure that the instance will have connectivity.
The *Resource claim* step belongs to the *Scheduling phase*, and the steps labeled *Image resize*, *Ephemeral disk creation*, *Data injection*, *Network configuration* belong to the aforementioned *Image preparation* phase. Observing Figure \[fig:bootchart-raw\] we can extract that there are three big contributors to the boot time, namely *Image download*, the *Image duplication* and the *Ephemeral disk creation* steps.
In this first test, raw images were used, meaning that the duplication involved the creation of a complete copy of the original image. This could be easily diminished by using Copy on Write (CoW) images.
The support for CoW images is implemented in all of the most common hypervisors (being the only difference the supported formats). Forcing the usage of CoW by the Cloud middleware reduces considerably the overhead, since it is not needed to duplicate the whole image container [@mcloughlin2008qcow2]. The ephemeral disk (if it exists) can be also created using CoW, so its contribution to the overhead will be diminished too. Therefore, one of the two biggest contributors to the boot time for an instance can be easily shrink with the adoption of CoW.
{width="1\linewidth"}
Figure \[fig:bootchart-cow\] shows the same request, when the cloud infrastructure has been configured to use CoW images. As it is seen, two of the three biggest penalties are reduced just by using Copy on Write images.
However, the *Image download* still introduces the biggest penalty and, unfortunately, this time is dependent on several factors:
- The image delivery method used will have a large impact on the final time. It is not the same to download an image from a single central location that transfer it using peer-to-peer techniques.
- The amount of data being transferred and obviously the image size: if several hundred gigabytes need to be transferred over the network each time a machine is booted, the delay will be difficult to shrink.
- The size of the request. It is not the same to swap just a few virtual machines that spawning hundreds of VMs.
- The load on the implied systems: the network usage, catalog server and compute hosts load have an influence on the overall process.
Virtual machine images range from a few hundreds of Megabytes to several Gigabytes [@Segal2010; @Femminella2014], hence an efficient image deliver method should try to tackle as much factors as possible. It should try to use a good image transfer method, should try to reduce the amount of data being transferred and thus reduce the load on the system. It should be also able to satisfy large requests, that are quite common on scientific workloads. For example, it is know that scientific communities often deploy a virtual cluster to support their users [@Keahey2008; @Afgan2010], sets of machines to execute a parallel application or workflow based applications [@Hardt2012].
Related work {#sec:related}
============
Several authors have also identified the image deployment phase as the biggest overhead to be solved when spawning VMs in a cloud infrastructure and have proposed different solutions. In the following subsections we will describe several of the proposals in the literature for addressing this issue.
One of the first approaches to reduce the image distribution time is to eliminate the step itself. This could be accomplished by the usage of of a shared storage (Section \[sec:distribution:shared\]) or by the pre-deployment or pre-fetch of images (Section \[sec:distribution:predeploy\] and Section \[sec:distribution:prefetch\] respectively). The election of a good delivery method (Section \[sec:distribution:ondemand\]) is also crucial. Finally, some authors point towards different and novel methods requiring further developments (Section \[sec:distribution:other\]), that seem promising.
Shared storage {#sec:distribution:shared}
--------------
This approach leverages the usage of a shared storage (such as access to a Network Attached Storage (NAS) or a Storage Area Network (SAN)) to eliminate at all image transfer. The catalog and the nodes share the same storage backend, thus once an image is uploaded to the system it will be directly available on the physical hosts. This method may seem ideal, however, it still has some drawbacks:
- The virtual machine disk is served over the network and nodes with an intensive Input/Output may underperform.
- It needs a dedicated and specialized storage system and network in order to not overload the instance’s network with the access to the disks. This network needs to be properly scaled, meaning that a good performance access and acceptable reliability and availability are a must: if the shared storage does not perform as expected, it will become a bottleneck for the cloud infrastructure and will impact negatively on the virtual machines performance.
- If the system is not reliable or has a low availability, the images could not be accessed. Therefore, the IaaS resource provider needs to invest in having a good shared storage solution.
- The access to the shared storage by the physical machines (i.e. the hypervisor nodes) will consume resources and create undesirable VMM Noise. This VMM noise has been shown to have impact in the virtualized guests running on those hosts and is something to avoid in scientific computing environments [@Menon2005; @Ferreira; @Petrini:2003:CMS:1048935.1050204; @gavrilovska2007high].
Image transfer improvements
---------------------------
The use of shared storage may eliminate the image transfer into the nodes, but as we explained it may not be desirable. In this Section we will discuss several possibilities to decrease the image transfer time.
### On demand downloading {#sec:distribution:ondemand}
If no shared storage is in place, the most common approach in many CMFs is to transfer the images on-the-fly into the compute nodes when a request to launch a specific machine is made.
As we already exposed in Section \[sec:problem\] the penalty introduced by this method will vary according to the size of the image, the size of the request, the network connectivity of the infrastructure, the load on the catalog servers and the transfer protocol being used.
If the on demand download is the chosen option, the objective should be reducing the image transfer time. In this line there is a clear trend towards studying Peer-to-Peer (P2P) mechanisms in cloud infrastructures and data-centers. Zhang Chen et al. [@ChenZ2009] proposed an effective approach for virtual images provisioning based on BitTorrent. Laurikainen et al. conducted a research focused on the OpenNebula cloud middleware, taking only into account the replacement of the native image transfer method by either BitTorrent or Multicast [@Laurikainen2012]. Their conclusions showed that the existent image transfer manager (based on SSH) was rather inefficient for large requests and therefore it needed to be modified.
Wartel et al. studied BitTorrent among other solutions as the image transfer method for their legacy CERN cloud infrastructure [@Wartel2010]. This study shown a significant performance gain when using BitTorrent over the other studied methods (that included multicasting). In the same line, Yang Chen et al. have proposed a solution based on multicasting the images instead of a direct download from the image catalog, in combination with a more efficient scheduling [@Chen2009] algorithm. However, transfer an image using multicast into the nodes implies that the server is initiating the transfer (i.e. the server pushes the image into the nodes) instead of the image being pulled from the hosts. This also forces that the deployment of the images is synchronized, therefore introducing extra complexity to the scheduling algorithms that must take this synchronization into account.
Once the image is downloaded into the node, this image can be cached and reused afterwards in a subsequent request. This feature opens the door to the pre-deployment of images and the image pre-fetch, that will be discussed in Section \[sec:distribution:predeploy\] and Section \[sec:distribution:prefetch\] respectively. Multicasting is an interesting option for these two cases, since the deployment could be done in a coordinated way, without interfering with the scheduling algorithms, but when compared with multicast, using a P2P method introduces another advantage: the nodes that have an image available are part of the P2P network, participating actively in the transfer when a new request is made.
### Pre-deployment of images {#sec:distribution:predeploy}
A different approach towards the elimination of the image transfer prior to the image boot consists on the pre deployment of the whole or a portion of the image catalog into the physical machines. In some environments this might be a valid solution, but it is not affordable in large setups for several reasons.
First, in an infrastructure with a large catalog a considerable amount of disk space will be wasted on the nodes. Considering that not all the images will be spawned into all the nodes at a time, this resource consumption is not affordable. Second, the pre-deployment process can overload the catalog server when it is triggered if it is not properly scheduled or if the catalog is too large.
A CMF using this method should also consider that a recently uploaded image may not be immediately available to the user, since it has to be pre-seeded into the nodes in advance, so an alternative, on-demand method should still be available.
### Smart pre-fetch {#sec:distribution:prefetch}
Another possibility, related with the previous one, is performing a selective pre-deployment of the images into the nodes (i.e. smart pre-fetch). Instead of the passive deployment of the whole catalog (or a large portion of it) into the nodes, the scheduler may chose to trigger a download of an image in advance, so that it anticipates a user request.
Image popularity (i.e. how often an image is instantiated) can be used as a parameter to decide which images to pre-fetch. A naive approach could be summing up how many virtual machines have been instantiated from a given image. Figure \[fig:pop\] shows the image *popularity* for a set of $13500$ VMs execute by $150$ different users on a production cloud infrastructure during one year in ascending order. The Y-axis shows the number of instances that were based on a given image.
![Image popularity based on the number of Virtual Machines spawned per image. Each bar represents a different image.[]{data-label="fig:pop"}](image_popularity.pdf){width="1\linewidth"}
As it can be seen, even if this popularity calculation is too naive, a large proportion of the spawned instances is spawned from a small number of images. Some other authors have observed the same behaviour in some related works, such as Peng et al. [@Peng2012]. Therefore, if the CMFs could take advantage of the image popularity making those VMs available on some nodes the efficiency of the image booting process will improve.
Other methods {#sec:distribution:other}
-------------
Lagar-Cavilla et al. have developed Snowflock [@Lagar-Cavilla2011], a new model for cloud computing that introduces VM forking in a way similar to the well known and familiar concept of process forking. This method permits the cloning of an already running VM into several identical copies. However this is not transparent, and the users need to be aware of its semantics and program their application accordingly.
Some other authors have chosen a totally different approach relying on the fact that the image is not needed completely at once, therefore it can be divided into smaller chunks that will be transferred when they are needed. Peng et al. propose the usage of a collaborative network based on the sharing of similar image chunks [@Peng2012]. In their studies, they found that this approach was more efficient than the usage of a P2P network, but it requires a long running preprocessing step. Moreover, this is true for the cases analyzed, where the number of different VMs requested at a time was not big but this may not apply to other cases, such as scientific cloud providers where the same image may need to be spawned into several nodes.
The work from Nicolae et al. is also based on this approach. They implemented a self adaptive mechanism, based on lazy downloads of image chunks, based on previously recorded access patterns [@Nicolae2011].
Transfer method evaluation {#sec:evaluation}
==========================
There is no silver bullet for solving the image distribution problems, since all of the presented schemes have their advantages and disadvantages. In some situations, the usage of a shared backend may be the best solution but it would not fit others. For example, sites deploying virtual machines that need high availability may already use a shared backend so that it is possible to quickly recover a running machine from a failure, whereas sites devoted to HTC and HPC computing may not find this deployment appropriate. In this Section we evaluate several image transfer methods in a Cloud Management Framework.
Experimental setup {#sec:testbed}
------------------
The tests were performed in a dedicated cloud testbed running only these workloads. It comprises a *head node* hosting all the required services to manage the cloud infrastructure, an *image catalog* server and 24 *compute nodes* that will eventually host the spawned virtual machines. All of them are identical machines, with two 4-core IntelXeonE5345 2.33GHz processors, 16GB of RAM and one 140GB, 10.000 rpm hard disk.
The network setup of the testbed consists on two 10GbE switches, interconnected with a 10GbE link. All the hosts are evenly connected to these switches using a 1GbE connection.
The operating system being used for these tests is an Ubuntu Server 14.04 LTS, running the Linux 3.8.0 Kernel. In order to implement the solution proposed we have used the OpenStack [@web:openstack] cloud middleware, in its Icehouse (2014.1) version.
In order to execute the same tests easily we used a benchmarking as a service product developed for OpenStack: Rally [@web:rally]. This tool allows for the definition and repetition of benchmarks, so that the benchmarking tests can be reproduced later on.
OpenStack’s default method for distributing the images into the nodes is an on-demand deployment: the images are fetched from the catalog when the new virtual machine is scheduled into a compute (physical) node and its image cannot be found on that host.
The catalog service component (whose codename is Glance) stores the images using one of the many available backends, but independently of the backend used, the default transfer method is HTTP. When Glance stores the images in a filesystem it is possible to setup a shared filesystem so that the space where the images are stored by glance are available on the compute nodes. Other backends make possible to distribute the images over the network using different protocols and methods (for example, using the Ceph Rados Block Devices (RBD)). However, since we wanted to test the influence of the transfer from the catalog to the nodes, the default method was used.
Test results
------------
In order to evaluate the effect of the image transfer method we decided to stress the system, making requests that involved fetching a large number of images, as described in Table \[tab:exp setup\], using several methods: HTTP, FTP and BitTorrent. We used 5GB images and the scheduler was configured to evenly distribute the images among the hosts in the cluster in order to maximize the effect of the image transfer on the nodes. All the tests were done by triplicate.
Name VMs per host Different images \# of VMs
------- -------------- ------------------ -----------
1x192 8 1 192
2x96 8 2 192
4x48 8 4 192
8x24 8 24 192
: Request characteristics.[]{data-label="tab:exp setup"}
### HTTP transfer {#sec:http}
In the first place we transferred the images using HTTP, since it is the default image transfer method available on OpenStack. Figure \[fig:http requests\], shows the required time to boot the virtual machines for each of the requests in Table \[tab:exp setup\].
![ Waiting time in function of the number of instances requested when the images are fetched using HTTP. 1x192 means 1 request of 192 machines using the same image; 2x96, 2 requests of 96 machines using two different images, 4x48, 4 requests of 48 machines with four different images; and 8x24 8 requests of 24 machines with eight different images. []{data-label="fig:http requests"}](delivery_method_http.pdf){width="1\linewidth"}
The best scenario in these tests is where a user requests a single image (1x192 in Figure \[fig:http requests\]). This is mainly because of the effect of the cache that is available in each of the nodes. Once the image is downloaded in a node, all the subsequent virtual machines can be spawned using that cached image (this fact is also true for the other studied methods). The worst scenario is when the user requested 8 groups if 24 virtual machines (8x24 in Figure \[fig:http requests\]), since all the 8 images had to be downloaded into each of the nodes.
### FTP transfer {#sec:ftp}
As a second step we decided to substitute the built-in HTTP server with a dedicated FTP server, and use the File Transfer Protocol (FTP) instead. Figure \[fig:ftp requests\] shows again the results for the requests in Table \[tab:exp setup\].
![ Waiting time in function of the number of instances requested when the images are fetched using FTP. 1x192 means 1 request of 192 machines using the same image; 2x96, 2 requests of 96 machines using two different images, 4x48, 4 requests of 48 machines with four different images; and 8x24 8 requests of 24 machines with eight different images.[]{data-label="fig:ftp requests"}](delivery_method_ftp.pdf){width="1\linewidth"}
As it can be seen, the boot time is almost the same for both methods, being FTP more homogeneous over HTTP, resulting in a most uniform boot time for the machines.
### BitTorrent deployment {#sec:bt}
Both the HTTP (Section \[sec:http\]) and FTP (Section \[sec:ftp\]) are based on a centralized client-server model. In order to see how the system performs using a peer-to-peer (P2P) model we adapted OpenStack image delivery method to use BitTorrent. We chose it for several reasons: it is a protocol designed for to reduce the impact of transferring large amounts of data over the network [@cohen2008bittorrent]; it is widely used in a daily basis and there is a wide range of libraries, clients and applications available; moreover, due to this lively implementation ecosystem, we found that it could be easily integrated into OpenStack.
We chose libtorrent [@web:libtorrent] as the implementation for our tests. libtorrent has Python bindings, and since OpenStack is written entirely in Python it was easily integrable. Our *swarm* used the BitTorrent Distributed Hash Table (DHT) extension, so that we could use tracker-less torrents, although it is perfectly feasible to run a tracker. We configured the clients to run only 3 concurrent active downloads, since in preliminary tests we observed this was the best choice for our infrastructure.
The results for serving the same requests as in the HTTP and FTP cases are show in Figure \[fig:bt requests\].
![ Waiting time in function of the number of instances requested when the images are fetched using BitTorrent. 1x192 means 1 request of 192 machines using the same image; 2x96, 2 requests of 96 machines using two different images, 4x48, 4 requests of 48 machines with four different images; and 8x24 8 requests of 24 machines with eight different images.[]{data-label="fig:bt requests"}](delivery_method_bittorrent_3_active.pdf){width="1\linewidth"}
In our implementation a new torrent is generated whenever a new image is uploaded to the catalog. The torrent metadata is stored along with the ordinary image metadata so that whenever a download of this image is requested, both the normal HTTP and the torrent’s magnet link are provided to the compute node. If the node needs to download the image, and a magnet link is available, this *peer* (i.e. a BitTorrent client) will join the *swarm* (i.e. all peers sharing a torrent). Due to the segmented file transfer that BitTorrent implements, this *peer* is able to *seed* (i.e. send its available data) the received data to the other peers. This way, the original seeder of the image (i.e. the catalog server) is freed from sending that portion to every peer of the network.
Result comparison {#sec:results}
-----------------
A comparison of the three methods evaluated (that is, transfer the images using HTTP, FTP and BitTorrent, and profit from the images caching) is shown in Figure \[fig:start\_time\_multiple\].
![Seconds elapsed from request until all the machines were available. The VMs were based on a 5GB image, and they were spawned on 24 hosts.[]{data-label="fig:start_time_multiple"}](start_time_last.pdf){width="1\linewidth"}
Both FTP and HTTP threw similar results, being those limited by the bandwidth of the server node. Using BitTorrent, there is a significant transfer time reduction. In the worst scenario (8x24: running 192 virtual machines, distributed in 8 different images in 24 nodes) it was possible to start the 192 machines at approximately one third of the time required to run those machines using HTTP or FTP.
![Seconds elapsed from request until the first machine of the request is available. The VMs were based on a 5GB image, and they were spawned on 24 hosts.[]{data-label="fig:start_time_multiple_first"}](start_time_first.pdf){width="1\linewidth"}
If we take into account the boot time for the first machine of the request we can find interesting results. Figure \[fig:start\_time\_multiple\_first\] shows the elapsed time until the first machine is available. In this case, BitTorrent also outperforms the other transfer methods, making possible to deliver the machines earlier to the users except in the case of transferring only one image into all the nodes. In this case, HTTP and BitTorrent throw similar results.
Another important fact is that the adoption of BitTorrent not only has the effect of reducing the transfer time, but it also reduces the load of the catalog server. Since the image distribution leverages the advantages of the P2P network, where all the nodes participate in the transfer, the catalog does not need to transfer all the data to all of the nodes.
[0.7]{} ![CPU usage for a 192 VMs request using 8 different images (8x24).[]{data-label="fig:cpu"}](cpu_test02_8x24_http.png "fig:"){width="\linewidth"}
\
[0.7]{} ![CPU usage for a 192 VMs request using 8 different images (8x24).[]{data-label="fig:cpu"}](cpu_test02_8x24_ftp.png "fig:"){width="\linewidth"}
\
[0.7]{} ![CPU usage for a 192 VMs request using 8 different images (8x24).[]{data-label="fig:cpu"}](cpu_test02_8x24_bittorrent.png "fig:"){width="\linewidth"}
[0.7]{} ![Network usage for a 192 VMs request using 8 different images (8x24).[]{data-label="fig:net"}](net_test02_8x24_http.png "fig:"){width="\linewidth"}
\
[0.7]{} ![Network usage for a 192 VMs request using 8 different images (8x24).[]{data-label="fig:net"}](net_test02_8x24_ftp.png "fig:"){width="\linewidth"}
\
[0.7]{} ![Network usage for a 192 VMs request using 8 different images (8x24).[]{data-label="fig:net"}](net_test02_8x24_bittorrent.png "fig:"){width="\linewidth"}
As it can be seen in Figure \[fig:cpu\] and Figure \[fig:net\], using BitTorrent makes possible to satisfy the same request at a fraction of the CPU usage and specially network bandwidth when compared with HTTP and FTP, resulting in a better utilization of the resources.
However, using BitTorrent has its drawbacks also. It needs another running service (a tracker, although it could be avoided using a Distributed Hash Table (DHT)). Moreover, the creation of a torrent file whenever a new machine image is added to the catalog takes a considerable amount of time and resources, growing with the size of the file. Therefore the torrent will not be available as soon as the image is uploaded, but a lapse of time will be introduced. Since this operation is done only once in the lifetime of a virtual machine it can be considered as part of the initial upload process.
Efficient image distribution {#sec:cache scheduler}
============================
In the previous section we have made emphasis in the effect of the image distribution method on the boot time for a virtual machine. In all of the presented these tests we have started from a clean environment, meaning that there were no images cached in the nodes. The tests were designed to stress the infrastructure so that the image transfer effects could be clearly noticed. In this section we will evaluate the effect of taking into account the images cached in a physical node when making scheduling decisions under more realistic scenarios.
The default scheduling process in OpenStack has two steps: filtering and weighting.
The first step is the filtering phase. The scheduler applies a concatenation of filter functions to the initial set of available hosts, based on the host properties. When the filtering process has concluded, all the hosts in the final set are able to satisfy the user request. At this point, the weighting process starts so that the best suited host is selected.
The scheduler will apply to each of the hosts the same set of weighers functions $\mathrm{w}_i(h)$ for each host $h$. Each of those weigher functions will return a value considering the characteristics of the host received as input parameter. Therefore, total weight $\Omega$ for a node $h$ is calculated as follows:
$$\Omega = \sum^n{m_i\cdot \mathrm{N}{(\mathrm{w}_i(h))}}$$
Where $m_i$ is the multiplier for a weighter function, $\mathrm{N}{(\mathrm{w}_i(h))}$ is the normalized weight between $[0, 1]$ calculated via a rescaling like:
$$\mathrm{N}{(\mathrm{w}_i(h))} = \frac{\mathrm{w}_i(h)-\min{W}}{\max{W} - \min{W}}$$
where $\mathrm{w}_i(h)$ is weight function, and $\min{W}$, $\max{W}$ are the minimum and maximum values that the weigher has assigned for the set of weighted hosts.
Once the set of hosts have weights assigned to them, the scheduler will select the host with the maximum weight and will schedule the request into it. Eventually, if several nodes have the same winner weight, the final host will be randomly selected from that set.
In order to evaluate how the cache could improve the boot time, we tested four different scenarios: using the OpenStack’s default scheduling algorithm and using a cache-aware scheduler; using both HTTP and BitTorrent as the transfer methods. This we we could asses not only the effect of the cache but also the transfer method.
In our test environment all the hosts have the same hardware characteristics, so when they are empty they are equally eligible for running a machine. As explained, the nodes will get the same weight and finally a random selection is done. Therefore it is possible that a machine is scheduled in a node that does not have the image available, when there is another node with the same weight with the image cached. In the best case, the image is transferred only once (that, is for the first request), whereas in the worst case the image will have to be transferred every time it is used.
By default OpenStack has an image cache in each of the nodes, but the scheduler does not take it into account when selecting the host that will execute a machine. We developed several modules for OpenStack, allowing to weight the hosts taking into account their cached images. First of all, the nodes have report their cached images back to the scheduler. Afterwards, the cache weigher will simply weight the nodes as follows:
$$\mathrm{w}_{\mathrm{cache}}(h) = \left\{
\begin{array}{ll}
1 \quad \text{if image is cached} \\
0 \quad \text{otherwise}
\end{array} \right.$$
We did not apply any other sanity check in the weigher since this is not the purpose of our function (there are specific weighers and filters that should prevent to overload a host).
Therefore, with the above configuration in the cache-aware tests, the images were only transferred the first time they are scheduled, since all the subsequent requests will be scheduled in any of those hosts.
Evaluation
----------
In order to make a realistic evaluation, we executed different simulated request traces for each of the scenarios described before: that is, an scheduler with and without cache, using HTTP and BitTorrent.
We generated two arrival patterns using an exponential distribution [@knuth1981art]: one for a rate of 80 machines per hour and a second one for 100 machines per hour. For each of the requests we assigned an image chosen randomly from a given set of 4 images. Finally, the two resulting traces were executed in each of the four scenarios.
Figure \[fig:scatter 80\] shows the scatter plot of the seconds needed to boot each of the requests and its respective request pattern for 80 machines at an arrival rate of 80 machines per hour. Figure \[fig:density 80\] shows the kernel density estimation of the test.
![Seconds elapsed to boot a machine for 80 requests during 1 hour, with the corresponding requests trace. `nocache http` and `nocache bt` refer to the default scheduling method using HTTP and BitTorrent respectively, whereas `cache http` and `cache bt` refer to the cache-aware scheduler, using HTTP and BitTorrent respectively. []{data-label="fig:scatter 80"}](random_req_scatter_80.pdf){width="1\linewidth"}
![Kernel density estimation for the time elapsed to boot the requests in Figure \[fig:scatter 80\]. `nocache http` and `nocache bt` refer to the default scheduling method using HTTP and BitTorrent respectively, whereas `cache http` and `cache bt` refer to the cache-aware scheduler, using HTTP and BitTorrent respectively. []{data-label="fig:density 80"}](random_req_density_80.pdf){width="1\linewidth"}
Besides, Figure \[fig:scatter 100\] contains the plot for 100 machines at an arrival rate of 100 machines per hour, with the corresponding density function shown in Figure \[fig:density 100\].
![Seconds elapsed to boot a machine for 100 requests during 1 hour, with the corresponding requests trace. `nocache http` and `nocache bt` refer to the default scheduling method using HTTP and BitTorrent respectively, whereas `cache http` and `cache bt` refer to the cache-aware scheduler, using HTTP and BitTorrent respectively. []{data-label="fig:scatter 100"}](random_req_scatter_100.pdf){width="1\linewidth"}
![Kernel density estimation for the time elapsed to boot the requests in Figure \[fig:scatter 100\]. `nocache http` and `nocache bt` refer to the default scheduling method using HTTP and BitTorrent respectively, whereas `cache http` and `cache bt` refer to the cache-aware scheduler, using HTTP and BitTorrent respectively. []{data-label="fig:density 100"}](random_req_density_100.pdf){width="1\linewidth"}
As it can be seen in both Figures \[fig:scatter 80\] and \[fig:scatter 100\], in all evaluated scenarios the minimum values are similar and very low due to the effect of the cache. In the cases when the scheduler did not have this feature available there is still a random chance that a machine is scheduled in a node with the image cached, thus the observed results. The probability of using a node with the image already available increases with time (more nodes have been used and therefore more nodes have the image cached) and as a consequence the boot times for the last images was lower. When the cache-aware scheduler was used, only the first machines started require transfer to the nodes, hence the boot times are reduced to the minimum early in the execution of the trace.
On the other hand, Figures \[fig:density 80\] and \[fig:density 100\] thrown interesting results, considering the size of the requests. The best results are always obtained when using BitTorrent and a cache-aware scheduler. However, the next best case depends on the request pattern. In the case of a rate request of 100 machines per hour, using BitTorrent without a cache is better than using HTTP with a cache, but in the case of a rate of 80 machines per hour it is better to use the later. This observation is due to the fact that in the 100 machines case there is a large initial portion of images that need to be transmitted if compared with the 80 machines case, (as depicted by the dots between time $0$ and $500$ in Figures \[fig:scatter 80\] and \[fig:scatter 100\]. Therefore BitTorrent outperforms HTTP, as already explained in Section \[sec:results\]. The cache does not consider the images that are being fetched, therefore the scheduler cannot take them into account. As the 100 machines case requests machines at a higher rate they are being scheduler when the images are not yet available, thus the observed results.
Image pre-fetch
---------------
As already explained, the usage of the cache with BitTorrent outperforms all of the other methods. In order to evaluate its effect regarding the tests shown in Section \[sec:evaluation\] we recreated the same requests from Table \[tab:exp setup\] with the images already cached on the nodes. Obviously, in this test we do not evaluate the penalty introduced by the image transfer since there is no transfer at all, but it is interesting in order to evaluate the overall performance of the system. As it can be seen in Figure \[fig:cache requests\], the booting time was dramatically reduced in all cases: booting all the 192 machines was done in less than 45 seconds as the only delays introduced where due to the scheduling algorithm and the different management operations.
![ Waiting time in function of the number of instances requested when the images are cached in the nodes. 1x192 means 1 request of 192 machines using the same image; 2x96, 2 requests of 96 machines using two different images, 4x48, 4 requests of 48 machines with four different images; and 8x24 8 requests of 24 machines with eight different images. []{data-label="fig:cache requests"}](delivery_method_cache.pdf){width="1\linewidth"}
Conclusions and future work {#sec:conclusions}
===========================
In this paper, we have evaluated several methods for the distribution of virtual machine images into the compute nodes of a cloud infrastructure. Although the work was performed using the OpenStack cloud middleware, the results can be extrapolated to other CMFs using similar transfer methods.
Our experiments showed that composing a P2P network based on a well established protocol such as BitTorrent is a simple, feasible and realistic solution to decrease the burden on the server and to reduce the transfer time to a smaller fraction of time.
Moreover, we have also evaluated the usage of an image cache in each of the compute nodes. Using an image cache obviously reduces the boot time to a minimum, since there is no transfer at all, therefore having a scheduler that takes this into account is a need. We obtained the best results when we adapted the scheduler to take into account this cache, coupled with the usage of BitTorrent as the image transfer method. Therefore, both solutions are complementary: on the one hand we reduce the image transfer time when it is needed, and on the other hand we profit from the cached images whenever possible.
Taking into account those results, we think that there is room for future work and improvements in the cloud scheduling algorithms so as to improve the boot time for virtual machines. Cloud schedulers should be adapted to be cache-aware, implementing at the same time policies that would ensure a compromise between a fast boot time (i.e. the usage of a node with an image cached) and a fair utilization of the resources (i.e. not constricting all request to be scheduled only in one node).
On the other hand and taking into account the fact that users tend to request images comprised in an small set of images (as shown in Figure \[fig:pop\] and explained in Section \[sec:distribution:prefetch\]) we think that the usage of popularity based distribution algorithms (so that the most used images are available in the hosts) together with the cache aware scheduling would introduce remarkable improvements in the deployment times. In this regard, cloud monitoring [@Aceto2012] plays a key role, since one of the premises for doing a proper pre-fetching is proper monitoring so as to get proper metrics to evaluate if an image needs to be deployed or not.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors acknowledge the financial support from the European Commission (via EGI-InSPIRE Grant Contract number RI-261323).
The authors want also to thank the IFCA Advanced Computing and e-Science Group.
|
---
abstract: 'A set-labeling of a graph $G$ is an injective function $f:V(G)\to \mathcal{P}(X)$ such that the induced function $f^{\oplus}:E(G)\to \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u){\oplus}f(v)$ for every $uv{\in} E(G)$, where $X$ is a non-empty finite set and $\mathcal{P}(X)$ be its power set. A set-indexer of $G$ is a set-labeling such that the induced function $f^{\oplus}$ is also injective. A set-indexer $f:V(G)\to \mathcal{P}(X)$ of a given graph $G$ is called a topological set-labeling of $G$ if $f(V(G))$ is a topology of $X$. An integer additive set-labeling is an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$, whose associated function $f^+:E(G)\to \mathcal{P}(\mathbb{N}_0)$ is defined by $f^+(uv)=f(u)+f(v), uv\in E(G)$, where $\mathbb{N}_0$ is the set of all non-negative integers. An integer additive set-indexer is an integer additive set-labeling such that the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ is also injective. In this paper, we extend the concepts of topological set-labeling of graphs to topological integer additive set-labeling of graphs.'
author:
- 'N. K. Sudev'
- 'K. A. Germina'
title: '**A Study on Topological Integer Additive Set-Labeling of Graphs**'
---
**Key words**: Set-labeling of graphs, integer additive set-labeling of graphs, topological integer additive set-labeling of graphs.
**AMS Subject Classification : 05C78**
Introduction
============
For all terms and definitions of graphs and graph classes, not defined specifically in this paper, we refer to [@BM1], [@BLS], [@FH] and [@DBW] and for graph labeling concepts, we refer to [@JAG]. For terms and definitions in topology, we further refer to [@JD], [@KDJ1] and [@JRM]. Unless mentioned otherwise, all graphs considered here are simple, finite and have no isolated vertices.
Research on graph labeling commenced with the introduction of $\beta$-valuations of graphs in [@AR]. Analogous to the number valuations of graphs, the concepts of set-assignments, set-labelings and set-indexers of graphs are introduced in [@BDA1] as follows.
Let $G(V,E)$ be a given graph. Let $X$ be a non-empty set and $\mathcal{P}(X)$ be its power sets. Then, the set-valued function $f:V(G)\to \mathcal{P}(X)$ is called the [*set-assignment*]{} of vertices of $G$ respectively. In a similar way, we can define a set assignment of edges of $G$ as a function $g:E(G)\to \mathcal{P}(Y)$ and a set assignment of elements (both vertices and edges) of $G$ as a function $h:V(G)\cup E(G)\to \mathcal{P}(Z)$, where $Y$ and $Z$ are non-empty sets. The term set assignment is used for set-assignment of vertices unless mentioned otherwise.
A set-assignment of a graph $G$ is said to be a [*set-labeling*]{} or a [*set-valuation*]{} of $G$ if it is injective. A graph with a set-labeling $f:V(G)\to \mathcal{P}(X)$ is denoted by $(G,f)$ and is referred to as a [*set-labeled graph*]{} or a [*set-valued graph*]{}.
For a graph $G(V,E)$ and a non-empty set $X$ of cardinality $n$, a [*set-indexer*]{} of $G$ is defined as an injective set-valued function $f:V(G) \to \mathcal{P}(X)$ such that the function $f^{\oplus}:E(G)\to \mathcal{P}(X)-\{\emptyset\}$ defined by $f^{\oplus}(uv) = f(u ){\oplus}f(v)$ for every $uv{\in} E(G)$ is also injective, where $\mathcal{P}(X)$ is the set of all subsets of $X$ and $\oplus$ is the symmetric difference of sets. A graph that admits a set-indexer is called a [*set-indexed graph*]{}. It is proved in [@BDA1] that every non-empty graph has a set-indexer.
More studies on set-labeled and set-indexed graphs have been done in [@GK1], [@BDA1], [@AGAS] and [@BDA2]. Then, the notion of topological set-labeling of a graph is defined in [@AGPR] as follows.
Let $G$ be a graph and let $X$ be a non-empty set. A set-labeling $f:V(G)\to \mathcal{P}(X)$ is called a [*topological set-labeling*]{} of $G$ if $f(V(G))$ is a topology of $X$. A graph $G$ which admits a topological set-labeling is called a [*topologically set-labeled graph*]{}. More studies on topological set-labeling of different graphs have been done subsequently.
The [*sumset*]{} of two non-empty sets $A$ and $B$, denoted by $A+B$, is the set defined by $A+B=\{a+b: a\in A, b\in B\}$. For every non-empty set $A$, we have $A+\{0\}=A$. Hence, $\{0\}$ and $A$ are said to be the [*trivial summands*]{} of the set $A$. If $C=A+B$, where $A$ and $B$ are non-trivial summands of $C$, then $C$ is said to be the [*non-trivial sumset*]{} of $A$ and $B$. In this paper, by the terms sumsets and summands, we mean non-trivial sumsets and non-trivial summands respectively.
If any either $A$ or $B$ is countably infinite, then their sumset $A+B$ will also be a countably infinite set. Hence, all sets mentioned in this paper are finite sets.We denote the cardinality of a set $A$ by $|A|$ and the power set of a set $A$ by $\mathcal{P}(A)$. We also denote, by $X$, the finite ground set of non-negative integers that is used for set-labeling the elements of $G$.
Using the terminology and concepts of sumset theory, a particular type of set-labeling, called integer additive set-labeling, was introduced as follows.
Let $\mathbb{N}_0$ be the set of all non-negative integers. An [*integer additive set-labeling*]{} (IASL, in short) is an injective function $f:V(G)\to \mathcal{P}(\mathbb{N}_0)$ such that the associated function $f^+:E(G)\to \mathcal{P}(X)$ is defined by $f^+ (uv) = f(u)+ f(v)$ for any two adjacent vertices $u$ and $v$ of $G$. A graph $G$ which admits an IASL is called an IASL graph.
An [*integer additive set-labeling*]{} $f$ is an integer additive set-indexer (IASI, in short) if the induced function $f^+:E(G) \to \mathcal{P}(\mathbb{N}_0)$ defined by $f^+ (uv) = f(u)+ f(v)$ is injective. A graph $G$ which admits an IASI is called an IASI graph (see [@GA],[@GS0]).
Cardinality of the set-label of an element (a vertex or an edge) of a graph $G$ is called the [*set-indexing number*]{} of that element. An IASL (or an IASI) is said to be a $k$-uniform IASL (or $k$-uniform IASI) if $|f^+(e)|=k ~ \forall ~ e\in E(G)$. The vertex set $V(G)$ is called [*$l$-uniformly set-indexed*]{}, if all the vertices of $G$ have the set-indexing number $l$.
Motivated by the studies on topological set-labeling of graphs, we introduce the notion of topological integer additive set-labeling of graphs and study the structural properties and characteristics of the graphs which admit this type of set-labeling.
Topological IASL-Graphs
=======================
Note that no vertex of a given graph $G$ has the empty set $\emptyset$ as its set-labeling with respect to a given integer additive set-labeling. Hence, in this paper, we consider only non-empty subsets of the ground set $X$ for set-labeling the elements of $G$.
Analogous to topological set-labeling of graphs, we introduce the notion of topological integer additive set-labeling of certain graphs as follows.
[Let $G$ be a graph and let $X$ be a non-empty set of non-negative integers. An integer additive set-labeling $f:V(G)\to \mathcal{P}(X)-\{\emptyset\}$ is called a [*topological integer additive set-labeling*]{} (TIASL, in short) of $G$ if $f(V(G))\cup \{\emptyset\}$ is a topology of $X$. A graph $G$ which admits a topological integer additive set-labeling is called a [*topological integer additive set-labeled graph*]{} (in short, TIASL-graph).]{}
The notion of a topological integer additive set-indexer of a given graph $G$ is introduced as follows.
[A topological integer additive set-labeling $f$ is called a [*topological integer additive set-indexer*]{} (TIASI, in short) if the associated function $f^+:E(G)\to \mathcal{P}(X)$ defined by $f^+(uv)=f(u)+f(v); ~ u,v\in V(G)$, is also injective. A graph $G$ which admits an integer additive set-graceful indexer is called an [*topological integer additive set-indexed graph*]{} (TIASI-graph, in short).]{}
\[R-IASGL1.1\][For a finite set $X$ of non-negative integers, let the given function $f:V(G)\to \mathcal{P}(X)-\{\emptyset\}$ be an integer additive set-labeling on a graph $G$. Since the set-label of every edge $uv$ is the sumset of the sets $f(u)$ and $f(v)$, it can be observed that $\{0\}$ can not be the set-label of any edge of $G$. More over, since $f$ is a TIASL defined on $G$, $X$ must be the set-label of some vertex, say $u$, of $G$ and hence the set $\{0\}$ will be the set-label of a vertex, say $v$, and the vertices $u$ and $v$ are adjacent in $G$.]{}
Let $f$ be a topological integer additive set-indexer of a given graph $G$ with respect to a non-empty finite ground set $X$. Then, $\mathcal{T}=f(V(G))\cup \{\emptyset\}$ is a topology on $X$. Then, the graph $G$ is said to be a [*$f$-graphical realisation*]{} (or simply *$f$-realisation*) of $\mathcal{T}$. The elements of the sets $f(V)$ are called *$f$-open sets* in $G$.
An interesting question that arises in this context is about the existence of an $f$-graphical realisation for a topology $\mathcal{T}$ of a given non-empty set $X$. Existence of graphical realisations for certain topologies of a given set $X$ is established in the following theorem.
Let $X$ be a non-empty finite set of non-negative integers. A topology $\mathcal{T}$ of $X$, consisting of the set $\{0\}$ is graphically realisable.
Let $X$ be a non-empty finite set of non-negative integers and let $0\in X$. Consider a topology $\mathcal{T}$ of $X$ consisting of the set $\{0\}$. We need to construct a graph $G$ such that the vertices of $G$ have the (non-empty) set-labels taken from $\mathcal{T}$ in an injective manner. Let us proceed in this direction as explained below. Take a star graph $K_{1,|\mathcal{T}|-2}$. Label its central vertex by $\{0\}$ and label the other vertices by the remaining $|\mathcal{T}|-2$ non-empty open sets in $\mathcal{T}$. Clearly, this labeling is a TIASL defined on the graph $K_{1,|\mathcal{T}|-2}$ and hence $K_{1,|\mathcal{T}|-2}$ is a graphical realisation of $\mathcal{T}$.
It can also be observed that if we join two vertices $u$ and $v$ of the above mentioned TIASL-graph $K_{1,|\mathcal{T}|-2}$ by an edge, subject to the condition that $f(u)+f(v)\subseteq X$, the resultant graph will also be a graphical realisation of $\mathcal{T}$. Hence, there may exist more than one graphical realisations for a given topology of $X$. In view of this fact, we have to address the questions regarding the structural properties of TIASL-graphs. Hence, we proceed to find out the structural properties of TIASL-graphs.
\[P-TIASI1\] If $f:V(G)\to \mathcal{P}(X)-\{\emptyset\}$ is a TIASL of a graph $G$, then $G$ must have at least one pendant vertex.
Let $f$ be a TIASL defined on a graph $G$. Then, clearly $X\in f(V)$. That is, for some vertex $v\in V(G),~ f(v)=X$. Then, by Remark \[R-IASGL1.1\], $v$ is adjacent to a vertex whose set-label is $\{0\}$. Now we claim that, the vertex $v$ can be adjacent to only one vertex that has the set-label $\{0\}$. This can be proved as follows.
Let $u$ be a vertex that is adjacent to the vertex $v$ and let $a$ be a non-zero element of $X$. Also, let $l$ be the maximal element of $X$. If possible, let $a\in f(u)$. Then, the element $a+l \in f^+(uv)$ and is greater than $l$, which leads to a contradiction to the fact that $f^+(uv)=f(u)+f(v)\subseteq X$, as $f$ is an IASL of $G$. Therefore, the vertex of $G$ having the set-label $X$ can be adjacent to a unique vertex that has the set-label $\{0\}$. That is, the vertex $v$ with $f(v)=X$ is definitely a pendant vertex of $G$. Any TIASL-graph $G$ has at least one pendant vertex.
Figure \[G-TIASL\] depicts the TIASL, say $f$, of a graph $G$, with respect to a ground set $X=\{0,1,2,3,4\}$ and a topology $\mathcal{T}=\{\emptyset,X, \{0\},\{1\},\{2\},\{0,1\},\{0,2\},\{1,2\}, \{0,1,2\}\}$ of $X$, where $f(V(G))=\mathcal{T}-\{\emptyset\}$ is the collection of the set-labels of the vertices in $G$.
![An example to a TIASL-graph.[]{data-label="G-TIASL"}](G-TIASL)
An interesting question in this context is about the number of pendant vertices required in a TIASL-graph. Clearly, the answer to this question depends on the ground set $X$ and the topology $\mathcal{T}$ of $X$ we choose for labeling the vertices of $G$. Our next objective is to determine the minimum number of pendant vertices required in a TIASL-graph.
\[P-TIASI2\] Let $f:V(G)\to \mathcal{P}(X)-\{\emptyset\}$ is a TIASL of a graph $G$. Then, the vertices whose set-labels containing the maximal element of the ground set $X$ are pendant vertices which are adjacent to the vertex having the set-label $\{0\}$.
For given ground set $X$ of non-negative integers, let $f:V(G)\to \mathcal{P}(X)-\emptyset$ be a TIASL of $G$. Let $l$ be the maximal element of the ground set $X$. Let $v$ be a vertex of $G$ whose set-label contains the element $l\in X$. Let $u$ be an adjacent vertex of $v$ whose set-label contains a non-zero element $b\in X$. Then, $b+l\not \in X$, contradicting the fact that $f$ is an IASL of $G$. If $l\in f(v) ~\text{for} ~ v\in V(G)$, then its adjacent vertices can have a set-label $\{0\}$. That is, all the vertices whose set-labels contain the maximal element of the ground set $X$ must be adjacent to a unique vertex whose set-label is $\{0\}$.
Invoking Proposition \[P-TIASI1\] and Proposition \[P-TIASI2\], we have
\[P-TIASI3\] Let $X$ be the ground set and $\mathcal{T}$ be the topology of $X$ which are used for set-labeling the vertices of a TIASL-graph $G$. Then, an element $x_r$ in $X$ can be an element of the set-label $f(v)$ of a vertex $v$ of $G$ if and only if $x_r+x_s\le l$, where $x_s$ is any element of the set-label of another vertex $u$ which is adjacent to $v$ in $G$ and $l$ is the maximal element in $X$.
The following result is an immediate consequence of the above propositions.
\[P-TIASI2a\] If $G$ has only one pendant vertex and if $G$ admits a TIASL, then $X$ is the only set-label of the vertices of $G$ containing the maximal element of $X$.
What is the minimum number of pendant vertices required for a graph which admits a TIASL with respect to a given topology $\mathcal{T}$ of the ground set $X$? The following proposition provides a solution to this question.
Let $\mathcal{T}$ be a given topology of the ground set $X$. Then,
1. the minimum number of pendant edges incident on a particular vertex of a TIASL-graph is equal to the number of $f$-open sets in $f(V(G))$ containing the maximal element of the ground set $X$
2. the minimum number of pendant vertices of a TIASL-graph $G$ is the number of $f$-open sets in $\mathcal{T}$, each of which is the non-trivial summand of at most one $f$-open set in $\mathcal{T}$.
Let $G$ be a graph which admits a TIASL $f$ with respect to a topology $\mathcal{T}$ of the ground set $X$.
[*Case (i):*]{} If an $f$-open set $X_i$ contains the maximal element of $X$, then by Proposition \[P-TIASI2\], $X_i$ can be the set-label of a pendant vertex, say $v_i$, which is adjacent to the vertex having set-label $\{0\}$. Hence, every $f$-open set containing the maximal element of $X$ must be the set-label of a pendant vertex that is adjacent to a single vertex whose set-label is $\{0\}$. Therefore, the minimum number of pendant edges incident on a single vertex is the number of $f$-open sets in $\mathcal{T}$ containing the maximal element of $X$.
[*Case (ii):*]{} If an $f$-open set $X_i$ is not a non-trivial summand of any $f$-open sets in $\mathcal{T}$, then the vertex with set-label $X_i$ can be adjacent only to the vertex with set-label $\{0\}$. If $X_i$ is the non-trivial summand of exactly one $f$-open set in $\mathcal{T}$, then the vertex $v_i$ with the set-label $X_i$ can be adjacent only to one vertex say $v_j$ with set-label $X_j$, where $X_i+X_j\subseteq X$. If $X_i$ is the non-trivial summand of more than one $f$-open sets in $\mathcal{T}$, then the vertex with set-label $X_i$ can be adjacent to more than one vertex of $G$ and hence $v_i$ need not be a pendant vertex. Therefore, the minimum number of pendant vertices in $G$ is the number of $f$-open sets in $\mathcal{T}$, each of which is the non-trivial summand of at most one $f$-open set in $\mathcal{T}$.
Does every graph with one pendant vertex admit a TIASL? The answer to this question depends upon the choice of the ground set $X$. Hence, let us verify the existence of TIASL for certain standard graphs having pendant vertices by choosing a ground set $X$ suitably. For this, first consider the following graphs.
Let $G$ be a graph on $n$ vertices and let $P_m$ be a path that has no common vertex with $G$. We call the graph obtained by identifying one vertex of $G$ and one end vertex of $P_m$ an [*$(n,m)$-ladle*]{}.
If $G$ is a cycle $C_n$, then this ladle graph is called an [*(n,m)-tadpole graph*]{} or a [*dragon graph*]{}. If $m=1$ in a tadpole graph, then $G$ is called an [*$n$-pan*]{}.
If $G$ is a complete graph on $n$ vertices, then the corresponding $(n,m)$-ladle graph is called an [*$(n,m)$-shovel*]{}.
Now, we proceed to discuss the admissibility of TIASL by these types of graphs. The following result establishes the admissibility of TIASL by a pan graph.
\[P-TIASL-p\] A pan graph admits a topological integer additive set-labeling.
Let $G$ be an $m$-pan graph. Let $v$ be the pendant vertex and $v_1,v_2, \ldots, v_n$ be the vertices of $C_n$. Without loss of generality, let $v_1$ be the unique vertex adjacent to $v$ in $G$. Label the vertices of the cycle $C_n$ of $G$ in such a way that we have $f(v_1)=\{0\}, f(v_i)=\{0,1,\ldots, i-1\}: 2\le i\le n$. Now, let $X=\{0,1,2,3,\ldots, m\}$, where $m\ge 2n-3$ and label the pendant vertex $v$ by the set $X$. Hence, the collection of the set-labels of the vertices of $G$ is $\mathcal{A}=\{\{0\}, \{0,1\}, \{0,1,2\}, \ldots, \{0,1,2,\ldots,n-1\}, X\}$. Clearly, the set $\mathcal{T}=\mathcal{A}\cup \{\emptyset\}$ is a topology on $X$. Therefore, this labeling of $G$ is a TIASL of $G$. Hence, the $n$-pan $G$ admits a TIASL.
Figure \[fig:G-TIASL4a\] illustrates the admissibility of TIASL by an $n$-pan with respect to the ground set $X=\{0,1,2,3,\ldots, 2n-3\}$.
![An $n$-pan graph with a TIASL defined on it.[]{data-label="fig:G-TIASL4a"}](G-TIASL4a){width="0.75\linewidth"}
We now proceed to verify the admissibility of TIASL by the general tadpole graphs.
\[P-TIASL-t\] A tadpole graph admits a topological integer additive set-labeling.
Let $G$ be an $(n,m)$-tadpole graph. Let $\{v_1,v_2, v_3, \ldots, v_n\}$ be the vertex set of $C_n$ and let $\{u_0,u_1,u_2,u_3, \ldots, u_m\}$ be the vertex set of $P_m$. Without loss of generality, let $u_0$ be the pendant vertex of $P_m$ in $G$. Identify the vertex $u_m$ of $P_m$ and the vertex $v_1$ of the cycle to form a tadpole graph. Let us define an IASL $f$ on $G$ as follows. Label the vertex $u_1$ by the set $\{0\}$, the vertex $u_2$ by the set $\{0,1\}$ and in general, the vertex $u_i$ by the set $\{0,1,2,\ldots, i-1\}$, for $1\le i\le m$. Therefore, the set-label of the vertex $u_m=v_1$ is $\{0,1,2,\ldots, m-1\}$. Now, label the remaining vertices of $C_n$ in $G$ as follows. Label the vertex $v_2$ by the set $\{0,1,2,\ldots, m\}$ and in general, label the vertex $v_j$ by the set $\{0,1,2,\ldots, m+j-2\}$. Now, choose the set $X=\{0,1,2,\ldots,l\}$, where $l\ge 2(m+n)-5$. Now, the only vertex of $G$ that remains to be labeled is the pendant vertex. Label the vertex $u_0$ by the set $X$. Then, the collection of set-labels of $G$ is $\mathcal{A}=\{\{0\}, \{0,1\}, \{0,1,2\}, \ldots, \{0,1,2,\ldots,m+n-2\},X\}$. Clearly, the set $\mathcal{T}=\mathcal{A}\cup \{\emptyset\}$ is a topology on $X$. Hence, this labeling is a TIASL defined on $G$.
Figure \[fig:G-TIASL4b\] illustrates the admissibility of TIASL by the $(m,n)$-tadpole graph with respect to the ground set $X=\{0,1,2,\ldots,2(m+n)-5\}$.
![An $(n,m)$-tadpole graph with a TIASL defined on it.[]{data-label="fig:G-TIASL4b"}](G-TIASL4b){width="0.9\linewidth"}
We can extend the above results to the shovel graphs also. The following result establishes the admissibility of TIASL by shovel graphs by properly choosing the ground set $X$.
\[P-TIASL-s\] The $(n,m)$-shovel graph admits a topological integer additive set-labeling.
Let $G$ be an $(n,m)$-shovel graph. Let $\{v_0,v_1,v_2,v_3, \ldots, v_m\}$ be the vertex set of $P_m$ and $\{v_m,v_{m+1}, v_{m+2}, \ldots, v_{m+n-1}\}$ be the vertex set of $K_n$ in the given shovel graph $G$, where $v_0$ is the pendant vertex of $P_m$ (and hence of $G$). Define an IASL $f$ on $G$ which assigns set-labels to the vertices of $G$ injectively in such a way that any vertex $v_i$ has the set-label $\{0,1,2,\ldots, i-1\}$, for $1\le i \le m+n-1$. Note that, the pendant vertex $v_0$ remains unlabeled at the moment. It can be noted that the maximal element of the set-label $f^+(v_{m+n-2}v_{m+n-1})$ is $2(m+n)-5$. Hence, choose the set $X=\{0,1,2,3,\ldots, 2(m+n)-5\}$ and label the pendant vertex $v_0$ by the set $X$ itself. Therefore, $f(V(G))=\{\{0\}, \{0,1\}, \{0,1,2\}, \ldots, \{0,1,2,\ldots,m+n-2\},X\}$ and $f(V(G))\cup \{\emptyset\}$ is a topology on $X$. Hence, $f$ is a TIASL on $G$.
Figure \[fig:G-TIASL5\] depicts the admissibility of TIASL by an $(n,m)$-shovel graph with ground set $X=\{0,1,2,3,\ldots, 2(m+n)-5\}$.
![An $(n,m)$-shovel graph with a TIASL defined on it.[]{data-label="fig:G-TIASL5"}](G-TIASL5){width="0.85\linewidth"}
The above propositions raise the question whether the existence of a pendant vertex in a given graph $G$ results in the admissibility of TIASL by it. The choice of $X$ in all the above results played a major role in establishing a TIASL for $G$. The following is a necessary and sufficient condition for a given graph with at least one pendant vertex to admit a TIASL.
\[T-TIASI1\] A graph $G$ admits a TIASL if and only if $G$ has at least one pendant vertices.
Let $G$ be a graph which admits a TIASL, say $f$. Then, the ground set $X\in f(V(G))$. Hence, by Proposition \[P-TIASI2\], the vertex with the set-label $X$ must be a pendant vertex. More over, by Proposition \[P-TIASI2\], if the set-label of a vertex $v_i$ contains the maximal element of $X$, then $v_i$ is a pendant vertex. Then, $G$ has at least one pendant vertex.
Conversely, assume that $G$ has at least one pendant vertex. Let $V(G)=\{v_1,v_2,v_3\ldots,v_n\}$. Without loss of generality, let $v_1$ be a pendant vertex of $G$. Now, label the vertex $v_i$ by the set $\{0,1,2,3, \ldots, i-1\}$ for $1\le i\le n$. Then, as explained in the above results, the maximal element in all set-labels of edges of $G$ is $2n-3$. Choose $X=\{0,1,2,\ldots, 2n-3\}$ and label the pendant vertex $v_1$ by the set $X$. Then, $f(V(G))=\{\{0\},\{0,1\},\{0,1,2\},\ldots, \{0,1,2,\ldots,n-1\}, X\}$. Therefore, $f(V(G))\cup \{\emptyset\}$ is a topology on $X$ and hence this labeling is a TIASL of $G$.
Theorem \[T-TIASI1\] gives rise to the following result.
\[T-TIASI1a\] Let $G$ be a graph with a pendant vertex $v$ which admits a TIASL, say $f$, with respect to a ground set $X$. Let $f_1$ be the restriction of $f$ to the graph $G-v$. Then, there exists a collection $\mathcal{B}$ of proper subsets of $X$ which together with $\{\emptyset\}$ form a topology of the union of all elements of $\mathcal{B}$.
Let $G$ be a graph with one pendant vertex, say $v$ and $X$ be the ground set for labeling the vertices of $G$. Choose the collection $\mathcal{B}$ of proper subsets of $X$ which contains the set $\{0\}$ and has the cardinality $n-1$ such that the sum of the maximal elements of any two sets in it is less than or equal to the maximal element of $X$ and the union of any two sets and the intersection of any two non-singleton sets in $\mathcal{B}$ are also in $\mathcal{B}$. Then, by Theorem \[T-TIASI1\], the set-labeling $f$ under which the pendant vertex $v$ is labeled by the set $X$ and other vertices of $G$ are labeled by the elements of $\mathcal{B}$ is a TIASL of $G$.
Let $f_1$ be the restriction of $f$ to the graph $G-v$. Therefore, $\mathcal{B}=f_1(V(G-v))$. Now let $B=\bigcup_{B_i\in \mathcal{B}}B_i$ and let $\mathcal{T'}=\mathcal{B}\cup \{\emptyset\}$. Since $G$ has only one end vertex, by Proposition \[P-TIASI2a\], no element of $\mathcal{A}$ contains the maximal element of $X$. Therefore, $B$ also does not contain the maximal element of $X$. Since the union of any number of sets in $\mathcal{B}$ is also in $\mathcal{B}$, the union of the elements in $\mathcal{T'}$. Then, $B$ belongs to $\mathcal{B}$ and to $\mathcal{T'}$ and $B$ is the maximal element of $\mathcal{T'}$. Since the intersection of any two non-singleton sets in $\mathcal{B}$ is also in $\mathcal{B}$ and $\emptyset \in \mathcal{T'}$, the finite intersection of elements in $\mathcal{T'}$ is also in $\mathcal{T'}$. The set $\mathcal{T'}=\mathcal{B}\cup \{\emptyset\}$ is a topology of the maximal set $B$ in $\mathcal{B}$.
[ If $v$ is the only pendant vertex of a given graph $G$, then the collection $\mathcal{B}=f(V(G-v))$, chosen as explained in Theorem \[T-TIASI1a\] does not induce a topological IASL on the graph $G-v$, since $f^+(uw)\neq f(u)+f(w)$, for some edge $uw\in E(G-v)$.]{}
TIASLs with respect to Certain Topologies
=========================================
The number of elements in the ground set $X$ is very important in all the studies of set-labeling of graphs. Keeping this in mind, we define
[ The minimum cardinality of the ground set $X$ required for a given graph to admit a topological IASL is known as the [*topological set-indexing number*]{} (topological set-indexing number) of that graph.]{}
In this section, we discuss the existence and admissibility of topological IASLs with respect to some standard topologies like indiscrete topologies and discrete topologies.
A topology $\mathcal{T}$ is said to be an indiscrete topology of $X$ if $\mathcal{T}=\{\emptyset,X\}$. Hence the following result is immediate.
\[T-TIASI4\] A graph $G$ admits a TIASL with respect to the indiscrete topology $\mathcal{T}$ if and only if $G\cong K_1$.
Let $v$ be the single vertex of the graph $G=K_1$. Let $X$ be the ground set for set-labeling $G$. Let $f(v)=X$. Then $f(V)=\{X\}$ and $f(V)\cup \{\emptyset\}=\{\emptyset, X\}$, which is the indiscrete topology on $X$. Conversely, assume that $G$ admits a TIASL with respect to the indiscrete topology $\mathcal{T}$ of the ground set $X$. Then, $f(V(G))=\mathcal{T}-\{\emptyset\}=\{X\}$, a singleton set. Therefore, $G$ can have only a single vertex. That is, $G\cong K_1$.
From Proposition \[T-TIASI4\], we have the following result.
\[P-TIASI4a\] The topological set-indexing number of $K_1$ is $1$.
Another basic topology of a set $X$ is the [*Sierpenski’s topology*]{}. If $X$ is a two point set, say $X=\{0,1\}$, then the topology $\mathcal{T}_1=\{\emptyset, \{0\},X\}$ and $\mathcal{T}_2=\{\emptyset, \{0\},X\}$ are the Sierpenski’s topologies. The following result establishes the conditions required for a graph to admit a TIASL with respect to the Sierpenski’s topology.
\[T-TIASI5\] A graph $G$ admits a TIASL with respect to the Sierpenski’s topology if and only if $G\cong K_2$.
Let $G$ be the given graph, with vertex set $V$, which admits a TIASL with respect to the Sierpenski’s topology. Let a two point set $X=\{0,1\}$ be the ground set used for set-labeling the graph $G$. Then, $f(V)=\{\{0\},X\}$. Therefore, $G$ can have exactly two vertices. That is, $G\cong K_2$.
Conversely, assume that $G\cong K_2$. Let $u$ and $v$ be the two vertices of $G$. Choose a two point set $X$ as the ground set to label the vertices of $G$. Label the vertex $u$ by $X$. Then by Proposition \[P-TIASI1\], $v$ must have the set-label $\{0\}$. Then $f(V(G))=\{\{0\},X\}$. Then, $f(v(G))\cup \{\emptyset\}$ is a topology on $X$, which is a Sierpenski’s topology of $X$. Therefore, $G\cong K_2$ admits a TIASL with respect to the Sierpenski’s topology.
From the above result, we observe the following.
The only Sierpenski’s topology of the two point set $X=\{0,1\}$ that induces a TIASL on the graph $K_2$ is $\mathcal{T}=\{\emptyset,\{0\},X\}$.
In view of Proposition \[T-TIASI5\], we claim that for any ground set $X$ containing two or more elements, one of which is $0$, induces a TIASL on $K_2$. Therefore, the following result is immediate.
The topological set-indexing number of $K_2$ is $2$.
The following results are the immediate consequences of \[T-TIASI1\].
\[P-TIASI6\] For $n\ge3$, no complete graph $K_n$ admits a TIASL.
The proof follows from Theorem \[T-TIASI1\] and from the fact that a complete graph on more than two vertices does not have any pendant vertex.
\[P-TIASI7\] For $m,n\ge 2$, no complete bipartite graph $K_{m,n}$ admits a TIASL.
The proof is immediate from the fact that a complete bipartite graph has no pendant vertices.
A path graph $P_m$ admits a TIASL.
Every path graph $P_m$ has two pendant vertices and hence satisfy the condition mentioned in Theorem \[T-TIASI1\]. Hence $P_m$ admits a TIASL.
Every tree admits a TIASL.
Since every tree $G$ has at least two pendant vertices, by Theorem \[P-TIASI1\], $G$ admits a TIASL.
\[P-TIASI8\] No cycle graph $C_n$ admits a TIASL.
A cycle does not have any pendant vertex. Then, the proof follows immediately by Theorem \[T-TIASI1\].
In view of the above results, we arrive at the following inference.
For $k\ge 2$, no $k$-connected graph admits a TIASL with respect to a ground set $X$.
No biconnected graph $G$ can have pendant vertices. Hence, by Theorem \[T-TIASI2\], $G$ can not admit a TIASL.
We have already discussed the admissibility of a TIASL by a graph with respect to the indiscrete topology of the ground set $X$. In this context, it is natural to ask whether a given graph admits the TIASL with respect to the discrete topology of a given set $X$. The following theorem establishes the condition required for $G$ to admit a TIASL with respect to the discrete topology of $X$.
\[T-TIASI2\] A graph $G$, on $n$ vertices, admits a TIASL with respect to the discrete topology of the ground set $X$ if and only if $G$ has at least $2^{|X|-1}$ pendant vertices which are adjacent to a single vertex of $G$.
Let $|X|=m$. Let the graph $G$ admits a TIASL $f$ with respect to the discrete topology $\mathcal{T}$ of $X$. Therefore, $f(V(G))=\mathcal{P}(X)-\{\emptyset\}$. Then, $|f(V(G))|=2^{|X|}-1$. Now, let $l$ be the maximal element in $X$. The number of subsets of $X$ containing $l$ is $2^{m-1}$. Since $f$ is a TIASL with respect to the discrete topology, all these sets containing $l$ must also be the set-labels of some vertices of $G$. By Proposition \[P-TIASI2\], all these vertices must be adjacent to the vertex whose set-label is $\{0\}$. By Proposition \[P-TIASI3\], no two vertices whose set-labels contain $l$ can be adjacent among themselves or to any other vertex which has a set-label with non-zero elements. Therefore, $G$ has $2^{m-1}$ pendant vertices which are adjacent to a single vertex whose set-label is $\{0\}$.
Conversely, let $G$ be a graph with $n=2^{|X|}-1$ vertices such that at least $2^{|X|-1}$ of them are pendant vertices incident on a single vertex of $G$. Label these pendant vertices by the $2^{|X|-1}$ subsets of $X$ containing the maximal element $l$ of $X$. Label remaining vertices of $G$ by the remaining $2^{|X|-1}-1$ subsets of $X$ which do not contain the element $l$, in such a way that the sum of the maximal elements of the set-labels of two adjacent vertices is less than or equal to $l$. This labeling is clearly a TIASL on $G$. That is, $G$ admits a TIASL with respect to the discrete topology of $X$.
Figure \[G-TIASL2a\] depicts the existence of a TIASL with respect to the discrete topology of a ground set $X=\{0,1,2,3\}$ for a graph $G$.
![*A TIASL of graph with respect to the discrete topology of $X$.*[]{data-label="G-TIASL2a"}](G-TIASL2a)
Since the necessary and sufficient condition for a graph to admit a TIASL with respect to the discrete topology of ground set $X$ is that $G$ has at least $2^{|X|-1}$ pendant vertices that incident at a single vertex of $G$, no paths $P_n; ~n\ge 3$, cycles, complete graphs and complete bipartite graphs can have a TIASL with respect to discrete topology of $X$.
Theorem \[P-TIASI2\] gives rise to the following results also.
A graph on even number of vertices does not admit a TIASL with respect to the discrete topology of the ground set $X$.
If a graph on $n$ vertices admits a TIASL with respect to the discrete topology of the ground set $X$, then by Theorem \[T-TIASI2\], $n=2^{|X|-1}$, which can never be an even integer. Therefore, $G$ on even number of vertices does not admit a TIASL with respect to the discrete topology of $X$.
A star graph $K_{1,r}$ admits a TIASL with respect to the discrete topology of the ground set $X$, if and only if $r=2^{|X|}-2$.
First assume that the star graph $G=K_{1,r}$ admits a TIASL $f$ with respect to the discrete topology of the ground set $X$. Then, $f(V(G))=\mathcal{P}(X)-\{\emptyset\}$. That is, $|f(V(G))|=2^{|X|}-1$. Hence, $G$ must have $2^{|X|}-1$ vertices. That is, $r+1=2^{|X|}-1$. Therefore, $r=2^{|X|}-2$.
Conversely, consider a star graph $G=K_{1,r}$, where $r=2^n-2$ for some positive integer $n$. Choose a set $X$ with cardinality $n$, which consists of the element $0$. Note that the number of non-empty subsets of $X$ is $2^n-1$. Define a set-labeling $f$ of $G$ which assigns $\{0\}$ to the central vertex of $G$ and the other non-empty subsets of $X$ to the pendant vertices of $G$. Clearly, this labeling is an IASL of $G$. Also, $f(V(G))=\mathcal{P}(X)-\{\emptyset\}$. Therefore, $f$ is a TIASL of $G$ with respect to the discrete topology of $X$.
Figure \[G-TIASL2b\] illustrates the existence of a TIASL with respect to the discrete topology of the ground set $X$ for a star graph.
Conclusion
==========
In this paper, we have discussed the concepts and properties of topological integer additive set-indexed graphs analogous to those of topological IASI graphs and have done a characterisation based on this labeling.
We note that the admissibility of topological integer additive set-indexers by the given graphs depends also upon the number and nature of the elements in $X$ and the topology $\mathcal{T}$ of $X$ concerned. Hence, choosing a ground set $X$ is very important in the process of checking whether a given graph admits a TIASL-graph.
Certain problems in this area are still open. Some of the areas which seem to be promising for further studies are listed below.
[Characterise different graph classes which admit topological integer additive set-labelings.]{}
[Estimate the topological set-indexing number of different graphs and graph classes which admit topological integer additive set-labelings.]{}
[Verify the existence of topological integer additive set-labelings for different graph operations and graph products.]{}
[Establish the necessary and sufficient condition for a graph to admit topological integer additive set-indexer.]{}
[Characterise the graphs and graph classes which admit TIASI.]{}
The integer additive set-indexers under which the vertices of a given graph are labeled by different standard sequences of non negative integers, are also worth studying. All these facts highlight a wide scope for further studies in this area.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors would like to thank the anonymous reviewer for his/her insightful suggestions and critical and constructive remarks which made the overall presentation of this paper better.
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---
abstract: 'The transition from the microscopic Heisenberg model to the macroscopic elastic theory is carried out for the chiral magnetics of MnSi-type with the $B20$ crystal structure. Both exchange and Dzyaloshinskii–Moriya (DM) interactions are taken into account for the first, second, and third magnetic neighbors. The particular components of the DM vectors of bonds are found, which are responsible for (i) the global magnetic twist and (ii) the canting between four different spin sublattices. A possible mechanism for effective reinforcement of the global magnetic twist is suggested: it is demonstrated that the components of the DM vectors normal to corresponding interatomic bonds become very important for the twisting power. The Ruderman–Kittel–Kasuya–Yosida (RKKY) theory is used for model calculation of exchange parameters. It is found that just the interplay between the exchange parameters of several magnetic shells rather than the signs of DM vectors can be responsible for the concentration-induced reverse of the magnetic chirality recently observed in the Mn$_{1-x}$Fe$_{x}$Ge crystals.'
author:
- 'Viacheslav A. Chizhikov[^1] and Vladimir E. Dmitrienko[^2]'
title: 'Multi-shell contribution to the Dzyaloshinskii–Moriya spiralling in MnSi-type crystals'
---
Introduction {#sec:intro}
============
Chiral spin textures are studied now very actively for possible spin self-organization, unusual quantum transport phenomena and spintronic applications. A well established mechanism of spin chirality is the spin-orbit Dzyaloshinskii–Moriya (DM) interaction which is responsible for intricate magnetic patterns in the MnSi-type crystals. Even half a century after the discovery of the strange magnetic properties of MnSi [@Williams1966; @Shinoda1966], the magnetics with the $B20$ crystal structure still amaze us with the variety and complexity of their magnetic phases and electronic properties [@Stishov2011] contrasting the simplicity of the crystalline arrangement (only four magnetic atoms per a unit cell). Among the magnetic phases, both experimentally observed and hypothetical, are simple and cone helices [@Ishikawa1976; @Motoya1976], the Skyrmions and their lattices associated with the recently found $A$-phase [@Rossler2006; @Grigoriev2006a; @Munzer2010; @Adams2011], possible 3D structures [@Tewari2006; @Binz2006; @Hamann2011] similar to the blue phases of liquid crystals, etc. This variety is due to, first, the lack of inverse and mirror symmetries, which gives rise to the chirality of the crystalline and spin structures; second, the frustrations [@Hopkinson2006; @Chizhikov2012] resulting from nontrivial topology of the trillium lattice, that introduces a competition of various interactions between different pairs of atoms.
Beginning from the discovery of the chiral magnetic properties of MnSi in 1976 [@Ishikawa1976; @Motoya1976] till present day the most used approach to describe and predict twisted magnetic structures remains the phenomenological theory based on the Ginsburg–Landau free energy with an additional term first introduced by Dzyaloshinskii [@Dzyaloshinskii58; @Bak80; @Nakanishi1980; @Tewari2006; @Binz2006]. However, the approach, which uses our knowledge of the system symmetry, is not able to say anything about the values of coefficients in the free energy, for instance, how they are connected with the real interactions between atoms.
The microscopic theories, e.g. the model of the classical Heisenberg ferromagnetics with a spin-orbit term originally developed by Moriya [@Moriya60b], have in their turn the shortcoming that the number of variables, including the spins of all the magnetic atoms, is infinite, and therefore they are difficult to use for any analytical computations. Nevertheless, in spite of some doubts about validity the Heisenberg model in the itinerant magnetics, it is often used for digital simulations [@Hamann2011; @Hopkinson2009]. For this model it is of great importance to know the parameters of different interactions between pairs of spins: $J_{ij}$ for exchange coupling of $i$th and $j$th atoms, and vectors $\mathbf{D}_{ij}$ for DM interactions. Those parameters can be obtained in two ways: from comparison of theoretical results with an experimental data and by means of [*ab intio*]{} calculations.
In our recent work [@Chizhikov2012], using a coarse-grained approximation, the phenomenological constants ${\cal J}$ and ${\cal D}$ of the elastic free energy have been connected with the corresponding parameters of the microscopic theory. A surprising detail was the recognition of inter-sublattice canting as a new microscopic feature of the magnetic structures in the MnSi-type crystals. Fig. [\[fig-canting\]]{} shows the difference between the twist and the canting by example of a 1D spin chain. An experimental confirmation of the canting could give us an argument for applicability of the Heisenberg model to the itinerant ferromagnetics of MnSi-type [@Dmitrienko2012]. Similar features could be observed in dielectric chiral magnetics like recently studied multiferroics Cu$_2$OSeO$_3$ [@Seki2012; @Adams2012; @GongXiang2012], BiFeO$_3$ [@Zvezdin2012] and langasite-type crystals [@Pikin2012].
Though the theory with only nearest neighbors interactions is frequently used to simulate spin structures [@Hopkinson2009; @Hamann2011; @Chizhikov2012], it does not always give an adequate physical description. Indeed, very often the coupling between second and next neighbors is comparable or even more considerable than that of the nearest neighbors. Thus, for example, in the weak ferromagnetic $\alpha$-Fe$_2$O$_3$, the antiferromagnetic exchange interactions and DM interactions are expected to be the most strong for third and fourth neighbors (see [*ab initio*]{} calculations in [@Mazurenko2005]); this is also supported by the experimental data for the Heisenberg exchange parameters [@Samuelsen1970].
It also could happen that only with the interactions between non-nearest neighbors one can describe an experimentally observed phenomenon. For example, if in a 1D spin chain with the ferromagnetic coupling between the nearest neighbors ($J_1 > 0$) one turns on the antiferromagnetic interaction with the second neighbors ($J_2 < 0$), then for $|J_2| > J_1/4$ a spiral magnetic structure appears owing to spontaneous breaking of the inverse symmetry. Such approach can explain the phase transition appearing at $\sim$28 K and inducing ferroelectricity in the multiferroic Tb(Dy)MnO$_3$ crystal [@Kenzelmann2005; @Cheong2007]. The approach works also in the itinerant magnetics of MnSi-type. Just the competition between ferromagnetic coupling of the nearest spins ($J_1
> 0$) and antiferromagnetic coupling of the second and third neighbors ($J_2 \approx J_3 <
0$) was utilized to explain the observed alignment of magnetic helices along crystallographic directions $\langle 110 \rangle$ in MnSi at high pressure [@Hopkinson2007].
In this paper, we suggest a possible mechanism for effective reinforcement of the twist terms in the chiral spin structures of the $B20$ magnetics, taking into account the interactions with non-nearest neighbors. The reinforcement is caused by an interplay between the exchange coupling with the second and third neighbors and the canting from DM interactions. In Sections \[sec:interactions\]–\[sec:energy\] the transition from the Heisenberg microscopic model to the continuous phenomenological one is performed, and the spin-orbit terms in the phenomenological energy are found up to the contributions of the order of $(D/J)^2$. In Sec. \[sec:extratwist\] the possibility of an extra twist induced by the canting is demonstrated. In Sec. \[sec:rkky\] the exchange interaction parameters are estimated within the RKKY theory. In Sec. \[sec:experimental\] the possibility of an experimental proof of the canting existence is discussed. The possible applications of the theory are suggested in Sec. \[sec:discussion\].
Dzyaloshinskii–Moriya interaction in microscopical and phenomenological approaches {#sec:interactions}
==================================================================================
Both phenomenological and microscopical theories describing twisted MnSi-type magnetics contain terms induced by chirality of the system and associated with the Dzyaloshinskii–Moriya interaction of the spin-orbit nature. In the Heisenberg model, the extra term being associated with an individual bond, e.g. connecting magnetic atoms 1 and 2, can be expressed as $$\label{eq:heisenberg}
\mathbf{D}_{12} \cdot [\mathbf{s}_1 \times \mathbf{s}_2] ,$$ where $\mathbf{s}_1$ and $\mathbf{s}_2$ are the classical spins of the atoms, $\mathbf{D}_{12}$ is the DM vector of the bond 1-2. In principle, there could be magnetic moments at Si atoms [@Brown] but this effect will be neglected below. Let us shortly describe the properties of the DM vector [@Moriya60b]. (i) As it is obvious from Eq. (\[eq:heisenberg\]), the sign of the DM vector depends on which atom of the bond we consider as the first. Indeed, because the cross product changes its sign when $\mathbf{s}_1$ and $\mathbf{s}_2$ are rearranged, then $\mathbf{D}_{21} = -\mathbf{D}_{12}$. (ii) The structure changes its chirality under inversion, whereas the energy remains unchanged, which means that $\mathbf{D}_{12}$ is a pseudovector, because $[\mathbf{s}_1 \times \mathbf{s}_2]$ is a pseudovector as well. (iii) The DM vector of a bond possesses the local symmetry of the bond. Thus, in MnSi the DM vectors are of their most general form, i.e. they have three independent components, because the Mn-Mn bonds in the $B20$ structure do not possess any internal symmetry element. (iv) The DM vectors vary from bond to bond; the DM vectors of the equivalent bonds have the same length but may be of different orientation (just the case of MnSi). In particular, if two bonds are connected by the rotation symmetry transformation of the space group $P2_13$ of the crystal, then their DM vectors do by the corresponding rotation of the point group $23$.
In the phenomenological theory, the chiral interaction is induced by the following extra term $$\label{eq:phenomenological}
{\cal D} \mathbf{M} \cdot [\boldsymbol{\nabla} \times \mathbf{M}]$$ in the expression for the magnetic energy density. Here $\mathbf{M}$ is a continuous field of the magnetic moment, ${\cal D}$ is a pseudoscalar constant of the interaction.
Because both the theories describe the same matter, there should be a relationship between them. In particular, the terms (\[eq:heisenberg\]) and (\[eq:phenomenological\]) of the different approaches should be somehow connected. Indeed, it was shown in Ref. [@Chizhikov2012] that in the nearest neighbors approximation the constant ${\cal D}$ of the phenomenological theory is proportional to the component $(D_x-2D_y-D_z)$ of the DM vector of the bond $(-2x, \frac{1}{2},
\frac{1}{2}-2x)$ between neighboring manganese atoms.
However, there is a problem here. According to different spin-orbit calculation schemes [@Mazurenko2005; @Cheong2007; @Moskvin1977; @Shekhtman93; @Sergienko2006; @Katsnelson10], the DM vector associated with a bond should be perpendicular or almost perpendicular to the bond. In the present case it means that the component $(D_x-2D_y-D_z)$, which lies almost along the bond, constitutes only a small part of the DM vector length. Taking into account that having the relativistic nature spin-orbit DM interaction serves as a small additive to the ferromagnetic exchange coupling, we can conclude that the twist observed in the MnSi-type crystals, particularly in MnGe, seems to be abnormally strong. A possible solution of this problem is that the nearest neighbors approximation is not sufficient and hereinafter in the paper we develop this idea in details.
In the following sections we will show, how to perform transformation from the microscopic Heisenberg model to the macroscopic elastic theory.
From the Heisenberg model to a continuous approximation {#sec:continuous}
=======================================================
In the classical Heisenberg model of magnetics with an extra interaction of the Dzyaloshinskii–Moriya type, the energy of the system is expressed as a sum of pair interactions between magnetic atoms and the interaction of individual atoms with an external magnetic field $\mathbf{H}$: $$\label{eq:energy1}
E = \sum_{cells} \left( \sum_{n,\{ij\}} \left\{ -\tilde{J}_n \mathbf{s}_i \cdot \mathbf{s}_j + \mathbf{D}_{n,ij} \cdot [\mathbf{s}_i \times \mathbf{s}_j] - \frac{(\mathbf{D}_{n,ij} \cdot \mathbf{s}_i)(\mathbf{D}_{n,ij} \cdot \mathbf{s}_j)}{2J_n} \right\} - g \mu_B \mathbf{H} \cdot \sum_{i=1}^4 \mathbf{s}_i \right) .$$ Here the external summation is over all the unit cells of the crystal, $n$ enumerates magnetic coordination shells, the sum $\{ij\}$ is over all the bonds of the shell $n$, the sum over $i$ is taken over all magnetic atoms in the unit cell; $\tilde{J}_n = J_n - (D_n^2/4J_n)$, $J_n$ is the exchange coupling interaction parameter of the $n$th shell, $\mathbf{D}_{n,ij}$ is the DM vector of the bond $ij$ of the $n$th shell, $\mathbf{s}_i$ are classical spins of magnetic atoms, $|\mathbf{s}|=1$, $g\mu_B$ is the effective magnetic moment of the atom. The terms of the order of $D_n^2$ are usually ignored but sometimes they can be important [@Shekhtman; @Yildirim].
The MnSi-type crystal has $B20$ structure ($P2_13$ space group) with four magnetic (Mn) atoms occupying crystallographically equivalent $4a$ positions within a unit cell, $\mathbf{r}_1
= ( x, x, x)$, $\mathbf{r}_2 = ( \frac{1}{2}-x, 1-x, \frac{1}{2}+x)$, $\mathbf{r}_3 = ( 1-x,
\frac{1}{2}+x, \frac{1}{2}-x)$, $\mathbf{r}_4 = ( \frac{1}{2}+x, \frac{1}{2}-x,
1-x)$ [@tables]. The shortest bonds between the magnetic atoms have the length $\sqrt{\frac{1}{2} - 2x + 8x^2}$. The next environment consists of two close magnetic shells with the radii $\sqrt{\frac{3}{2} - 6x + 8x^2}$ and $\sqrt{\frac{1}{2} + 2x + 8x^2}$. The parameter $x$ defines which of them is closer to the initial atom. When $x > \frac{1}{8}$ ($x\approx 0.138$ in the case of MnSi), then the shell with the radius $\sqrt{\frac{3}{2} - 6x +
8x^2}$ is closer, so we will refer to it as the second one. When $x = \frac{1}{8}$, the 2nd and 3rd neighbors are at the same distance so that the manganese sublattice gains the space group $P4_332$ [@Dmitriev2012], connecting the 2nd and 3rd shells by a symmetry transformation. This could establish a linkage between the DM vectors of the shells, but the silicon sublattice (having in its turn the space group $P4_132$, when $x_{Si} =
\frac{7}{8}$) breaks the symmetry and the DM vectors of the shells are different even in this case. If $x < \frac{1}{8}$, then the 3rd neighbors become closer than the 2nd ones.
Every magnetic atom possesses six neighbors in each of its three nearest magnetic shells, so there are 36 bonds in a unit cell (in twelves for each kind). Fig. \[fig-neighbors\] shows the magnetic environment of a manganese atom in MnSi. The atom lies on a 3-fold axis. Each of the first three magnetic shells consists of two equilateral triangles. All the atoms of the 2nd shell are on one side of the plane perpendicular to the 3-fold axis, all the atoms of the 3rd shell are situated on the other side of the plane.
Let $( D_{1x}, D_{1y}, D_{1z})$ be the coordinates of the vector $\mathbf{D}_{1,13}$ corresponding to the bond $\mathbf{b}_{1,13} = (
-2x, \frac{1}{2}, \frac{1}{2}-2x)$ directed from $\mathbf{r}_1$ to $\mathbf{r}_3-(1,0,0)$, $(
D_{2x}, D_{2y}, D_{2z})$ be the coordinates of the vector $\mathbf{D}_{2,13}$ corresponding to the bond $\mathbf{b}_{2,13} = ( 1-2x,
\frac{1}{2}, \frac{1}{2}-2x)$ directed from $\mathbf{r}_1$ to $\mathbf{r}_3$, and $( D_{3x},
D_{3y}, D_{3z})$ be the coordinates of the vector $\mathbf{D}_{3,13}$ corresponding to the bond $\mathbf{b}_{3,13} = ( -2x, \frac{1}{2},
-\frac{1}{2}-2x)$ directed from $\mathbf{r}_1$ to $\mathbf{r}_3-(1,0,1)$.
We specify directions for all twelve bonds of a shell in the way that all directed distances $\mathbf{b}_{ij}$ between neighboring atoms connect to each other by the symmetry transformations of the point group $23$, so corresponding DM vectors are connected by the same transformations. In the MnSi-type crystals there is only one type of magnetic bonds directed from a central atom of the type $t$ to an atom of the type $t^\prime \neq t$ at a given shell ($n =
1, 2, 3$). We can take advantage from this fact and introduce for each pair $ij$ ($i, j = 1, 2,
3, 4$, $i \neq j$) a local triad $$\label{eq:triad}
( \boldsymbol{\tau}_i - \boldsymbol{\tau}_j ) \perp ( \boldsymbol{\tau}_i + \boldsymbol{\tau}_j ) \perp [\boldsymbol{\tau}_i \times \boldsymbol{\tau}_j] ,$$ and then use it as a basis for vectors $\mathbf{b}_{ij}$ and $\mathbf{D}_{ij}$. Here $\boldsymbol{\tau}_i$ is a vector directed along the 3-fold axis passing through the position $\mathbf{r}_i$: $$\label{eq:tau}
\begin{array}{l}
\boldsymbol{\tau}_1 = ( 1, 1, 1) , \\
\boldsymbol{\tau}_2 = ( -1, -1, 1) , \\
\boldsymbol{\tau}_3 = ( -1, 1, -1) , \\
\boldsymbol{\tau}_4 = ( 1, -1, -1) .
\end{array}$$ Then the bond vector and the DM vector of an arbitrary bond in three first magnetic shells can be written as
\[eq:bij\] $$\begin{aligned}
\mathbf{b}_{1,ij} & = & (-8x+1) (\boldsymbol{\tau}_i - \boldsymbol{\tau}_j) / 8 + (\boldsymbol{\tau}_i + \boldsymbol{\tau}_j) / 4 + [\boldsymbol{\tau}_i \times \boldsymbol{\tau}_j] / 8 , \\
\mathbf{b}_{2,ij} & = & (-8x+3) (\boldsymbol{\tau}_i - \boldsymbol{\tau}_j) / 8 + (\boldsymbol{\tau}_i + \boldsymbol{\tau}_j) / 4 - [\boldsymbol{\tau}_i \times \boldsymbol{\tau}_j] / 8 , \\
\mathbf{b}_{3,ij} & = & (-8x-1) (\boldsymbol{\tau}_i - \boldsymbol{\tau}_j) / 8 + (\boldsymbol{\tau}_i + \boldsymbol{\tau}_j) / 4 - [\boldsymbol{\tau}_i \times \boldsymbol{\tau}_j] / 8 ,\end{aligned}$$
$$\label{eq:Dij}
\mathbf{D}_{n,ij} = \frac{D_{n+}}{4} (\boldsymbol{\tau}_i - \boldsymbol{\tau}_j) + \frac{D_{ny}}{2} (\boldsymbol{\tau}_i + \boldsymbol{\tau}_j) - \frac{D_{n-}}{4} [\boldsymbol{\tau}_i \times \boldsymbol{\tau}_j] ,$$
with $D_{n\pm} = D_{nx} \pm D_{nz}$; here $\mathbf{D}_{n,ji} \neq -\mathbf{D}_{n,ij}$ because the indices $ij$ and $ji$ designate two different bonds.
Notice that the energy (\[eq:energy1\]) does not depend on the parameter $x$, so it can be chosen arbitrarily, making convenient transition to the continuous approximation (Fig. \[fig-xid\](b)). Indeed, to provide the transition the continuous unimodular vector functions will be defined, $\hat{\mathbf{s}}_1$, $\hat{\mathbf{s}}_2$, $\hat{\mathbf{s}}_3$, $\hat{\mathbf{s}}_4$, which coincide in special points with the spin values in corresponding atomic positions. If the parameter $x$ is equal to its experimental value, e.g. $x=0.138$ in MnSi, then we should take the functions in the real atomic positions in order to obtain the real spins. In the case of an arbitrary choice of $x$ we should take the function values in the points shifted from the real atomic positions.
It is convenient to choose the parameter $x$ from the condition $$\label{eq:x}
\sum_{n,\{ij\}} \tilde{J}_n (\boldsymbol{\tau}_{i}-\boldsymbol{\tau}_{j}) \otimes \mathbf{b}_{n,ij} = 0 ,$$ where $\otimes$ means the direct product of two vectors. Notice that this $3 \times 3$-tensor is proportional to the unit one due to the averaging over the symmetry elements of the point group $23$ (see Appendix, Eq. (\[eq:aijbij\])).
Thus, for example, in the nearest neighbors approximation the condition gives $x =
\frac{1}{8}$, which is shown in Ref. [@Chizhikov2012] to be necessary in order to obtain smooth spin functions.
For $n = 1, 2, 3$ using Eqs. (\[eq:sumtaub\]) we find from Eq. (\[eq:x\]) $$\label{eq:xexch}
x_{exch} = \frac{\tilde{J}_1 + 3 \tilde{J}_2 - \tilde{J}_3}{8(\tilde{J}_1 + \tilde{J}_2 + \tilde{J}_3)} ,$$ where the index “exch” means that $x_{exch}$ is not a real coordinate, but some physical parameter expressed trough the exchange interaction constants. In untwisted spin structures, the canting is determined wholly by the DM interactions. In twisted states, an extra canting arises induced by a disagreement of phases of the helices connected with different magnetic sublattices. The physical meaning of the “exchange” coordinates is that the spin shift to the new positions removes the canting appearing due to the spiralling.
The bond $ij$ of the $n$th shell connects the function values $\hat{\mathbf{s}}_i(\mathbf{r})$ and $\hat{\mathbf{s}}_j(\mathbf{r} +
\mathbf{b}_{n,ij})$. Using continuity of the functions $\hat{\mathbf{s}}$ we can write $$\label{eq:taylor}
\hat{\mathbf{s}}_j (\mathbf{r} + \mathbf{b}_{n,ij}) = {\cal B}_{n,ij} \hat{\mathbf{s}}_j (\mathbf{r}) ,$$ where $$\label{eq:calB}
{\cal B}_{n,ij} \equiv \exp(\mathbf{b}_{n,ij} \cdot \boldsymbol{\nabla}) = 1 + (\mathbf{b}_{n,ij} \cdot \boldsymbol{\nabla}) + \frac12 (\mathbf{b}_{n,ij} \cdot \boldsymbol{\nabla})^2 + \ldots$$ is the operator representing the Taylor series expansion, and pass from the summation over the unit cells to the integration over the crystal volume (in the chosen units, the lattice parameter $a=1$ and the unit cell volume is equal to 1): $$\label{eq:energy2}
\begin{array}{ll}
E = & \int_{V} {\cal E} d\mathbf{r} ,
\end{array}$$ $$\label{eq:energy2b}
\begin{array}{ll}
{\cal E} = & \sum_{n,\{ij\}} \{ -\tilde{J}_n \hat{\mathbf{s}}_i \cdot {\cal B}_{n,ij} \hat{\mathbf{s}}_j + \mathbf{D}_{n,ij} \cdot [\hat{\mathbf{s}}_i \times {\cal B}_{n,ij} \hat{\mathbf{s}}_j] \\
& - (\mathbf{D}_{n,ij} \cdot \hat{\mathbf{s}}_i)(\mathbf{D}_{n,ij} \cdot {\cal B}_{n,ij} \hat{\mathbf{s}}_j)/(2J_n) \} - g \mu_B \mathbf{H} \cdot \sum_{i=1}^4 \hat{\mathbf{s}}_i .
\end{array}$$
Magnetic moment density, calculus of variations and perturbation theory {#sec:moment}
=======================================================================
In the phenomenological theory, the magnetic moment density $\mathbf{M}$ is used as an order parameter and considered as an experimentally observed physical quantity. Nevertheless, when we try to express a smooth continuous function $\mathbf{M}$ through a discrete distribution of spins of the magnetic atoms, an ambiguity arises from the fact that we can determine the weight function in different ways when averaging the spins on a local volume. In this case the ambiguity results in that the smooth functions $\hat{\mathbf{s}}_i$ having specified values in a discrete set of points can be defined by infinite number of ways (Fig. \[fig-xid\](a)). We will avoid the problem, supposing that the functions $\hat{\mathbf{s}}_i$ are defined in the most convenient way.
It would be natural to define the magnetization as $$\label{eq:momentum}
\mathbf{M} = 4 g \mu_B \mathbf{m} ,$$ $$\label{eq:momentumb}
\mathbf{m} = \frac14 \sum_{i=1}^4 \hat{\mathbf{s}}_i .$$ In analogy with Eq. (\[eq:momentumb\]) we introduce the canting tensor $$\label{eq:udef2}
u_{\sigma\alpha} = \frac14 \sum_{i=1}^4 \tau_{i\sigma} s_{i\alpha} , \phantom{x} \alpha, \sigma = x, y, z ,$$ so that $$\label{eq:udef}
\hat{s}_{i\alpha} = m_\alpha + \tau_{i\sigma} u_{\sigma\alpha} .$$ Hereinafter the summation on repeated greek indices is implied. Four conditions $|\hat{\mathbf{s}}_i| =
1, i=1,2,3,4$, can be rewritten using the invariants $$\label{eq:invariantI0}
I_0 \equiv \frac14 \sum_{i=1}^4 \hat{s}_i^2 = m^2 + u_{\sigma\alpha} u_{\sigma\alpha} = 1 ,$$ $$\label{eq:invariantsI}
I_\sigma \equiv \frac14 \sum_{i=1}^4 \tau_{i\sigma} \hat{s}_i^2 = 2 m_\alpha u_{\sigma\alpha} + |\varepsilon_{\sigma\beta\gamma}| u_{\beta\alpha} u_{\gamma\alpha} = 0 .$$
In order to express the energy as a functional of the magnetic moment $\mathbf{m}$, the energy should be minimized by the canting tensor components using calculus of variations. The variation of the energy is $$\label{eq:varenergy}
\delta\tilde{E} = \int_{V} \delta\tilde{\cal E} d\mathbf{r} = \int_{V} \delta u_{\sigma\alpha} \Psi_{\sigma\alpha} d\mathbf{r} ,$$ where the integrand now includes the Lagrange terms: $$\label{eq:varenergyb}
\tilde{\cal E} = {\cal E} + \lambda_0 I_0 + \lambda_x I_x + \lambda_y I_y + \lambda_z I_z .$$ Taking into account Eq. (\[eq:udef\]), $$\label{eq:varcale}
\begin{array}{rl}
\Psi_{\sigma\alpha} = & \sum_{n,\{ij\}} \{ -\tilde{J_n} [ \tau_{i\sigma} {\cal B}_{n,ij} + \tau_{j\sigma} {\cal B}_{n,ij}^{-1} ] m_\alpha \\
& -\tilde{J}_n [ \tau_{i\sigma} \tau_{j\rho} {\cal B}_{n,ij} + \tau_{j\sigma} \tau_{i\rho} {\cal B}_{n,ij}^{-1} ] u_{\rho\alpha} \\
& -\varepsilon_{\alpha\beta\gamma} D_{n,ij\beta} [ \tau_{i\sigma} {\cal B}_{n,ij} - \tau_{j\sigma} {\cal B}_{n,ij}^{-1} ] m_\gamma \\
& -\varepsilon_{\alpha\beta\gamma} D_{n,ij\beta} [ \tau_{i\sigma} \tau_{j\rho} {\cal B}_{n,ij} - \tau_{j\sigma} \tau_{i\rho} {\cal B}_{n,ij}^{-1} ] u_{\rho\gamma} \\
& - (1/2J_n) {D}_{n,ij\alpha} {D}_{n,ij\beta} [ \tau_{i\sigma} {\cal B}_{n,ij} + \tau_{j\sigma} {\cal B}_{n,ij}^{-1} ] m_\beta \\
& - (1/2J_n) {D}_{n,ij\alpha} {D}_{n,ij\beta} [ \tau_{i\sigma} \tau_{j\rho} {\cal B}_{n,ij} + \tau_{j\sigma} \tau_{i\rho} {\cal B}_{n,ij}^{-1} ] u_{\rho\beta} \} \\
& + 2 \lambda_0 u_{\sigma\alpha} + 2 \lambda_\sigma m_\alpha + 2 |\varepsilon_{\sigma\beta\gamma}| \lambda_\beta u_{\gamma\alpha} .
\end{array}$$ Here $$\label{eq:calB2}
{\cal B}_{n,ij}^{-1} = \exp(-\mathbf{b}_{n,ij} \cdot \boldsymbol{\nabla})$$ and the evident equation was used, which corresponds to the integration in infinite volume, $$\label{eq:parts}
\int f(\mathbf{r}) {\cal B} g(\mathbf{r}) d\mathbf{r} = \int g(\mathbf{r}) {\cal B}^{-1} f(\mathbf{r}) d\mathbf{r} .$$
The minimum of the energy is determined from the condition $\delta\tilde{\cal E} = 0$ with arbitrary functions $\delta
u_{\sigma\alpha}$, so the problem is reduced to the system of 9 equations $$\label{eq:Psi}
\Psi_{\sigma\alpha} = 0 , \phantom{x} \sigma, \alpha = x, y, z ,$$ with the extra conditions (\[eq:invariantI0\]), (\[eq:invariantsI\]) determining the functions $\lambda_0(\mathbf{r})$, $\lambda_x(\mathbf{r})$, $\lambda_y(\mathbf{r})$, $\lambda_z(\mathbf{r})$.
The general solution of the problem is too difficult, if possible, but we can use the perturbation theory with a small parameter $$\label{eq:param}
D/J \ll 1 ,$$ that is the ratio of typical absolute values of spin-orbit ($D$) and exchange ($J$) interactions. We assume that this parameter also describes the typical order of magnitude of spatial derivatives and the spin components responsible for the canting: $$\label{eq:param2} |\boldsymbol{\nabla}| \sim |u_{\sigma\alpha}|
\sim D/J .$$ Another quantity that can be connected with the small value of canting is $\sqrt{1-m^2}$. In a weak magnetic field, when a spiral structure still exists, $\sqrt{1-m^2} \sim D/J$. But if the field is very strong ($g\mu_B H \gg 8 J$ [@Dmitrienko2012]), it induces a ferromagnetic alignment, and $\sqrt{1-m^2} \sim
D/(g\mu_B H) \ll D/J$. In that case we can take the maximal of two parameters, $D/J$, as constitutive one.
In the next section, we will use the consecutive approximations in order to find a solution of the system (\[eq:Psi\]).
Canting in the first approximation {#sec:1stapprox}
==================================
Assuming $\lambda_\alpha^{(0)}=0$, $\alpha=x,y,z$, the zeroth order equations on $D/J$ have the view $$\label{eq:approx0}
\sum_{n,\{ij\}} \tilde{J}_n (\tau_{i\sigma} + \tau_{j\sigma}) m_\alpha = 0$$ and become trivial after the summation on bonds, see Eq. (\[eq:aij\]). The first order equations on $D/J$ are $$\label{eq:approx1a}
\begin{array}{rl}
\sum_{n,\{ij\}} & \{ -\tilde{J}_n (\tau_{i\sigma} - \tau_{j\sigma}) b_{n,ij\mu} \nabla_\mu m_\alpha -\tilde{J}_n (\tau_{i\sigma} \tau_{j\rho} + \tau_{j\sigma} \tau_{i\rho}) u_{\rho\alpha}^{(1)} \\
& -\varepsilon_{\alpha\beta\gamma} D_{n,ij\beta} (\tau_{i\sigma} - \tau_{j\sigma}) m_\gamma \} + 2 \lambda_0^{(0)} u_{\sigma\alpha}^{(1)} + 2 \lambda_\sigma^{(1)} m_\alpha = 0 ,
\end{array}$$ where upper index $(p)$ means that the corresponding term is of the order of $(D/J)^p$. The first summand in curly brackets gives zero in accordance with Eq. (\[eq:x\]), two other can be calculated using Eqs. (\[eq:sumtautau\]f) and (\[eq:tD\]). Then, $$\label{eq:approx1b}
2 (\lambda_0^{(0)} + 4 (\tilde{J}_1 + \tilde{J}_2 + \tilde{J}_3)) u_{\sigma\alpha}^{(1)} + 8 (D_{1+} + D_{2+} + D_{3+}) \varepsilon_{\sigma\alpha\gamma} m_\gamma + 2 \lambda_\sigma^{(1)} m_\alpha = 0 .$$ The normalization conditions are $$\label{eq:norm1a}
u_{\sigma\alpha}^{(1)} u_{\sigma\alpha}^{(1)} = u_{\sigma\alpha} u_{\sigma\alpha} = 1 - m^2 ,$$ $$\label{eq:norm1b}
m_\alpha u_{\sigma\alpha}^{(1)} = 0 ,$$ where the first equation is of the second order on $D/J$.
Multiplication of Eq. (\[eq:approx1b\]) by $m_\alpha$ and summation on $\alpha$ with use of Eq. (\[eq:norm1b\]) give $$\label{eq:lambdas1}
\lambda_\sigma^{(1)} = 0 ,$$ therefore $$\label{eq:u1a}
u_{\sigma\alpha}^{(1)} = - \frac{4 (D_{1+} + D_{2+} + D_{3+}) \varepsilon_{\sigma\alpha\gamma} m_\gamma}{\lambda_0^{(0)} + 4 (\tilde{J}_1 + \tilde{J}_2 + \tilde{J}_3)} .$$ The substitution of Eq. (\[eq:u1a\]) into Eq. (\[eq:norm1a\]) gives $$\label{eq:lambda01}
\lambda_0^{(0)} + 4 (\tilde{J}_1 + \tilde{J}_2 + \tilde{J}_3) = \frac{4\sqrt{2} |D_{1+} + D_{2+} + D_{3+}| m}{\sqrt{1-m^2}} ,$$ and, finally, $$\label{eq:u1b}
u_{\sigma\alpha}^{(1)} = - sign (D_{1+} + D_{2+} + D_{3+}) \frac{\sqrt{1-m^2}}{\sqrt{2} m} \varepsilon_{\sigma\alpha\gamma} m_\gamma ,$$ where the sign of the right parts of Eqs. (\[eq:lambda01\]) and (\[eq:u1b\]) is chosen from the condition of the minimum of the canting energy.
The substitution of Eq. (\[eq:u1b\]) into Eq. (\[eq:udef\]) gives $$\label{eq:shat}
\hat{\mathbf{s}}_{i} = \mathbf{m} + \varkappa [\boldsymbol{\tau}_i \times \mathbf{m}] ,$$ with $$\label{eq:kappa}
\varkappa = sign (D_{1+} + D_{2+} + D_{3+}) \frac{\sqrt{1-m^2}}{\sqrt{2} m} .$$
From Eqs. (\[eq:approx1b\]), (\[eq:u1a\]) and (\[eq:u1b\]) we conclude that responsible for the canting is the combination $D_{1+} + D_{2+} + D_{3+}$ of the DM vectors components.
Energy density {#sec:energy}
==============
The contributions to the energy density from the magnetic moment $\mathbf{m}$ and its derivatives are $$\label{eq:calem0}
{\cal E}_m^{(0)} = - \sum_{n,\{ij\}} \tilde{J}_n m_\alpha m_\alpha = -12 (\tilde{J}_1 + \tilde{J}_2 + \tilde{J}_3) m^2 , \\$$
$$\label{eq:calem1}
{\cal E}_m^{(1)} = \sum_{n,\{ij\}} \{ -\tilde{J}_n m_\alpha (\mathbf{b}_{n,ij} \cdot \boldsymbol{\nabla}) m_\alpha + \varepsilon_{\alpha\beta\gamma} D_{n,ij\alpha} m_\beta m_\gamma \} = 0 , \\$$
$$\label{eq:calem2}
\begin{array}{lll}
{\cal E}_m^{(2)} & = & \sum_{n,\{ij\}} \{ - \frac{1}{2} \tilde{J}_n m_\alpha (\mathbf{b}_{n,ij} \cdot \boldsymbol{\nabla})^2 m_\alpha + \varepsilon_{\alpha\beta\gamma} D_{n,ij\alpha} m_\beta (\mathbf{b}_{n,ij} \cdot \boldsymbol{\nabla}) m_\gamma \\
& & - (2J_n)^{-1} D_{n,ij\alpha} D_{n,ij\beta} m_\alpha m_\beta \} .
\end{array}$$
Using Eqs. (\[eq:sumbb\]), (\[eq:sumDb2\]), and (\[eq:DD\]) with $x=x_{exch}$ we obtain $$\label{eq:calem2-2}
{\cal E}_m^{(2)} = - {\cal J} \mathbf{m} \cdot \Delta \mathbf{m} + {\cal D} \mathbf{m} \cdot [\boldsymbol{\nabla} \times \mathbf{m}] - \left( \frac{2D_1^2}{J_1} + \frac{2D_2^2}{J_2} + \frac{2D_3^2}{J_3} \right) m^2 ,$$ with $$\label{eq:calJ}
{\cal J} = \frac{3 \tilde{J}_1^2 + 3 \tilde{J}_2^2 + 3 \tilde{J}_3^2 + 10 \tilde{J}_1 \tilde{J}_2 + 10 \tilde{J}_1 \tilde{J}_3 + 22 \tilde{J}_2 \tilde{J}_3 }{4 (\tilde{J}_1 + \tilde{J}_2 + \tilde{J}_3)} ,$$ $$\label{eq:calD}
\begin{array}{ll}
{\cal D} & = - 4 (\mathbf{D}_{1,13} \cdot \mathbf{b}_{1,13} + \mathbf{D}_{2,13} \cdot \mathbf{b}_{2,13} + \mathbf{D}_{3,13} \cdot \mathbf{b}_{3,13}) \\
& = 8 x_{exch} (D_{1+} + D_{2+} + D_{3+}) - (D_{1+} - D_{1-} + 2 D_{1y}) - (3 D_{2+} + D_{2-} + 2 D_{2y}) - (-D_{3+} + D_{3-} + 2 D_{3y}).
\end{array}$$
The first term in Eq. (\[eq:calem2-2\]) can be rewritten as $$\label{eq:efer}
{\cal J} \frac{\partial m_\alpha}{\partial r_\beta} \frac{\partial m_\alpha}{\partial r_\beta} - \frac{1}{2} {\cal J} \boldsymbol{\nabla} \cdot (\boldsymbol{\nabla} m^2) ,$$ where the second term gives a contribution to the surface energy only: $$\label{eq:esuf}
- \frac{1}{2} {\cal J} \int_V d\mathbf{r} \boldsymbol{\nabla} \cdot (\boldsymbol{\nabla} m^2) = - \frac{1}{2} {\cal J} \oint_S d\mathbf{f} \cdot \boldsymbol{\nabla} m^2 .$$ Far from the transition between paramagnetic and ferromagnetic states, the absolute value $m$ of the magnetic moment changes slowly, and this contribution to the energy can be neglected. However, near the phase transition, $m$ can undergo considerable changes, and in the crystals with a significant surface (nanocrystals, thin films) the term (\[eq:esuf\]) could play an important role. In particular, it could be important for the stabilization of the A-phase observed in thin films of Fe$_{0.5}$Co$_{0.5}$Si and FeGe [@Yu2010; @Yu2011].
The contributions from the canting into the energy density are $$\label{eq:caleu1}
{\cal E}_u^{(1)} = - \sum_{n,\{ij\}} \tilde{J}_n (\tau_{i\sigma} + \tau_{j\sigma}) u_{\sigma\alpha}^{(1)} m_\alpha = 0 ,$$
$$\label{eq:caleu2}
\begin{array}{lll}
{\cal E}_u^{(2)} & = & \sum_{n,\{ij\}} \{ - \tilde{J}_n (\tau_{i\sigma} - \tau_{j\sigma}) u_{\sigma\alpha}^{(1)} (\mathbf{b}_{n,ij} \cdot \boldsymbol{\nabla}) m_\alpha - \tilde{J}_n (\tau_{i\sigma} + \tau_{j\sigma}) u_{\sigma\alpha}^{(1)} m_\alpha \\
& & - \tilde{J}_n \tau_{i\sigma} \tau_{j\rho} u_{\sigma\alpha}^{(1)} u_{\rho\alpha}^{(1)} + \varepsilon_{\alpha\beta\gamma} D_{n,ij\alpha} (\tau_{i\sigma} - \tau_{j\sigma}) u_{\sigma\beta}^{(1)} m_\gamma \} \\
& = & 4 (\tilde{J}_1 + \tilde{J}_2 + \tilde{J}_3) u_{\sigma\alpha}^{(1)} u_{\sigma\alpha}^{(1)} + 8 (D_{1+} + D_{2+} + D_{3+}) \varepsilon_{\sigma\beta\gamma} u_{\sigma\beta}^{(1)} m_\gamma \\
& = & 4 (\tilde{J}_1 + \tilde{J}_2 + \tilde{J}_3) (1-m^2) - 8 \sqrt{2} |D_{1+} + D_{2+} + D_{3+}| m \sqrt{1-m^2} .
\end{array}$$
Thus, we can finally rewrite the bulk energy density as a function of the magnetic moment $\mathbf{m}$ accurate within the second order terms on $D/J$: $$\label{eq:cale}
\begin{array}{ll}
{\cal E} = & {\cal J} \frac{\partial m_\alpha}{\partial r_\beta} \frac{\partial m_\alpha}{\partial r_\beta} + {\cal D} \mathbf{m} \cdot [\boldsymbol{\nabla} \times \mathbf{m}] + (\tilde{J}_1 + \tilde{J}_2 + \tilde{J}_3) (4 - 16 m^2) \\
& - 8 \sqrt{2} |D_{1+} + D_{2+} + D_{3+}| m \sqrt{1-m^2} - \left( \frac{2D_1^2}{J_1} + \frac{2D_2^2}{J_2} + \frac{2D_3^2}{J_3} \right) m^2 - 4 g \mu_B \mathbf{H} \cdot \mathbf{m}.
\end{array}$$
The first two terms with derivatives are nothing but the deformation energy of the conventional phenomenological theory of the chiral magnetics. However here the values ${\cal J}$ and ${\cal D}$ are expressed through the parameters of the microscopic theory accordingly to Eqs. (\[eq:calJ\]) and (\[eq:calD\]). The following term, on conditions that $m \approx
1$, gives $-12(\tilde{J}_1 + \tilde{J}_2 +
\tilde{J}_3)$, that is the energy of 36 ferromagnetic bonds in first three shells in the unit cell. Then, the term with $\sqrt{1-m^2}$ is the contribution of the canting. It is always negative, which means that the canting is an important and unavoidable peculiarity of the magnetic structures of the MnSi-type crystals. When minimizing the energy on $m$, this term gives the contribution to the derivative proportional to $1 / \sqrt{1-m^2}$, which becomes dominating, when $m \rightarrow 1-0$, impeding $m$ to exceed 1.
Extra twist induced by canting {#sec:extratwist}
==============================
Now we return to Eq. (\[eq:calD\]), which defines the DM parameter $\cal D$ of the phenomenological theory. As it was mentioned above, from the physical point of view, the DM vectors are almost perpendicular to the corresponding bonds and, consequently, the expected value of ${\cal D}$ is small. In Ref. [@Dmitrienko2012] some speculative estimation has been performed with use of the well known Keffer rule [@Keffer] based on the Moriya theory [@Moriya60b], which gives us following expression for the DM vector $$\label{eq:keffer}
\mathbf{D}_{12} = D [\mathbf{r}_{1i} \times \mathbf{r}_{2i}] ,$$ where $D$ is unknown coefficient, and the vectors $\mathbf{r}_{1i}$ and $\mathbf{r}_{2i}$ are directed from the positions of 1st and 2nd magnetic (Mn) atoms to that of an intermediate nonmagnetic (Si) atom realizing the spin-orbit interaction. It is evident that $\mathbf{D}_{12}
\perp \mathbf{r}_{12} = \mathbf{r}_{2i} -
\mathbf{r}_{1i}$.
For the Keffer rule one needs the constants $D$ and coordinates of different intermediate atoms. Surprisingly, a considerable result can be achieved with a more general rule, namely that the DM vectors are perpendicular to the bonds. The condition can be written as $$\label{eq:Dperpb}
\mathbf{D} \cdot \mathbf{b} (x=x_{real}) = 0 ,$$ and, comparing with Eq. (\[eq:calD\]), we easily find in this case that $$\label{eq:calD2}
{\cal D} = 8 (D_{1+} + D_{2+} + D_{3+}) (x_{exch}-x_{real}) ,$$ which gives us a new definition of the exchange coordinate as the manganese atom position inhibiting spiralling when combining with the Keffer rule. The combination $D_{1+} + D_{2+} +
D_{3+}$ is responsible for the canting in accordance with Sec. \[sec:1stapprox\], and $x_{exch}$ is a combination of the exchange interaction constants. Eq. (\[eq:calD2\]) can be interpreted as an evidence of the canting direct participation in the magnetic structure spiralling. Besides, the sign of ${\cal D}$ and consequently the magnetic chirality are determined both by the sign of the canting component $D_{1+} +
D_{2+} + D_{3+}$ of the DM vectors and that of the difference $x_{exch}-x_{real}$, depending on the exchange constants $\tilde{J}_1$, $\tilde{J}_2$, and $\tilde{J}_3$.
Therefore, the canting, initially considered as a supplementary microscopic peculiarity of the chiral magnetics [@Chizhikov2012; @Dmitrienko2012], can be in fact an essential cause of the twisting power. Let us demonstrate by a simple, albeit not very realistic, example how the canting between different magnetic sublattices can result in an essential twist gain. We consider a periodical 1D chain of spins with a local interaction between them, composed of unit cells containing two spins, say A and B (Fig. \[fig-chain\]). All the spins can rotate in a plane, and the energy of the chain is a function of differences of the angles, $$\label{eq:Echain}
E = \sum_n \left\{ C_R (\varphi_{n}^A-\varphi_{n}^B+\beta)^2 + C_L (\varphi_{n}^A-\varphi_{n-1}^B+\beta)^2 + J (\varphi_{n}^A-\varphi_{n}^B)^2 + J (\varphi_{n}^A-\varphi_{n-1}^B)^2 \right\} .$$ Here $\varphi_{n}^A$ and $\varphi_{n}^B$ are orientation angles of the spins A and B of the $n$th cell, laid off from an arbirary direction; $C_R$ and $C_L$ are positive constants. The condition $C_R \neq C_L$ determines the chirality of the structure.
When $J = 0$, Eq. (\[eq:Echain\]) describes a periodical magnetic structure with the angle $\beta$ of canting between the spin sublattices A and B (Fig. \[fig-chain\](a)). We suppose that this structure is a result of the competition of two ordering interactions between neighboring spins, a ferromagnetic one and a twisting chiral one; besides, there is a reason, which is not considered here, eliminating the twist in this state.
The question arises: what happens with the structure after introducing of an additional ferromagnetic ($J > 0$) interaction with the nearest neighbors from the complementary sublattice? The minimization of Eq. (\[eq:Echain\]) gives the solution in the helix form $$\label{eq:phi1phi2}
\begin{array}{l}
\varphi_{n}^A = n \delta , \\
\varphi_{n}^B = n \delta + \alpha ,
\end{array}$$ where $$\label{eq:alpha2}
\alpha = \frac{C_R \beta}{C_R + J}$$ is a tilt angle between spins in the unit cell, and $$\label{eq:ddelta}
\delta = \frac{J (C_R - C_L) \beta}{(C_R + J)(C_L + J)}$$ is the twist angle per one period of the chain (Fig. \[fig-chain\](b)).
At the first sight it seems to be paradoxical that the introducing of an additional aligning interaction induces a twist, but a simple analysis of the problem shows that there is no contradiction here. When $|J| \rightarrow \infty$, the angles $\alpha$ and $\delta$ go to zero as expected. In order to understand the system behavior for the finite values of $J$, consider the numerator of Eq. (\[eq:ddelta\]). The factor $(C_L
- C_R)$ reflects the degree of the internal chirality of the structure. The product $J \beta$ reminds of the lever torque, with $\beta$ playing the role of the lever arm and $J$ being analogous to the rotating force. We can imagine that the “aligning force” $J$, being applied to the initially tilted by the angle $\beta$ sublattices, results in structure distortions, which in their turn induce the twist due to the potential chirality of the structure ($C_R \neq C_L$). Notice that the change of the sign of the constant $J$, corresponding to the transition from ferromagnetic ($J > 0$) to antiferromagnetic ($J < 0$) coupling, increases degree of the twist (reduces the helix pitch) and changes its handedness.
We expect that the similar effect takes place in the chiral magnetic MnSi. Indeed, there is an evident similarity of Eqs. (\[eq:calD2\]) and (\[eq:ddelta\]). In this case the role of a lever is played by the canting induced by the DM interactions, whereas the “force” is the ferro- or antiferromagnetic aligning interaction with neighboring atoms.
Estimation of the exchange parameters and the RKKY theory {#sec:rkky}
=========================================================
In analogy with the Dzyaloshinskii–Moriya interaction, the exchange one is described differently in microscopical and phenomenological approaches. Although it is evident that exchange constants of both the theories should be connected to each other, the connection is found in the nearest neighbors approximation [@Chizhikov2012] to be not so trivial as we could expect. Indeed, the expression (\[eq:calJ\]) for ${\cal J}$ in the approximation of three magnetic shells is strongly different from the simple proportionality in the former case. However, the situation seems to be even more intricate, because ${\cal J}$ is not sufficient to induce the ferromagnetic alignment, and another exchange parameter is also needed, namely $$\label{eq:Jsum}
J_{\Sigma} = \sum_j \tilde{J}_j ,$$ where summation is taken over all the neighbors contributing to the exchange interaction. Then the condition of the validity of our approximation can be written as two inequalities, $$\label{eq:Jsum0}
J_{\Sigma} > 0 ,$$ $$\label{eq:Jcal0}
{\cal J} > 0 ,$$ both equally important. Here $J_{\Sigma}$ is nothing but taken with an opposite sign energy of an individual spin interaction with its magnetic surroundings in the untwisted (ferromagnetic) state, and Eq. (\[eq:Jsum0\]) guarantees stability of the state relative to the change of the spin sign. Eq. (\[eq:Jcal0\]) in its turn guarantees stability relative to the spin small rotations, providing smallness of the magnetic moment gradients. Therefore, only combined use of the conditions leads to a ferromagnetic ordering with a weak spiralling.
Neglecting the contribution of the DM interaction into symmetrical exchange (i.e. using $J_n$ instead of $\tilde{J}_n$), we can estimate $J_\Sigma$ and ${\cal J}$ in the frame of the RKKY theory [@Ruderman1954; @Kasuya1956; @Yosida1957; @VanVleck1962], which is applicable to the itinerant magnetics. Indeed, in this model $J_n$ is a simple function of the distance between interacting atoms, $$\label{eq:Jrkky}
J_n \equiv J(b_n) \sim -F(2 k_F a b_n) ,$$ $$\label{eq:F}
F(x) = \frac{x \cos x - \sin x}{x^4} .$$ Here $k_F$ is the Fermi wave number, $b_n$ is the dimensionless distance to the $n$th magnetic shell.
It follows from Eqs. (\[eq:Jrkky\]), (\[eq:F\]) that atoms situated at approximately the same distances, e.g. $b_2$ and $b_3$, make similar contributions to the exchange energy. Therefore, together with the 2nd and 3rd shells it is necessary to take into account the 4th shell corresponding to the atoms separated by lattice periods. Because the atoms of the 4th shell belong to the same magnetic sublattice, they do not affect the canting between different sublattices, i.e. they leave $u_{\sigma\alpha}$ and $x_{exch}$ unchanged and give simple additive contributions to $J_\Sigma$ and ${\cal J}$.
All the magnetic shells have 6 atoms, so $$\label{eq:Jsum-2}
J_{\Sigma} = 6 (J_1 + J_2 + J_3 + J_4) .$$ In order to calculate the additive from the 4th shell to ${\cal
J}$, we can write, for example, the interaction energy of two spins separated by the period $(1,0,0)$ of the lattice, $$\label{eq:J4part}
-J_4 \mathbf{s}(\mathbf{r}) \cdot \mathbf{s}(\mathbf{r}+(1,0,0)) \approx -J_4 \left( 1 + \mathbf{s} \cdot \frac{\partial\mathbf{s}}{\partial x} + \frac12 \mathbf{s} \cdot \frac{\partial^2\mathbf{s}}{\partial x^2} \right) .$$ If we take the sum over 6 bonds, multiply it by the number of magnetic atoms in the unit cell and take into account that all the bonds in that sum are taken twice, then the correction to the energy density can be written as $$\label{eq:J4toE}
\Delta{\cal E} = -12 J_4 - 2 J_4 \mathbf{m} \cdot \Delta\mathbf{m} .$$ It is obvious from the comparison with Eq. (\[eq:calem2-2\]) that the additive to ${\cal J}$ from the 4th magnetic shell is $2
J_4$.
The parameters $F_n \equiv F(2 k_F a b_n)$ are oscillating functions of the argument $k_F a$, which are easy to calculate using real distances between magnetic atoms. In MnSi for $x=0.138$ we find $b_1=0.613$, $b_2=0.908$, $b_3=0.964$ and $b_4=1$. Fig. \[fig-J\](a) shows the functions $F_1$–$F_4$ plotted in the area $10 < k_F a < 20$, where the value $k_F a =
16.4$ for MnSi is situated ($k_F = 3.6$Å$^{-1}$ [@Kirkpatrick2009], $a = 4.56$Å). The graphs show that Eqs. (\[eq:Jsum0\]), (\[eq:Jcal0\]) can be satisfied together in the areas of negative values of $F_1$. Fig. \[fig-J\](b) represents dependences of $J_\Sigma$ and ${\cal J}$ on $k_F a$, calculated for the same area. Both $J_\Sigma$ and ${\cal J}$ are oscillating alternating-sign functions; the jumps of ${\cal J}$ correspond to the zeros of $J_1 + J_2 + J_3$. The nearest to the value $k_F a = 16.4$ “plateau” with positive $J_\Sigma$ and ${\cal J}$ is in the area $17.1 < k_F a < 18.9$.
Notice that the behavior of ${\cal J}$ at some $k_Fa$ could seem paradoxical. For example, the divergence of ${\cal J}
\rightarrow +\infty$ would mean a strong suppression of long-wavelength fluctuations. However, the paradox can be resolved by taking into account the behavior of $J_\Sigma$. Indeed, if $J_1 + J_2 + J_3 = 0$, then $J_\Sigma \sim J_4$, where $J_4$ is nothing but the coupling constant of the spins belonging to the same sublattice. This means that, when ${\cal J}
\rightarrow +\infty$, the connection between sublattices gets broken. Thus, when the spins of three sublattices are aligned in the same direction, the spins of the fourth one can have an arbitrary direction, even if the condition $J_4 > 0$ guarantees the ferromagnetic order within the sublattice. The foregoing means that in addition to Eqs. (\[eq:Jsum0\]), (\[eq:Jcal0\]), we should introduce another inequality, $$\label{eq:Jsum0-2}
J_{\Sigma}^\prime = \left. \sum_j \right.^\prime \tilde{J}_j > 0 ,$$ determining the ferromagnetic connection of the four magnetic sublattices. Here the sum is taken over the bonds connecting atoms belonging to different sublattices. In the approximation of four magnetic shells, $J_{\Sigma}^\prime = 6 (J_1 + J_2 + J_3) = J_{\Sigma} - 6 J_4$, therefore the zeros of $J_{\Sigma}^\prime$ coincide with the jumps of ${\cal J}$.
The RKKY model allows as well to estimate the exchange coordinate $x_{exch}$. Because, according to Eq. (\[eq:calD2\]), the degree of the twist is determined by the difference $x_{exch} -
x_{real}$, it is useful to have an idea about how much this difference could be. In the nearest neighbors approximation $x_{exch} = \frac18$, which is close to the real value $x_{real} =
0.138$ for MnSi. However, when taking into account the contributions from the 2nd and 3rd shells, $x_{exch}$ can have arbitrary large positive and negative values near the zeros of $J_1 + J_2 + J_3$ (Fig. \[fig-xexch\]). Close to the minima of $F_1$ $J_2$ and $J_3$ are small in comparison with $J_1$ and, therefore, $x_{exch} \approx \frac18$, see inset in Fig. \[fig-xexch\]. Notice that the zeros of $J_1 + J_2 + J_3$ do not result in a divergence of the wave number $k={\cal
D}/2{\cal J}$, because ${\cal J}$ in the denominator and $x_{exch}$ in the numerator increase simultaneously.
As it is seen from Fig. \[fig-xexch\] and the inset in it, the difference $x_{exch} - x_{real}$ can change its sign depending on $k_F$. It gives a possibility to control the magnetic structure chirality by varying the concentration of different elements in the crystal.
Eqs. (\[eq:Jsum0\]), (\[eq:Jcal0\]) are evident preconditions of the experimentally observed ferromagnetic order in MnSi. Nevertheless, we can not preclude that one of the constants $J_\Sigma$, ${\cal J}$, or both these parameters can have negative values. For example, the condition $J_\Sigma < 0$ does not surely result in an antiferromagnetic order. The strong frustrations intrinsic for the system, e.g. the triangles of bonds, and nonsymmetric DM interactions can induce a small magnetic moment and lead to a ferri- or a weak ferromagnetic order. It could explain the weak magnetic moment observed in MnSi ($g \approx
0.4$). When ${\cal J} < 0$, the contributions to the energy density with higher spatial derivatives should be taken into account, which can stabilize the helix pitch.
Canting and magnetic diffraction {#sec:experimental}
================================
In Sec. \[sec:1stapprox\], we obtain Eq. (\[eq:shat\]), which says that in the first approximation the canting can be described as spin rotations by the same small angle around corresponding 3-fold axes. A similar expression for the ferromagnetic state caused by a strong magnetic field was found in Ref. [@Dmitrienko2012], where an approach had been suggested to measure the canting using neutron or X-ray magnetic diffraction. In Ref. [@Dmitrienko2012], the angle of spin tilt was proportional to $\delta = D_{1+} / 4 J_1$. In order to obtain that result we can find the coefficient $\varkappa \sim
\sqrt{1-m^2} / \sqrt{2} m$ for untwisted state. The minimization of Eq. (\[eq:cale\]) on magnetic moment modulus $m$, assuming that $\mathbf{H} \parallel \mathbf{m} = const$, gives $$\label{eq:kappa2}
\varkappa = \frac{D_{1+} + D_{2+} + D_{3+}}{4 (J_1 + J_2 + J_3 + g \mu_B H / 8)}$$ Notice that $\varkappa =\delta$, when $D_2 = D_3 = J_2 = J_3 = H
= 0$. The corresponding expressions for the structure factors of purely magnetic reflections $00\ell(\ell=2n+1)$ can be easily found from Ref. [@Dmitrienko2012].
Notice that there is another contribution to the reflections, induced by the anisotropy of magnetic susceptibility tensor [@Gukasov2002; @Cao2009]. The contributions can be distinguished, because (i) the tilts have different directions, (ii) the canting effect does not depend on the magnetic field modulus, whereas the tilts induced by the susceptibility tensor anisotropy are proportional to $H$.
The fact that both the twist and canting are determined by the same DM vectors components gives an additional possibility of numerical verification of the theory. Indeed, excluding $D_{1+} +
D_{2+} + D_{3+}$ from Eqs. (\[eq:kappa2\]) and (\[eq:calD2\]), we can connect the canting angle in unwound state and the wave number $k = {\cal D} / 2 {\cal J}$ of magnetic helices: $$\label{eq:kappa3}
\varkappa = \frac{\cal J}{16 (J_1 + J_2 + J_3 + g \mu_B H / 8) (x_{exch} - x_{real})} k .$$ Excepting observable physical values, Eq. (\[eq:kappa3\]) contains only exchange interaction constants, which are easier to calculate using [*ab initio*]{} methods than the DM vectors.
Summary and discussion {#sec:discussion}
======================
An essential difference of the microscopic description of the magnetic properties of the MnSi-type crystals from the phenomenological one is the usage of the pseudovector $\mathbf{D}$ instead of pseudoscalar ${\cal D}$ when describing the Dzyaloshinskii–Moriya interaction. The presence of the extra parameters (pseudovector components) results in the existence of a local canting, the feature not studied yet in magnetic twisted structures. An important problem solved in the present work is how to distinguish the components of $\mathbf{D}$ responsible for the twist and the canting. Hopkinson and Kee in Ref. [@Hopkinson2009] showed numerically (in the nearest neighbors approximation) that responsible for the twist were the components of the $\mathbf{D}$-vectors lying along the bonds, whereas for the canting did the $\mathbf{D}$ components directed perpendicular to the bonds and lying in the planes of bond triangles in the trillium lattice. In Ref. [@Chizhikov2012] we specified the result and showed that real components of $\mathbf{D}$ inducing the twist and the canting lay along crystallographic directions closed to those found in Ref. [@Hopkinson2009]. Nevertheless, the problem remained that in accordance with the quantum mechanical description the DM vectors should be perpendicular or almost perpendicular to the bonds. In other words, the $\mathbf{D}$ components responsible for the twist accordingly to Refs. [@Hopkinson2009; @Chizhikov2012] could be diminutive. In order to solve the problem we take into account the contribution of non-nearest neighbors in the magnetic interaction. Surprisingly, it appears that in the case, when all the DM vectors are perpendicular to the bonds, the spiralling is determined by the same $\mathbf{D}$ components as the canting. It leads to the conclusion that the canting, initially being considered as an additional microscopic effect in relation to the global twist, can in fact serve as a cause of the abnormal twist experimentally found in the MnSi-type crystals, particularly in MnGe. It is also important that the contribution of non-nearest magnetic neighbors should be taken into account.
In the simplest phenomenological theory describing twisted magnetic structures, it is supposed that $|\mathbf{M}| = const$, which is roughly true at low temperatures, far below the phase transition from the paramagnetic state. However, this condition makes energetically unfavorable such structures as the Skyrmions and their lattices, associated with the A-phase observed close by the transition point. In order to overcome this problem as well as to describe the critical phenomena, two additional terms, $M^2$ and $M^4$, limiting the value of magnetic moment are included into the free energy [@Bak80]. The presence of the terms decreases $M$ in the regions with a large density of the magnetic energy, thereby decreasing the energy of the whole structure. Thus, in Ref. [@Roessler2011] the terms $M^2$ and $M^4$ are used in order to calculate the energy of Skyrmions. It is found that $M$ decreases considerably nearby the core of the Skyrmion, where the magnetic moment $\mathbf{M}$ has the opposite direction to the external magnetic field, and has a maximum at some distance from the core, where the energy gain from the double twist is maximal. However, in the latter case the magnetic moment exceeds its saturation value $M_0$, which is not acceptable for physical reasons. In the present work we show that there should be a contribution from the canting $\sim m \sqrt{1 - m^2}$ into the energy density, which can play the same role as $M^2$ and $M^4$, but does not allow the magnetic moment modulus to exceed the saturation threshold. Besides the canting, the thermal fluctuations of spins also contribute to the reduction of the magnetic moment $M$. Nearby the transition point the amplitude of the fluctuations can be comparable with the canting. Moreover, the less is the effective local field $\mathbf{h}_{eff,i} =
-\partial E / \partial \mathbf{s}_i$, acting on the individual spin $\mathbf{s}_i$, the more are fluctuations and cantings. Therefore, the reinforcement both of the thermal fluctuations and the canting have the same cause, so they should give a similar effect.
If the canting would be observed in MnSi, the direct confirmation of non-nearest neighbors effect could be possible using Eq. (\[eq:kappa2\]), connecting the propagation number of the magnetic helices with the magnitude of the residual canting in the unwound state in a strong magnetic field. The equation involves only the exchange constants $J_n$ and does not depend on the DM vectors. The possibility of an experimental proof of the theory should stimulate [*ab intio*]{} calculations of the interaction constants. Some semi-quantitative estimations are made in the present work with use of the RKKY model. However problems still remain. For example, the RKKY parameter $k_F a = 16.4$ for MnSi corresponds to the area, where $J_\Sigma < 0$ and $ {\cal J} <
0$. More realistic calculations would give a tip about the direction of the further search.
It has been found in Ref. [@Grigoriev2009] that the propagation number $k$ of the magnetic helix in the alloy Fe$_{1-x}$Co$_x$Si is strongly dependent on the concentration $x$ of the cobalt atoms. Thus, when $x$ changes from 0.05 to 0.15, the helix period decreases abruptly in several times. It can be explained, in particular, by the strong dependence of the Fermi wave number $k_F$ on the cobalt impurity concentration, which also effects on the exchange parameters $J_\Sigma$, ${\cal J}$ and $x_{exch}$. This phenomenon could be also responsible for the recently observed [@GrigorievFKS-2013] change of the sign of the magnetic chirality in Mn$_{1-x}$Fe$_{x}$Ge alloys. This is drastically different from the usually supposed change of the sign of the DM interaction. Indeed, Eq. (\[eq:calD2\]) shows that the chirality can change even if the microscopic DM interaction, defined by the vectors $\mathbf{D}$, remains constant. In this case the sign change is due to the interplay between the exchange parameters $J_1$, $J_2$, and $J_3$ of three magnetic shells. Thus a potential possibility arises to control the sign of the magnetic chirality by varying the concentration of the different components and therefore affecting the Fermi wave number $k_F$. Notice that the possibility of such effect becomes evident only when the interactions with non-nearest neighbors are taken into account.
Another interesting fact, not being yet explained within the framework of microscopic theories, is the helix ordering along some special directions, e.g. $\langle 111 \rangle$ or $\langle
100 \rangle$. Usually this ordering is associated with a weak anisotropic exchange [@Bak80], but in fact the ordinary DM interaction also can result in the appearing of cubic anisotropic terms in the energy of spiral orientation in the lattice with a cubic space group. These contributions of the order $(D/J)^4$ will determine the critical magnetic field $H_{c1}$, at which the helix comes off from its preferable zero-field direction. This is in a good agreement with the observed ratio of the first and second critical fields $H_{c1} \ll H_{c2}$, because it follows from our estimations that $H_{c1} / H_{c2} \sim (D/J)^{2}$. Indeed, e.g. the period of the magnetic helix in MnSi makes about 40 unit cell parameters, which gives the value $2\pi / 40$ for the propagation number modulus $|k|$. On the other part, it follows from the phenomenological description that $k = {\cal D} / (2
{\cal J})$, or $D/J \sim \pi / 10 \approx 0.3$. This gives us the estimation $H_{c1} / H_{c2} \sim 0.1$, which is in a good agreement with the experimental data. In our previous work [@Chizhikov2012] we proposed a coarse grain approximation, which allowed us to calculate only the contributions to the energy proportional to $(D/J)^2$. In the present work a new approach has been developed permitting more precise calculations of the energy. In particular, the terms of the order of $(D/J)^4$ would give us a contribution of the Dzyaloshinskii–Moriya interaction into the energy of the cubic anisotropy responsible for (i) the ordering of the spiral axes along selected crystallographic directions, e.g. $\langle 111 \rangle$ in MnSi, in the absence of external magnetic field; (ii) the orientation of the triangle Skyrmion lattice in the A-phase observed in these crystals [@Grigoriev2006a; @Munzer2010; @Adams2011].
Acknowledgements {#acknowledgements .unnumbered}
================
We are grateful to M. V. Gorkunov, S. V. Demishev, S. V. Maleyev, S. V. Grigoriev, V. A. Dyadkin and S. S. Doudoukine for useful discussions and encouragement. This work was supported by two projects of the Presidium of the Russian Academy of Sciences: “Matter at high energy densities; Substance under high static compression” and “Diffraction of synchrotron radiation in multiferroics and chiral magnetics”.
{#appendixA}
The vectors $\boldsymbol{\tau}_i$, $\boldsymbol{\tau}_j$, $(\boldsymbol{\tau}_i - \boldsymbol{\tau}_j)$, $(\boldsymbol{\tau}_i + \boldsymbol{\tau}_j)$, $[\boldsymbol{\tau}_i \times \boldsymbol{\tau}_j]$, $\mathbf{b}_{n,ij}$, $\mathbf{D}_{n,ij}$ ($n = 1, 2, 3$) change with use of the same symmetry transformations of the point group $23$, when the index $ij$ passes trough 12 possible values $\{12, 13, 14, 21, 23, 24, 31, 32, 34, 41, 42, 43\}$. Therefore we can use the formulae $$\label{eq:aij}
\sum_{\{ij\}} \mathbf{a}_{ij} = 0 ,$$
$$\label{eq:aijbij}
\sum_{\{ij\}} a_{ij\alpha} b_{ij\beta} = 4 (\mathbf{a}_{12} \cdot \mathbf{b}_{12}) \delta_{\alpha\beta} ,$$
where $\mathbf{a}_{ij}$ and $\mathbf{b}_{ij}$ are the vectors from the above-listed set. Thereby the following summations can be easily made:
\[eq:sumtautau\] $$\begin{aligned}
\sum_{\{ij\}} (\boldsymbol{\tau}_i - \boldsymbol{\tau}_j)_{\alpha} (\boldsymbol{\tau}_i + \boldsymbol{\tau}_j)_{\beta} = 0 , \\
\sum_{\{ij\}} (\boldsymbol{\tau}_i \pm \boldsymbol{\tau}_j)_{\alpha} [\boldsymbol{\tau}_i \times \boldsymbol{\tau}_j]_{\beta} = 0 , \\
\sum_{\{ij\}} (\boldsymbol{\tau}_i - \boldsymbol{\tau}_j)_{\alpha} (\boldsymbol{\tau}_i - \boldsymbol{\tau}_j)_{\beta} = 32 \delta_{\alpha\beta} , \\
\sum_{\{ij\}} (\boldsymbol{\tau}_i + \boldsymbol{\tau}_j)_{\alpha} (\boldsymbol{\tau}_i + \boldsymbol{\tau}_j)_{\beta} = 16 \delta_{\alpha\beta} , \\
\sum_{\{ij\}} [\boldsymbol{\tau}_i \times \boldsymbol{\tau}_j]_{\alpha} [\boldsymbol{\tau}_i \times \boldsymbol{\tau}_j]_{\beta} = 32 \delta_{\alpha\beta} , \\
\sum_{\{ij\}} (\tau_{i\alpha} \tau_{j\beta} + \tau_{j\alpha} \tau_{i\beta}) = -8\delta_{\alpha\beta} .\end{aligned}$$
Using Eqs. (\[eq:sumtautau\]a)-(\[eq:sumtautau\]e) we can calculate the sums containing products of the vectors $\mathbf{b}_{n,ij}$, $\mathbf{D}_{n,ij}$ ($n = 1, 2, 3$):
\[eq:sumtaub\] $$\begin{aligned}
\sum_{\{ij\}} (\boldsymbol{\tau}_i - \boldsymbol{\tau}_j)_{\alpha} b_{1,ij\beta} = 4 (-8x+1) \delta_{\alpha\beta} , \\
\sum_{\{ij\}} (\boldsymbol{\tau}_i - \boldsymbol{\tau}_j)_{\alpha} b_{2,ij\beta} = 4 (-8x+3) \delta_{\alpha\beta} , \\
\sum_{\{ij\}} (\boldsymbol{\tau}_i - \boldsymbol{\tau}_j)_{\alpha} b_{3,ij\beta} = 4 (-8x-1) \delta_{\alpha\beta} ,\end{aligned}$$
$$\label{eq:tD}
\sum_{\{ij\}} (\boldsymbol{\tau}_i - \boldsymbol{\tau}_j)_{\alpha} D_{n,ij\beta} = 8D_{n+}\delta_{\alpha\beta} ,$$
$$\label{eq:DD}
\sum_{\{ij\}} D_{n,ij\alpha} D_{n,ij\beta} = 4 D_n^2 \delta_{\alpha\beta} ,$$
\[eq:sumbb\] $$\begin{aligned}
\sum_{\{ij\}} b_{1,ij\alpha} b_{1,ij\beta} = (32 x^2 - 8 x + 2) \delta_{\alpha\beta} , \\
\sum_{\{ij\}} b_{2,ij\alpha} b_{2,ij\beta} = (32 x^2 - 24 x + 6) \delta_{\alpha\beta} , \\
\sum_{\{ij\}} b_{3,ij\alpha} b_{3,ij\beta} = (32 x^2 + 8 x + 2) \delta_{\alpha\beta} ,\end{aligned}$$
\[eq:sumDb2\] $$\begin{aligned}
\sum_{\{ij\}} D_{1,ij\alpha} b_{1,ij\beta} = ((-8x+1) D_{1+} - D_{1-} + 2 D_{1y}) \delta_{\alpha\beta} , \\
\sum_{\{ij\}} D_{2,ij\alpha} b_{2,ij\beta} = ((-8x+3) D_{2+} + D_{2-} + 2 D_{2y}) \delta_{\alpha\beta} , \\
\sum_{\{ij\}} D_{3,ij\alpha} b_{3,ij\beta} = ((-8x-1) D_{3+} + D_{3-} + 2 D_{3y}) \delta_{\alpha\beta} .\end{aligned}$$
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Figures {#figures .unnumbered}
=======
![\[fig-canting\] (color online). (a) The 1D twisted magnetic structure; the spins rotate in the plane perpendicular to the propagation vector. (b) The canting in the absence of a twist; the lattice is divided into two sublattices (blue and sand-coloured arrows) with the spins tilted by a constant angle. (c) The canting and the twist; the canting of the sublattices leads to the combination of two helices with the same propagation vector and a constant phase shift.](fig-canting){width="7cm"}
![\[fig-neighbors\] (color online). Nearest magnetic neighborhood of a manganese atom in MnSi. The central atom (magenta) belongs to the first manganese sublattice, its neighbors are in 2nd (green), 3rd (orange) and 4th (blue) ones. The shortest bonds are given by solid lines, the bonds with the atoms from the 2nd and 3rd magnetic shells are shown by dashes and dots, correspondingly. (a) The 3-fold axis $[111]$ is vertical. The horizontal lines designate equidistant atomic planes containing the manganese atoms of 2nd, 3rd and 4th sublattices. (b) The projection on the plane perpendicular to the axis $[111]$. Two atomic triangles from (a), the upper and the lowest, belonging to different shells are situated just one above the other.](fig-neighbors){width="7cm"}
![\[fig-xid\] (color online). (a) The ambiguity of the transition from discrete to continuous model: two different smooth functions having the equal values in a discrete set of points. (b) The gain from the arbitrary choice of the parameter $x$. The initial curve (orange) is plotted with the use of the function values given in the discrete set of points with the coordinates $x$ and $\bar{x}$ within the cells of a 1D crystal. When the points are shifted to the center of the cells without a change of function values, the view of the curve changes (blue).](fig-xid){width="7cm"}
![\[fig-chain\] (color online). Periodical 1D chain of magnetic atoms with two spins within a unit cell. The spins can rotate in a plane, and their directions are fully determined by the only variable $\varphi$. The spin interaction is described by the classic Hamiltonian (\[eq:Echain\]). (a) When $J = 0$, the chirality is hidden; the lattice is divided onto two sublattices with the ferromagnetic ordering of spins and the canting angle $\beta$ between them. (b) When an additional ferromagnetic interaction with the atoms of the alternate kind is included ($J >
0$), the chirality becomes apparent in the magnetic helix with the angle $\delta$ between neighboring unit cells; besides, the angle $\alpha$ between the atoms $A$ and $B$ within a cell remains the same over the chain.](fig-chain){width="7cm"}
![\[fig-J\] (color online). (a) The $F_n \equiv F(2 k_F
a b_n)$ dependences on the Fermi wave number for four magnetic shells. The distances $b_1$–$b_4$ are chosen as for MnSi. The minima of $F_1$ give the areas with $J_\Sigma, {\cal J} > 0$. (b) The $J_\Sigma$ (solid line) and ${\cal J}$ (dash line) dependences on the Fermi wave number. The plots show strong oscillations. The jumps of ${\cal J}$ correspond to the zeros of $J_1+J_2+J_3$. The areas with $J_\Sigma, {\cal J} > 0$ determine weakly twisted ferromagnetic states. The most close to the known for MnSi value $k_F a = 16.4$ plateau is in the area $17.1 < k_F a < 18.9$.](fig-J){width="7cm"}
![\[fig-xexch\] (color online). Exchange coordinate $x_{exch}$ calculated in the RKKY model. The jumps of the function correspond to the zeros of $J_1+J_2+J_3$. The dot line identifies the real value $x_{real}=0.138$ for MnSi. The inset shows the $x_{exch}$ dependence in the corresponding to positive $J_\Sigma$ and ${\cal J}$ area $17.1 < k_F a < 18.9$, where $-0.28 < x_{exch}
- x_{real} < 0.16$. The chirality of the magnetic structure change its sign, when $x_{exch} = x_{real}$.](fig-xexch){width="7cm"}
[^1]: email: chizhikov@crys.ras.ru
[^2]: email: dmitrien@crys.ras.ru
|
---
abstract: |
The [*$k$-forest problem*]{} is a common generalization of both the [*$k$-MST*]{} and the [*dense-$k$-subgraph*]{} problems. Formally, given a metric space on $n$ vertices $V$, with $m$ demand pairs $\subseteq V \times V$ and a “target” $k\le m$, the goal is to find a minimum cost subgraph that connects *at least* $k$ demand pairs. In this paper, we give an $O(\min\{\sqrt{n},\sqrt{k}\})$-approximation algorithm for $k$-forest, improving on the previous best ratio of $O(n^{2/3}\log n)$ by Segev & Segev [@ss].
We then apply our algorithm for $k$-forest to obtain approximation algorithms for several [*Dial-a-Ride*]{} problems. The basic problem is the following: given an $n$ point metric space with $m$ objects each with its own source and destination, and a vehicle capable of carrying *at most* $k$ objects at any time, find the minimum length tour that uses this vehicle to move each object from its source to destination.
We prove that an $\alpha$-approximation algorithm for the $k$-forest problem implies an $O(\alpha\cdot\log^2n)$-approximation algorithm for . Using our results for $k$-forest, we get an $O(\min\{\sqrt{n},\sqrt{k}\}\cdot\log^2 n)$-approximation algorithm for . The only previous result known for was an $O(\sqrt{k}\log
n)$-approximation by Charikar & Raghavachari [@cr]; our results give a different proof of a similar approximation guarantee—in fact, when the vehicle capacity $k$ is large, we give a slight improvement on their results.
The reduction from to the $k$-forest problem is fairly robust, and allows us to obtain approximation algorithms (with the same guarantee) for the following generalizations: (i) Non-uniform , where the cost of traversing each edge is an arbitrary non-decreasing function of the number of objects in the vehicle; and (ii) Weighted , where demands are allowed to have different weights. The reduction is essential, as it is unclear how to extend the techniques of Charikar & Raghavachari to these generalizations.
author:
- 'Anupam Gupta[^1]'
- 'MohammadTaghi Hajiaghayi[^2]'
- 'Viswanath Nagarajan[^3]'
- 'R. Ravi$^{\ddagger}$'
bibliography:
- 'dial-ride.bib'
title: 'Dial a Ride from $k$-forest'
---
Introduction
============
In the Steiner forest problem, we are given a set of vertex-pairs, and the goal is to find a forest such that each vertex pair lies in the same tree in the forest. This is a generalization of the Steiner tree problem, where all the pairs contain a common vertex called the root; both the tree and forest versions are well-understood fundamental problems in network design [@akr; @gw]. An important extension of the Steiner tree problem studied in the late 1990s was the $k$-MST problem, where one sought the least-cost tree that connected any $k$ of the terminals: several approximations algorithms were given for the problem, culminating in the $2$-approximation of Garg [@garg]; the $k$-MST problem proved crucial in many subsequent developments in network design and vehicle routing [@cgrt; @fhr; @bcklmm; @bbcm]. One can analogously define the $k$-forest problem where one needs to connect *only $k$ of the pairs* in some Steiner forest instance: surprisingly, very little is known about this problem, which was first studied formally as recently as last year [@hj; @ss]. In this paper, we give a simpler and improved approximation algorithm for the .
Moreover, just like the $k$-MST variant, the seems to be useful in applications to network design and vehicle routing. In the second half of the paper, we show a (somewhat surprising) reduction of a well-studied vehicle routing problem called the problem to the . In the problem, we are given a metric space with people having sources and destinations, and a bus of some capacity $k$; the goal is to find a route for this bus so that each person can be taken from her source to destination without exceeding the capacity of the bus at any point, such that the length of the bus route is minimized. We show how the results for the slightly improve upon existing results for the problem; in fact, they give the first approximation algorithms for some generalizations of which do not seem amenable to previous techniques.
The $k$-Forest Problem
----------------------
Our starting point is the $k$-forest problem, which generalizes both the $k$-MST and the dense-$k$-subgraph problems.
\[kF-defn\] Given an $n$-vertex metric space $(V,d)$, and *demands* $\{s_i,
t_i\}_{i=1}^m \sse V \times V$, find the least-cost subgraph that connects at least $k$ demand-pairs.
Note that the is a generalization of the (minimization version of the) well-studied problem, for which nothing better than an $O(n^{1/3 - \delta})$ approximation is known. The was first defined in [@hj], and the first non-trivial approximation was given by Segev and Segev [@ss], who gave an algorithm with an approximation guarantee of $O(n^{2/3} \log n)$ for the case when $k
= O(\poly(n))$. We improve the approximation guarantee of the $k$-forest problem to $O(\min\{\sqrt{n},\sqrt{k}\})$; formally, we prove the following theorem in Section \[kF-section\].
\[th:intro2\] There is an $O(\min\{\sqrt{n}\cdot\frac{\log k}{\log n},
\sqrt{k}\})$-approximation algorithm for the $k$-forest problem. For the case when $k$ is less than a polynomial in $n$, the approximation guarantee improves to $O(\min\{\sqrt{n},\sqrt{k}\})$.
Apart from giving an improved approximation guarantee, our algorithm for the is arguably simpler and more direct than that of [@ss] (which is based on Lagrangian relaxations for the problem, and combining solutions to this relaxation). Indeed, we give two algorithms, both reducing the to the in different ways and achieving different approximation guarantees—we then return the better of the two answers. The first algorithm (giving an approximation of $O(\sqrt{k})$) uses the $k$-MST algorithm to find good solutions on the sources and the sinks independently, and then uses the Erdős-Szekeres theorem on monotone subsequences to find a “good” subset of these sources and sinks to connect cheaply; details are given in Section \[alg2\]. The second algorithm starts off with a single vertex as the initial solution, and uses the $k$-MST algorithm to repeatedly find a low-cost tree that satisfies a large number of demands which have one endpoint in the current solution and the other endpoint outside; this tree is then used to greedily augment the current solution and proceed. Choosing the parameters (as described in Section \[alg1\]) gives us an $O(\sqrt{n})$ approximation.
The Problem
-----------
In this paper, we use the to give approximation algorithms for the following vehicle routing problem.
\[dr-defn\] Given an $n$-vertex metric space $(V,d)$, a starting vertex (or *root*) $r$, a set of $m$ demands $\{(s_i, t_i)\}_{i = 1}^m$, and a vehicle of capacity $k$, find a minimum length tour of the vehicle starting (and ending) at $r$ that moves each object $i$ from its source $s_i$ to its destination $t_i$ such that the vehicle carries at most $k$ objects at any point on the tour.
We say that an object is [*preempted*]{} if, after being picked up from its source, it can be left at some intermediate vertices before being delivered to its destination. In this paper, we will not allow this, and will mainly be concerned with the *non-preemptive* problem.[^4]
The approximability of the problem is not very well understood: the previous best upper bound is an $O(\sqrt{k} \log
n)$-approximation algorithm due to Charikar and Raghavachari [@cr], whereas the best lower bound that we are aware of is APX-hardness (from TSP, say). We establish the following (somewhat surprising) connection between the and $k$-forest problems in Section \[dr-section\].
\[th:intro1\] Given an $\alpha$-approximation algorithm for $k$-forest, there is an $O(\alpha\cdot\log^2 n)$-approximation algorithm for the problem.
In particular, combining Theorems \[th:intro2\] and \[th:intro1\] gives us an $O(\min\{\sqrt{k},\sqrt{n}\}\cdot \log^2n)$-approximation guarantee for . Of course, improving the approximation guarantee for $k$-forest would improve the result for as well.
Note that our results match the results of [@cr] up to a logarithmic term, and even give a slight improvement when the vehicle capacity $k \gg n$, the number of nodes. Much more interestingly, our algorithm for easily extends to generalizations of the problem. In particular, we consider a substantially more general vehicle routing problem where the vehicle has no *a priori* capacity, and instead the cost of traversing each edge $e$ is an arbitrary non-decreasing function $c_e(l)$ of the number of objects $l$ in the vehicle; setting $c_e(l)$ to the edge-length $d_e$ when $l \leq k$, and $c_e(l) = \infty$ for $l > k$ gives us back the classical setting. In Section \[arbit-cost-section\], we show that this general [*non-uniform* ]{} problem admits an approximation guarantee that matches the best known for the classical problem. Another extension we consider is the *weighted* problem. In this, each object may have a different size, and total size of the items in the vehicle must be bounded by the vehicle capacity; this has been earlier studied as the [*pickup and delivery*]{} problem [@savs]. We show in Section \[wt-section\] that this problem can be reduced to the (unweighted) problem at the loss of only a constant factor in the approximation guarantee.
As an aside, we consider the effect of preemptions in the problem (Section \[pmt-section\]). It was shown in Charikar & Raghavachari [@cr] that the gap between the optimal preemptive and non-preemptive tours could be as large as $\Omega(n^{1/3})$. We show that the real difference arises between *zero* and *one* preemptions: allowing multiple preemptions does not give us much added power. In particular, we show in Section \[1-pmt\] that for any instance of the problem, there is a tour that preempts each object *at most once* and has length at most $O(\log^2 n)$ times an optimal preemptive tour (which may preempt each object an arbitrary number of times). Motivated by obtaining a better guarantee for on the Euclidean plane, we also study the preemption gap in such instances. We show that even in this case, there are instances having a gap of $\tilde{\Omega}(n^{1/8})$ between optimal preemptive and non-preemptive tours.
Related Work
------------
[**The $k$-forest problem:**]{} The $k$-forest problem is relatively new: it was defined by Hajiaghayi & Jain [@hj]. An $\tilde{O}(k^{2/3})$-approximation algorithm for even the directed $k$-forest problem can be inferred from [@ccc]. Recently, Segev & Segev [@ss] gave an $O(n^{2/3}\log n)$ approximation algorithm for $k$-forest.
[**Dense $k$-subgraph:**]{} The *$k$-forest* problem is a generalization of the problem [@fpk], as shown in [@hj]. The best known approximation guarantee for the problem is $O(n^{1/3-\delta})$ where $\delta>0$ is some constant, due to Feige et al. [@fpk], and obtaining an improved guarantee has been a long standing open problem. Strictly speaking, Feige et al. [@fpk] study a potentially harder problem: the *maximization* version of , where one wants to pick $k$ vertices to maximize the number of edges in the induced graph. However, nothing better is known even for the *minimization* version of (where one wants to pick the minimum number of vertices that induce $k$ edges), which is a special case of $k$-forest. The $k$-forest problem is also a generalization of $k$-MST, for which a 2-approximation is known (Garg [@garg]).
[**:**]{} While the problem has been studied extensively in the operations research literature, relatively little is known about its approximability. The currently best known approximation ratio for is $O(\sqrt{k}\log n)$ due to Charikar & Raghavachari [@cr]. We note that their algorithm assumes instances with unweighted demands. Krumke et al. [@krw] give a 3-approximation algorithm for the problem on a [*line metric*]{}; in fact, their algorithm finds a non-preemptive tour that has length at most 3 times the preemptive lower bound. (Clearly, the cost of an optimal preemptive tour is at most that of an optimal non-preemptive tour.) A $2.5$-approximation algorithm for *single source* version of (also called the “capacitated vehicle routing” problem) was given by Haimovich & Kan [@hk]; again, their algorithm output a non-preemptive tour with length at most 2.5 times the preemptive lower bound. For the *preemptive* problem, Charikar & Raghavachari [@cr] gave the current-best $O(\log n)$ approximation algorithm, and G[ø]{}rtz [@g] showed that it is hard to approximate this problem to better than $\Omega(\log^{1/4-\epsilon} n)$. Recall that no super-constant hardness results are known for the non-preemptive problem.
The $k$-forest problem {#kF-section}
======================
In this section, we study the $k$-forest problem, and give an approximation guarantee of $O(\min\{\sqrt{n},\sqrt{k}\})$. This result improves upon the previous best $O(n^{2/3}\log
n)$-approximation guarantee [@ss] for this problem. The algorithm in Segev & Segev [@ss] is based on a Lagrangian relaxation for this problem, and suitably combining solutions to this relaxation. In contrast, our algorithm uses a more direct approach and is much simpler in description. Our approach is based on approximating the following “density” variant of $k$-forest.
\[min-k-ratio\] Given an $n$-vertex metric space $(V,d)$, $m$ pairs of vertices $\{s_i, t_i\}_{i=1}^m$, and a target $k$, find a tree $T$ that connects [*at most*]{} $k$ pairs, and minimizes the ratio of the length of $T$ to the number of pairs connected in $T$.[^5]
We present two different algorithms for *minimum-ratio $k$-forest*, obtaining approximation guarantees of $O(\sqrt{k})$ (Section \[alg2\]) and $O(\sqrt{n})$ (Section \[alg1\]); these are then combined to give the claimed result for the . Both our algorithms are based on subtle reductions to the , albeit in very different ways.
As is usual, when we say that our algorithm [*guesses*]{} a parameter in the following discussion, it means that the algorithm is run for each possible value of that parameter, and the best solution found over all the runs is returned. As long as only a constant number of parameters are being guessed and the number of possibilities for each of these parameters is polynomial, the algorithm is repeated only a polynomial number of times.
An $O(\sqrt{k})$ approximation algorithm {#alg2}
----------------------------------------
In this section, we give an $O(\sqrt{k})$ approximation algorithm for minimum ratio $k$-forest, which is based on a simple reduction to the $k$-MST problem. The basic intuition is to look at the solution $S$ to minimum-ratio $k$-forest and consider an Euler tour of this tree $S$—a theorem of Erdős & Szekeres on increasing subsequences implies that there must be at least $\sqrt{|S|}$ sources which are visited in the same order as the corresponding sinks. We use this existence result to combine the source-sink pairs to create an instance of $\sqrt{|S|}$-MST from which we can obtain a good-ratio tree; the details follow.
Let $S$ denote an optimal ratio tree, that covers $q$ demands & has length $B$, and let $D$ denote the largest distance between any demand pair that is covered in $S$ (note $D\le B$). We define a new metric $l$ on the set $\{1,\cdots,m\}$ of demands as follows. The distance between demands $i$ and $j$, $l_{i,j} =
d(s_i,s_j)+d(t_i,t_j)$, where $(V,d)$ is the original metric. The $O(\sqrt{k})$ approximation algorithm first guesses the number of demands $q$ & the largest demand-pair distance $D$ in the optimal tree $S$ (there are at most $m$ choices for each of $q$ & $D$). The algorithm discards all demand pairs $(s_i,t_i)$ such that $d(s_i,t_i)>D$ (all the pairs covered in the optimal solution $S$ still remain). Then the algorithm runs the unrooted $k$-MST algorithm [@garg] with target $\lfloor\sqrt{q}\rfloor$, in the metric $l$, to obtain a tree $T$ on the demand pairs $P$. From $T$, we easily obtain trees $T_1$ (on all sources in $P$) and $T_2$ (on all sinks in $P$) in metric $d$ such that $d(T_1)+d(T_2)=l(T)$. Finally the algorithm outputs the tree $T'=T_1\cup T_2\cup \{e\}$, where $e$ is any edge joining a source in $T_1$ to its corresponding sink in $T_2$. Due to the pruning on demand pairs that have large distance, $d(e)\le D$ and the length of $T'$, $d(T')\le l(T)+D\le
l(T)+B$.
We now argue that the cost of the solution $T$ found by the $k$-MST algorithm $l(T)\le 8B$. Consider the optimal ratio tree $S$ (in metric $d$) that has $q$ demands $\{(s_1,t_1),\cdots,(s_q,t_q)\}$, and let $\tau$ denote an Euler tour of $S$. Suppose that in a traversal of $\tau$, the [*sources*]{} of demands in $S$ are seen in the order $s_1,\cdots,s_q$. Then in the same traversal, the [*sinks*]{} of demands in $S$ will be seen in the order $t_{\pi(1)},\cdots,t_{\pi(q)}$, for some permutation $\pi$. The following fact is well known (see, e.g., [@steele-survey]).
[**(Erdős & Szekeres)**]{}\[perm\] Every permutation on $\{1,\cdots,q\}$ has either an increasing subsequence of length $\lfloor\sqrt{q}\rfloor$ or a decreasing subsequence of length $\lfloor\sqrt{q}\rfloor$.
Using Theorem \[perm\], we obtain a set $M$ of $p=\lfloor\sqrt{q}\rfloor$ demands such that (1) the sources in $M$ appear in increasing order in a traversal of the Euler tour $\tau$, and (2) the sinks in $M$ appear in increasing order in a traversal of either $\tau$ or $\tau^R$ (the reverse traversal of $\tau$). Let $j_0 < j_1<\cdots <j_{p-1}$ denote the demands in $M$ in increasing order. From statement (1) above, $\sum_{i=0}^{p-1}
d(s(j_i),s(j_{i+1})) \le d(\tau)$, where the indices in the summation are modulo $p$. Similarly, statement (2) implies that $\sum_{i=0}^{p-1} d(t(j_i),t(j_{i+1})) \le \max\{d(\tau),d(\tau^R)\}
=d(\tau)$. Thus we obtain: $$\sum_{i=0}^{p-1} [d(s(j_i),s(j_{i+1})) + d(t(j_i),t(j_{i+1}))] \le
2d(\tau)\le 4B$$ But this sum is precisely the length of the tour $j_0,j_1,\cdots,j_{p-1},j_0$ in metric $l$. In other words, there is a tree of length $4B$ in metric $l$, that contains $\lfloor\sqrt{q}\rfloor$ vertices. So, the cost of the solution $T$ found by the $k$-MST approximation algorithm is at most $8B$.
Now the final solution $T'$ has length at most $l(T)+B\le 9B$, and ratio that at most $9\sqrt{q}\frac{B}{q}\le 9\sqrt{k}\frac{B}{q}$. Thus we have an $O(\sqrt{k})$ approximation algorithm for minimum ratio $k$-forest.
An $O(\sqrt{n})$ approximation algorithm {#alg1}
----------------------------------------
In this section, we show an $O(\sqrt{n})$ approximation algorithm for the minimum ratio $k$-forest problem. The approach is again to reduce to the $k$-MST problem; the intuition is rather different: either we find a vertex $v$ such that a large number of demand-pairs of the form $(v,*)$ can be satisfied using a small tree (the “high-degree” case); if no such vertex exists, we show that a repeated greedy procedure would cover most vertices without paying too much (and since we are in the “low-degree” case, covering most vertices implies covering most demands too). The details follow.
Let $S$ denote an optimal solution to minimum ratio $k$-forest, and $q\le k$ the number of demand pairs covered in $S$. We define the [ *degree*]{} $\Delta$ of $S$ to be the maximum number of demands (among those covered in $S$) that are incident at any vertex in $S$. The algorithm first guesses the following parameters of the optimal solution $S$: its length $B$ (within a factor 2), the number of pairs covered $q$, the degree $\Delta$, and the vertex $w\in S$ that has $\Delta$ demands incident at it. Although, there may be an exponential number of choices for the optimal length, a polynomial number of guesses within a binary-search suffice to get a $B$ such that $B\le d(S)\le 2\cdot B$. The algorithm then returns the better of the two procedures described below.
[**Procedure 1 (high-degree case):**]{} Since the degree of vertex $w$ in the optimal solution $S$ is $\Delta$, there is tree rooted at $w$ of length $d(S)\le 2B$, that contains at least $\Delta$ demands having one end point at $w$. We assign a weight to each vertex $u$, equal to the number of demands that have one end point at this vertex $u$ and the other end point at $w$. Then we run the $k$-MST algorithm [@garg] with root $w$ and a target weight of $\Delta$. By the preceding argument, this problem has a feasible solution of length $2B$; so we obtain a solution $H$ of length at most $4B$ (since the algorithm of [@garg] is a 2-approximation). The ratio of solution $H$ is thus at most $4B/\Delta=\frac{4q}{\Delta}\frac{B}{q}$.
[**Procedure 2 (low-degree case):**]{} Set $t=\frac{q}{2\Delta}$; note that $q\le \frac{\Delta\cdot n}{2}$ and so $t\le n/4$. We maintain a current tree $T$ (initially just vertex $w$), which is updated in iterations as follows: shrink $T$ to a supernode $s$, and run the $k$-MST algorithm with root $s$ and a target of $t$ new vertices. If the resulting $s$-tree has length at most $4B$, include this tree in the current tree $T$ and continue. If the resulting $s$-tree has length more than $4B$, or if all the vertices have been included, the procedure ends. Since $t$ new vertices are added in each iteration, the number of iterations is at most $\frac{n}{t}$; so the length of $T$ is at most $\frac{4n}{t}B$. We now show that $T$ contains at least $\frac{q}{2}$ demands. Consider the set $S\setminus T$ (recall, $S$ is the optimal solution). It is clear that $|S\setminus T|<t$; otherwise the $k$-MST instance in the last iteration (with the current $T$) would have $S$ as a feasible solution of length $\le 2B$ (and hence would find one of length at most $4B$). So the number of demands covered in $S$ that have at least one end point in $S\setminus T$ is at most $|S\setminus T|\cdot \Delta\le t\cdot \Delta = q/2$ (as $\Delta$ is the degree of solution $S$). Thus there are at least $q/2$ demands [*contained*]{} in $S\cap T$, in particular in $T$. Thus $T$ is a solution having ratio at most $\frac{4n}{t}B\cdot
\frac{2}{q}=\frac{8n}{t}\frac{B}{q}$.
The better ratio solution among $H$ and $T$ from the two procedures has ratio at most $\min\{\frac{4q}{\Delta},
\frac{8n}{t}\}\cdot\frac{B}{q} =
\min\{8t,\frac{8n}{t}\}\cdot\frac{B}{q}\le
8\sqrt{n}\cdot\frac{B}{q}\le 8\sqrt{n}\cdot\frac{d(S)}{q}$. So this algorithm is an $O(\sqrt{n})$ approximation to the minimum ratio $k$-forest problem.
Approximation algorithm for $k$-forest {#ratio-to-kforest}
--------------------------------------
Given the two algorithms for minimum ratio $k$-forest, we can use them in a standard greedy fashion (i.e., keep picking approximately minimum-ratio solutions until we obtain a forest connecting at least $k$ pairs); the standard set cover analysis can be used to show an $O(\min\{\sqrt{n},\sqrt{k}\}\cdot\log k)$-approximation guarantee for $k$-forest. A tighter analysis of the greedy algorithm (as done, e.g., in Charikar et al. [@ccc]) can be used to remove the logarithmic terms and obtain the guarantee stated in Theorem \[th:intro2\].
Applications to problems {#dr-section}
=========================
In this section, we study applications of the $k$-forest problem to the problem (Definition \[dr-defn\]), and some generalizations. A natural solution-structure for involves servicing demands in batches of at most $k$ each, where a batch consisting of a set $S$ of demands is served as follows: the vehicle starts out being empty, picks up each of the $|S|\le k$ objects from their sources, then drops off each object at its destination, and is again empty at the end. If we knew that the optimal solution has this structure, we could obtain a greedy framework for by repeatedly finding the best ‘batch’ of $k$ demands. However, the optimal solution may involve carrying almost $k$ objects at every point in the tour, in which case it can not be decomposed to be of the above structure. In Theorem \[pick-drop\], we show that there is always a near optimal solution having this ‘pick-drop in batches’ structure. Building on Theorem \[pick-drop\], we obtain approximation algorithms for the classical problem (Section \[basic-dr-section\]), and two interesting extensions: non-uniform (Section \[arbit-cost-section\]) & weighted (Section \[wt-section\]).
\[Structure Theorem\] \[pick-drop\] Given any instance of , there exists a feasible tour $\tau$ satisfying the following conditions:
1. $\tau$ can be split into a set of segments $\{S_1,\cdots ,S_t\}$ (i.e., $\tau=S_1\cdot
S_2\cdots S_t$) where each segment $S_i$ services a set $O_i$ of at most $k$ demands such that $S_i$ is a path that first picks up each demand in $O_i$ and then drops each of them.
2. The length of $\tau$ is at most $O(\log m)$ times the length of an optimal tour.
Consider an optimal non-preemptive tour $\sigma$: let $c(\sigma)$ denote its length, and $|\sigma|$ denote the number of edge traversals in $\sigma$. Note that if in some visit to a vertex $v$ in $\sigma$ there is no pick-up or drop-off, then the tour can be short-cut over vertex $v$, and it still remains feasible. Further, due to triangle inequality, the length $c(\sigma)$ does not increase by this operation. So we may assume that each vertex visit in $\sigma$ involves a pick-up or drop-off of some object. Since there is exactly one pick-up & drop-off for each object, we have $|\sigma|\le 2m+1$. Define the [*stretch*]{} of a demand $i$ to be the number of edge traversals in $\sigma$ between the pick-up and drop-off of object $i$. The demands are partitioned as follows: for each $j=1,\cdots ,\lceil\log (2m)\rceil$, group $G_j$ consists of all the demands whose stretch lie in the interval $[2^{j-1},2^j)$. We consider each group $G_j$ separately.
\[grp-tour\] For each $j=1,\cdots ,\lceil\log (2m)\rceil$, there is a tour $\tau_j$ that serves all the demands in group $G_j$, satisfies condition 1 of Theorem \[pick-drop\], and has length at most $6\cdot c(\sigma)$.
Consider tour $\sigma$ as a line $\mathcal{L}$, with every edge traversal in $\sigma$ represented by a distinct edge in $\mathcal{L}$. Number the vertices in $\mathcal{L}$ from 0 to $h$, where $h=|\sigma|$ is the number of edge traversals in $\sigma$. Note that each vertex in $V$ may be represented multiple times in $\mathcal{L}$. Each demand is associated with the numbers of the vertices (in $\mathcal{L}$) where it is picked up & dropped off.
Let $r=2^{j-1}$, and partition $G_j$ as follows: for $l=1,\cdots,
\lceil\frac{h}{r}\rceil$, set $O_{l,j}$ consists of all demands in $G_j$ that are picked up at a vertex numbered between $(l-1)r$ and $lr-1$. Since every demand in $G_j$ has stretch in the interval $[r,2r]$, every demand in $O_{l,j}$ is dropped off at a vertex numbered between $lr$ and $(l+2)r-1$. Note that $|O_{l,j}|$ equals the number of demands in $G_j$ carried over edge $(lr-1,lr)$ by tour $\sigma$, which is at most $k$. We define segment $S_{l,j}$ to start at vertex number $(l-1)r$ and traverse all edges in $\mathcal{L}$ until vertex number $(l+2)r-1$ (servicing all demands in $O_{l,j}$ by first picking up each demand between vertices $(l-1)r$ & $lr-1$; then dropping off each demand between vertices $lr$ & $(l+2)r-1$), and then return (with the vehicle being empty) to vertex $lr$. Clearly, the number of objects carried over any edge in $S_{l,j}$ is at most the number carried over the corresponding edge traversal in $\sigma$. Also, each edge in $\mathcal{L}$ participates in at most 3 segments $S_{l,j}$, and each edge is traversed at most twice in any segment. So the total length of all segments $S_{l,j}$ is at most $6\cdot c(\sigma)$. We define tour $\tau_j$ to be the concatenation $S_{1,j}\cdots S_{\lceil h/r\rceil,j}$. It is clear that this tour satisfies condition 1 of Theorem \[pick-drop\].
Applying this claim to each group $G_j$, and concatenating the resulting tours, we obtain the tour $\tau$ satisfying condition 1 and having length at most $6\log(2m)\cdot c(\sigma)=O(\log m)\cdot
c(\sigma)$.\
[**Remark:**]{} The ratio $O(\log m)$ in Theorem \[pick-drop\] is almost best possible. There are instances of (even on an unweighted line), where every solution satisfying condition 1 of Theorem \[pick-drop\] has length at least $\Omega(\max\{\frac{\log m}{\log\log m},\frac{k}{\log k}\})$ times the optimal non-preemptive tour. So, if we only use solutions of this structure, then it is not possible to obtain an approximation factor (just in terms of capacity $k$) for that is better than $\Omega(k/\log k)$. The solutions found by the algorithm for in [@cr] also satisfy condition 1 of Theorem \[pick-drop\]. It is interesting to note that when the underlying metric is a hierarchically well-separated tree, [@cr] obtain a solution of such structure having length $O(\sqrt{k})$ times the optimum, whereas there is a lower bound of $\Omega(\frac{k}{\log k})$ even for the simple case of an unweighted line.
Classical Dial-a-Ride {#basic-dr-section}
---------------------
Theorem \[pick-drop\] suggests a greedy strategy for Dial-a-Ride, based on repeatedly finding the best batch of $k$ demands to service. This greedy subproblem turns out to be the minimum ratio $k$-forest problem (Definition \[min-k-ratio\]), for which we already have an approximation algorithm. The next theorem sets up the reduction from $k$-forest to .
\[Reducing to minimum ratio $k$-forest\] \[dar-to-kforest\] A $\rho$-approximation algorithm for minimum ratio $k$-forest implies an $O(\rho \log^2 m)$-approximation algorithm for Dial-a-Ride.
The algorithm for Dial-a-Ride is as follows.
1. $\mathcal{C}=\phi$.
2. Until there are no uncovered demands, do:
1. Solve the minimum ratio $k$-forest problem, to obtain a tree $C$ covering $k_C\le k$ new demands.
2. Set $\mathcal{C}\leftarrow \mathcal{C}\cup C$.
3. For each tree $C\in\mathcal{C}$, obtain an Euler tour on $C$ to locally service all demands (pick up all $k_C$ objects in the first traversal, and drop them all in the second traversal). Then use a 1.5-approximate TSP tour on the sources, to connect all the local tours, and obtain a feasible non-preemptive tour.
Consider the tour $\tau$ and its segments as in Theorem \[pick-drop\]. If the number of uncovered demands in some iteration is $m'$, one of the segments in $\tau$ is a solution to the minimum ratio $k$-forest problem of value at most $\frac{d(\tau)}{m'}$. Since we have a $\rho$-approximation algorithm for this problem, we would find a segment of ratio at most $O(\rho)\cdot \frac{d(\tau)}{m'}$. Now a standard set cover type argument shows that the total length of trees in $\mathcal{C}$ is at most $O(\rho\log m)\cdot d(\tau)\le O(\rho\log^2 m)\cdot OPT$, where $OPT$ is the optimal value of the Dial-a-Ride instance. Further, the TSP tour on all sources is a lower bound on $OPT$, and we use a 1.5-approximate solution [@c]. So the final non-preemptive tour output in step 5 above has length at most $O(\rho\log^2 m)\cdot
OPT$.
This theorem is in fact stronger than Theorem \[th:intro1\] claimed earlier: it is easy to see that any approximation algorithm for $k$-forest implies an algorithm with the same guarantee for minimum ratio $k$-forest. Note that, $m$ and $k$ may be super-polynomial in $n$. However, we show in Section \[wt-section\] that with the loss of a constant factor, the general problem can be reduced to one where the number of demands $m\le n^4$. Based on this and Theorem \[dar-to-kforest\], a $\rho$ approximation algorithm for minimum ratio $k$-forest actually implies an $O(\rho\log^2 n)$ approximation algorithm for . Using the approximation algorithm for minimum ratio $k$-forest (Section \[kF-section\]), we obtain an $O(\min\{\sqrt{n},\sqrt{k}\}\cdot \log^2 n)$ approximation algorithm for the problem.
[**Remark:**]{} If we use the $O(\sqrt{k})$ approximation for $k$-forest, the resulting non-preemptive tour is in fact feasible even for a $\sqrt{k}$ capacity vehicle! As noted in [@cr], this property is also true of their algorithm, which is based on an entirely different approach.
Non-uniform Dial-a-Ride {#arbit-cost-section}
-----------------------
The greedy framework for described above is actually more generally applicable than to just the classical problem. In this section, we consider the problem under a substantially more general class of cost functions, and show how the $k$-forest problem can be used to obtain an approximation algorithm for this generalization as well. In fact, the approximation guarantee we obtain by this approach matches the best known for the classical problem. Our framework for is well suited for such a generalization since it is a ‘primal’ approach, based on directly approximating a near-optimal solution; this approach is not too sensitive to the cost function. On the other hand, the Charikar & Raghavachari [@cr] algorithm is a ‘dual’ approach, based on obtaining a good lower bound, which depends heavily on the cost function. Thus it is unclear whether their techniques can be extended to handle such a generalization.
Given an $n$ vertex undirected graph $G=(V,E)$, a root vertex $r$, a set of $m$ demands $\{(s_i,t_i)\}_{i=1}^m$, and a non-decreasing cost function $c_e:\{0,1,\cdots ,m\}\rightarrow \mathbb{R}^+$ on each edge $e\in E$ (where $c_e(l)$ is the cost incurred by the vehicle in traversing edge $e$ while carrying $l$ objects), find a non-preemptive tour (starting & ending at $r$) of minimum [*total cost*]{} that moves each object $i$ from $s_i$ to $t_i$.
Note that the classical problem is a special case when the edge costs are given by: $c_e(l)=d_e$ if $l\le k$ & $c_e(l)=\infty$ otherwise, where $d_e$ is the edge length in the underlying metric. We may assume (without loss in generality) that for any fixed value $l\in [0,m]$, the edge costs $c_e(l)$ induce a metric on $V$. Similar to Theorem \[pick-drop\], we have a near optimal solution with a ‘batch’ structure for the non-uniform problem as well, which implies the algorithm in Theorem \[non-unif-dr\]. The proof of the following corollary is almost identical to that of Theorem \[pick-drop\], and is omitted.
\[pick-drop-g\] Given any instance of non-uniform , there exists a feasible tour $\tau$ satisfying the following conditions:
1. $\tau$ can be split into a set of segments $\{S_1,\cdots ,S_t\}$ (i.e., $\tau=S_1\cdot
S_2\cdots S_t$) where each segment $S_i$ services a set $O_i$ of demands such that $S_i$ is a path that first picks up each demand in $O_i$ and then drops each of them.
2. The cost of $\tau$ is at most $O(\log m)$ times the cost of an optimal tour.
\[non-unif-dr\] A $\rho$-approximation algorithm for minimum ratio $k$-forest implies an $O(\rho \log^2 m)$-approximation algorithm for non-uniform . In particular, there is an $O(\sqrt{n}\log^2
m)$-approximation algorithm.
Corollary \[pick-drop-g\] again suggests a greedy algorithm for non-uniform based on the following [*greedy subproblem*]{}: find a set $T$ of uncovered demands and a path $\tau_0$ that first picks up each object in $T$ and then drops off each of them, such that the ratio of the cost of $\tau_0$ to $|T|$ is minimized. However, unlike in the classical problem, in this case the cost of path $\tau_0$ does not come from a single metric. Nevertheless, the minimum ratio $k$-forest problem can be used to solve this subproblem as follows.
1. For every $k=1,\cdots ,m$:
1. Define length function $d^{(k)}_e = c_e(k)$ on the edges.
2. Solve the minimum ratio $k$-forest problem on metric $(V,d^{(k)})$ with target $k$, to obtain tree $T'_{k}$ covering $n_k\le k$ demands.
3. Obtain an Euler tour $T_{k}$ of $T'_{k}$ that services these $n_k$ demands, by picking up all demands in one traversal and then dropping them all in a second traversal.
2. Return the tour $T_{k}$ having the smallest ratio $\frac{c(T_k)}{n_k}$ (over all $1\le k\le m$).
Assuming a $\rho$-approximation algorithm for minimum ratio $k$-forest (for all values of $k$), we now show that the above algorithm obtains a $16\rho$-approximate solution to the greedy subproblem. The cost of tour $T_k$ in step 3 is $c(T_k)\le 4\cdot
d^{(k)}(T'_{k})$, since $T_k$ involves traversing a tour on tree $T'_k$ twice and the vehicle carries at most $n_k\le k$ objects at every point in $T_k$. So the ratio of tour $T_k$ is $\frac{c(T_k)}{n_k}\le 4\frac{d^{(k)}(T'_{k})}{n_k}=4\cdot
\textrm{ratio}(T'_k)$. Let $\tau$ denote the optimal path for the greedy subproblem, $T$ the set of demands that it services, and $t=|T|$. Let $T_1$ denote the last $\frac{3}{4}t$ demands that are picked up, and $T_2$ denote the first $\frac{3}{4}t$ demands that are dropped off. It is clear that $T'=T_1\cap T_2$ has at least $t/2$ demands; let $T''\subset T'$ be any subset with $|T''|=t/4$. Let $\tau'$ denote the portion of $\tau$ between the $\frac{t}{4}$-th pick up and the $\frac{3t}{4}$-th drop off. Note that when path $\tau$ is traversed, there are at least $\frac{t}{4}$ objects in the vehicle while traversing each edge in $\tau'$. So the cost of $\tau$, $c(\tau)\ge
\sum_{e\in \tau'} c_e(t/4)$. Since $\tau'$ contains the end points of all demands in $T'\supset T''$, it is a feasible solution (covering the demands $T''$) to minimum ratio $k$-forest with target $k=t/4$ in the metric $d^{(t/4)}$, having ratio $(\sum_{e\in\tau'}c_e(t/4))/\frac{t}{4}\le \frac{4c(\tau)}{t}$. So the ratio of tour $T_{t/4}$ (obtained from the $\rho$-approximate tree $T'_{t/4}$) is at most $4\cdot\textrm{ratio}(T'_k)\le
4\rho\frac{4c(\tau)}{t}=16\rho\frac{c(\tau)}{t}$. Thus we have a $16\rho$-approximation algorithm for the greedy subproblem.
Based on Corollary \[pick-drop-g\], it can now be shown (as in Theorem \[dar-to-kforest\]) that a $\rho'$-approximation algorithm for the greedy subproblem implies an $O(\rho'\cdot \log^2
m)$-approximation algorithm for non-uniform . Using the above $16\rho$-approximation for the greedy subproblem, we have the theorem.
Weighted Dial-a-Ride {#wt-section}
--------------------
So far we worked with the unweighted version of , where each object has the same weight. In this section, we extend our greedy framework for to the case when objects have different sizes, and the total size of objects in the vehicle must be bounded by the vehicle capacity. Here we only extend the classical problem and not the generalization of Section \[arbit-cost-section\]. The problem studied in this section has been studied earlier as the [*pickup and delivery*]{} problem [@savs].
Given a vehicle of capacity $Q\in\mathbb{N}$, an $n$-vertex metric space $(V,d)$, a root vertex $r$, and a set of $m$ objects $\{(s_i,
t_i,w_i)\}_{i = 1}^m$ (with object $i$ having source $s_i$, destination $t_i$ & an integer size $1\le w_i\le Q$), find a minimum length (non-preemptive) tour of the vehicle starting (and ending) at $r$ that moves each object $i$ from its source to its destination such that the total size of objects carried by the vehicle is at most $Q$ at any point on the tour.
The classical problem is a special case when $w_i=1$ for all demands and the vehicle capacity $Q=k$. The following are two lower bounds for weighted : a TSP tour on the set of all sources & destinations (Steiner lower bound); and $\sum_{i=1}^{m}
\frac{w_i\cdot d(s_i,t_i)}{Q}$ (flow lower bound). In fact, as can be seen easily, these two lower bounds are valid even for the preemptive version of weighted ; so they are termed [*preemptive lower bounds*]{}.
The main result of this section (Theorem \[wt-algo\]) reduces weighted to the classical problem with the additional property that the number of demands ($m$) is small (polynomial in the number of vertices $n$). This shows that in order to approximate weighted , it suffices to consider instances of the classical problem with a small number of demands. The next lemma shows that even if the vehicle is allowed to split each object over multiple deliveries, the resulting tour is [*not*]{} much shorter than the tour where each object is required to be served in a single delivery (as is the case in weighted ). This lemma is the main ingredient in the proof of Theorem \[wt-algo\]. In the following, for any instance of weighted , we define the [*unweighted instance*]{} corresponding to it as a classical instance with vehicle capacity $Q$, and $w_i$ (unweighted) demands each having source $s_i$ and destination $t_i$ (for each $1\le i\le m$).
\[unwt-to-wt\] Given any instance $\mathcal{I}$ of weighted , and a solution $\tau$ to the unweighted instance corresponding to $\mathcal{I}$, there is a polynomial time computable solution to $\mathcal{I}$ having length at most $O(1)\cdot d(\tau)$.
Let $\mathcal{J}$ denote the unweighted instance corresponding to $\mathcal{I}$. Define line $\mathscr{L}$ as in the proof of Theorem \[pick-drop\] by traversing $\tau$ from $r$: for every edge traversal in $\tau$, add a new edge of the same length at the end of $\mathscr{L}$. For each unweighted object in $\mathcal{J}$ corresponding to demand $i$ in $\mathcal{I}$, there is a segment in $\tau$ (correspondingly in $\mathscr{L}$) where it is moved from $s_i$ to $t_i$. So each demand $i\in\mathcal{I}$ corresponds to $w_i$ segments in $\tau$ (each being a path from $s_i$ to $t_i$). For each demand $i$ in $\mathcal{I}$, we assign $i$ to one of its $w_i$ segments picked uniformly at random: call this segment $l_i$. For an edge $e\in \mathscr{L}$, let $N_e=\sum_{i:e\in
l_i} w_i$ denote the random variable which equals the [*total weight*]{} of demands whose assigned segments contain $e$. Note that the expected value of $N_e$ is exactly the number of unweighted objects carried by $\tau$ when traversing the edge corresponding to $e$. Since $\tau$ is a feasible tour for $\mathcal{J}$, $E[N_e]\le
Q$ for all $e\in \mathscr{L}$.
Consider a random instance $\mathcal{R}$ of on line $\mathscr{L}$ with vehicle capacity $Q$ and demands as follows: for each demand $i$ in $\mathcal{I}$, an object of weight $w_i$ is to be moved along segment $l_i$ (chosen randomly as above). Clearly, any feasible tour for $\mathcal{R}$ corresponds to a feasible tour for $\mathcal{I}$ of the same length. Note that the flow lower bound for instance $\mathcal{R}$ is $F=\sum_{e\in \mathscr{L}} d_e\lceil
\frac{N_e}{Q}\rceil$, and the Steiner lower bound is $\sum_{e\in
\mathscr{L}} d_e = d(\tau)$. Using linearity of expectation, $E[F]\le \sum_{e\in \mathscr{L}} d_e(\frac{E[N_e]}{Q}+1)\le 2\cdot
d(\tau)$. Let $R^*$ denote the instance (on line $\mathscr{L}$) obtained by assigning each demand $i$ in $\mathcal{I}$ to its shortest length segment (among the $w_i$ segments corresponding to it). Clearly this assignment minimizes the flow lower bound (over all assignments of demands to segments). So $R^*$ has flow bound $\le E[F]\le 2\cdot d(\tau)$, and Steiner lower bound $d(\tau)$.
Finally, we note that the 3-approximation algorithm for on a line [@krw] extends to a constant factor approximation algorithm for the case with weighted demands as well (this can be seen directly from [@krw]). Additionally, this approximation guarantee is relative to the preemptive lower bounds. Thus, using this algorithm on $R^*$, we obtain a feasible solution to $\mathcal{I}$ of length at most $O(1)\cdot d(\tau)$.
\[wt-algo\] Suppose there is a $\rho$-approximation algorithm for instances of classical with at most $O(n^4)$ demands. Then there is an $O(\rho)$-approximation algorithm for weighted (with any number of demands). In particular, there is an $O(\sqrt{n}\log^2
n)$ approximation for weighted .
Let $\mathcal{I}$ denote an instance of weighted with objects $\{(w_i,s_i,t_i) : 1\le i\le m\}$, and $\tau^*$ an optimal tour for $\mathcal{I}$. Let $\mathcal{P}=\{(s_1,t_1),\cdots
,(s_l,t_l)\}$ be the distinct pairs of vertices that have some demand between them, and let $T_i$ denote the total size of [*all*]{} objects having source $s_i$ and destination $t_i$. Note that $l\le n(n-1)$. Let $\mathcal{P}_{high} = \{i\in\mathcal{P}:T_i\ge
\frac{Q}{2}\}$, $\mathcal{P}_{low} = \{i\in\mathcal{P}:T_i\le
\frac{Q}{l}\}$, and $\mathcal{P}'=\mathcal{P}\setminus
(\mathcal{P}_{high}\cup\mathcal{P}_{low})$. We now show how to separately service objects in $\mathcal{P}_{low}$, $\mathcal{P}_{high}$ & $\mathcal{P}'$.
[**Servicing $\mathcal{P}_{low}$:**]{} The total size in $\mathcal{P}_{low}$ is at most $Q$; so we can service all these pairs by traversing a single 1.5-approximate tour [@c] on the sources and destinations. Note that the length of this tour is at most 1.5 times the Steiner lower bound, hence at most $1.5\cdot
d(\tau^*)$.
[**Servicing $\mathcal{P}_{high}$:**]{} Let $C$ be a 1.5-approximate minimum tour on all the sources. The pairs in $\mathcal{P}_{high}$ are serviced by a tour $\tau_1$ as follows. Traverse along $C$, and when a source $s_i$ in $\mathcal{P}_{high}$ is visited, traverse the direct edge to the corresponding destination $t_i$ & back, as few times as possible so as to move all the objects between $s_i$ and $t_i$, as described next. Note that every object to be moved between $s_i$ and $t_i$ has size (the original $w_i$ size) at most $Q$, and the total size of such objects $T_i\ge Q/2$. So these objects can be partitioned such that the size of each part (except possibly the last) is in the interval $[\frac{Q}{2},Q]$. So the number of times edge $(s_i,t_i)$ is traversed to service the demands between them is at most $2\lceil\frac{2T_i}{Q}\rceil\le 2(\frac{2T_i}{Q}+1)\le
8\frac{T_i}{Q}$. Now, the length of tour $\tau_1$ is at most $d(C)+
\sum_{(s_i,t_i)\in \mathcal{P}_{high}} 8 d(s_i,t_i)\frac{T_i}{Q}\le
d(C)+ 8\sum_{i=1}^m \frac{w_i\cdot d(s_i,t_i)}{Q}$. Note that $d(C)$ is at most 1.5 times the minimum tour on all sources (Steiner lower bound), and the second term above is the flow lower bound. So tour $\tau_1$ has length at most $O(1)$ times the preemptive lower bounds for $\mathcal{I}$, which is at most $O(1)\cdot d(\tau^*)$.
[**Servicing $\mathcal{P}'$:**]{} We know that the total size $T_i$ of each pair $i$ in $\mathcal{P}'$ lies in the interval $(Q/l,Q/2)$. Let $\mathcal{I}'$ denote the instance of weighted with demands $\{(s_i,t_i,T_i):i\in \mathcal{P}'\}$ and vehicle capacity $Q$; note that the number of demands in $\mathcal{I}'$ is at most $l$. The tour $\tau^*$ restricted to the objects corresponding to pairs in $\mathcal{P}'$ is a feasible solution to the [*unweighted instance*]{} corresponding to $\mathcal{I}'$ (but it may not feasible for $\mathcal{I}'$ itself). However Lemma \[unwt-to-wt\] implies that the optimal value of $\mathcal{I}'$, $opt(\mathcal{I})\le
O(1)\cdot d(\tau^*)$.
Next we reduce instance $\mathcal{I}'$ to an instance $\mathcal{J}$ of weighted satisfying the following conditions: [**(i)**]{} $\mathcal{J}$ has at most $l$ demands, [**(ii)**]{} each object in $\mathcal{I}$ has size at most $2l$, [**(iii)**]{} any feasible solution to $\mathcal{J}$ is feasible for $\mathcal{I}'$, and [**(iv)**]{} the optimal value $opt(\mathcal{J})\le O(1)\cdot
opt(\mathcal{I}')$. If $Q\le 2l$, $\mathcal{J}=\mathcal{I}'$ itself satisfies the required conditions. Suppose $Q\ge 2l$, then define $p=\lfloor \frac{Q}{l}\rfloor$; note that $Q\ge l\cdot p\ge Q-l\ge
\frac{Q}{2}$. Round up each size $T_i$ to the smallest integral multiple $T'_i$ of $p$, and round down the capacity $Q$ to $Q'=l\cdot p$. Since each size $T_i\in (\frac{Q}{l},\frac{Q}{2})$, all sizes $T'_i\in \{p,2p,\cdots, lp\}$. Now let $\mathcal{I}''$ denote the weighted instance with demands $\{(s_i,t_i,T'_i):i\in
\mathcal{P}'\}$ and vehicle capacity $Q'=lp$. One can obtain a feasible solution for $\mathcal{I}''$ from any feasible solution $\sigma$ for $\mathcal{I}'$ by traversing $\sigma$ a constant number of times: this follows from $Q'\ge \frac{Q}{2}$ & $T'_i\le \max\{
2T_i,Q'\}$.[^6] So the optimal value of $\mathcal{I}''$ is at most $O(1)\cdot opt(\mathcal{I}')$. Now note that all sizes and the vehicle capacity in $\mathcal{I}''$ are multiples of $p$; scaling down each of these quantities by $p$, we get an instance $\mathcal{J}$ [*equivalent*]{} to $\mathcal{I}''$ where the vehicle capacity is $l$ (and every demand size is at most $l$). This instance $\mathcal{J}$ satisfies all the four conditions claimed above.
Now observe that the instance $\mathcal{J}$ can be solved using $\rho$-approximation algorithm assumed in the theorem. Since $\mathcal{J}$ has at most $l$ demands (each of size $\le 2l$), the unweighted instance corresponding to $\mathcal{J}$ has at most $2l^2\le 2n^4$ demands. Thus, this unweighted instance can be solved using the $\rho$-approximation algorithm for such instances, assumed in the theorem. Then using the algorithm in Lemma \[unwt-to-wt\], we obtain a solution to $\mathcal{J}$, of length at most $O(\rho)\cdot opt(\mathcal{J})\le O(\rho)\cdot opt(\mathcal{I}')\le
O(\rho)\cdot d(\tau^*)$. Since any feasible solution to $\mathcal{J}$ corresponds to one for $\mathcal{I}'$, we have a tour servicing $\mathcal{P}'$ of length at most $O(\rho)\cdot d(\tau^*)$.
Finally, combining the tours servicing $\mathcal{P}_{low}$, $\mathcal{P}_{high}$ & $\mathcal{P}'$, we obtain a feasible tour for $\mathcal{I}$ having length $O(\rho)\cdot d(\tau^*)$, which gives us the desired approximation algorithm.\
Theorem \[wt-algo\] also justifies the assumption $\log
m=O(\log n)$ made at the end of Section \[dr-section\]. This is important because in general $m$ may be super-polynomial in $n$.
The Effect of Preemptions {#pmt-section}
=========================
In this section, we study the effect of the number of preemptions in the problem. We mentioned two versions of the problem (Definition \[dr-defn\]): in the preemptive version, an object may be preempted any number of times, and in the non-preemptive version objects are not allowed to be preempted even once. Clearly the preemptive version is least restrictive and the non-preemptive version is most restrictive. One may consider other versions of the problem, where there is a specified upper bound $P$ on the number of times an object can be preempted. Note that the case $P=0$ is the non-preemptive version, and the case $P=n$ is the preemptive version. We show that for any instance of the problem, there is a tour that preempts each object at most once (i.e., $P=1$) and has length at most $O(\log^2 n)$ times an optimal preemptive tour (i.e., $P=n$). This implies that the real gap between preemptive and non-preemptive tours is between zero and one preemption per object. A tour that preempts each object at most once is called a [*1-preemptive tour*]{}.
\[1-pmt\] Given any instance of the problem, there is a 1-preemptive tour of length at most $O(\log^2 n)\cdot OPT_{pmt}$, where $OPT_{pmt}$ is the length of an optimal preemptive tour. Such a tour can be found in randomized polynomial time.
Using the results on probabilistic tree embedding [@frt], we may assume that the given metric is a [*hierarchically well-separated*]{} tree $T$. This only increases the expected length of the optimal solution by a factor of $O(\log n)$. Further, tree $T$ has $O(\log \frac{d_{max}}{d_{min}})$ levels, where $d_{max}$ and $d_{min}$ denote the maximum and minimum distances in the original metric. We first observe that using standard scaling arguments, it suffices to assume that $\frac{d_{max}}{d_{min}}$ is polynomial in $n$. Without loss of generality, any preemptive tour involves at most $2m\cdot n$ edge traversals: each object is picked or dropped at most $2n$ times (once at each vertex), and every visit to a vertex involves picking or dropping at least one object (otherwise the tour can be shortcut over this vertex at no increase in length). By retaining only vertices within distance $OPT_{pmt}/2$ from the root $r$, we preserve the optimal preemptive tour and ensure that $d_{max}\le OPT_{pmt}$. Now consider modifying the original metric by setting all edges of length smaller than $OPT_{pmt}/2mn^3$ to length 0; the new distances are shortest paths under the modified edge lengths. So any pairwise distance decreases by at most $\frac{OPT_{pmt}}{2mn^2}$. Clearly the length of the optimal preemptive tour only decreases under this modification. Since there are at most $2mn$ edge traversals in any preemptive tour, the increase in tour length in going from the new metric to the original metric is at most $2mn\cdot \frac{OPT_{pmt}}{2mn^2}\le
\frac{OPT_{pmt}}{n}$. Thus at the loss of a constant factor, we may assume that $d_{max}/d_{min}\le 2mn^3$. Further, the reduction in Theorem \[unwt-to-wt\] also holds for preemptive ; so we may assume (at the loss of an additional constant factor) that the number of demands $m\le O(n^4)$. So we have $d_{max}/d_{min}\le
O(n^7)$ and hence tree $T$ has $O(\log n)$ levels.
The tree $T$ resulting from the probabilistic embedding has several Steiner vertices that are not present in the original metric; so the tour that we find on $T$ may actually preempt objects at Steiner vertices, in which case it is not feasible in the original metric. However as shown by Gupta [@gupta], these Steiner vertices can be simulated by vertices in the original metric (at the loss of a constant factor). Based on the preceding observations, we assume that the metric is a tree $T$ on the original vertex set having $l=O(\log n)$ levels, such that the expected length of the optimal preemptive tour is $O(\log n)\cdot OPT_{pmt}$.
We now partition the demands in $T$ into $l$ sets with $D_i$ (for $i=1,\cdots,l$) consisting of all demands having their least common ancestor (lca) in level $i$. We service each $D_i$ separately using a tour of length $O(OPT_{pmt})$. Then concatenating the tours for each level $i$, we obtain the theorem.
[**Servicing $D_i$:**]{} For each vertex $v$ at level $i$ in $T$, let $L_v$ denote the demands in $D_i$ that have $v$ as their lca. Consider an optimal [*preemptive*]{} tour that services the demands $D_i$. Since the subtrees under any two different level $i$ vertices are disjoint and there is no demand in $D_i$ across such subtrees, we may assume that this optimal tour is a concatenation of disjoint preemptive tours servicing each $L_v$ separately. If $OPT_{pmt}(v)$ denotes the length of an optimal preemptive tour servicing $L_v$ with $v$ as the starting vertex, $\sum_v
OPT_{pmt}(v) \le OPT_{pmt}$.
Now consider an optimal preemptive tour $\tau_v$ servicing $L_v$. Since the $s_j-t_j$ path of each demand $j\in L_v$ crosses vertex $v$, at some point in tour $\tau_v$ the vehicle is at $v$ with object $j$ in it. Consider the tour $\sigma_v$ obtained by modifying $\tau_v$ so that it drops each object $j$ at $v$ when the vehicle is at $v$ with object $j$ in it. Clearly $d(\sigma_v)=d(\tau_v)=OPT_{pmt}(v)$. Note that $\sigma_v$ is a feasible preemptive tour for the [*single source* ]{} problem with sink $v$ and all sources in $L_v$. Thus the algorithm of [@hk] gives a non-preemptive tour $\sigma'_v$ that moves all objects in $L_v$ from their sources to $v$, having length at most $2.5d(\sigma_v)=2.5OPT_{pmt}(v)$. Similarly, we can obtain a non-preemptive tour $\sigma''_v$ that moves all objects in $L_v$ from $v$ to their destinations, having length at most $2.5OPT_{pmt}(v)$. Now $\sigma'_v\cdot \sigma''_v$ is a 1-preemptive tour servicing $L_v$ of length at most $5\cdot OPT_{pmt}(v)$.
We now run a DFS on $T$ to visit all vertices in level $i$, and use the algorithm described above for servicing demands $L_v$ when $v$ is visited in the DFS. This results in a tour servicing $D_i$, having length at most $2d(T) + 5\sum_v OPT_{pmt}(v)$. Here $2d(T)$ is the Steiner lower bound, and $\sum_v OPT_{pmt}(v) \le OPT_{pmt}$. Thus the tour servicing $D_i$ has length at most $6\cdot OPT_{pmt}$.
Finally concatenating the tours for each level $i=1,\cdots,l$, we obtain a 1-preemptive tour on $T$ of length $O(\log n)\cdot
OPT_{pmt}$, which translates to a 1-preemptive tour on the original metric having length $O(\log^2 n)\cdot OPT_{pmt}$.
Motivated by obtaining an improved approximation for on the Euclidean plane, we next consider the worst case gap between an optimal non-preemptive tour and the preemptive lower bounds. As mentioned earlier, [@cr] showed that there are instances of where the ratio of the optimal non-preemptive tour to the optimal preemptive tour is $\Omega(n^{1/3})$. However, the metric involved in this example was the uniform metric on $n$ points, which can not be embedded in the Euclidean plane. The following theorem shows that even in this special case, there can be a polynomial gap between non-preemptive and preemptive tours, and implies that just preemptive lower bounds do not suffice to obtain a poly-logarithmic approximation guarantee.
\[plane-lb\] There are instances of on the Euclidean plane where the optimal non-preemptive tour has length $\Omega(\frac{n^{1/8}}{\log^3 n})$ times the optimal preemptive tour.
Consider a square of side 1 in the Euclidean plane, in which a set of $n$ demand pairs are distributed uniformly at random (each demand point is generated independently and is distributed uniformly at random in the square). The vehicle capacity is set to $k=\sqrt{n}$. Let $\mathcal{R}$ denote a random instance of obtained as above. We show that in this case, the optimal non-preemptive tour has length $\tilde{\Omega}(n^{1/8})$ with high probability. We first show the following claim.
\[rand-k-tree\] The minimum length of a tree containing $k$ pairs in $\mathcal{R}$ is $\Omega(\frac{n^{1/8}}{\log n})$, w.h.p.
Take any set $S$ of $k=\sqrt{n}$ demand pairs. Note that the number of such sets $S$ is ${n \choose k}$. This set $S$ has $2k$ points each of them generated uniformly at random. It is known that there are $p^{p-2}$ different labeled trees on $p$ vertices (see e.g. [@vw], Ch.2). The term [*labeled*]{} emphasizes that we are not identifying isomorphic graphs, i.e., two trees are counted as the same if and only if exactly the same pairs of vertices are adjacent. Thus there are at most $(2k)^{2k- 2}$ such trees just on set $S$. Consider any tree $T$ among these trees and root it at the source point with minimum label. Here we assume that $T$ has been generated using the “Principle of Deferred Decisions”, i.e., nodes will be generated one by one according to some breadth-first ordering of $T$. We say that an edge is [*short*]{} if its length is at most $\frac{c}{\alpha k}$ ($c$ and $\alpha\in (0,\frac{1}{2})$ will be fixed later).
If $T$ has length at most $c$, it is clear that at most an $\alpha$ fraction of its edges are [*not*]{} short. So $Pr[length(T)\le c]
\le \sum_{H} Pr[edges~in~H~are~short]$, where $H$ in the summation ranges over all edge-subsets in $T$ with $|H|\ge (1-\alpha)2k$. For a fixed $H$, we bound $Pr[edges~in~H~are~short]$ as follows. For any edge $(v,\text{parent}(v))$ (note $\text{parent}(v)$ is well-defined since $T$ is rooted), assuming that $\text{parent}(v)$ is fixed, the probability that this edge is short is $p=\pi(\frac{c}{\alpha
k})^2$. So we can upper bound the probability that edges $H$ are short by $p^{|H|}\le p^{(1-\alpha)2k}$. So we have $Pr[length(T)\le
c] \le 2^{2k}\cdot p^{(1-\alpha)2k}$, as the number of different edge sets $H$ is at most $2^{2k}$.
By a union bound over all such labeled trees $T$, the probability that the length of the minimum spanning tree on $S$ is less than $c$ is at most $(2k)^{2k}\cdot2^{2k}\cdot p^{(1-\alpha)2k}$. Now taking a union bound over all $k$-sets $S$, the probability that the minimum length of a tree containing $k$ pairs is less than $c$ is at most ${n \choose k}(2k)^{2k}2^{2k}p^{(1-\alpha)2k}$. Since $k=\sqrt{n}$, this term can be bounded as follows: $$(ek)^{k} (4k)^{2k} \pi^{(1-\alpha)2k} (\frac{c}{\alpha
k})^{{(1-\alpha)4k}} \le 500^k k^{3k} (\frac{c}{\alpha
k})^{(1-\alpha)4k} =[500\cdot (\frac{c}{\alpha})^{4-4\alpha}
(\frac{1}{k})^{1-4\alpha}]^k \le 2^{-k}$$ The last inequality above holds when $c\le \frac{\alpha}{1000}\cdot
k^{1/4-3\alpha/(1-4\alpha)}$. Setting $\alpha = \frac{1}{\log k}$, we get $$Pr[\exists~~\frac{k^{1/4}}{8000\cdot\log k} \textrm{ length
tree containing $k$ pairs in $\mathcal{R}$}] \le 2^{-k}$$ So, with probability at least $1-2^{-\sqrt{n}}$, the minimum length of a tree containing $k$ pairs in $\mathcal{R}$ is at least $\Omega(\frac{n^{1/8}}{\log n})$.
From Theorem \[pick-drop\], we obtain that there is a near optimal non-preemptive tour servicing all the demands in segments, where each segment (except possibly the last) involves servicing a set of $\frac{k}{2}\le t \le k$ demands. Although the lower bound of $k/2$ is not stated in Theorem \[pick-drop\], it is easy to extend the statement to include it. This implies that any solution of this structure has at least $\frac{n}{k}=k$ segments. Since each segment covers at least $k/2$ pairs, Claim \[rand-k-tree\] implies that each of these segments has length $\Omega(n^{1/8}/\log n)$. So the best solution of the structure given in Theorem \[pick-drop\] has length $\Omega(\frac{n^{1/8}}{\log n}k)$. But since there is a near-optimal solution of this structure, the optimal non-preemptive tour on $\mathcal{R}$ has length $\Omega(\frac{n^{1/8}}{\log^2
n}k)$.
On the other hand, the flow lower bound for $\mathcal{R}$ is at most $\frac{n}{k}=k$, and the Steiner lower bound is at most $O(\sqrt{n})=O(k)$ (an $O(\sqrt{n})$ length tree on the $2n$ points can be constructed using a $\sqrt{2n}\times \sqrt{2n}$ gridding). So the preemptive lower bounds are both $O(k)$; now using the algorithm of [@cr], we see that the optimal preemptive tour has length $O(k\log n)$. Combined with the lower bound for non-preemptive tours, we obtain the Theorem.
[**Acknowledgements:**]{} We thank Alan Frieze for his help in proving Theorem \[plane-lb\].
[^1]: Computer Science Department, Carnegie Mellon University. Supported in part by an NSF CAREER award CCF-0448095, and by an Alfred P. Sloan Fellowship.
[^2]: Computer Science Department, Carnegie Mellon University. Supported in part by NSF ITR grant CCR-0122581 (The ALADDIN project).
[^3]: Tepper School of Business, Carnegie Mellon University. Supported in part by NSF grants CCF-0430751 and ITR grant CCR-0122581 (The ALADDIN project).
[^4]: A note on the parameters: a feasible non-preemptive tour can be short-cut over vertices that do not participate in any demand, and we can assume that every vertex is an end point of some demand, and $n\le 2m$. We may also assume, by preprocessing some demands, that $m\le n^2\cdot k$. However in general, the number of demands $m$ and the vehicle capacity $k$ may be much larger than the number of vertices $n$.
[^5]: Even if we relax the solution to be any forest, we may assume (by averaging) that the *optimal ratio* solution is a tree.
[^6]: In particular, consider simulating a traversal along $\sigma$ of a capacity $Q$ vehicle ($T_0$) by 8 capacity $Q'$ vehicles $T'_1,\cdots,T'_8$, each running in parallel along $\sigma$. Whenever vehicle $T_0$ picks-up an object $i$, one of the vehicles $\{T'_g\}_{g=1}^8$ picks-up $i$: if $w_i\le \frac{Q}{4}$, any vehicle $\{T'_g\}_{g=1}^4$ that has free capacity picks-up $i$; if $w_i>\frac{Q}{4}$, any vehicle $\{T'_g\}_{g=5}^8$ that is empty picks-up $i$. It is easy to see that if at some point none of the vehicles $\{T'_g\}_{g=1}^8$ picks-up an object, there must be a capacity violation in $T_0$.
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abstract: 'We present an anti-ferromagnetically ordered ground state of Na$_{2}$IrO$_{3}$ based on density-functional-theory calculations including both spin-orbit coupling and on-site Coulomb interaction $U$. We show that the splitting of $e_{g}''$ doublet states by the strong spin-orbit coupling is mainly responsible for the intriguing nature of its insulating gap and magnetic ground state. Due to its proximity to the spin-orbit insulator phase, the magnetic ordering as obtained with finite $U$ is found to exhibit a strong in-plane anisotropy. The phase diagram of Na$_{2}$IrO$_{3}$ suggests a possible interplay between spin-orbit insulator and Mott anti-ferromagnetic insulator phases.'
author:
- Hosub
- Heungsik
- Hogyun
- 'Choong H.'
- Jaejun
title: 'Mott Insulating Ground State and its Proximity to Spin-Orbit Insulators in Na$_{2}$IrO$_{3}$'
---
Recently, the role of spin-orbit coupling (SOC) has attracted great attention in many fields of condensed matter physics. In multiferroic materials, for example, SOC combined with a large electron-lattice interaction has been suggested to be responsible for the multiferroic behavior which exhibit both non-collinear magnetic ordering and lattice polarization [@Kimura03; @Hur04]. SOC is also indispensable to anomalous Hall and spin Hall effects where Hall and spin Hall currents are generated by an external electric field, respectively [@PhysRev.95.1154; @ShuichiMurakami09052003; @PhysRevLett.92.126603]. In particular, the quantum spin Hall effect has led to the notion of topological insulators, new states of quantum matter [@zhang09; @xia09]. While they have bulk energy gaps generated by the SOC, topological insulators are characterized by the presence of gapless surface states which are protected by time-reversal symmetry [@kane:146802].
Another manifestation of strong SOC combined with on-site Coulomb interactions is the $j_{\mathrm{eff}}$=1/2 Mott insulator discovered in Sr$_{2}$IrO$_{4}$, one of the 5$d$ transition-metal oxides [@kim:076402; @moon:226402]. The novel spin-orbit integrated state with $j_{\mathrm{eff}}$=1/2 arises from the combined action of both strong SOC and intermediate on-site Coulomb interactions within the Ir 5$d$ $t_{2g}$ manifold. In addition, there has been a theoretical proposal on the room temperature quantum spin Hall effect in Na$_2$IrO$_3$ based on the $j_{\mathrm{eff}}$=1/2 physics [@shitade-2008], where the honeycomb lattice consisting of edge-shared IrO$_6$ octahedra in each Ir-O layer was considered to be an ideal realization of the Kane-Mele model, where hopping integrals between the $j_{\mathrm{eff}}$=1/2 states at the Fermi level was assumed to be an essential ingredient for the quantum spin Hall effect [@kane:146802; @PhysRevLett.95.226801]. Since the crystal structure and local environment of Ir atoms in Na$_2$IrO$_3$ are different from those of Sr$_{2}$IrO$_{4}$, however, it is necessary to clarify the electronic and magnetic structures of the Ir 5$d$ manifold in this Na$_2$IrO$_3$ compound with hexagonal lattice.
In this paper, we present novel electronic structure and magnetic properties of Na$_2$IrO$_3$ by carrying out density-functional-theory (DFT) calculations including both SOC $\lambda_{\mathrm{SO}}$ and on-site Coulomb interaction. We observe that a new form of spin-orbit coupled states emerges from the $e_{g}'$ doublet states near the Fermi level ($E_{\mathrm{F}}$) and determines the intriguing nature of its insulating gap. With an effective on-site Coulomb interaction parameter $U=2.0$ eV, the ground state of Na$_2$IrO$_3$ is found to be an antiferromagnetic (AFM) insulator with the ordered moments lying down within the honeycomb lattice of Ir atoms. The large splitting of the $e_{g}'$ doublet by the strong SOC is related to the strong in-plane anisotropy of magnetic ordering. Considering the role of SOC, we propose a phase diagram in the $\lambda_{\textrm{SO}}$–$U$ parameter space which features a phase boundary between AFM Mott insulators and SO insulators. By estimating the exchange couplings between neighboring Ir atoms, we suggest a possible frustration of magnetic ordering in its ground state, which is consistent with a recent experiment [@Takagi].
In order to examine the effects of both SOC and on-site Coulomb interaction on the electronic structure of Na$_2$IrO$_3$, it is necessary to treat both SOC and $U$ on an equal footing in the description of Ir 5$d$ states. To identify the role of each contribution as well as the interplay between them, we carried out DFT calculations within the local-density approximation (LDA), LDA including SOC (LDA+SO), and LDA+$U$ including SOC (LDA+$U$+SO) respectively. For the calculations, we used the DFT code, OpenMX [@openmx], based on the linear-combination-of-pseudo-atomic-orbitals method [@PhysRevB.67.155108], where both the LDA+$U$ method [@han:045110] and the SOC contribution were included via a relativistic $j$-dependent pseudo-potential scheme in the non-collinear DFT formalism. Double valence and single polarization orbitals were used as basis sets, which were generated by a confinement potential scheme with cutoff radii of 7.0, 7.0 and 5.0 a.u. for Na, Ir, and O atoms respectively. We used a (14$\times$14$\times$14) **k**-point grid for the k-space integration.
Up to our knowledge there is no crystal structure data for Na$_2$IrO$_3$ published yet. Thanks to the preliminary information provided by Takagi[@Takagi], we were able to construct a minimal unit-cell containing two formula units based on the hexagonal structure of Na$_{2}$RuO$_{3}$ [@Kailash04], a sibling compound of Na$_2$IrO$_3$. The crystal structure of Na$_2$IrO$_3$ can be viewed as an alternate stacking of (Ir$_{2/3}$Na$_{1/3}$)O$_2$ and Na layers. Edge-shared IrO$_6$ octahedra form a honeycomb lattice of Ir atoms. Na atoms are placed at the center of each hexagon. Upper and lower triangle oxygens are rotated by 3.5$^{\circ}$ to shorten the Ir-O distance. The positions of atoms in the unit cell were determined through the full structural optimization by the LDA calculations with 0.5$\times 10^{-3}$ Hatree/[Å]{} of force criterion. There is a possible stacking disorder in the types of the Na-layers relative to the (Ir$_{2/3}$Na$_{1/3}$)O$_2$ layers. We have checked the effect of different stacking sequences and observed a negligible change in the energy dispersions. Since the basic electronic structure is dominated by the in-plane Ir-O hybridization and remains intact regardless of the stacking sequence, we will focus on the electronic structure without structural disorder hereafter.
We investigated the electronic and magnetic structures of Na$_2$IrO$_3$ by performing LDA, LDA+SO, and LDA+$U$+SO calculations. Calculated electronic band structure near $E_{\mathrm{F}}$ are shown in Fig. \[fig:1\]. The LDA band structure in Fig. \[fig:1\](a) features the Ir 5$d$ bands of $e_{g}$ and $t_{2g}$ components separated by a large cubic crystal field $\Delta_{\textrm{cubic}}\sim$ 4 eV. While narrow $e_{g}$ bands are located at 3 eV above $E_{\mathrm{F}}$, the top of $t_{2g}$ bands are pinned at $E_{\mathrm{F}}$ and spread out to -2.0 eV below $E_{\mathrm{F}}$. Due to the extended nature of Ir 5$d$ orbitals, there are large contributions to the band structure from both the indirect hopping via the Ir 5$d$-O 2$p$ hybridization and the direct hopping between the neighboring Ir 5$d$ orbitals. From the tight-binding analysis [@Choong], even the next-nearest-neighbor hopping terms through oxygen and sodium atoms make significant contributions to the LDA band structure.
The trigonal crystal field ($\Delta_{\textrm{trigonal}}$) splits the $t_{2g}$ bands into $a_{1g}$ and $e_{g}'$ states. In addition, there is a strong hybridization between neighboring Ir 5$d$ orbitals which gives rise to the bonding and anti-bonding of $e_{g}'$ orbitals. The bonding and anti-bonding doublet states consist of $e_{g}'$ orbital pairs of two Ir atoms per unit cell. At the $\Gamma$ point of the LDA band structure, the $e_{g}'$ anti-bonding states, to be called by $e_{AB}$, are close to $E_{\mathrm{F}}$ while the $e_{g}'$ bonding states, to be called by $e_{B}$, are at about $-$0.8 eV. The $a_{1g}$ bands located at $-$1 eV have a negligible effect of the hybridization between neighboring Ir atoms but show a relatively large $c$-axis dispersion, which may be derived from the character of $a_{1g}$ orbitals pointing toward the Na atoms in the next layers. Here it is noted that the appearance of the $e_{AB}$ doublet at $E_{\mathrm{F}}$ in the LDA band structure of Na$_{2}$IrO$_{3}$ is in contrast to the presence of almost degenerate $t_{2g}$ state in Sr$_{2}$IrO$_{4}$ which serves as a basis for the $j_{\mathrm{eff}}$=1/2 state when SOC is introduced [@kim:076402].
![(Color online) Electronic band structures of Na$_2$IrO$_3$ within (a) LDA, (b) LDA+SO, and (c) LDA+$U$+SO schemes. Green, red, and blue colored energy dispersions in (a) are indicating $e_g$, $e_g'$, and $a_{1g}$ bands respectively, induced by the largest cubic and the next largest trigonal crystal fields.[]{data-label="fig:1"}](Fig1.pdf){width="8cm"}
In the LDA band structure, the doubly degenerate $e_{AB}$ states form a narrow band and cross $E_{\mathrm{F}}$. The introduction of SOC breaks the degeneracy of $e_{AB}$ by preserving the time-reversal symmetry so that the $e_{AB}$ bands split off over the whole Brillouin zone (BZ) as shown in Fig. \[fig:1\](b). Despite the split of $e_{AB}$ bands, the LDA+SO band structure is still metallic with a small electron pocket at the $A$ point and hole pockets off the $k_{c}=0$ plane near $M$. From the tight-binding analysis of the Na$_{2}$IrO$_{3}$ band structure [@Choong], we obtained $\Delta_{\textrm{trigonal}}\sim$ 0.6 eV, which is larger than the SOC parameter $\lambda_{\textrm{SO}}\sim$ 0.4 eV [@kim:076402; @PhysRevB.13.2433]. Thus the band structure of Na$_{2}$IrO$_{3}$ near $E_{\mathrm{F}}$ is characterized by the bonding $e_{B}$ and anti-bonding $e_{AB}$ states with $\Delta_{\textrm{cubic}}>
\Delta_{\textrm{trigonal}}>\lambda_{\textrm{SO}}$. Since $\Delta_{\textrm{trigonal}} > \lambda_{\textrm{SO}}$, however, the $e_{AB}$ character of the bands are maintained. Contrary to the layered perovskite Sr$_2$IrO$_4$ system, where the SOC entangles almost degenerate $t_{2g}$ orbitals with spin states and produces the spin-orbit integrated $j_{\mathrm{eff}}$=1/2, the strong trigonal field in Na$_{2}$IrO$_{3}$ suppresses the mixing of $a_{1g}$ and $e_{g}'$ states. Instead, the SOC acting on the $e_{g}'$ subspace plays a role of effective Zeeman coupling, the details of which will be discussed later. The presence of the effective Zeeman coupling is manifested in the parallel splitting of $e_{AB}$ and $e_{B}$ bands.
Similarly to the case of Sr$_{2}$IrO$_{4}$, both the on-site Coulomb interaction and the SOC are expected to be important in the description of Ir 5$d$ states. The LDA+$U$+SO band structure shown in Fig. \[fig:1\](c) was calculated with an effective $U=2.0$ eV, which was found to be consistent with angle-resolved photoemission and optical spectroscopy experiments [@kim:076402]. As a result of the combined action of both on-site Coulomb interaction and SOC, a small band gap arises between the SO-split $e_{AB}$ bands. Two $e_{AB}$ bands form valence and conduction bands with nearly the same dispersion above and below $E_{\mathrm{F}}$, respectively. Contrary to the non-magnetic metallic solution of the LDA and LDA+SO calculations, the LDA+$U$+SO solution predicts an AFM ordering with local magnetic moments lying within the $ab$ plane. The magnitude of total moment is 0.47 $\mu_B$ per each Ir atom, which is decomposed into the spin moment of 0.12 $\mu_B$ and the orbital moment of 0.35 $\mu_B$.
![Electronic band structures from LDA+SO calculations with the scaling factors of SOC strength $\lambda_{\textrm{SO}}/\lambda_0$ are (a) 0.5, (b) 1.0, (c) 1.5, and (d) 2.0, where $\lambda_0$ is the SOI magnitude of a real Ir atom. Gap opens when $\lambda_{\textrm{SO}}/\lambda_0$ is increasing from 1.0 to 1.5.[]{data-label="fig:2"}](Fig2.pdf){width="8cm"}
Despite that the importance of both $U$ and $\lambda_{\mathrm{SO}}$, the nature of the insulating ground state of Na$_{2}$IrO$_{3}$ is quite distinct from that of Sr$_{2}$IrO$_{4}$. In Sr$_{2}$IrO$_{4}$, the $j_{\mathrm{eff}}$=1/2 degeneracy can not be lifted by the SOC and the Mott-Hubbard gap can be attained only when the on-site $U$ is introduced. Thus breaking the time-reversal symmetry is essential to get the insulating ground state of Sr$_{2}$IrO$_{4}$. In the case of Na$_{2}$IrO$_{3}$, however, the broken time-reversal symmetry is not required to acquire the insulating state. As shown in Fig. \[fig:1\](b), the SO-split $e_{AB}$ bands are separated over the whole BZ so that the increase of the SOC strength can enlarge the already present gap between two $e_{AB}$ bands. To probe this idea, we carried out DFT calculations by controlling the SOC strength, which can be achieved by changing the scaling factor when generating the $j$-dependent pseudo-potential [@openmx]. Calculated results for the scale factors $\lambda_{\mathrm{SO}}/\lambda_{0}$= 0.5, 1.0, 1.5, and 2.0 are shown in Fig. \[fig:2\]. Taking the original SOC in the real Ir atom as $\lambda_{0}$ as a reference, $\lambda_{\mathrm{SO}}/\lambda_{0}$= 1.5 was found to be enough to open a full insulating gap. We call these insulating ground states as spin-orbit (SO) insulators, which have energy gaps generated by the SOC. SO insulators have no local moment and preserve the time-reversal symmetry and thus are distinct from the Mott-Hubbard insulator.
To understand the origin of SO insulators, we consider the SOC matrix elements within the $e_g'$ subspace. Since the degenerate $e_g'$ states can be written by $$\begin{aligned}
\mid e_1'\rangle=\frac{1}{\sqrt3}(\mid d_{xy}\rangle+e^{\imath\theta}\mid d_{yz}\rangle
+e^{-\imath\theta}\mid d_{zx}\rangle) \nonumber \\
\mid e_2'\rangle=\frac{1}{\sqrt3}(\mid d_{xy}\rangle+e^{-\imath\theta}\mid d_{yz}\rangle
+e^{\imath\theta}\mid d_{zx}\rangle)\end{aligned}$$ where $\theta=2\pi/3$, the on-site SOC term becomes $$\langle \mathcal{H}_{\textrm{SO}}\rangle_{e_g'}=
\langle \lambda_{\textrm{SO}}\mathbf{L}\cdot\mathbf{S}\rangle_{e_g'}=\frac{\lambda_{\textrm{SO}}}{2}
\left(\begin{array}{c|c}
\hat{n}\cdot \vec{\sigma} & \\ \hline
& \; -\hat{n}\cdot \vec{\sigma}
\end{array}\right)$$ where the basis sets are $\mid e_g'\rangle\otimes\mid
S=\frac{1}{2}\rangle=\{ \mid e_1'\alpha\rangle ,\mid e_1'\beta\rangle,\mid
e_2'\alpha\rangle,\mid e_2'\beta\rangle\}$ and $\hat{n}$ is the unit vector along the $c$-axis, i.e., the \[111\] direction in the local coordinate of IrO$_6$ octahedron. This block-diagonal form comes from the fact that $\langle \mathbf{L} \rangle$ is simultaneously diagonalized within $e_g'$ manifold and its eigenvalues are $\hat{n}$ and $-\hat{n}$, respectively. Here the SOC terms in $e_g'$ act as an internal magnetic field perpendicular to the $ab$-plane. The internal field gives rise to an effective Zeeman splitting, but the field direction in the $e_{1}'$ component is opposite to that in the $e_{2}'$ component. Thus, the effective Zeeman coupling does not break the time-reversal symmetry and $|e_1'\alpha\rangle$–$|e_2'\beta\rangle$ and $|e_1'\beta\rangle$–$|e_2'\alpha\rangle$ remain as time-reversal partners. Since the $e_{AB}$ states are the anti-bonding combination of the $e_{g}'$ orbitals of neighboring Ir atoms, the splitting of $e_{AB}$ bands by the effective Zeeman coupling is proportional to the SOC strength as shown in Fig. \[fig:2\], especially at the $\Gamma$ point.
The ground states of Na$_{2}$IrO$_{3}$ with the large SOC strength are SO insulators. The band gaps are induced by the effective Zeeman coupling of the SOC within the $e_{g}'$ subspace. Their characters are different from other types of band insulators such as covalent or ionic ones. The Fermi level is placed between bonding and anti-bonding bands in covalent solids and between different ionic configurations in ionic solids. In SO insulators, the gap is not driven by bonding characters, but mainly related to the symmetry of the states at $E_{\mathrm{F}}$. In a sense that their band gaps are generated by the SOC, SO insulators share the same ground with topological insulators though it is necessary to prove the non-trivial topology of its ground state.
One important consequence of the SO insulating phase is the proximity of the AFM ground state to the SO insulator state. In the LDA+$U$+SO calculation, the AFM ordered local moments are aligned in the $ab$-plane. Due to the huge internal field along \[111\] direction, it is hard to break the time-reversal symmetry and to develop local magnetic moments along that direction. Thus, transverse magnetic moments which are perpendicular to the internal field can be easily developed. Strong magnetic anisotropy originated from the internal magnetic fields might be seen in magnetic susceptibility measurements.
![Phase diagram in the $\lambda_{\textrm{SO}}$–$U$ parameter space depicting four different phases from LDA+$U$+SO calculations with varying $U$ and $\lambda_{\textrm{SO}}$ values. Paramagnetic metallic phase appears near the origin, Mott insulator in the region of $U > 1$, and SO insulator in the region of $\lambda_{\textrm{SO}}>1$ and $U <
1$. The real ground state is located inside Mott insulating territory.[]{data-label="fig:3"}](Fig3.pdf){width="8cm"}
To elucidate the relation between SO insulator and AFM Mott insulator phases, we explored a possible phase diagram of Na$_{2}$IrO$_{3}$ in an extended $\lambda_{\textrm{SO}}$–$U$ parameter space and present the result in Fig. \[fig:3\]. When $U$ is small and $\lambda_{\textrm{SO}}/\lambda_{0}$ is less than 1.5, the ground state remains as a paramagnetic metal. When there is no SOC, i.e., $\lambda_{\mathrm{SO}} = 0$, a ferromagentic metallic phase develops in a narrow range of the parameter space with $\lambda_{\mathrm{SO}} = 0$ upto $U=5.0$ eV. This ferromagnetic state becomes unstable in the presence of the SOC. On the other hand, for the value of $U$ smaller than about 1 eV, the SO insulator phase emerges as a non-magnetic insulator. Since the band gap is induced by the effective Zeeman coupling of the SOC within the $e_{g}'$ subspace, the Kramers degeneracy of the valence states holds up and the time-reversal symmetry remains unbroken. For the finite $\lambda_{\mathrm{SO}}$, Mott insulating AFM states develop as $U$ becomes larger than about 1.0 eV. The difference between two insulating phases, i.e., the criterion for the boundary is the existence of local magnetic moments. The Mott insulating phase has AFM ordering where on-site Coulomb repulsion breaks the symmetry developing local moments during the correlation gap opens. Our LDA+$U$+SO calculation predicts that the real ground state of Na$_2$IrO$_3$ is located in the Mott AFM region with $U=2.0 \sim 3.0$ eV and $\lambda_{\textrm{SO}}/\lambda_0=1$. However, the strongly anisotropic nature of its AFM ordering originates from its proximity to the SO insulator phase.
![(Color online) Schematic drawing of the (Ir$_{2/3}$Na$_{1/3}$)O$_2$ plane and magnetic configuration of the AFM insulating ground state of Na$_2$IrO$_3$. Magnetic moments are ordered anti-ferromagnetically lying on the $ab$-plane due to the strong internal field along the $c$-axis. Not only the NN exchange $J$ (dotted arrow) but the NNN exchange $J'$ (dashed arrow) are significant and may give rise to magnetic frustration.[]{data-label="fig:4"}](Fig4.pdf){width="8cm"}
Another important aspect in Na$_2$IrO$_3$ is magnetic frustration indicated in large $\theta_{\textrm{CW}}/T_{\textrm{N}}$ ratio from susceptibility measurements [@Takagi]. To reveal the origin of frustration, we have estimated exchange interactions $J$ and $J'$ between nearest-neighbor (NN) and next-nearest-neighbor (NNN) Ir atoms respectively.(Fig. \[fig:4\]) Calculation scheme is based on the perturbation formalism. $$J_{ij}=\frac{1}{2\pi}\int^{\epsilon_{F}}d\epsilon \left[
\hat{G}^{\uparrow}_{ij} \hat{V}_j \hat{G}^{\downarrow}_{ji} \hat{V}_i \right],$$ where $\hat{G}$ is the one-particle Green’s function and $\hat{V}$ is on-site exchange interaction potential [@PhysRevB.70.184421]. The result is $J'/J=0.47$, which means that NN and NNN exchange coupling strength are comparable and they might be a source of frustration. Above result is mainly attributed to the extended nature of Ir 5$d$ orbitals. Large direct overlap between NN Ir atoms gives FM direct exchange interaction, competing with AFM superexchange from oxygen mediated hopping channels and finally reducing AFM exchange $J$. On the other hand, the NNN hopping integrals are not negligible that the NNN AFM interaction $J'$ can be comparable and frustrate long range AFM ordering.
In conclusion we have shown that the spin-orbit entangled $e_g'$ states under the strong internal Zeeman field driven by the SOC lead to an unusual band gap. The predicted AFM ground state is in close proximity to the SO insulator phase where the AFM ordering in Na$_2$IrO$_3$ becomes strongly anisotropic with quenched moments along the $c$-axis. The highly anisotropic AFM state in Na$_2$IrO$_3$ may serve as a model system for the two-dimensional XY model with frustrated exchange interactions. One may be able to drive a crossover between AFM and SO insulators through the modulation of structural parameters or chemical substitution, though we need more study on the role of SOC in the Mott AFM phase in connection with the topological nature of SO insulators.
We are grateful to H. Takagi for sharing information prior to publication. This work was supported by the KOSEF through the ARP (R17-2008-033-01000-0). We also acknowledge the support from KISTI under the Supercomputing Application Support Program.
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---
abstract: |
*Syntax-Guided Synthesis (SyGuS)* is the computational problem of finding an implementation $f$ that meets both a semantic constraint given by a logical formula $\varphi$ in a background theory $T$, and a syntactic constraint given by a grammar $G$, which specifies the allowed set of candidate implementations. Such a synthesis problem can be formally defined in SyGuS-IF, a language that is built on top of SMT-LIB.
The *Syntax-Guided Synthesis Competition ([SyGuS-Comp]{})* is an effort to facilitate, bring together and accelerate research and development of efficient solvers for SyGuS by providing a platform for evaluating different synthesis techniques on a comprehensive set of benchmarks. In this year’s competition we added a new track devoted to *programming by examples*. This track consisted of two categories, one using the theory of bit-vectors and one using the theory of strings. This paper presents and analyses the results of [SyGuS-Comp]{}’16.
author:
- Rajeev Alur
- Dana Fisman
- Rishabh Singh
- 'Armando Solar-Lezama'
bibliography:
- 'bib-wdoi.bib'
nocite: '[@*]'
title: 'SyGuS-Comp 2016: Results and Analysis'
---
Introduction {#sec:intro}
============
Competition Settings {#sec:setting}
====================
Competition Results and Analysis {#sec:results}
================================
Discussion {#sec:discussion}
==========
|
---
abstract: 'We describe the design and construction of the low rate neutron calibration sources used in the Daya Bay Reactor Anti-neutrino Experiment. Such sources are free of correlated gamma-neutron emission, which is essential in minimizing induced background in the anti-neutrino detector. The design characteristics have been validated in the Daya Bay anti-neutrino detector.'
address:
- 'Kellogg Radiation Laboratory, California Institute of Technology, Pasadena, California, USA'
- 'Department of Physics, Shanghai Jiao Tong University, Shanghai, China'
- 'Department of Physics, College of William and Mary, Williamsburg, Virginia, USA'
- 'Lawrence Berkeley National Laboratory, Berkeley, California, USA'
- 'Brookhaven National Laboratory, Upton, New York, USA'
- 'Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Alabama 35487, USA'
author:
- 'J. Liu'
- 'R. Carr'
- 'D. A. Dwyer'
- 'W. Q. Gu'
- 'G. S. Li'
- 'R. D. McKeown'
- 'X. Qian'
- 'R. H. M. Tsang'
- 'F. F. Wu'
- 'C. Zhang'
bibliography:
- '<your-bib-database>.bib'
title: Neutron Calibration Sources in the Daya Bay Experiment
---
neutron sources ,$^{241}$Am-$^{13}$C ,reactor neutrinos ,$\theta_{13}$ ,Daya Bay ,calibration
Introduction {#sec:intro}
============
Neutron sources are important calibration sources with a wide range of applications. In modern reactor neutrino experiments such as Daya Bay [@tdr], Double Chooze [@DCTDR] and RENO [@RENOTDR], the electron anti-neutrinos are detected by liquid scintillator detectors via the inverse beta decay (IBD) reaction $\bar{\nu_e} + p \rightarrow e^+ + n$ with the time-correlated prompt positron signal ranging from 1 to 10 MeV, and the delayed neutron capture signal of $\sim$8 MeV on the gadolinium dopant or 2.2 MeV on hydrogen. In the Daya Bay experiment, which is located in South China, the regular deployment of neutron sources allows a thorough characterization of the detector response to IBD neutrons, which contributes to the recent discovery of the neutrino mixing angle $\theta_{13}$ [@DYB12].
The automated calibration units (ACUs) of the Daya Bay anti-neutrino detectors (ADs) are detailed in [@ref:ACU]. Each AD is submerged in a water pool and is equipped with three ACUs on top. Neutrino interactions are rare, so minimizing potential background created by the neutron sources is a top consideration. In this article, we discuss the design and construction of low rate ($\sim<$ 1 Hz) neutron sources that are free of correlated gamma emission. Although used in a specific experiment, such a source could also be potentially useful in other occasions where ultra low background is desired, or where one seeks a neutron source with no associated gamma rays.
Physical requirements to the neutron sources {#sec:requirements}
============================================
As discussed in [@DYB12], the Daya Bay ADs are arranged in a 4-near and 4-far configuration to the nuclear reactor cores, with the near detectors sampling the reactor neutrino flux and far detectors detecting the $\bar{\nu_e}$ disappearance due to $\theta_{13}$. The rates of the IBD in the near (far) detectors are approximately 700 (70)/day/AD [@DYB12]. For each AD module, three ACUs are instrumented, each of which is capable of deploying radioactive sources vertically into the detector. There is one neutron source in each ACU [@tdr][@ref:ACU]. During normal neutrino data taking, the neutron sources are “parked” inside the ACUs right above the AD. Although these neutrons rarely leak directly into the neutrino target (gadolinium loaded liquid scintillator), they can get captured on surrounding materials, in particular, stainless steel, i.e. Fe, Mn, Cr, Ni, etc, and emit gamma rays ranging from 6 to 10 MeV. These high energy gamma rays, later referred to as SS-capture gammas, are difficult to shield, leading to two kinds of background due to accidental and correlated coincidences: a) the SS-capture gammas can be in random delayed coincidence with ambient gamma background, mimicking the IBD signals, and b) if multiple neutrons are emitted per decay, or a neutron is emitted with a correlated gamma ray, or a neutron is producing gammas via inelastic scattering, they can form real correlated background to the IBDs. The signal to background ratio at the far site drives the requirements to the neutron sources:
- the accidental background to be less than 5% of the IBD signal at the far site. Such a background can be statistically subtracted;
- the correlated background to be less than 0.5% of the IBD signal at the far site, i.e. $<0.35$ per day. Such a background can be estimated via Monte Carlo (MC) simulation and benchmarked by special control data.
For a typical neutron source inside the ACU, a GEANT4 [@G4] simulation with realistic detector geometry predicts that the SS-capture gamma ray leaking into the detector satisfying the IBD delay energy cut is approximately $2\times10^{-3}$ per neutron. Taking into account the $\sim$70 Hz singles rate [@DYB12] and 200 $\mu$s coincidence window, the first requirement translates into a limit of the neutron rate per source $<$1 Hz.
Design of the neutron source {#sec:design}
============================
Selection of neutron source
---------------------------
There are several types of commonly used compact neutron calibration sources, e.g. fission sources such as $^{252}$Cf, ($\alpha$,n) sources such as $^{241}$Am-Be, and photo-neutron sources such as $^{124}$Sb-Be . For the third type, photo-neutron cross section is of the order millibarns, implying the need for a rather strong driving gamma source. Due to low background considerations, this option was rejected early on.
A $^{252}$Cf source emits multiple neutrons per fission together with gammas. Typical ($\alpha$,n) sources have correlated gamma neutron emission when the final state nucleus is in an excited state. As mentioned earlier, such sources would inevitably lead to correlated background in the AD, in addition to the accidental background. Just to set the scale, the predicted correlated background for $^{252}$Cf and $^{241}$Am-Be sources are 2.6/day and 1.3/day, respectively, assuming a 0.5 Hz neutron rate.
The alphas from $^{241}$Am are $\sim$5.5 MeV. To eliminate correlated gamma rays emission, $^{7}$Li would be a good candidate target since such alpha energy can only produce ground state $^{10}$B. However, the most common and chemically inert Li compound is LiF, and ($\alpha$,n) on $^{19}$F creates a significant amount of high energy gamma rays.
$^{13}$C is the final candidate. We note that $^{241}$Am-$^{13}$C can produce neutrons with the final state $^{16}$O either in the ground state, and the first or second excited state. The first excited state of $^{16}$O decays into a e$^+$e$^-$ pair, which will be stopped by the source enclosure and surrounding materials [^1]. The second excited state of $^{16}$O will emit a 6.13 MeV gamma ray, producing correlated background together with the neutron. However if the energy of $\alpha$ is attenuated to below 5.11 MeV, this correlated $\gamma$-neutron process can be eliminated entirely. Based on all considerations above, we selected $^{241}$Am-$^{13}$C as the neutron source, further requiring that $E_{\alpha} < 5.11$ MeV.
Physical Design
===============
$^{241}$Am Sources {#sec:alpha}
------------------
$^{241}$Am discs from NRD Inc., 5-mm in diameter, were procured with 4.5 MeV alpha energy as a key specification (custom energy was achieved by varying the thickness of the electrodeposited gold coating). The activity of $^{241}$Am is approximately 28 $\mu$Ci, and is deposited on one side of the disc only. Measurements of the emitting alpha energy were performed at Caltech in a vacuum chamber with a Si detector. The raw energy spectrum from the $^{241}$Am source is shown in Fig. \[fig:nrd\_attenuated\]. Also overlaid is the energy spectrum from a standard $^{241}$Am source [^2]. It was discovered that although the alpha energy for the NRD sources is peaked around 4.6 MeV, the distribution is rather broad, all the way up to 5.5 MeV. The same measurements were performed on multiple discs and the results were consistent. In order to further reduce the alpha energy, 1 $\mu$m thick gold foil was purchased from Alfa Aesar and attached to the front surface of a NRD source. The attenuated energy spectrum is overlaid in Fig. \[fig:nrd\_attenuated\].
![*Measured alpha spectra with (blue) and without (red) the 1 $\mu$m gold foil. The spectrum from a standard $^{241}$Am $\alpha$ source (5.5MeV) (black) is overlaid for reference.*[]{data-label="fig:nrd_attenuated"}](NRD_attenuated.pdf){width="4in"}
Compared to that without the gold foil, the entire energy spectrum was shifted by about 0.5 MeV, as expected. Out of about 20000 total alphas, no events were observed beyond 5.11 MeV threshold.
Expected neutron rate and energy spectrum
-----------------------------------------
A GEANT4 program was developed to calculate the neutron rate from the source. To simulate the one-sided NRD source, the alphas were generated in random directions in the active hemisphere with energy sampled from the spectrum in Fig. \[fig:nrd\_attenuated\] (the red histogram). The energy loss of the alphas in 1 $\mu$m gold and $^{13}$C was simulated by GEANT4. GEANT4 tracks alphas until they are stopped. When an alpha enters $^{13}$C, for each step $i$ along the track, one computes a step weight (which gets summed at the end of the event) $$\label{eq:weight0}
\mathrm{weight_{i}} = \sigma(E_{\alpha,i})\times d_i\,,$$ where $E_{\alpha,i}$ is the mean alpha energy in this step, $d_i$ is the step length and $\sigma(E_{\alpha,i})$ is the $^{13}$C($\alpha$,n) reaction cross section. The total neutron rate can then be calculated as $$\label{eq:rate}
R_n = R_{\alpha} \times \displaystyle\sum_{i} \mathrm{weight_i}
\times \displaystyle\frac{\rho_{^{13}C} \cal{N}_A}{13}\,$$ where $R_{\alpha}$ is the alpha emission rate, $\rho_{^{13}C}$ is the density of $^{13}$C, and $\cal{N}_A$ is the Avogadro’s number. The outgoing neutron energy spectrum was obtained by generating a neutron in random direction (but not tracked) with respect to the alpha momentum in each tracking step in $^{13}$C, with its energy calculated based on 2-body elastic scattering kinematics. Each neutron was assigned a step weight as in Eqn. \[eq:weight0\]. The resulting neutron energy spectrum is shown in Fig. \[fig:neutronEnergy\]. One sees that on average the neutron energy is 4 MeV, with a tail extending to 6.5 MeV or so. The neutron energy-angle (where $\theta$ is the angle relative to the normal of the active alpha surface) correlated distribution is shown in Fig. \[fig:energy\_angle\]. Clearly, neutrons heading opposite to the “active” side of the alpha source have significantly lower energy.
[ \[fig:energy\_angle\] ![ *a) Simulated neutron energy spectrum from the $^{241}$Am-$^{13}$C source: black=total, red=heading out from the active side of $^{241}$Am (“up”), blue=heading opposite to the active side of $^{241}$Am (“down”). The fraction of neutrons with energy $>$4.4 MeV is 33%, 50%, and 16% for “total”, “up”, and “down” distributions, respectively. b) Simulated neutron energy vs. angle distribution. See text for details.* []{data-label="fig:neutron_energy_energy"}](neutron_energy_angle.pdf "fig:"){width="45.00000%"} ]{}
High energy neutrons could inelastically scatter with materials surrounding and inside the detector, e.g. stainless steel, $^{12}$C, etc, and produce gamma rays along the way, introducing correlated background when being detected in delayed coincidence with the final SS-capture gammas. To reduce such a residual background, we chose to use the source with the active side of the $^{241}$Am facing up to reduce the energy of the downward going neutrons.
Mechanical Design and Fabrication {#sec:fab}
=================================
Pure $^{13}$C is immediately available in powder form. To get a good neutron flux, it is important to have the $^{241}$Am in close and uniform contact with $^{13}$C. Alpha leakage is a serious contamination, so such a source should also be very safely sealed.
The mechanical design [^3] of the $^{241}$Am-$^{13}$C source is shown in Fig. \[fig:enclosure\].
![*Mechanical drawing of the $^{241}$Am-$^{13}$C source assembly. See text for more details.*[]{data-label="fig:enclosure"}](AmC_design_zoom.pdf){width="3.5in"}
A stainless cup with 5 mm inner diameter is the holder for $^{13}$C, which is pressed into a solid form. $^{241}$Am source, Au foil, and $^{13}$C are sandwiched tightly. The stainless cup will be enclosed in an acrylic enclosure made out of three pieces: the bottom cup which houses the stainless cup; a plunger that presses $^{241}$Am, the Au foil, and $^{13}$C together, and the top flange that seals the entire assembly. The seals between the acrylic pieces are made using the Weld-on 3 (volatile, water thin) acrylic cement [@weld_on]. Such a design went through a standard test procedure and obtained the State of California Certificate of Sealed Sources, which was a key requirement to transport these sources to Daya Bay.
Quality controls {#sec:meas}
================
Twenty-eight $^{241}$Am-$^{13}$C sources were fabricated at Caltech, twenty-four of which were transported to Daya Bay after a quality control (QC) process. During QC, we set up a neutron detector in a low background environment, and measured the neutron rates from these sources. The detector assembly consists of an array of 4 NaI detectors (15$\times$15$\times$30 cm), and neutrons are detected via $${}^{127}I + n \rightarrow {}^{128}I + \gamma \rm (\sim 6 MeV)\,.$$ Schematic views of the setup are shown in Fig. \[fig:ndet\_schematic\].
[ \[fig:topview\] ![ *Schematic views of the neutron detector setup.* []{data-label="fig:ndet_schematic"}](topview.pdf "fig:"){width="45.00000%"} ]{} [ \[fig:sideview\] ![ *Schematic views of the neutron detector setup.* []{data-label="fig:ndet_schematic"}](sideview.pdf "fig:"){width="45.00000%"} ]{}
Due to the relatively low neutron emission rate in our $^{241}$Am-$^{13}$C sources, external background had to be significantly reduced to achieve a tolerable signal to background ratio. We placed a detector into a subbasement lab on Caltech campus, with an overburden of about 6 meters consisting of 1 meter of concrete and 5 meters of dirt to shield against cosmogenic neutron background. The assembly was shielded against ambient gamma rays by lead bricks of about 2 inch (5.08 cm) thick. Two scintillator paddles ($\sim$ 60 cm $\times$ 60 cm $\times$ 2.5 cm) were placed on the top and one on the bottom of the assembly to serve as a muon veto with an efficiency measured to be $\sim$ 95%. The $^{241}$Am-$^{13}$C neutron source to be assayed was placed at the center of the detector assembly to maximize the acceptance. The threshold of each NaI was set at $\sim$3 MeV. Any over-threshold hit in a NaI generated a first level trigger which, in combination with the veto signals from the muon paddles (30$\mu$s window), formed the main trigger to read out the ADCs/TDCs. The residual background neutron-like rate, summed over all four NaI detectors, is about 1.4 (0.25) Hz without (with) the muon veto enforced.
The neutron detection efficiency was calibrated with a standard 2.7 kHz $^{252}$Cf neutron source. The energy spectrum of the capture gammas is shown in Fig. \[fig:n\_det\_demo\].
![*Detected energy spectra for neutron sources with background subtracted. Black: spectrum from standard $^{252}$Cf, Red: a typical $^{241}$Am-$^{13}$C source, scaled for visual clarity.*[]{data-label="fig:n_det_demo"}](ncap_NaI.pdf){width="4in"}
One clearly observes the iodine capture gamma “bump”. When setting an energy cut between 4.5 and 8 MeV, the neutron detection efficiency is calibrated to be $5\pm1\%$, where the uncertainty is dominated by the absolute neutron rate. For our weak $^{241}$Am-$^{13}$C sources, a measurement cycle consists of a 24-hour overnight background counting run, followed by a 24-hour overnight “signal” run. Statistical subtraction was made to extract the neutron source signals. The shape of the extracted neutron signals is roughly consistent with that from the calibrated sources in Fig. \[fig:n\_det\_demo\]. The measured neutron rates of all 24 neutron sources are summarized in Fig. \[fig:nrate\].
![*Summary of the rate of the neutron sources measured on the bench. Only statistical errors are shown. The larger error bars of the last two points were due to an increase of background.*[]{data-label="fig:nrate"}](nrate.pdf){width="4in"}
The average neutron rate per source is 0.56 Hz, with a maximum variation of $\pm$15% from source to source and an overall systematic uncertainty of 20% due to detection efficiency. This clearly satisfies our low rate requirement highlighted in Sec. \[sec:requirements\].
Performance of the neutron sources at Daya Bay {#sec:dayaanalysis}
==============================================
Neutron sources were shipped to Daya Bay and installed in the ACUs during the detector assembly (detailed in [@ref:ACU]). To achieve a minimum background, at the parking location inside the ACU the neutron source is surrounded by a borated polyethylene cylinder (BPE) with 5 inch height and 2.25 inch wall thickness. Below the cylinder, there is another BPE disk with 3.25 inch in diameter and 2.5 inch in thickness acting as further shield.
Neutron sources are used extensively in calibrating the detector response, which has been reported in [@DYBNIM12; @DYBCPC]. We limit the discussions here to some basic performance of these sources. When the sources are deployed into the detector, the proton recoil and the following capture gammas form time-correlated pairs, similar to IBD signals. The average neutron rate as measured in the detector is 0.7 Hz, consistent with the QC measurements at Caltech. In Fig. \[fig:AmC\_on-site\], the prompt-delayed energy spectrum, as well as the time separation in between for an $^{241}$Am-$^{13}$C source deployed in an AD is shown, overlaid with the expected distribution from MC. All distributions agree well with MC expectations. We observe no evidence of the 6.1 MeV gamma emission, confirming that the key design specification is met. The residual background introduced by these neutrons is quite small [@DYB12]. The details of the evaluation will be discussed in a separate article [@ref:AmC].
[ \[fig:Ep\] ![ *Reconstructed prompt (a) and delayed (b) energy spectra, as well as the time separation between the two (c). The data and MC simulation are normalized to equal area and overlaid.* []{data-label="fig:AmC_on-site"}](ep.pdf "fig:"){width="40.00000%"} ]{} [ \[fig:dt\] ![ *Reconstructed prompt (a) and delayed (b) energy spectra, as well as the time separation between the two (c). The data and MC simulation are normalized to equal area and overlaid.* []{data-label="fig:AmC_on-site"}](dt.pdf "fig:"){width="40.00000%"} ]{}
Summary
=======
We have discussed the design and construction of the special compact and low rate neutron sources in the Daya Bay experiment. The performance of the low background neutron sources has been proven to satisfy the design specifications at Daya Bay. Such sources could be used in other experiments requiring low backgrounds.
\[sec:summary\]
Acknowledgments
===============
This work was done with support from the US DoE, Office of Science, High Energy Physics, the US National Science Foundation, the Natural Science Foundation of China Grants 11175116, the Chinese MOST grant 2013CB834306, and Shanghai Laboratory for Particle Physics and Cosmology at the Shanghai Jiao Tong University. This work is supported in part by the CAS Center for Excellence in Particle Physics (CCEPP).
We gratefully thank Dick Hahn from BNL for his critical guidance in making these sources. We also thank David Jaffe from BNL for his intellectual input. The technical support from the Kellogg technical team Ray Cortez and Jim Pendlay, particularly Ray’s design work, is truly indispensable. We also appreciate the safety guidance from the Caltech radiation safety officers Haick Issaian and Andre Jefferson, and on-site logistical support from Xiaonan Li of IHEP.
[00]{}
X. H. Guo et al. (Daya Bay Collaboration), arXiv:hep-ex/0701029. F. Ardellier et al. (Double Chooz Collaboration), arXiv:hep-ex/0606025. J. K. Ahn et al. (RENO Collaboration), arXiv:1003.1391
F. P. An et al. (Daya Bay Collaboration), Phys. Rev. Lett. 108, 171803 (2012).
K. H. Ansell and E. G. Hall, Neutron Sources and Applications, Proceedings of the American Nuclear Society National Topical Meeting, CONF-710402, II-90-99 (1971).
http://www.weldon.com.
J. Liu et al, Nucl. Instrum. Meth. A750, 19-37 (2014).
S. Agostineli et al. (GEANT4 Collaboration), Nucl. Instrum. Meth. A 506, 250-303 (2003)
F. P. An et al. (Daya Bay Collaboration), Nucl. Instrum. Meth. A 685, 78-97 (2012)
F. P. An et al. (Daya Bay Collaboration), Chin. Phys. C 37, 011001 (2013)
W. Q. Gu et al., in preparation.
[^1]: The stopped e$^+$ would annihilate into two back-to-back 0.511 MeV gammas, which can hardly deposit enough energy in the AD to cross the trigger threshold.
[^2]: A typical alpha calibration source has a thin front window of 100$\mu$g/cm$^2$, which attenuates the 5.5 MeV alpha energy only by about 22 keV.
[^3]: A switchable source design [@switchable_source] was considered to further reduce background, but it was given up due to the complexity in automating the switch.
|
---
abstract: 'The popularity of social media platforms such as Twitter has led to the proliferation of automated bots, creating both opportunities and challenges in information dissemination, user engagements, and quality of services. Past works on profiling bots had been focused largely on malicious bots, with the assumption that these bots should be removed. In this work, however, we find many bots that are benign, and propose a new, broader categorization of bots based on their behaviors. This includes *broadcast*, *consumption*, and *spam* bots. To facilitate comprehensive analyses of bots and how they compare to human accounts, we develop a systematic profiling framework that includes a rich set of features and classifier bank. We conduct extensive experiments to evaluate the performances of different classifiers under varying time windows, identify the key features of bots, and infer about bots in a larger Twitter population. Our analysis encompasses more than 159K bot and human (non-bot) accounts in Twitter. The results provide interesting insights on the behavioral traits of both benign and malicious bots.'
author:
- 'Richard J. Oentaryo'
- Arinto Murdopo
- 'Philips K. Prasetyo'
- 'Ee-Peng Lim'
bibliography:
- 'main.bib'
title: On Profiling Bots in Social Media
---
Introduction {#sec:intro}
============
In recent years, we have seen a dramatic growth of people’s activities taking place in social media. Twitter, for example, has evolved from a personal microblogging site to a news and information dissemination platform. The openness of the Twitter platform, however, has made it easy for a user to set up an automated social program called *bot*, to post tweets on his/her behalf.
The proliferation of bots has both good and bad consequences [@Chu2012; @Ferrara2014]. On the one hand, bots can generate benign, informative tweets (e.g., news and blog updates), which enhance information dissemination. Bots can also be helpful for the account owners, e.g., bots that aggregate contents from various sources based on the owners’ interests. On the other hand, spammers may exploit bots to attract regular accounts as their followers, enabling them to hijack search engine results or trending topics, disseminate unsolicited messages, and entice users to visit malicious sites [@Ghosh2012; @Hu2013; @Ferrara2014]. In addition to deteriorating user experience and trust, malicious bots may cause more severe impacts, e.g., creating panic during emergencies, biasing political views, or damaging corporate reputation [@Wang2010; @Ferrara2014].
It is thus important to characterize different types of bots and understand how they compare with human users. Recent studies have shown the importance of profiling bots in social media [@Wang2010; @Stringhini2010; @Lee2011; @Chu2012; @Hwang2012; @Wagner2012; @Ghosh2012; @Hu2013; @Boshmaf2013; @Ferrara2014; @Abokhodair2015; @Subrahmanian2016], but these works have focused mainly on malicious (e.g., spam) bots, failing to account for other types of benign bots. With the rise of new services and intelligent apps in Twitter, benign bots are increasingly becoming prominent as well.
Comprehensive profiling of both malicious and benign bots would offer several major benefits. In information dissemination and retrieval, knowing the activity traits of both bot types and the nature of their tweet contents can improve search and recommendation services by separating tweets of bots from those of humans, returning more relevant, personalized search results, and promoting certain products/services more effectively. For social science research, a more accurate understanding of human interactions and information diffusion patterns [@Ferrara2014; @Freitas2014] can also be obtained by filtering out activity biases generated by bots. In turn, these would benefit the overall user community as well.
{width="100.00000%"}
\[fig:examples\]
To illustrate the usefulness of profiling bots, consider the examples in Fig. \[fig:examples\], of different types of benign and malicious bots (which we further describe in Section \[sec:definition\]). The first example is a user who utilizes the IFTTT service[^1] to gather contents from diverse sources for her own consumption. Knowing that she uses a consumption bot, Twitter can provide a new service to organize the unstructured contents, or recommend new contents that match her interest. The second example involves a broadcast bot managed by a job agency to advertise job openings. Twitter recently introduced a new feature called *promoted tweets*[^2] and, knowing it is a (benign) broadcast bot, Twitter can recommend the feature to help the agency reach a wider audience. The last example shows a malicious, spam bot that lures users to visit adult websites, posssibly containing harmful malware. For such a bot, Twitter may develop a strategy to demote—or even filter out—its posts, so that the followers do not see them on their tweet streams.
**Contributions**. In this paper, we present a new categorization of bots based on long-term observations on the behaviors of various automated accounts in Twitter. To our best knowledge, this work is the first extensive study on both *benign* and *malicious* Twitter bots, with detailed analyses on both their static and dynamic patterns of activity. In recent years, Twitter bots have evolved rapidly, and so our work also provides a more timely study that offers updated insights on the bot characteristics. Our findings should also benefit social science and network mining researches. We summarize our key contributions below:
- We propose a new categorization of Twitter bots based on their behavioral traits. In contrast to past studies that focus largely on malicious bots, our study encompasses more detailed examinations of both malicious and benign bots, as well as how they compare to human accounts. For this, we have studied a large dataset of more than 159K Twitter accounts, out of which we have manually labeled 1.6K bot and human accounts.
- To facilitate comprehensive analyses on bots, we develop a systematic profiling framework that includes a rich set of numeric, categorical, and series features. This enables us to examine both the static and dynamic patterns of bots, which span various user profile, tweet, and follow network entities. Our framework also features a classifier bank that includes prominent classification algorithms, thus allowing us to comprehensively evaluate various algorithms so as to identify the best approach for bot profiling.
- We carry out extensive empirical studies to evaluate the performance of our classifiers under different time windows and to identify the most relevant, discriminating features that characterize both benign and malicious bots. We also conduct a novel study to assess the generalization ability of our method on unseen, unlabeled Twitter accounts, based on which we infer the behavioral traits of bots in a larger Twitter population.
Background and Related Work {#sec:related}
===========================
A number of studies have been conducted to identify and profile bots in social media. To detect spam bots, Wang [@Wang2010] utilized content- and graph-based features, derived from the tweet posts and follow network connectivity respectively. Chu *et al.* [@Chu2012] investigated whether a Twitter account is a human, bot, or cyborg. Here a bot was defined as an aggresive or spammy automated account, while cyborg refers to a bot-assisted human or human-assisted bot. Different from our work, the bots defined in [@Chu2012] are more of malicious nature, and the study did not provide further categorization/analysis of benign and malicious bots in Twitter.
To investigate on spam bots, Stringhini *et al.* [@Stringhini2010] created honey profiles on Facebook, Twitter and MySpace. By analyzing the collected data, they identified anomalous accounts who contacted the honey profiles and devised features for detecting spam bots. Going further, Lee *et al.* [@Lee2011] conducted a 7-month study on Twitter by creating 60 social honeypots that try to lure “content polluters” (a.k.a. spam bots). Users who follow or message two or more honeypot accounts are automatically assumed to be content polluters. There are also related works on spam bot detection based on social proximity [@Ghosh2012] or both social and content proximities [@Hu2013]. Tavares and Faisal [@Tavares2013] distinguished between personal, managed, and bot accounts in Twitter, according to their tweet time intervals.
Ferrara *et al.* [@Ferrara2014] built a web application to test if a Twitter account behaves like a bot or human. They used the list of bots and human accounts identified by [@Lee2011], and collected their tweets and follow network information. This study, however, covers only malicious bots. Dickerson *et al.* [@Dickerson2014] used network, linguistic, and application-oriented features to distinguish between bots and humans in the 2014 Indian election. Abokhodair *et al.* [@Abokhodair2015] studied on a network of bots that collectively tweet about the 2012 Syrian civil war. This study covers both malicious (e.g., phishing) and benign (e.g., testimonial) bots. In contrast to our work, however, their findings are tailored to a specific event (i.e., the civil war) and may not be applicable to other bot types in a larger Twitter population.
There are also studies aiming to quantify the susceptibility of social media users to the influence of bots [@Hwang2012; @Wagner2012; @Boshmaf2013]. By embedding their bots into the Facebook network, Boshmaf *et al.* [@Boshmaf2013] demonstrated that users are vulnerable to phishing (e.g., exposing their phone number or address). The susceptibility of users is also evident in Twitter [@Hwang2012; @Wagner2012]. Freitas *et al.* [@Freitas2014] tried to reverse-engineer the infiltration strategies of malicious Twitter bots in order to understand their functioning. Most recently, Subrahmanian *et al.* [@Subrahmanian2016] reported the winning solutions of the DARPA Twitter Bot Detection Challenge. Again, however, all these studies deal mainly with malicious bots and ignore benign bots.
New Categorization of Bots {#sec:definition}
==========================
We define a bot as a Twitter account that generates contents and interacts with other users automatically—at least according to human judgment. Our definition thus includes *both* benign and malicious bots. Based on long-term observations on Twitter data, we propose to categorize Twitter bots into three main types:
{width="0.97\columnwidth"}
\[fig:cartoon\]
- **Broadcast bot**. This bot aims at disseminating information to general audience by providing, e.g., benign links to news, blogs or sites. Such bot is often managed by an organization or a group of people (e.g., bloggers).
- **Consumption bot**. The main purpose of this bot is to aggregate contents from various sources and/or provide update services (e.g., horoscope reading, weather update) for personal consumption or use.
- **Spam bot**. This type of bots posts malicious contents (e.g., to trick people by hijacking certain account or redirecting them to malicious sites), or promotes harmless but invalid/irrelevant contents aggressively.
Fig. \[fig:cartoon\] illustrates the three bot types, where the arrow direction represents the flow of information. It is worth noting that our proposed categorization is more general than the taxonomy put forward in [@Mitter2013], which covers mainly malicious bots. Our categorization is also general enough to cater for new, emerging types of bot (e.g., chatbots can be viewed as a special type of broadcast bots).
Dataset {#sec:data}
=======
**Data collection**. Our study involves a Twitter dataset generated by users in Singapore and collected from 1 January to 30 April 2014 via the Twitter REST and streaming APIs[^3]. Starting from popular seed users (i.e., users having many followers), we crawled their follow, retweet, and user mention links. We then added those followers/followees, retweet sources, and mentioned users who state Singapore in their profile location. With this, we have a total of 159,724 accounts.
[|c|c|c|c|c|]{} & **Unlabeled data**\
*Consumption bot* & *Broadcast bot* & *Spam bot* & *Human account* &\
313 & 171 & 105 & 1,024 & 158,111\
\[tab:label\_dist\]
---------------------------------------------------------------------
{width="0.45\columnwidth"}
\(a) Cumulative distribution functions
---------------------------------------------------------------------
-------------------------------------------------------------------------
{width="0.54\columnwidth"}
\(b) Temporal dynamics
-------------------------------------------------------------------------
\[fig:results\]
To identify bots, we first checked active accounts who tweeted at least 15 times within the month of April 2014. We then manually labeled these accounts and found 589 bots. As many more human users are expected in the Twitter population, we randomly sampled the remaining accounts, manually checked them, and identified 1,024 human accounts. In total, we have 1,613 labeled accounts, as summarized in Table \[tab:label\_dist\]. The labeling was done by four volunteers, who were carefully instructed on the definitions in Section \[sec:definition\]. The volunteers agree on more than $90\%$ of the labels, and any labeling differences in the remaining accounts are resolved by consensus. Also, if an account exhibits both human and bot characteristics, we determine the label based on the majority posting patterns.
**Exploratory analysis**. We conducted a preliminary study on our 1,613 labeled data to get a glimpse of the activity patterns of bots as well as human accounts. Fig. \[fig:results\](a) shows the cumulative distribution functions (CDF) of several key attributes. An early increase in CDF value means a more skewed distribution. We focus on key attributes that reflect a user’s social and posting patterns: $popularity = \frac{|F|}{|E| + |F|}$, $follow\_ratio = \frac{|E|}{|F|}$, $reciprocity = \frac{|E \cap F|}{|E \cup F|}$, $retweet\_unique\_ratio = \frac{|R|}{|T|}$, $url\_unique\_ratio = \frac{|U|}{|T|}$, $mention\_unique\_ratio = \frac{|M|}{|T|}$, $hashtag\_unique\_ratio = \frac{|H|}{|T|}$, where $E$, $F$, $R$, $T$, $U$, $M$, $H$ are the set of followees, followers, retweets, tweets, URLs, user mentions, and hashtags for a given account, respectively. We also define $readership = \frac{retweeted}{|T|}$, where $retweeted$ is the number of times a user’s tweets get retweeted (by others). Fig. \[fig:results\](b) shows heatmaps of tweet counts $|T|$ for different days and hours over 4 months.
*How do humans compare with bots and how do bots differ from one another*? The $popularity$, $follow\_ratio$, and $reciprocity$ results in Fig. \[fig:results\](a) suggest that bots (except for consumption bots) generally have more followers than followees, but are less reciprocal (i.e., follow each other) than humans. Based on the $retweet\_unique\_ratio$ and $readership$ results, humans are more likely to reshare contents from others and have their contents reshared than bots, respectively. Similarly, the $mention\_unique\_ratio$ result suggests that humans are more likely to mention (i.e., talk to) others than bots. Meanwhile, the $url\_unique\_ratio$ and $hashtag\_unique\_ratio$ results show the bots tend to include more diverse web links and topics than humans, respectively. Finally, comparisons among the three bot types show that broadcast bots are the most popular and post the most diverse URLs and hashtags, but they are the least reciprocal and rarely mention others. A plausible reason is that broadcast bots are typically used by organizations solely for information dissemination, and not for interaction with others.
*How do activities of humans and bots change over time?* Fig. \[fig:results\](b) shows that seasonality exists in the tweet activities of human and bot accounts[^4]. That is, humans seldom tweet in early morning (from 2am to 7am) and post moderately from 7am to 8pm. Afterwards, their tweet traffic increases significantly between 8pm and midnight, suggesting that Singapore users are more active after dinner time and before they sleep. Meanwhile, consumption bots tweet more actively than humans from 3am to 7am (i.e., sleep hours), but are less active from 9am to 3pm (i.e., busy working/school hours). Also, consumption bots are less active in the weekends than in the weekdays. While broadcast bots have generally similar patterns to consumption bots, the former is less active during sleep hours (3am–7am) whereas the latter during busy hours (9am–3pm). We can attribute this to the intuition that broadcast bots aim to reach a wider audience during their non-sleep hours. Lastly, unlike broadcast and consumption bots, spam bots are active all days/hours, and they exhibit very random timings. In summary, different bots serve different purposes and their temporal signatures reflect these.
Profiling Framework {#sec:method}
===================
We develop a systematic profiling framework to facilitate comprehensive analyses of bots. Below we describe each component of the framework in turn.
**Database**. Our framework takes as input three types of database: *profile*, *tweet*, and *follow* databases. The profile database contains user information such as the Twitter user id, screenname, location, and profile description. The tweet database contains all the tweets posted by different users, which may include various entities such as hashtags, URLs, user mentions, videos/images, retweet information, and tweet sources/devices. We collectively refer to these as *tweet entities*. Finally, the follow database contains the snapshots of users’ relationship network over time, which include both followers and followees of the users at different time periods. We collectively call these *follow entities*.
**Feature extraction**. This component serves to construct a *feature vector* that represents a Twitter account. It takes three types of feature: *numeric*, *categorical*, and *series*. We describe the extraction steps for each type below:
- For **numeric features**, we perform *standarization* by scaling each feature to a unit range $[0, 1]$. This would allow us to mitigate feature scaling issues, particularly for classification methods that rely on some distance metric. Examples of numeric features are count and ratio attributes (see Table \[tab:features\]).
- For **categorical features**, we first select the top $K$ categories based on their frequencies in each data point, and then filter out the remaining categories. Next, we perform *one-hot encoding* by transforming the top $K$ categories into a binary vector with $K$ elements. For example, a categorical attribute with four possible values: “A”, “B”, “C”, and “D” is encoded as $[1, 0, 0, 0]$, $[0, 1, 0, 0]$, $[0, 0, 1, 0]$, and $[0, 0, 0, 1]$, respectively.
- For **series features**, we first count the frequency of every (discrete) number in the series. For instance, given a series $[a, a, b, a, c, b, c, a, b]$, we can compute the histogram bins: $(a,4), (b,3), (c,2)$. To ensure a moderate feature size, we keep only top $100$ bins with the highest count frequencies. Subsequently, we normalize the frequencies such that they sum to 1, thus forming a probability distribution. For the previous histogram bins $(a,4)$, $(b,3)$, $(c,2)$, the normalization will result in $(a,\frac{4}{9})$, $(b,\frac{3}{9})$, $(c,\frac{2}{9})$.
**Classifier bank**. Finaly, to learn the association between the extracted features and different bot types (or human), our framework includes a classifier bank that comprises a rich collection of classification algorithms. In our study, we employ four prominent classifiers: *naïve Bayes* (NB) [@Domingos1997], *random forest* (RF) [@Breiman2001], and two instances of generalized linear model, i.e., *support vector machine* (SVM) and *logistic regression* (LR) [@Fan2008]. These algorithms represent the state-of-the-art methods previously used for (malicious) bot classification. For instance, RF was utilized in [@Chu2012; @Lee2011; @Ferrara2014; @Dickerson2014], while SVM and NB were used in [@Wang2010; @Dickerson2014].
Feature Engineering {#sec:features_extracted}
===================
We have crafted a rich set of features based on the feature extraction component in our bot profiling framework. Our feature set consists of three groups: *tweet*, *follow* and *profile* features. For tweet features, we also distinguish between *static* (i.e., time-independent) and *dynamic* (i.e., time-dependent) tweet features. Table \[tab:features\] provides a listing of all the features used in our empirical study.
[|l|l|l|]{} **Group** & **Entity** & **Features**\
Static & tweet\_word & count (N), unique\_count (N), unique\_ratio (N), basic\_stats (N)\
tweet & retweet & retweeted (N), readership (N), count (N), unique\_count (N), ratio (N),\
features & & unique\_ratio (N), basic\_stats (N)\
& hashtag & count (N), unique\_count (N), ratio (N), unique\_ratio (N), basic\_stats (N)\
& mention & count (N), unique\_count (N), ratio (N), unique\_ratio (N), basic\_stats (N)\
& url & count (N), unique\_count (N), ratio (N), unique\_ratio (N), basic\_stats (N)\
& media & count (N), unique\_count (N), ratio (N), unique\_ratio (N), basic\_stats (N)\
& source & sources (S)\
Dynamic & tweet & hours (S), days (S), weekdays (S), timeofdays (S), extended\_stats (N)\
tweet & retweet & hours (S), days (S), weekdays (S), timeofdays (S), extended\_stats (N)\
features & hashtag & hours (S), days (S), weekdays (S), timeofdays (S), extended\_stats (N)\
& mention & hours (S), days (S), weekdays (S), timeofdays (S), extended\_stats (N)\
& url & hours (S), days (S), weekdays (S), timeofdays (S), extended\_stats (N)\
& media & hours (S), days (S), weekdays (S), timeofdays (S), extended\_stats (N)\
Follow & followees\_count & basic\_stats (N)\
features & followers\_count & basic\_stats (N)\
& mutual\_count & basic\_stats (N)\
& reciprocity & basic\_stats (N)\
& in\_reciprocity & basic\_stats (N)\
& out\_reciprocity & basic\_stats (N)\
& popularity & basic\_stats (N)\
& follow\_ratio & basic\_stats (N)\
Profile & profile & is\_geo\_enabled (C), lang (C), time\_zone (C), account\_age (N),\
features & & favourites\_count (N), listed\_count (N), statuses\_count (N), utc\_offset (N)\
\
\
\
\[tab:features\]
**Static tweet features**. We generate static tweet features based on the combination of entities and statistical metrics, as shown in Table \[tab:features\]. For instance, to generate the hashtag features of a user, we treat each hashtag as a “bag” and count how many times the word occurs in all of $x$’s tweets. This yields a bag-of-hashtag vector, from which we can compute first-order statistics (i.e., $count$, $unique\_count$, $mean$, $median$, $min$, and $max$) as well as second-order metrics (i.e., standard deviation ($std$) and Shannon entropy [@Shannon1963] ($entropy$)). We note that the second-order metrics serve to quantify the *diversity* of the entities. We also compute the $ratio = \frac{count}{|T|}$ and $unique\_ratio = \frac{unique\_count}{|T|}$, where $|T|$ is the total number of tweets posted by a user. For the retweet entity, we additionally consider $retweeted$ and $readership$ features, as described in Section \[sec:data\]. Finally, we consider a series feature to represent the source entity, whereby each source maps to a histogram bin containing the normalized frequency of the source.
**Dynamic tweet features**. For these features (cf. Table \[tab:features\]), we introduce additional time dimensions that capture the dynamics of tweet activities, namely: *hours* $\in \{0,\ldots,23\}$, *days* $\in \{1,\ldots,31\}$, *weekdays* $\in \{Monday,\ldots,Sunday\}$, *timeofdays* $\in \{morning$ (4am–12pm), $afternoon$ (12pm–5pm), $evening$ (5pm–8pm), $night$ (8pm–4am)$\}$, and *timegaps*. The timegap dimension refers to the gap (in milliseconds) between two *consecutive* entity timestamps, e.g., for $N$ tweets posted by a user $x$, we can compute a timegap vector with length $(N-1)$. For each time dimension, we can then generate the series features based on the histogram binning described in Section \[sec:method\], as well as compute the statistical metrics such as $mean$, $median$, $min$, $max$, $std$ and $entropy$.
**Follow features**. These features are derived by computing metrics that summarize snapshots of the follow network at different time points (cf. Table \[tab:features\]). Let $E$ and $F$ be the set of followees and followers of a given user. In turn, we compute the $followees\_count = |E|$, $followers\_count = |F|$, $mutual\_count = |E \cap F|$. as well as ratio metrics such as $reciprocity = \frac{|E \cap F|}{|E \cup F|}$, $in\_reciprocity = \frac{|E \cap F|}{|F|}$, $out\_reciprocity = \frac{|E \cap F|}{|E|}$, $popularity = \frac{|F|}{|E| + |F|}$, and $follow\_ratio = \frac{|E|}{|F|}$. We calculate these metrics for every snapshot of the follow network at a given time point, and then compute the statistics $mean$, $median$, $min$, $max$, $std$ and $entropy$ to summarize the metrics over all time points.
**Profile features**. Finally, we also consider several basic user profile features, as per Table \[tab:features\]. Here, $account\_age$ refers to the lapse between the time a user first joined Twitter and the current reference time. Further details on the definitions of the other profile features can be found in `https://dev.twitter.com/`.
Results and Findings {#sec:experiment}
====================
This section elaborates our empirical study on bots. We first describe our experiment setup, and then address several research questions in Sections \[sec:RQ1\]–\[sec:RQ4\].
**Evaluation metrics**. To evaluate our classifiers, we utilize three metrics popularly used in information retrieval [@Manning2008]: $Precision$, $Recall$ and $F1$. We report, for each class $c \in \{broadcast, consumption, spam, human\}$, the $Precision(c) = \frac{TP(c)}{TP(c) + FP(c)}$, $Recall = \frac{TP(c)}{TP(c) + FN(c)}$, and $\text{\emph{F1}(c)} = \frac{2 Precision(c) Recall(c)}{Precision(c) + Recall(c)}$, where $TP(c)$, $FP(c)$ and $FN(c)$ are the true positives, false positives, and false negatives respectively. Based on these, we also report the macro-averaged $Precision = \frac{1}{4} \sum_{c=1}^4 Precision(c)$, $Recall = \frac{1}{4} \sum_{c=1}^4 Recall(c)$, and $\text{\emph{F1}} = \frac{1}{4} \sum_{c=1}^4 \text{\emph{F1}(c)}$.
**Experiment protocols**. In this work, we consider two sets of experiment:
- **Experiment** $E_1$: This set of experiment involves evaluation on our **1,613** *labeled data* (see Table \[tab:label\_dist\]). For this evaluation, we use a *stratified* 10-fold cross-validation (CV), whereby we split the labeled data into 10 mutually exclusive groups, each retaining the class proportion as per the original data. This stratification serves to ensure that each fold is a good representative of the whole, i.e., it retains the (unbalanced) class distribution as in the original data. For each CV iteration $f$, we then use group $f$ ($10\%$) for testing and the remaining groups $f' \neq f$ ($90\%$) for training. We report the results averaged over 10 iterations, which include $Precision(c)$, $Recall(c)$ and $F1(c)$ for each class $c$, as well as the macro-averaged $Precision$, $Recall$ and $F1$.
- **Experiment** $E_2$: This set of experiment serves to evaluate predictions on the remaining **158,111** *unlabeled data* (see again Table \[tab:label\_dist\]). Based on this, we can infer the behavioral traits of bots in a larger Twitter population. For this experiment, we are unable to compute $Recall$, as we would have to manually verify one by one a large number of unlabeled data. Instead, we evaluate based on $Precision$ at top $K$ for each class ($K \ll$ 158,111).
**Model parameters**. We configured our classifier bank as follows: For the NB classifier, we use the smoothing parameter $\alpha=1$. For RF, we use $N=100$ decision trees. Finally, for SVM and LR, we set the cost parameter $C=1$ and $\texttt{class\_weight}=$“balanced”; the latter is for automatically handling the imbalanced class distribution. We performed grid search to determine all these parameters, which give the optimal performances for each classifier. In particular, we varied the NB parameter from the range $\alpha \in \{0.1, 1, 10\}$. For RF, we tried $N \in \{10,20,\ldots,100\}$, and for SVM and LR, we tried $C \in \{0.01, 0.1, 1, 10, 100\}$.
**Significance test**. Finally, we use *Wilcoxon signed-rank test* [@Wilcoxon1945] to test for the statistical significance of our results. When comparing between two performance vectors, we look at the $p$-value at a significance level of $0.01$. If the $p$-value is less than $0.01$, we say that the performance difference is indeed significant.
How Well Can the Classifiers Predict for Bots? {#sec:RQ1}
----------------------------------------------
To answer this research question, we first conduct a sensitivity study by varying the time duration for which features (cf. Table \[tab:features\]) are generated. For this study, we use the CV procedure on our labeled data (i.e., Experiment $E_1$), whereby the classifiers were trained using all features listed in Table \[tab:features\]. Fig. \[fig:time\_window\] shows the macro-averaged $Precision$, $Recall$, and $F1$ over 10 CV folds, with the duration varied from 1 week, 2 weeks and 1 month to 2 months and 4 months (up to 30 April 2014). Based on the $F1$ results, we can conclude that 2 weeks is the best duration and that LR outperforms the other classifiers. In this case, RF gives higher $Precision$ than LR, but its $Recall$ is much lower, and so is its $F1$. It is also shown that a tradeoff exists in choosing the duration; an overly short duration degrades the performance, which can be attributed to data scarcity. The same goes for an overly long duration, due to inclusion of outdated data.
Table \[tab:benchmark\] shows further breakdown of the CV results for the best time duration (i.e., 2 weeks). Overall, LR and SVM give the best results, and outperform the more complex RF and simpler NB methods (except for $Precision$ of the “spam” class). For spam bots, RF yields higher $Precision$, but much lower $Recall$ and $F1$ than LR and SVM. While SVM and LR perform very similarly, we decided to use LR as our main classifier for two reasons: (i) LR outputs more meaningful probabilitic scores than the unbounded decision scores in SVM; and (ii) LR is more robust than SVM against variation in time duration, as we saw in Fig. \[fig:time\_window\].
{width="0.6\columnwidth"}
\[fig:time\_window\]
[|l|l|rl|rl|rl|rl|rl|]{} & & &\
**Metric** & **Method** & & & & &\
Precision & NB & 0.6519 & $(-)$ & 0.7206 & $(-)$ & 0.7069 & $(+)$ & 0.9929 & & 0.7681 & $(-)$\
& RF & 0.5880 & $(-)$ & **0.9462** & & **0.8636** & $(+)$ & 0.9750 & $(-)$ & **0.8432** & $(+)$\
& SVM &**0.6952** & & 0.9278 & & 0.6574 & $(-)$ & **0.9961** & & 0.8191 &\
& LR & 0.6798 & & 0.9366 & & 0.6869 & & 0.9942 & & 0.8244 &\
Recall & NB & 0.6901 & $(-)$ & **0.8818** & $(+)$ & 0.3905 & $(-)$ & 0.9609 & $(-)$ & 0.7308 & $(-)$\
& RF & **0.8596** & $(+)$ & 0.8435 & & 0.3619 & $(-)$ & 0.9902 & & 0.7638 & $(-)$\
& SVM & 0.7602 & $(-)$ & 0.8626 & & **0.6762** & $(+)$ & **0.9990** & & 0.8245 &\
& LR & 0.8070 & & 0.8498 & & 0.6476 & & 0.9971 & & **0.8254** &\
F1-score & NB & 0.6705 & $(-)$ & 0.7931 & $(-)$ & 0.5031 & $(-)$ & 0.9767 & $(-)$ & 0.7358 & $(-)$\
& RF & 0.6983 & $(-)$ & 0.8919 & & 0.5101 & $(-)$ & 0.9826 & $(-)$ & 0.7707 & $(-)$\
& SVM & 0.7263 & & **0.8940** & & **0.6667** & & **0.9976** & & 0.8211 &\
& LR & **0.7380** & & 0.8911 & & **0.6667** & & 0.9956 & & **0.8228** &\
\
\[tab:benchmark\]
Based on the individual $Precision(c)$, $Recall(c)$ and $F1(c)$ of each class $c$, we can conclude that, among the bots, consumption bots are the easiest to detect, followed by broadcast and spam bots. This is expected, owing to the imbalanced class distribution as per Table \[tab:label\_dist\]. We can also compare the results of our classifiers with that of a random guess[^5]. Based on the statistics in Table \[tab:label\_dist\], the expected $F1$ scores of a random guess for broadcast bot, consumption bot, spam bot, and human classes are $10.6\%$, $19.40\%$, $6.51\%$ and $63.49\%$, respectively. Our four classifiers thus outperform the random guess baseline by a large margin.
For spam bots, several studies [@Lee2011; @Chu2012; @Ferrara2014] have reported high classification accuracies, while our results are modest by comparison, largely due to the lack of spam bot accounts in our data. However, it must be noted that these works focused largely on distinguishing between (malicious) bots vs. other accounts, whereas our study deals with a much more challenging and fine-grained categorization of broadcast, consumption and spam bots. Also, the lack of spam bots in our data can be attributed to several factors, such as our relatively strict definition of spam bot (whereby the majority of its postings need to have malicious or irrelevant contents), or our data collection process that begins with popular seed users and their connections (thus possibly missing unpopular spam bots). Nevertheless, our main focus is to analyze benign bots, which has been largely ignored in the past studies. Further studies on less prominent spam bots that post malicious contents at a sparse rate is beyond the scope of our current study.
Which Features are the Most Indicative of Each Bot Type? {#sec:RQ2}
--------------------------------------------------------
In light of this research question, we trained our best classifier (i.e., LR) using all 1,613 labeled data, and look at the weight coefficients $w_{i,c}$ of each class in the trained LR. Here we use the raw weights $w_{i,c}$ instead of the absolute values $|w_{i,c}|$ or squared values $w_{i,c}^2$, as the raw weights allow us to distingush between features that correlate positively with a class label (which are our main interest) and those that correlate negatively. Fig. \[fig:feature\_importance\] shows the top 15 positively-correlated features for each class. In general, we find that the top features are dominated by the *source* (i.e., where the tweets come from) and *entropy-based dynamic tweet* features. Below we elaborate our feature analysis for each class further.
{width="1.0\columnwidth"}
\[fig:feature\_importance\]
**Broadcast bots**. Among the top features for broadcast bots, certain sources that are popularly used for blogging (such as WordPress and Twitterfeed) or brand management (such as HootSuite) are found to be highly indicative. It is also shown that the entropy-based features for the url entity correlate strongly with broadcast bots. Recall from Section \[sec:features\_extracted\] that entropy is a second-order metric that quantifies how diverse a distribution is. Accordingly, as broadcast bots generally aim to disseminate information about certain sites/brands, we can expect that they would have more concentrated url distribution (i.e., low entropy). We will further verify this in Section \[sec:RQ4\]. Fig. \[fig:feature\_importance\] also suggests that certain critical timings of the url postings are highly indicative of broadcast bots.
**Consumption bots**. From Fig. \[fig:feature\_importance\], we firstly find that the top three sources for consumption bots (i.e., Unfollowers, Twittascope, and Buffer) are service apps that allow users to track their followers/followees status, horoscope readings, and scheduled postings, respectively. Secondly, we discover that the diversity (entropy) of tweet postings is a strong indicator for consumption bots. Lastly, Fig. \[fig:feature\_importance\] shows that certain timezones and timings (weekday and day) of the hashtag and url activities constitute yet another important set of indicators. All these led us to conclude that consumption bots post tweets in a way that follows certain timings/schedules. We will further analyze this in Section \[sec:RQ4\].
**Spam bots**. The result in Fig. \[fig:feature\_importance\] suggests that there are certain sources that can be exploited by spammers to post irrelevant or unsolicited tweets. For example, TwittBot is an application that allows multiple users (and thus spammers) to post to a single Twitter account. In addition, the timing diversities of the url, mention, tweet and hashtag activities are found to be the key signatures of spam bots. As also shown in Fig. \[fig:results\](b) (of Section \[sec:data\]), the temporal patterns of spam bots are highly irregular. Altogether, these suggest that spam bots have highly diverse timings (i.e., high entropy), which we will again verify in Section \[sec:RQ4\].
**Humans**. The top three features in Fig. \[fig:feature\_importance\] suggest that human accounts typically use credible sources such as “web” (i.e., Twitter website) and the official Twitter mobile apps. Next, the $account\_age$ and $isGeoEnabled$ features suggest that human accounts have lived relatively long in Twitter and usually have his/her tweets’ location enabled, respectively. Also, high timing diversity (entropy) of the tweet, retweet and mention activities are indicative of human accounts, although it is not as high as that of spam bots. Again, Section \[sec:RQ4\] analyzes this further. Lastly, the $media\_median$ and $media\_mean$ features suggest that human accounts like to attach media files (e.g., photos) in their tweets.
What Can We Tell about Bots in a Larger Twitter Population? {#sec:RQ4}
-----------------------------------------------------------
To address this question, we performed Experiment $E_2$ by deploying our trained LR classifier to predict for the unlabeled $158,111$ accounts. We then picked the top $K$ accounts with the highest probability scores for each class, and manually assessed the class assignments of these accounts. The assessment results can be found in Appendix \[sec:predictions\] (Table \[tab:top\_pred\_sg\]). We found that the prediction results generally match well with our manual judgments. Based on this, we can make inference on the behavior of bots in a larger Twitter population, i.e., the entire population of Singapore Twitter users. We focus our analyses on the entropy-based dynamic tweet features, which dominate the top features as shown in Fig. \[fig:feature\_importance\]. That is, we analyze the entropy distributions of the tweet, retweet, mention, hashtag and url activities. The complete distributions can be found in Appendix \[sec:predictions\] (Fig. \[fig:entropy\_sg\_cdf\]), which reveals several interesting insights as elaborated below.
**Tweet patterns**. We first compared the distributions of the tweet timings, and discovered that consumption and spam bots exhibit higher diversity (entropy) than that of humans. In contrast, broadcast bots were found to have more concentrated timings. These suggest that broadcast bots post tweets at more specific timings than humans and other types of bots. We also found that consumption and spam bots are very similar in terms of daily timings (i.e., weekday and day entropies), but the former is less diverse than the latter in terms of hourly timings. We can thus conclude that consumption and spam bots tweet equally regularly on a daily basis, but the latter tend to post at random hours.
**Retweet and mention patterns**. Retweet and mention activities can be used to gauge how much a bot (or human) cares about other accounts. Comparing the distributions of the retweet and mention timings in Fig. \[fig:entropy\_sg\_cdf\], we can see again that spam bots have the most random patterns compared to humans and other bot types. But unlike the results for tweet timings, consumption bots have the lowest diversity in terms of daily and hourly timings for the retweet and mention activities. This suggests that consumption bots reshare contents and mention other users at more specific timings, respectively. Such regularity makes sense, especially for consumption bots that provide update services to their users, e.g., Unfollowers and Twittascope (cf. Section \[sec:RQ2\]).
**Hashtag patterns**. In Twitter, a hashtag can be viewed as representing a topic of interest. As shown in Fig. \[fig:entropy\_sg\_cdf\], humans and consumptions bots have very similar diversities of hashtag timings. It is also shown that spam bots have the most diverse hashtag timings (as expected), whereas broadcast bots exhibit very focused hashtag timings. The latter suggests that broadcast bots tend to talk about different topics at more regular time intervals. This is intuitive, especially if we consider the nature of the account owners of broadcast bots (e.g, news/blogger sites), which aim to disseminate various information on a regular basis.
**URL patterns**. For the URL timings, we find that in general humans and broadcast bots use URLs at more specific timings than consumption and spam bots. Interestingly, however, we observe that consumption bots exhibit higher diversity in daily timings than spam bots, but the reverse is true for hourly timings. This suggests that consumption bots use URLs on a more regular daily basis than spam bots, but the latter post URLs at more random hours.
**Comparisons**. It is also interesting to see how our results in Figs. \[fig:feature\_importance\] and \[fig:entropy\_sg\_cdf\] put little emphasis on the importance of the follow network features in the classification task. This is different from previous studies on (malicious) bots [@Lee2011; @Stringhini2010; @Wagner2012; @Chu2012; @Dickerson2014], whereby the follow features play a key role. We can attribute this to the evolution of bot activities as well as stricter regulations imposed by Twitter (especially for spam bots). Also, to our best knowledge, no attempt has been made in the previous works to infer on a larger population. Thus, our work offers more comprehensive insights on the behavioral traits of bots.
Conclusion {#sec:conclusion}
==========
In this paper, we present a new categorization of bots, and develop a systematic bot profiling framework with a rich set of features and classification methods. We have carried out extensive empirical studies to analyze on broadcast, consumption and spam bots, as well as how they compare with regular human accounts. We discovered that the diversities of timing patterns for posting activities (i.e., tweet, retweet, mention, hashtag and url) constitute the key features to effectively identify the behavioral traits of different bot types.
This study hopefully will benefit social science studies and help create better user services. In the future, we plan to examine the prevalence of our findings across multiple countries, beyond our current Singapore data. We also wish to study information diffusion and user interaction in Twitter with the aid of bots.
**Acknowledgments**. This research is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its International Research Centres in Singapore Funding Initiative.
Predictions on Unlabeled Twitter Accounts {#sec:predictions}
=========================================
To facilitate our study on a larger Twitter population, we first examined how well our best classfier (i.e., LR) can predict for unlabeled data that it never sees in the (labeled) CV data. Table \[tab:top\_pred\_sg\] summarizes the top $K$ prediction results, whereby we varied $K$ from $10$ to $50$ to verify the robustness of the predictions. For each class, we computed the number of correctly predicted instances ($TP$) as well as precision at top $K$, i.e., $Precision = \frac{TP}{K}$.
As shown in Table \[tab:top\_pred\_sg\], our LR classifier produces fairly accurate and consistent predictions across different $K$ values. With respect to human accounts, our LR classifier achieved perfect $Precision$ for all $K$ values. Unsurprisingly, we can expect that human accounts constitute the largest proportion of the Twitter population, and thus they should be the easiest to classify. We also obtained good results for the broadcast and consumption bots, with precision scores greater than $75\%$ and $95\%$ respectively. On the other hand, we observe rather modest $Precision$ scores for spam bots (i.e., $40$–$47.5\%$). We can attribute this to the insufficient number of instances for spam bots, which form only $\frac{105}{1,613} = 6.51\%$ of our labeled data (cf. Table \[tab:label\_dist\]). This may (again) be due to our data collection procedure that involved popular users as seeds and/or due to our relatively strict criteria for the characterization of spam bot accounts (cf. Section \[sec:RQ1\]). Nevertheless, the $Precision$ scores of $40$–$47.5\%$ remain relatively good, if we compare with that of a random guess for our labeled data (i.e., $6.51\%$).
All in all, we find our top $K$ predictions on unlabeled data to be satisfactory. Based on this, we can use our predictions to infer the behavioral profiles of bots in a larger Twitter population, which in this case spans the overall Singapore users. In particular, we analyze the entropy-based dynamic tweet features, namely the entropy distributions of the tweet, retweet, mention, hashtag and url activities, which constitute the majority group of the top discriminative features in Fig. \[fig:feature\_importance\]. Fig. \[fig:entropy\_sg\_cdf\] presents the cumulative distribution functions of these features. The detailed analysis of the distributions can be found in Section \[sec:RQ4\].
[|l|c|c|c|c|c|c|c|c|c|c|]{} & & & & &\
**Label** & **TP** & **Precision** & **TP** & **Precision** & **TP** & **Precision** & **TP** & **Precision** & **TP** & **Precision**\
Broadcast bot & 9 & 0.80 & 18 & 0.90 & 27 & 0.90 & 34 & 0.85 & 38 & 0.76\
Consumption bot & 10 & 1.00 & 20 & 1.00 & 30 & 1.00 & 38 & 0.95 & 48 & 0.96\
Spam bot & 4 & 0.40 & 9 & 0.45 & 12 & 0.43 & 19 & 0.475 & 23 & 0.48\
Human & 10 & 1.00 & 20 & 1.00 & 30 & 1.00 & 40 & 1.00 & 40 & 1.00\
\[tab:top\_pred\_sg\]
![Distribution of entropy-based features for 158,111 Twitter accounts[]{data-label="fig:entropy_sg_cdf"}](figures/bot-entropy-sg-cdf.png){width="100.00000%"}
[^1]: https://ifttt.com
[^2]: https://business.twitter.com/solutions/promoted-tweets
[^3]: https://dev.twitter.com/overview/
[^4]: The exceptionally low tweet frequencies in the first week of January and 12-14 February are due to major downtime of our servers.
[^5]: Random guess w.r.t. a class $c$ refers to a classifier that assigns a proportion $p_c\%$ of the instances to class $c$, and $(1-p_c)\%$ to classes other than $c$. In this case, $Precision(c) = Recall(c) = F1(c) = p_c$, where $p_c = \frac{P(c)}{P(c)+N(c)} = \frac{TP(c) + FN(c)}{TP(c) + FN(c) + TN(c) + FP(c)}$.
|
---
abstract: 'In a recent work we derived the kinematic Hamiltonian and primary constraints of the new general relativity class of teleparallel gravity theories and showed that these theories can be grouped in 9 classes, based on the presence or absence of primary constraints in their Hamiltonian. Here we demonstrate an alternative approach towards this result, by using differential forms instead of tensor components throughout the calculation. We prove that also this alternative derivation yields the same results and show how they are related to each other.'
address: |
Laboratory of Theoretical Physics, Institute of Physics, University of Tartu\
W. Ostwaldi 1, 50411 Tartu, Estonia\
manuel.hohmann@ut.ee
author:
- Manuel Hohmann
bibliography:
- 'friedmann.bib'
title: Hamiltonian of new general relativity using differential forms
---
Introduction
============
We recently derived the kinematic Hamiltonian and primary constraints of new general relativity[@Hayashi:1979qx] using the language of tensor components.[@Blixt:2018znp; @Blixt:2019ene] For two particular examples from this class of theories, a toy model[@Okolow:2011np] as well as the teleparallel equivalent of general relativity,[@Okolow:2011nq; @Okolow:2013lwa] the Hamiltonian has also been derived using differential forms. Here we demonstrate how the kinematic Hamiltonian and primary constraints are derived in this latter formalism in the general case.
Geometric setting
=================
We assume a globally hyperbolic spacetime manifold $M \cong \mathbb{R} \times \Sigma$, on which the dynamical fields are given by the coframe $\boldsymbol{\theta}^a \in \Omega^1(M)$ with $a = 0, \ldots, 3$ and spin connection $\boldsymbol{\omega}^a{}_b \in \Omega^1(M)$. They define the metric $\mathbf{g}$ and the torsion $\mathbf{T}^a$ via $$\mathbf{g} = \eta_{ab}\boldsymbol{\theta}^a \otimes \boldsymbol{\theta}^b\,, \quad
\mathbf{T}^a = {\boldsymbol{{\mathrm{D}}}}\boldsymbol{\theta}^a = {\boldsymbol{{\mathrm{d}}}}\boldsymbol{\theta}^a + \boldsymbol{\omega}^a{}_b \wedge \boldsymbol{\theta}^b\,.$$ Further, we denote by $\mathbf{e}_a$ the frame dual to $\boldsymbol{\theta}^a$, so that $\mathbf{e}_a \intprod \boldsymbol{\theta}^b = \delta_a^b$, and by $\star$ the Hodge star of $\mathbf{g}$. Note that we use bold symbols to denote objects defined on $M$.
Together with the projectors ${\boldsymbol{\mathfrak{t}}}: M \to \mathbb{R}$ and ${\boldsymbol{\mathfrak{s}}}: M \to \Sigma$ we define the time translation vector field ${\boldsymbol{\partial}_{{\boldsymbol{\mathfrak{t}}}}}$ such that it satisfies ${\boldsymbol{\partial}_{{\boldsymbol{\mathfrak{t}}}}}\intprod {\boldsymbol{{\mathrm{d}}}}{\boldsymbol{\mathfrak{t}}}= 1$ and ${\boldsymbol{\mathfrak{s}}}_*{\boldsymbol{\partial}_{{\boldsymbol{\mathfrak{t}}}}}= 0$. It allows us to decompose the coframe, and any other differential form on $M$, in the form $$\boldsymbol{\theta}^a = \hat{\boldsymbol{\theta}}^a\,{\boldsymbol{{\mathrm{d}}}}{\boldsymbol{\mathfrak{t}}}+ \vec{\boldsymbol{\theta}}^a\,, \quad
\hat{\boldsymbol{\theta}}^a = {\boldsymbol{\partial}_{{\boldsymbol{\mathfrak{t}}}}}\intprod \boldsymbol{\theta}^a\,, \quad
{\boldsymbol{\partial}_{{\boldsymbol{\mathfrak{t}}}}}\intprod \vec{\boldsymbol{\theta}}^a = 0\,.$$ We denote the pullbacks of $\hat{\boldsymbol{\theta}}^a$ and $\vec{\boldsymbol{\theta}}^a$ to $\Sigma$ by $\hat{\theta}^a$ and $\vec{\theta}^a$, i.e., we use non-bold font for objects defined on $\Sigma$. The latter defines the metric and musical isomorphisms $$q = \eta_{ab}\vec{\theta}^a \otimes \vec{\theta}^b\,, \quad
\tau^{\sharp} = q^{-1}(\cdot, \tau)\,, \quad
\zeta^{\flat} = q(\cdot, \zeta)$$ for one-forms $\tau$ and vector fields $\zeta$. Writing $\ast$ for the hodge star of $q$, we define $$\xi^a = -\frac{1}{6}\eta^{ae}\epsilon_{ebcd}\ast(\vec{\theta}^b \wedge \vec{\theta}^c \wedge \vec{\theta}^d)\,, \quad
\hat{\theta}^a = \alpha\xi^a + \beta \intprod \vec{\theta}^a\,,$$ where the lapse function $\alpha$ and shift vector field $\beta$ are uniquely defined from the latter equation, which expands $\hat{\theta}^a$ in the basis spanned by $\xi^a$ and $\vec{\theta}^a$. Finally, we use $q$ and $\xi^a$ to decompose any one-form $\sigma_a$ into irreducible components $$\begin{gathered}
{\overset{\otimes}{\sigma}\vphantom{\sigma}}_a = \frac{1}{3}q^{-1}(\vec{\theta}^b, \sigma_b)\vec{\theta}_a\,, \quad
{\overset{\ominus}{\sigma}\vphantom{\sigma}}_a = \frac{1}{2}\left[q^{-1}(\vec{\theta}_a, \vec{\theta}^b)\sigma_b - q^{-1}(\vec{\theta}_a, \sigma_b)\vec{\theta}^b\right]\,,\nonumber\\
{\overset{\oplus}{\sigma}\vphantom{\sigma}}_a = \frac{1}{2}\left[q^{-1}(\vec{\theta}_a, \vec{\theta}^b)\sigma_b + q^{-1}(\vec{\theta}_a, \sigma_b)\vec{\theta}^b\right] - \frac{1}{3}q^{-1}(\vec{\theta}^b, \sigma_b)\vec{\theta}_a\,, \quad
{\overset{\odot}{\sigma}\vphantom{\sigma}}_a = -\xi_a\xi^b\sigma_b\,,\label{eqn:irreddecomp}\end{gathered}$$ which satisfy $\sigma_a = {\overset{\otimes}{\sigma}\vphantom{\sigma}}_a + {\overset{\ominus}{\sigma}\vphantom{\sigma}}_a + {\overset{\oplus}{\sigma}\vphantom{\sigma}}_a + {\overset{\odot}{\sigma}\vphantom{\sigma}}_a$.
Kinematic Hamiltonian of new general relativity
===============================================
The action of new general relativity[@Hayashi:1979qx] can be written in the form $$S[\boldsymbol{\theta}^a, \boldsymbol{\omega}^a{}_b] = \int_M\boldsymbol{\mathcal{L}} = \int_M\left(C_T{\boldsymbol{\mathcal{T}}}^a \wedge \star{\boldsymbol{\mathcal{T}}}_a + C_V{\boldsymbol{\mathcal{V}}}^a \wedge \star{\boldsymbol{\mathcal{V}}}_a + C_A{\boldsymbol{\mathcal{A}}}^a \wedge \star{\boldsymbol{\mathcal{A}}}_a\right)\,.$$ Here $C_T, C_V, C_A$ are free constants, and the torsion components are given by $${\boldsymbol{\mathcal{V}}}^a = \frac{1}{3}\boldsymbol{\theta}^a \wedge (\mathbf{e}_b \intprod \mathbf{T}^b)\,, \quad
{\boldsymbol{\mathcal{A}}}^a = \frac{1}{3}\eta^{ab}\mathbf{e}_b \intprod (\eta_{cd}\boldsymbol{\theta}^c \wedge \mathbf{T}^d)\,, \quad
{\boldsymbol{\mathcal{T}}}^a = \mathbf{T}^a - {\boldsymbol{\mathcal{V}}}^a - {\boldsymbol{\mathcal{A}}}^a\,.$$ From now on we will work in the Weitzenböck gauge $\boldsymbol{\omega}^a{}_b \equiv 0$, without loss of generality.[@Blixt:2019mkt]. In order to derive the Hamiltonian, we first write the Lagrangian in the form $\boldsymbol{\mathcal{L}} = {\boldsymbol{{\mathrm{d}}}}{\boldsymbol{\mathfrak{t}}}\wedge \hat{\boldsymbol{\mathcal{L}}}$, where the pullback $\hat{\mathcal{L}}$ of the spatial part $\hat{\boldsymbol{\mathcal{L}}}$ is given by $$\begin{split}
\hat{\mathcal{L}} &= \frac{2C_T + C_V}{3}\left\{\alpha{\mathrm{d}}\vec{\theta}^a \wedge \ast {\mathrm{d}}\vec{\theta}_a - \frac{1}{\alpha}\left[\dot{\vec{\theta}}^a - {\mathrm{d}}(\alpha\xi^a) - {\pounds}_{\beta}\vec{\theta}^a\right] \wedge \ast\left[\dot{\vec{\theta}}_a - {\mathrm{d}}(\alpha\xi_a) - {\pounds}_{\beta}\vec{\theta}_a\right]\right\}\\
&+ \frac{C_T - C_V}{3}\left[\alpha({\mathrm{d}}\vec{\theta}^a \wedge \vec{\theta}_b) \wedge \ast({\mathrm{d}}\vec{\theta}^b \wedge \vec{\theta}_a) - \frac{1}{\alpha}\left(\dot{\vec{\theta}}^a \wedge \vec{\theta}_b + E^a{}_b\right) \wedge \ast\left(\dot{\vec{\theta}}^b \wedge \vec{\theta}_a + E^b{}_a\right)\right]\\
&+ \frac{C_A - C_T}{3}\left[\alpha({\mathrm{d}}\vec{\theta}^a \wedge \vec{\theta}_a) \wedge \ast({\mathrm{d}}\vec{\theta}^b \wedge \vec{\theta}_b) - \frac{1}{\alpha}\left(\dot{\vec{\theta}}^a \wedge \vec{\theta}_a + E^a{}_a\right) \wedge \ast\left(\dot{\vec{\theta}}^b \wedge \vec{\theta}_b + E^b{}_b\right)\right]\,.
\end{split}$$ Here dots denote time derivatives, and we used the abbreviation $$E^b{}_a = -{\mathrm{d}}(\alpha\xi^b) \wedge \vec{\theta}_a + \alpha\xi_a{\mathrm{d}}\vec{\theta}^b - ({\pounds}_{\beta}\vec{\theta}^b) \wedge \vec{\theta}_a\,.$$ Note that there are no time derivatives of the temporal part $\hat{\theta}^a$ of the tetrad, or equivalently the lapse $\alpha$ and shift $\beta$. The next step is to derive the canonical momenta $p_a$ conjugate to the spatial tetrad components $\vec{\theta}^a$. Varying the Lagrangian with respect to the velocities $\dot{\vec{\theta}}^a$ and using the relation $\delta_{\dot{\theta}}\hat{\mathcal{L}} = \delta\dot{\vec{\theta}}^a \wedge p_a$ we find $$\begin{gathered}
p_a = -\frac{2}{3\alpha}\Big\{(2C_T + C_V)\ast\left[\dot{\vec{\theta}}_a - {\mathrm{d}}(\alpha\xi_a) - {\pounds}_{\beta}\vec{\theta}_a\right]\\
+ (C_T - C_V)\vec{\theta}_b \wedge \ast\left(\dot{\vec{\theta}}^b \wedge \vec{\theta}_a + E^b{}_a\right) + (C_A - C_T)\vec{\theta}_a \wedge \ast\left(\dot{\vec{\theta}}^b \wedge \vec{\theta}_b + E^b{}_b\right)\Big\}\,.\end{gathered}$$ To invert the relation between velocities $v^a \equiv \dot{\vec{\theta}}^a$ and momenta, we split the momenta in the form $\alpha\ast p_a = s_a - \pi_a$ into a part $\pi_a$ linear in the velocities and $s_a$ independent of the velocities. These are given by $$\pi_a = \frac{2}{3}\left[(2C_T + C_V)v_a - (C_T - C_V)\vec{\theta}^{\sharp}_b \intprod \left(v^b \wedge \vec{\theta}_a\right) - (C_A - C_T)\vec{\theta}^{\sharp}_a \intprod \left(v^b \wedge \vec{\theta}_b\right)\right]$$ and $$s_a = \frac{2}{3}\left\{(2C_T + C_V)\left[{\mathrm{d}}(\alpha\xi_a) + {\pounds}_{\beta}\vec{\theta}_a\right] + (C_T - C_V)\vec{\theta}^{\sharp}_b \intprod E^b{}_a + (C_A - C_T)\vec{\theta}^{\sharp}_a \intprod E^b{}_b\right\}\,.$$ By applying the irreducible decomposition , one finds the relations $$\label{eqn:velmom}
{\overset{\odot}{\pi}\vphantom{\pi}}_a = \frac{2}{3}(2C_T + C_V){\overset{\odot}{v}\vphantom{v}}_a\,, \quad
{\overset{\ominus}{\pi}\vphantom{\pi}}_a = \frac{2}{3}(2C_A + C_T){\overset{\ominus}{v}\vphantom{v}}_a\,, \quad
{\overset{\oplus}{\pi}\vphantom{\pi}}_a = 2C_T{\overset{\oplus}{v}\vphantom{v}}_a\,, \quad
{\overset{\otimes}{\pi}\vphantom{\pi}}_a = 2C_V{\overset{\otimes}{v}\vphantom{v}}_a\,.$$ Note that depending on the vanishing or non-vanishing of the constant factors the terms $\pi_a$ vanish, giving rise to another primary constraint, or do not vanish, and contribute to the momenta.[@Blixt:2018znp; @Blixt:2019ene] The kinematic Hamiltonian $\hat{\mathcal{H}}_0 = v^a \wedge p_a - \hat{\mathcal{L}}$ is then given by $$\begin{gathered}
\hat{\mathcal{H}}_0 = \frac{C_A - C_T}{3}\alpha\left[\xi_a\xi_b{\mathrm{d}}\vec{\theta}^a \wedge \ast{\mathrm{d}}\vec{\theta}^b - {\mathrm{d}}\vec{\theta}^a \wedge \theta_a \wedge \ast({\mathrm{d}}\vec{\theta}^b \wedge \theta_b)\right] - C_T\alpha{\mathrm{d}}\vec{\theta}^a \wedge \ast{\mathrm{d}}\vec{\theta}_a\\
+ \frac{C_T - C_V}{3}\alpha(\vec{\theta}^{\sharp}_a \intprod {\mathrm{d}}\vec{\theta}^a) \wedge \ast(\vec{\theta}^{\sharp}_b \intprod {\mathrm{d}}\vec{\theta}^b) - (\alpha\xi^a + \beta \intprod \vec{\theta}^a){\mathrm{d}}p_a - {\mathrm{d}}\vec{\theta}^a \wedge (\beta \intprod p_a)\\
+ \hat{\mathcal{H}}_0[{\overset{\odot}{p}\vphantom{p}}] + \hat{\mathcal{H}}_0[{\overset{\ominus}{p}\vphantom{p}}] + \hat{\mathcal{H}}_0[{\overset{\oplus}{p}\vphantom{p}}] + \hat{\mathcal{H}}_0[{\overset{\otimes}{p}\vphantom{p}}] + {\mathrm{d}}\left[(\alpha\xi^a + \beta \intprod \vec{\theta}^a)p_a\right]\,.\end{gathered}$$ Here the last term is a total derivative and hence does not contribute to the dynamics. The remaining terms in the last line depend on the presence or absence of constraints and read $$\begin{gathered}
\hat{\mathcal{H}}_0[{\overset{\otimes}{p}\vphantom{p}}] = \begin{cases}
0 & \text{for } C_V = 0\,,\\
-\frac{\alpha}{4C_V}{\overset{\otimes}{c}\vphantom{c}}_a \wedge \ast{\overset{\otimes}{c}\vphantom{c}}^a & \text{otherwise,}
\end{cases} \quad
\hat{\mathcal{H}}_0[{\overset{\oplus}{p}\vphantom{p}}] = \begin{cases}
0 & \text{for } C_T = 0\,,\\
-\frac{\alpha}{4C_T}{\overset{\oplus}{c}\vphantom{c}}_a \wedge \ast{\overset{\oplus}{c}\vphantom{c}}^a & \text{otherwise,}
\end{cases}\nonumber\\
\hat{\mathcal{H}}_0[{\overset{\ominus}{p}\vphantom{p}}] = \begin{cases}
0 & \text{for } 2C_A + C_T = 0\,,\\
-\frac{3\alpha}{4(2C_A + C_T)}{\overset{\ominus}{c}\vphantom{c}}_a \wedge \ast{\overset{\ominus}{c}\vphantom{c}}^a & \text{otherwise,}
\end{cases}\\
\hat{\mathcal{H}}_0[{\overset{\odot}{p}\vphantom{p}}] = \begin{cases}
0 & \text{for } 2C_T + C_V = 0\,,\\
-\frac{3\alpha}{4(2C_T + C_V)}{\overset{\odot}{c}\vphantom{c}}_a \wedge \ast{\overset{\odot}{c}\vphantom{c}}^a & \text{otherwise,}
\end{cases}\nonumber\end{gathered}$$ where we used the abbreviations $$\begin{aligned}
{\overset{\odot}{c}\vphantom{c}}_a &= \ast{\overset{\odot}{p}\vphantom{p}}_a - \frac{2}{3}(C_T - C_V)\xi_a\vec{\theta}^{\sharp}_b \intprod {\mathrm{d}}\vec{\theta}^b\,, &
{\overset{\oplus}{c}\vphantom{c}}_a &= \ast{\overset{\oplus}{p}\vphantom{p}}_a\,,\nonumber\\
{\overset{\ominus}{c}\vphantom{c}}_a &= \ast{\overset{\ominus}{p}\vphantom{p}}_a - \frac{2}{3}(C_A - C_T)\vec{\theta}^{\sharp}_a \intprod \left(\vec{\theta}^b \wedge {\mathrm{d}}\xi_b\right)\,, &
{\overset{\otimes}{c}\vphantom{c}}_a &= \ast{\overset{\otimes}{p}\vphantom{p}}_a\,.\label{eqn:ifconstr}\end{aligned}$$ Note that the result is linear in lapse $\alpha$ and shift $\beta$, so that these quantities are Lagrange multipliers, corresponding to primary constraints arising from diffeomorphism invariance of the theory. If any of the constant factors in the relations vanishes, additional constraints appear, which force the corresponding term to vanish. This reproduces our result derived using tensor components.[@Blixt:2018znp; @Blixt:2019ene]
Conclusion
==========
We have derived the kinematic Hamiltonian and primary constraints of new general relativity in the language of differential forms. Our result agrees with the result of a previous calculation performed using tensor components.[@Blixt:2018znp; @Blixt:2019ene] This consolidates our result, which is an important step towards counting the degrees of freedom in these theories. The latter will be achieved after deriving the constraint algebra, as it has been done for the teleparallel equivalent of general relativity.[@Okolow:2013lwa]
Acknowledgments {#acknowledgments .unnumbered}
===============
The author thanks Daniel Blixt, Viktor Gakis and Christian Pfeifer for discussions. He gratefully acknowledges the full support by the Estonian Ministry for Education and Science through the Personal Research Funding project PRG356, as well as the European Regional Development Fund through the Center of Excellence TK133 “The Dark Side of the Universe”.
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---
abstract: 'An increasing array of biomedical and computer vision applications requires the predictive modeling of complex data, for example images and shapes. The main challenge when predicting such objects lies in the fact that they do not comply to the assumptions of Euclidean geometry. Rather, they occupy non-linear spaces, a.k.a. manifolds, where it is difficult to define concepts such as coordinates, vectors and expected values. In this work, we construct a non-parametric predictive methodology for manifold-valued objects, based on a distance modification of the Random Forest algorithm. Our method is versatile and can be applied both in cases where the response space is a well-defined manifold, but also when such knowledge is not available. Model fitting and prediction phases only require the definition of a suitable distance function for the observed responses. We validate our methodology using simulations and apply it on a series of illustrative image completion applications, showcasing superior predictive performance, compared to various established regression methods.'
author:
- Dimosthenis Tsagkrasoulis
- 'Giovanni Montana [^1]'
title: 'Random Forest regression for manifold-valued responses'
---
Introduction {#intro}
============
Predictive modeling is an integral part of data analysis. It encompasses the process of developing models which can accurately predict yet-to-be-seen data. A multitude of regression models exist that can be used for prediction of univariate, or multivariate vectorial, responses. Nevertheless, less work has been done on the difficult problem of modeling and predicting more complex objects that possess additional structure, be it morphological, directional, or otherwise [@Rahman2005].
Images [@peyre2009], shapes [@Small1996], graphs [@Tsochantaridis2004], deformation tensors [@Zhu2009] are examples of complex data types that appear naturally as responses in image analysis, computer vision, medical imaging and other application areas. While such objects are typically represented as points on very high-dimensional spaces, they can be meaningfully represented as points lying on smooth non-linear hyper-surfaces of lower dimensionality, a.k.a manifolds [@Chavel2006]. Manifolds can be understood as surfaces that locally resemble the Euclidean plane, but have different global structures [@robbin2011]. Ideally, a suitable predictive modeling methodology should work under the additional constrains imposed by the data’s inherent manifold structure instead of simply treating these complex objects as points on Euclidean spaces. Major difficulties of course arise by forgoing the assumption of a Euclidean space, such as lack of coordinates, vectors and no analytical definitions of expected values [@Fletcher2013]. In this work, we are additionally interested in experimental settings where the input observations may be very-high dimensional, which poses further requirements in the construction of a predictive algorithm.
Three main families of methodologies concerning with regression for manifold-valued responses can be found in the literature; intrinsic manifold regression, kernel-based and Manifold Learning (ML)-based methods. Intrinsic manifold regression models are generalizations of linear regression on manifolds [@Fletcher2013; @Zhu2009]. They require the analytical definition of the data manifold, since they fit a parametrized curve on the data.The assumption of an underlying manifold drives the choice of geometric elements employed during model definition and parameter estimation. Unfortunately, this requirement can be rarely satisfied, either due to the nature of data or the inability to define a suitable manifold. Furthermore, this methodology is not ideal for regression analysis with highly dimensional input observations.
Kernel methods first appeared in the literature as methodologies tailored for complex input objects, such as trees or graphs [@shawe2004]. The basic idea behind these methods is that, if the input data lie on a non-linear space, then they can be implicitly mapped on a very high (or infinite) dimensional inner-product space, in which standard regression methods can be applied [@shawe2004]. This implicit mapping is achieved through a kernel function that defines the objects’ inner-product in that space [@shawe2004]. Kernel methods for complex responses have also been proposed [@Tsochantaridis2004; @Geurts2006], but suffer from a number of issues. First, depending on the data at hand, a meaningful kernel function must be found or constructed, a process that is not always intuitive. Second, another problem is the formulation of a prediction methodology, which has to be described as a kernel minimization problem over the response space. This is most commonly solved by reducing the search space to the training dataset.
The last family of manifold regression methods is based on non-linear dimensionality reduction, a.k.a Manifold Learning. Given a set of observed points, ML methods aim to project these onto a space of lower dimensionality, whilst retaining as well as possible the original geometrical structure of the data. ML-based methods first apply ML on the response data, and subsequently use standard regression models trained on the learned response embedding. A suitable methodology must be formulated to map predicted points from the embedding to the manifold, a process that is referred to as backscoring [@wahba1992]. Appropriate selection of the ML technique is largely based on intuition and previous experience. A decrease in model fitting accuracy, compared to an intrinsic manifold model, is to be expected, since the response embedding is not guaranteed to completely capture the structure of the underlying manifold. Furthermore, ML techniques generate discrete and one way maps from the manifold to the embedding, and the existence of an inverse and continuous map cannot always be guaranteed [@wahba1992].
Here, we present a new methodology for predictive modeling of manifold-valued responses, which addresses a number of key issues listed above. Our objective was to propose a unified framework for regression and prediction of complex objects that is accurate, computationally efficient and can be readily deployed in a variety of applications. In the training phase, we employ our modified Random Forest (RF) regression algorithm that can be trained using only pairwise distances between response observations [@Sim2013]. The non-parametric RF methodology is efficient and can handle high dimensional input spaces. In contrast to intrinsic manifold regression, no analytical definition of the underlying response manifold is required, apart from the construction of an appropriate distance function for the responses. Comparing our distance-based approach to kernel methods, we identify further advantages. First, a vast library of readily available distance metrics exists in the literature (see for example [@deza2009]). Second, when the manifold metric is not known, distances can be intuitively approximated, using for example the Isomap distance formulation [@Tenenbaum2000].
In the prediction phase, a response point estimate for a new input observation is found as follows. Pairwise distances between the unseen response and all training observations are estimated. Using this set of distances, the response is predicted on a Euclidean embedding computed using the ML technique of metric multi-dimensional scaling (MDS) and is finally mapped to the response manifold through a standardized backscoring procedure. Our prediction methodology follows a two-step approach akin to ML-based methods, whilst offering two additional advantages. First, our regression model is trained on the original manifold, which enhances the quality of the fitted model. Second, due to the fact that manifold distances are known, MDS is employed for ML and predictions can be analytically computed and backscored to the response manifold.
The rest of this article is organized as follows. In section \[sec:methods\] we present the details of our manifold regression methodology. Our simulation and application experiments are included in section \[sec:exp\]. We conclude this work in section \[sec:disc\].
Random Forest Predictive Methodology for Manifold-Valued Objects {#sec:methods}
================================================================
Problem Definition
------------------
Let $S$ be a dataset of $N$ observed input-response pairs $\{(\mathbf{x}_i, \mathbf{y}_i)\}_{i=1}^N$, with inputs $\mathbf{x}_i = (x_{i1}, \ldots,x_{ip}) \in \mathbb{R}^p$ corresponding to responses $\mathbf{y}_i \in \mathcal{M} \subset \mathbb{R}^q$. $\mathcal{M}$ is a possibly unknown manifold, equipped with a distance metric $d(\cdot,\cdot)$. We refer to $\mathbb{R}^q$ as the response representation space. Let $\mathbf{D}$ be the $N \times N$ matrix of pairwise distances between the observed response points, with elements $D_{ij} = d(\mathbf{y}_i,\mathbf{y}_j)$. See Fig. \[man:pred:fig1\](a) for a graphical illustration of the described data. We want to construct a predictive methodology that leverages these distances in order to ensure that, for any given input $\mathbf{x}_{new}$, the predicted $\mathbf{\hat{y}}_{new}$ lies on $\mathcal{M}$.
![Illustration of the dRF prediction methodology. (a) The dRF model is fitted using the distance matrix $\mathbf{D}$ of manifold distances between the observed responses (section \[sec:drf\]). (b) When a new input $\mathbf{x}_{new}$ is observed, it is passed through the forest and a similarity vector, $\mathbf{a}$, between the training responses and the yet to be predicted response is extracted from the model (section \[sec:dist\]). (c) Based on $\mathbf{a}$, the set of distances between the new point and the training responses on the manifold are predicted (section \[sec:dist\]). (d) Knowledge of these distances allows prediction of the response on a Euclidean embedding of the manifold, extracted through multi-dimensional Scaling (section \[sec:outofsample\]). (e) A backscoring method is used to project the predicted point back to the original manifold (section \[sec:back\]).[]{data-label="man:pred:fig1"}](drawing6_c.pdf){width="60.00000%"}
Distance Random Forest Regression {#sec:drf}
---------------------------------
We first concentrate on the construction of a suitable manifold regression learning algorithm. Random Forest (RF) is a non-parametric, non-linear regression and classification algorithm [@Breiman2001]. In more detail, RF is a collection of Classification and Regression Trees (CARTs). A CART splits the input space recursively, according to a predefined split function, to small rectangular regions and then fits a simple model, commonly a constant value, in each one of them. See [@Breiman2001] for a detailed description of RF. In [@Sim2013], we presented a modified distance Random Forest (dRF) algorithm, where the split criterion was formulated to depend only on pairwise distances between responses: $$\label{man:drf:eq:cost_function}
\begin{split}
G_{\mathcal{M}}(S_w) = \frac{1}{2N_w} \sum_{\mathbf{y}_i \in S_w} \sum_{\mathbf{y}_i \in S_w} d^2 (\mathbf{y}_i,\mathbf{y}_j) - \frac{1}{2N_{wl}} \sum_{\mathbf{y}_i \in S_{wl}} \sum_{\mathbf{y}_i \in S_{wl}} d^2 (\mathbf{y}_i,\mathbf{y}_j) - \frac{1}{2N_{wr}} \sum_{\mathbf{y}_i \in S_{wr}} \sum_{\mathbf{y}_i \in S_{wr}} d^2 (\mathbf{y}_i,\mathbf{y}_j) \enspace ,
\end{split}$$ with $\mathbf{y}_i \in \mathcal{M}$. Here $N_w, N_{wl}, N_{wr} $ are the cardinalities of the data subsets $S_w$, $S_{wl}$, $S_{wr}$ belonging to a node $w$ and its children nodes (left and right), respectively. The objective is a generalization of the cost function used in standard regression RF, decoupled from the use of Euclidean norms and means, dependent only on pairwise response distances. As such, it enables the RF algorithm to be applied in general metric spaces. Previously, we used dRF for regression applications with graph and covariance-based response objects [@Sim2013]. dRF lends itself naturally to manifold-valued data, whether the manifold in question is analytically defined or implied by the use of a specific distance metric.
Predictive Methodology for dRF {#sec:pred}
------------------------------
Traditionally, when working with responses lying in Euclidean spaces, an RF prediction in made simply by averaging response points; a new input observation follows the split decision rules learned during training and reaches a terminal node -leaf- in the tree. It is then assigned a suitable value on the response space, most often the average response of that leaf’s responses in the training set. This approach though is not valid under the assumption of non-vectorial manifold responses [@Pennec2006].
Our proposed methodology uses dRF for learning a family of trees from the data, which only requires an appropriate distance metric for the responses, as described above. The prediction phase is substantially different from standard RF. For a new input point $\mathbf{x}_{new}$, we use the trained dRF model to predict all pairwise distances between the unknown response, $\mathbf{y}_{new}$, and all observed responses in the training dataset, $\mathbf{y}_i$. Having estimated these distances, we then utilize them to to predict the response on an Euclidean embedding of the manifold, learned from the observed response set using metric MDS. Subsequently, a backscoring method is employed to project the point back to the original response manifold $\mathcal{M}$. Figure \[man:pred:fig1\] provides an illustration of the proposed methodology, and the details are in order.
### RF-based Distance Prediction {#sec:dist}
The first step is to estimate the set of distances $\{\hat{d}(\mathbf{y}_{new},\mathbf{y}_i)\}_{i=1}^N$. We exploit the inherent ability of the dRF model to provide a measure of similarity between pairs of response observations (see Fig. \[man:pred:fig1\](b)). When the new input point $\mathbf{x}_{new}$ is ‘dropped’ through each tree in the forest, it reaches a leaf associated with a subset of the training data. The $\mathbf{y}_{new}$ can be considered similar to that leaf’s training responses. To compute the dRF-based similarities, a vector $\mathbf{a} = (a_{1},\ldots,a_{N} )$ is used, with each element $a_i$ corresponding to the similarity between $\mathbf{y}_{new}$ and $\mathbf{y}_i$. Initially $a_i$ is set to zero, for all $i=1, \ldots, N$. For each tree in RF, $a_{i}$ is incremented by one, each time $\mathbf{x}_{new}$ ends in the same node as $\mathbf{x}_{i}$. Normalization of similarities is performed by division with the number of trees.
Based on the similarity vector $\mathbf{a}$ and the training distance matrix $\mathbf{D}$, the response distances are predicted using the algorithmic procedure \[man:pred:algo1\] ( see Fig. \[man:pred:fig1\](c) ), which guarantees that the new point will lie in close proximity to at least its closest neighbor based on the dRF affinities, and that predicted distance values respect the triangular inequality property of metric $d$. In detail, the algorithm initially identifies the closest training point to the new response, according to $\mathbf{a}$, and assigns the minimum distance observed in the training set as the distance between these two responses. Subsequently, we iterate over the remaining responses, in decreasing order of distance from the first point, and assign a value for the distance to the new observation as follows: For each triplet including the considered point, the new point and a point for which the distance to $\mathbf{y}_{new}$ has been already estimated, we keep the maximum of the two known distances. Subsequently, we assign the minimum of all identified maximums as the predicted distance.
$\mathbf{a}$ $\{ D_{ij} \}_{i,j=1}^N$ $S = \{1, \ldots ,N \}, \enspace Q = \emptyset$ $l \gets \underset{i \in S}{\mbox{argmax}} \enspace\ a_i $ $\hat{d}(\mathbf{y}_{new},\mathbf{y}_{l}) \gets \min \{ D_{ij}\}_{i,j=1}^N$ $S \gets S \setminus \{ l \}, \enspace Q \gets Q \cup \{ l \}$ $p \gets \underset{i \in S}{\mbox{argmax}} \enspace D_{il} $ $\hat{d}(\mathbf{y}_{new},\mathbf{y}_{p}) \gets min \{ max \{\hat{d}(\mathbf{y}_{new},\mathbf{y}_{i}), D_{ip} \}| i \in Q \} $ $S \gets S \setminus \{ p \}, \enspace Q \gets Q \cup \{ p \}$ $\{\hat{d}(\mathbf{y}_{new},\mathbf{y}_i)\}_{i=1}^N$
### Prediction on the Response Embedding {#sec:outofsample}
Given $\{\hat{d}(\mathbf{y}_{new},\mathbf{y}_i)\}_{i=1}^N$, we can predict a point estimate of the response on a Euclidean embedding of the manifold using MDS [@Cox2010] (see Fig. \[man:pred:fig1\](d) ).
MDS computes an approximation $\{ \mathbf{z}_i| \mathbf{z}_i \in \mathbb{R}^m \}_{i=1}^N$ of the manifold-valued dataset that resides on a $m$-dimensional Euclidean space, by minimizing a stress function of the form $\sum_{i=1}^N\sum_{j=1}^N \left( d^2(\mathbf{y}_i,\mathbf{y}_j) - \| \mathbf{z}_i - \mathbf{z}_j\|^2 \right)$. Let $\mathbf{D}^{(2)}$ be the $N \times N $ matrix of squared manifold distances and $\mathbf{K} = - \frac{1}{2} \mathbf{H} \mathbf{D}^{(2)} \mathbf{H}$, where $\mathbf{H}=\mathbf{I}_N-\frac{1}{N}\mathbf{e} \mathbf{e}^T$, with $\mathbf{I}_N$ the $N \times N $ identity matrix and $\mathbf{e}$ an $N \times 1$ column vector of all ones. Individual elements of $\mathbf{K}$ are given by $K_{ij} = - \frac{1}{2} (D^{(2)}_{ij} - \frac{1}{n}S_i - \frac{1}{n}S_j + \frac{1}{n^2}S), i,j = 1, \ldots ,N ,$ where $S_i = \sum_j D^{(2)}_{ij}, S_j = \sum_i D^{(2)}_{ij}, S = \sum_{i,j} D^{(2)}_{ij}$ are the $i^{th}$ row, $j^{th}$ column and overall element-wise sums of $\mathbf{D}^{(2)}$, respectively. In [@mardia1978], it was shown that, if $\mathbf{K}$ has rank $p \enspace (p \leq N-1)$, with $\lambda_1, \ldots , \lambda_p$ the $p$ ordered non-zero eigenvalues of $\mathbf{K}$ with corresponding eigenvectors $\mathbf{u}_1, \ldots, \mathbf{u}_p$, and $\mathbf{u}_l = (u_{l1},\ldots, u_{lN})$ for $l = 1, \ldots, p$, then the solution to the MDS problem, assuming $m \leq p$, is given by $z_{il} = \sqrt{\lambda_l}u_{li}$, with $l = 1, \ldots, m, \enspace i = 1, \ldots, N$, and $\mathbf{z}_i = (z_{i1},\ldots,z_{im}) $.
The result of the MDS decomposition is a discrete and one-way mapping $\mathbf{z}_i = f_{MDS}(\mathbf{y}_i)$ defined on the training data set. An Out-Of-Sample (OOS) method which allows mapping of a new manifold observation on the learned space of the embedding was constructed in [@Bengio2004]. Let $Y \in \mathcal{M}$ be a random variable defined on the manifold surface and $\mathbf{y}_a,\mathbf{y}_b\in \mathcal{M}$ two observations of $Y$. A continuous kernel function $k_{MDS}$, which gives rise to $\mathbf{K}$ under the training observations, is defined as $k_{MDS}(\mathbf{y}_a,\mathbf{y}_b) = - \frac{1}{2} (d^2(\mathbf{y}_a,\mathbf{y}_b) - E[d^2(Y,\mathbf{y}_a)] - E[d^2(Y,\mathbf{y}_b)] + E[d^2(Y,Y)] ) , $ where $E(\cdot)$ denotes expectation. Then, an OOS prediction $\hat{\mathbf{z}}_{new} = (\hat{z}_{new,1},\ldots,\hat{z}_{new,m}) $ is given by [@Bengio2004]: $$\label{man:pred:eq:oos}
\hat{z}_{new,k} = \frac{1}{\sqrt{\lambda_k}} \sum^N_{i=1} u_{ik} \hat{k}_{MDS}(\mathbf{y}_{new},\mathbf{y}_i), \enspace k = 1,\ldots,m \enspace ,$$ with $\hat{k}_{MDS}$ denoting the mean estimator of $k_{MDS}$ under the augmented dataset $\{ \mathbf{y}_i\}_{i=1}^N \cup \{ \mathbf{y}_{new} \}$: $$\begin{split}
\hat{k}_{MDS}(\mathbf{y}_{new},&\mathbf{y}_i) = \enspace -\frac{1}{2} d^2(\mathbf{y}_{new},\mathbf{y}_i) + \frac{1}{2(N+1)} \left( \sum_{j=1}^N d^2(\mathbf{y}_j,\mathbf{y}_{new}) \right) \\
& + \frac{1}{2(N+1)} \left( \left( \sum_{j=1}^N d^2(\mathbf{y}_j,\mathbf{y}_i) \right) + d^2(\mathbf{y}_{new},\mathbf{y}_i) \right) \\
& - \frac{1}{2(N+1)^2} \left( \left( \sum_{j,l=1}^N d^2(\mathbf{y}_{j},\mathbf{y}_l) \right) + 2\sum_{j=1}^N d^2(\mathbf{y}_{new},\mathbf{y}_j) \right) \enspace .
\end{split}$$
The OOS formula does not depend on the actual value of $\mathbf{y}_{new}$, but rather on the distances between the new response and points on the training dataset. We can thus leverage the predicted $\{\hat{d}(\mathbf{y}_{new},\mathbf{y}_i)\}_{i=1}^N$ on equation in order to get a point estimate of our response $\hat{\mathbf{z}}_{new}$ on the embedding space $\mathbb{R}^m$. Now we are left with the task of mapping $\hat{\mathbf{z}}_{new}$ to the original manifold ( see figure \[man:pred:fig1\](e) ).
### Mapping from the Embedding to the Manifold {#sec:back}
The mapping of the response from the embedding to the manifold - backscoring - can be formulated as an interpolation problem. Specifically, we are looking for a smooth continuous function $g: \mathbb{R}^m \rightarrow \mathbb{R}^q $, that minimizes the cost function $\enspace \gamma_g \sum^{N}_{i=1} (g(\mathbf{z}_i)- \mathbf{y}_i)^2 + \|g\|^2_{G} $. $\gamma_g$ is a weight parameter balancing the smoothness of the interpolation and the adherence to the data and $G$ is a space of smooth functions equipped with the norm $\|\cdot\|_{G}$, which will be constructed in the following.
A solution to the interpolation problem was presented in [@wahba1992]. Let $\mathcal{V}$ be a closed subset of $\mathbb{R}^m$ and $k_G$ a kernel function of the form $$\label{man:pred:back:eq2}
k_G(r,\mathbf{v},t,\mathbf{w}), \enspace r,t \in T^q = \{1,\dots, q \}, \mathbf{v}, \mathbf{w} \in \mathcal{V} \enspace .$$ Furthermore, assume that $k_G$ is semi-positive definite on $(T^q \times \mathcal{V}) \times (T^q \times \mathcal{V})$, with $$\sum_{i=1}^N \sum_{j=1}^N a_{i} a_{j} \sum_{r=1}^q \sum_{t=1}^q k_G(r,\mathbf{v}_i,t, \mathbf{v}_j) \geq 0 \enspace,$$ for any finite set of points $\{\mathbf{v}_i| \mathbf{v}_i \in \mathcal{V}\}_{i=1}^N$ and real numbers $\{a_i| a_i \in \mathbb{R}\}_{i=1}^N$. For a fixed $(r,\mathbf{v})$, equation defines a function from $\mathbb{R}^m$ to $\mathbb{R}^q$ by the formula $$\label{man:pred:back:eq4}
g_{r\mathbf{v}} (\mathbf{w}) = \left( k_G(r,\mathbf{v},1,\mathbf{w}), \ldots, k_G(r,\mathbf{v},q,\mathbf{w}) \right)^T \enspace .$$
Based on the above, let $G$ be the space of all finite linear combinations of functions of the form , as $(r,\mathbf{v})$ varies in $T^q \times \mathcal{V}$, and its closure w.r.t the scalar inner product $\langle g_{r\mathbf{v}}, g_{t\mathbf{w}}\rangle = k_G(r,\mathbf{v},t,\mathbf{w})$. It follows that $\| g_{r\mathbf{v}} \|_G = \sqrt{\langle g_{r\mathbf{v}}, g_{r\mathbf{v}} \rangle }$. The interpolation problem can be solved over the space of functions $G$ as follows [@wahba1992]. Let $\mathbf{Y}$ be the $N \times q$ matrix of training responses with rows $Y_{i\cdot} = \mathbf{y}_i$, and $\mathbf{K}^G$ the $Nq \times Nq$ matrix of kernel values $K_{N(r-1)+i,N(t-1)+j}^G = k_G(r,\mathbf{z}_i,t,\mathbf{z}_j)$. The minimizing function can be estimated as: $$\label{man:pred:back:eq5}
\hat{\mathbf{y}}_{new} = \hat{g}(\hat{\mathbf{z}}_{new}) = \sum_{i=1}^{N} \sum_{r=1}^{q} C_{ir} g_{r\mathbf{z}_i}(\hat{\mathbf{z}}_{new}) \enspace ,$$ where $C_{ir}$ are elements of the $N \times q$ coefficient matrix $\mathbf{C}$ given by $$\label{man:pred:back:eq6}
vec(\mathbf{C}) = ( \mathbf{K}^{G} + \frac{N}{\gamma_g}\mathbf{I})^{-1} vec(\mathbf{Y}) \enspace ,$$ with $vec(\cdot)$ denotes the vectorization of a matrix into a column vector.
It is clear, from equations and , that the estimation of $\hat{g}(\hat{\mathbf{z}}_{new})$ requires just the knowledge of pairwise kernel values between the $N+1$ points $\mathbf{z}_1. \ldots, \mathbf{z}_N, \hat{\mathbf{z}}_{new}$.
In our studies, we opted to simplify the minimizer , in favor of having just one tuning parameter, by choosing $$k_G(r,\mathbf{v},t,\mathbf{w}) = \begin{cases}
\exp \left( - \frac{\| \mathbf{v} - \mathbf{w} \|^2}{\sigma_G} \right), \enspace r,t = 1, \ldots , q, \enspace r = t\\
0, r \neq t
\end{cases}
\enspace ,$$ where $\sigma_G$ is a free parameter adjusting the bandwidth of the kernels.
We notice that the backscoring formulation does not take into explicit consideration the response manifold and $\hat{g}(\hat{\mathbf{z}}_{new})$ is not guaranteed to lie exactly on $\mathcal{M}$. Nevertheless, we justify our choice by pointing that since $\hat{g}$ is a smooth interpolating function from the embedding to the training responses, predictions should also adhere well to the manifold.
Experiments {#sec:exp}
===========
In this section, we present a comparative simulation study to assess the ability of our methodology to cast predictions that adhere to the response manifold. Subsequently, we use our method in two illustrative image completion applications, where the objective is to predict one half of an image from the other half.
Simulation Study - Prediction on a Swiss-roll Manifold {#man:sim}
------------------------------------------------------
We simulated $N = 900$ paired input-response points $\{(\mathbf{x}_i, \mathbf{y}_i)\}_{i=1}^N$, with inputs $\mathbf{x}_i \in \mathbb{R}^6$ corresponding to responses $\mathbf{y}_i \in \mathcal{M}_{sr} \subset \mathbb{R}^3$, where $\mathcal{M}_{sr}$ denotes the 2-dimensional swiss roll manifold embedded in $\mathbb{R}^3$. Only the first two input dimensions were built to be predictive of the output. In detail, we first sampled $900$ points $\{t_i\}_{i=1}^N$ from the uniform distribution $\mathcal{U}(\pi,3\pi)$ and $\{u_i\}_{i=1}^N$ from $\mathcal{U}(0,21)$. The response points on the swiss-roll were computed as $y_{i1} = t_i \cos t_i, \enspace y_{i2} = u_i, \enspace y_{i3} = t_i \sin t_i$, while the input variables $x_{i1}$ and $x_{i2}$ were computed by mapping $u_i$ and $t_i$ in the unit circle: $x_{i1} = \frac{t_i -\bar{t_i}}{\max t_i} \sqrt{1 - \frac{1}{2} \left( \frac{u_i -\bar{u_i}}{\max u_i} \right)^2 }, \enspace x_{i2} = \frac{u_i -\bar{u_i}}{\max u_i} \sqrt{1 - \frac{1}{2} \left( \frac{t_i -\bar{t_i}}{\max t_i} \right)^2 }$ . The values for the remaining input coordinate variables were drawn from the standard normal Gaussian distribution. Gaussian Noise, with $\Sigma=0.5\mathbf{I}_3$, was added on the response points.
We compared our dRF prediction methodology to $k$NN regression and standard RF, which do not take into consideration the special form of the response space, as well as kernel RF (kRF) [@Geurts2006], an RF method that employs a kernel function to capture the structure of the response space during training. The simulated dataset was split into $S_{train}$, comprising of $600$ input-response pairs and $S_{test}$, consisting of the remaining $300$. For $k$NN regression, the value of $k$ was selected to be $5$. All RFs were built with 150 trees, no pruning and the number of candidate split features in each node was set to $3$. For kRF, the training gram matrix was calculated using the Gaussian kernel $g(\mathbf{y}_i, \mathbf{y}) = \exp{\left(- \frac{ \|\mathbf{y}_i - \mathbf{y}_j\|^2}{2 \sigma^2} \right) }$ with $\sigma = 2.5$. We followed the prediction methodology as described in [@Geurts2006], with $\mathbf{y}_{new} = \operatorname*{arg\,min}_{\mathbf{y} \in S_{train}} \left( g(\mathbf{y}, \mathbf{y}) -2 \sum_{i=1}^N a(\mathbf{x}_{new}, \mathbf{x}_{i}) g(\mathbf{y}, \mathbf{y}_i) \right)$, where $a(\cdot,\cdot)$ is the RF-based affinity. The minimization problem was solved over the training set.
For dRF, the backscoring parameters were $\sigma_{\mathcal{G}} = 100$ and $\gamma_{\mathcal{G}} = 200$. Since there is no analytical form for computing distances on a swiss-roll manifold, we approximated manifold distances using the Isomap distance formulation [@Tenenbaum2000]: A neighborhood graph $G$ was constructed, in which each point $\mathbf{y}_i$ was connected to its $k=7$ nearest neighbors in $\mathbb{R}^q$ . Graph edges were assigned weights equal to the Euclidean distances between the connected points in $\mathbb{R}^q$. For any two points $\mathbf{y}_i$ and $\mathbf{y}_j$ in $S$, $d(\mathbf{y}_i,\mathbf{y}_j)$ was then estimated by the shortest path connecting $\mathbf{y}_i$ and $\mathbf{y}_j$ in graph $G$.
Figure \[man:sim:figure9\] shows test error vectors for the various methodologies used, projected on the $y_{\cdot 1}y_{\cdot 3}$ plane. We can see that $5$NN missed the goal of regressing on the manifold. For standard RF, it is clear that the model constantly underestimated the radius of the predicted points around the $y_{\cdot 1}$ axis. This can be justified by considering that predictions are taken as average points on the euclidean space $\mathbb{R}^3$, unaware of a possible structure in the response space. kRF preserved the manifold structure of the predicted points better than RF, but suffered significantly from the added noise in the responses. dRF outperformed the other methods both in terms of compliance to the underlying response manifold and regarding good robustness to the addition of noise.
![Test error vectors projected on the $y_{\cdot 1}y_{\cdot 3}$ plane of the various regression models for the simulated swiss-roll dataset. (a) $5$NN (b) RF (c) kRF (d) dRF. The figure highlights compliance to the response manifold.[]{data-label="man:sim:figure9"}](vecerrors.pdf){width="60.00000%"}
Applications on Image Completion {#man:app}
--------------------------------
In imaging analysis, it is common to assume that a collection of similar images lies on a manifold [@peyre2009]. This assumption is guided by the complex nature of images as data objects, as well as the observation that the Euclidean metric and the corresponding geometric structure that it imposes on the space do not bode well with the human perception of similarity and difference between images [@peyre2009]. Here, we used our manifold regression methodology to predict the bottom half of handwritten digits and human faces from their upper half.
### Handwritten digits
For this application, we extracted $1000$ gray-scale images of handwritten digits, from the UCI Machine Learning Repository [@Lichman2013]. Each digit class, from $0$ to $9$, was represented in the dataset by $100$ $8 \times 8$ pixel images. Input data were constructed by vectorization of the $8 \times 4$ upper half pixel intensities. The dataset was split into training and testing subsets with $N_{train} = 800$ and $ N_{test} = 200$. The upper part of each images was taken as input for the predictive models. Responses comprised of the images’ bottom parts. The test set can be seen in Fig. \[man:app:figureall\](a).
![Test and reconstructed images from the digit completion application. Reconstructed digits were generated by concatenating the predicted responses with the upper -input- half of the test images. Bad reconstructions are enclosed in red squares. (a) Test digits. (b)-(e) Reconstructions using predictions from (b) $1$NN, (c) RF, (d) kRF and (e) dRF.[]{data-label="man:app:figureall"}](digits_all.pdf){width="40.00000%"}
We compared predictions from $1$NN, RF, kRF and dRF models. All RFs were built with $300$ trees and $5$ candidate split features in each node. For dRF, the distance matrix was constructed using the Isomap distance, with the number of neighbors set to $5$. A $25$-dimensional embedding space was used and the backscoring parameters were $\sigma_{\mathcal{G}} = 3$ and $\gamma_{\mathcal{G}} = 20$. For kRF, the training gram matrix was calculated using the Gaussian kernel with $\sigma = 5$. The reconstructed test digits for the various models are shown in Fig. \[man:app:figureall\](b)-(e).
Table \[man:app:table1\] summarizes the prediction results for the test data. In the first column we include the test Euclidean Mean Squared Errors (EMSE) for all models. In the case of $1$NN and kRF, which draw predictions from the training dataset, we are also able to report misclassification errors, a.k.a the percentage of predicted lower parts that mismatched the ground truth, which are shown on the second column of Table \[man:app:table1\]. Finally, in the last two columns, we report on the number and percentage of badly reconstructed test images from visual inspection, based on the following criteria: blurriness of the reconstructed image, smooth transition between the upper and bottom image parts, correct digit reconstruction. This qualitative performance measure is important due to the non-Euclidean nature of the data, which, as will be discussed below, makes the EMSE unsuitable for judging the predictive performance.
MSE Clas. Error Bad Rec. No Bad Rec. $\%$
----- -------- ------------- ------------- --------------- --
1NN 3.3665 0.165 40 20
RF 2.7651 - 54 27
kRF 3.2675 0.21 48 24
dRF 3.3287 - 37 18.5
: Test errors from the digit completion application. Classification Error was not applicable for RF and dRF. The number and percentage of badly reconstructed images was visually ascertained from Fig. \[man:app:figureall\].
\[man:app:table1\]
1NN and kRF cast predictions drawn directly from the training images. As such, no blurriness existed on the reconstructed digits of Fig. \[man:app:figureall\](b) and \[man:app:figureall\](d). Bad reconstructions were either misclassifications or reconstructions where the transition between the upper (input) and lower (predicted) image parts was not smooth. Surprisingly, $1$NN outperformed kRF in terms of classification error, as well as upon visual inspection of the images.
It is obvious from Fig. \[man:app:figureall\](c) that standard RF is ill-suited for the specific application. The RF prediction approach of averaging pixel intensities from various images resulted in a high number of blurry and nonsensical digit reconstructions. It is important to notice that while RF performed the worst, based on visual assessment of the reconstructed images, it had the lowest test EMSE, as shown in table \[man:app:table1\]. This observation highlights the unsuitability of EMSE as a measure of performance, in the case of manifold-valued responses.
Reconstructed digits from our dRF model are shown in Fig. \[man:app:figureall\](e). Our predictions were not drawn directly from the training set. Nevertheless, we notice that the large majority of reconstructions had no fuzziness and the transition from upper to lower parts was smooth. For cases where some blurriness could be noticed in the reconstructions, its effect was significantly less severe than in the RF predictions, resulting in the overall lowest number of badly reconstructed digits, based on visual inspection.
### Faces {#man:app2}
In the second application, we used $400$ images of faces from the Olivetti dataset, as included in the scikit-learn python package [@scikitlearn]. The dataset consisted of ten gray-scale $64 \times 64$ pixel images for each of 40 distinct subjects, with same subject images taken at different times and with varying pose and facial expressions. Input data were constructed by vectorization of the $64 \times 32$ upper half pixel intensities. The dataset was split into training and testing subsets with $N_{train} = 300$ and $N_{test} = 100$. Images of the same subject were only allocated either to the training or the testing set. Again, the upper parts of the images were taken as inputs, while bottom parts as responses.
We compared results from $1$NN, linear Regression (LR), RF and dRF. RFs were built with $300$ trees and $8$ candidate split features in each node. For dRF, the training distance matrix was constructed using the Isomap metric with the number of neighbors set to $5$, a $25$-dimensional embedding space was used and the backscoring parameters were $\sigma_{\mathcal{G}} = 9$ and $\gamma_{\mathcal{G}} = 50$.
Four test images and their reconstructions for the various models are included in Fig. \[man:app:figure4\]. As we noted in the previous application, EMSE does not provide a suitable descriptor of performance. We rely again on visual inspection of the reconstructed images. 1NN exhibited a hits and miss behavior, with some predictions being similar to the original face, whilst others, such as the second and third depicted faces, being completely different. In addition, there was minimal smoothness in the transitions from the input to the predicted parts, accentuated specifically on the nasal and zygomatic edges. LR predictions were extremely blurry. RF also suffered from a large amount of blurriness, although transitions between the two parts of the faces looked more natural. Finally, dRF reconstructions exhibited the best transition smoothness of all methods, with the predicted half-images being well aligned to their inputs. Although the predictions were not completely free of blur artifacts, the effect was less severe and facial details, such as nasolabial folds and lips, were clearly portrayed.
![Example test and reconstructed images from the face completion application. (a) Test images. (b)-(e) Reconstructions using predictions from (b) $1$NN, (c) LR, (d) RF and (e) dRF. []{data-label="man:app:figure4"}](oliall.pdf){width="40.00000%"}
Discussion and Conclusion {#sec:disc}
=========================
In this work we presented a predictive modeling approach for response objects occupying non-linear manifold spaces. The regression methodology is based on a distance modification of the RF algorithm that we previously published [@Sim2013], which decouples the model’s training from the problem of response representation. For prediction purposes, we constructed a framework in which point estimates are first predicted on a Euclidean embedding of the response manifold, learned from the training dataset, and then projected back on the original space.
Our methodology, in contrast to intrinsic manifold methods, necessitates just the definition of a meaningful distance metric in the response space. This can be especially useful for real-life applications, such as image analysis, where the underlying manifolds are usually not known. Our distance-based regression algorithm draws similarities to the family of kernel-based methodologies. One benefit over kernel methods is the vast library of readily available distance metrics for a plethora of data objects. Furthermore, our methodology presents a unified framework, which deals with backscoring to the original response space, an issue that a lot of presented kernel methods do not tackle sufficiently well.
In the performed experiments, our method showed superior predictive performance in comparison to various regression methods, whilst being able to handle high-dimensional inputs and noise on the response observations. In the future, we aim to investigate the problem of automatic estimation of the Euclidean embedding’s dimensionality, as well as the use of more elaborate kernel functions in the backscoring formulation.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. The article is currently under consideration for publication at Patter Recognition Letters.
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[^1]: Corresponding author: giovanni.montana@kcl.ac.uk
|
---
author:
- Julien Provillard
- Enrico Formenti
- Alberto Dennunzio
bibliography:
- 'rnuca.bib'
date: |
Laboratoire I3S\
Université Nice Sophia Antipolis,\
2000, route des Lucioles - Les Algorithmes - bât. Euclide B - BP 12\
06903 Sophia Antipolis Cedex - France
title: |
[DRAFT]{}\
Non-uniform cellular automata and distributions of rules
---
Introduction
============
Cellular automata (CA) are discrete dynamical systems consisting in an infinite number of finite automata arranged on a regular lattice. Each node of the lattice contains a variable which can only take a finite number of different values. At any time, the state of the CA, also called its current configuration, is specified by the values of those variables. A CA make evolve its configuration in discrete time by applying a local rule simultaneously to all the variables in the lattice. The value of a variable in a new configuration is computed from the values of a finite number of variables of its neighborhood in the previous configuration according to the local rule. CA have been studied and used in a number of disciplines (computer science, mathematics, physic, biology, chemistry) with different purposes (representing natural phenomena, pseudo-random number generation, cryptography).
However the fact that a same rule is applied everywhere in the lattice can be a constraint in some cases (singular behavior, boundary conditions, non-resiliency). Then variants of CA have been introduced to allow the local rule used to compute the value of a variable to depend of its position. Those variants are called non-uniform cellular automata ($\nu$-CA) or hybrid cellular automata (HCA) [@Dennunzio2011].
In this paper we study $\nu$-CA on one-dimensional lattice defined over a finite set of local rules. The main goal is to determine how the local rules can be mixed to ensure the produced $\nu$-CA has some properties. In a first part, we give some background for the study of $\nu$-CA. Then surjectivity and injectivity are studied using a variant of DeBruijn graphs. The next part is dedicated to the number-conserving property. In a last section, we will be interested in the dynamical properties for linear $\nu$-CA.
Definitions
===========
Automata
--------
Let $A$ be a finite set ($|A| \geq 2$). A (one-dimensional) configuration over $A$ is a mapping from $\mathbb{Z}$ to $A$. If $c$ is a configuration and $i$ an integer $c(i)$ is the state of the configuration at index $i$, this state is more often written $c_i$ to simplify notation.
$A$ is called the alphabet of the configuration. $\mathcal{C}_A$ denotes the set of all configurations on the alphabet $A$. For a configuration $c$, $c_{[i,j]}$ denotes the sequence of states $c_i c_{i+1} \dots c_j$.
A local rule of radius $r$ on the alphabet $A$ is a mapping from $A^{2r+1}$ to $A$. $\mathcal{R}_{A,r}$ denotes the set of all local rules of radius $r$ on $A$ and $\mathcal{R}_A = \cup_{r \in \mathbb{N}} \mathcal{R}_{A,r}$ the set of all local rules on $A$. Local rules are central in the definition both of cellular automata and non-uniform cellular automata.
\[def\_CA\] A mapping $F : \mathcal{C}_A \rightarrow \mathcal{C}_A$ is a cellular automaton if $$\exists r \in \mathbb{N}, \exists f \in \mathcal{R}_{A,r}, \forall i \in \mathbb{Z}, \forall x \in \mathcal{C}_A, H(x)_i = f(x_{[i-r,i+r]}) \enspace.$$
\[def\_NuCA\] A mapping $F : \mathcal{C}_A \rightarrow \mathcal{C}_A$ is a non-uniform cellular automaton if $$\forall i \in \mathbb{Z}, \exists r \in \mathbb{N}, \exists f \in \mathcal{R}_{A,r}, \forall x \in \mathcal{C}_A, H(x)_i = f(x_{[i-r,i+r]}) \enspace.$$
\[def\_NuCA\] A mapping $F : \mathcal{C}_A \rightarrow \mathcal{C}_A$ is a non-uniform cellular automaton with fixed radius if $$\exists r \in \mathbb{N}, \forall i \in \mathbb{Z}, \exists f \in \mathcal{R}_{A,r}, \forall x \in \mathcal{C}_A, H(x)_i = f(x_{[i-r,i+r]})\enspace.$$
The definition \[def\_CA\] is the classical definition for CA. A CA is fully determined by its local rule which is applied simultaneously at all sites. The definition of $\nu$-CA allows different rules at different sites, each rule can have its own radius but all share the same alphabet. This generic definition is quite strong. Indeed even if each rule accesses to a finite number of data of the configuration, this number can be unbounded. This behavior is not expected in classical studies because the notion of locality makes less sense. The r$\nu$-CA are an intermediary model which allow different rules at different sites but each rule accesses to the same range of data according to its position.
\[shift\] The shift automaton $\sigma$ on an alphabet $A$ is defined as follow $$\forall i \in \mathbb{Z}, \forall x \in \mathcal{C}_A, \sigma(x)_i = x_{i+1} \enspace.$$ It is a CA of local rule $f$ where $$\begin{array}{rccl}
f : & A^3 & \rightarrow & A \\
& (x,y,z) & \rightarrow & z\\
\end{array}\enspace.$$
Distributions
-------------
Let $\mathcal{R}$ be a subset of $\mathcal{R}_A$, a distribution on $\mathcal{R}$ is an application $\theta$ from $\mathbb{Z}$ to $\mathcal{R}$. As for configurations, if $\theta$ is a distribution and $i$ an integer $\theta(i)$, or $\theta_i$, is the local rule of the distribution at index $i$.
$\Theta_{\mathcal{R}}$ denotes the set of all distributions on $\mathcal{R}$. For a distribution $\theta$, $\theta_{[i,j]}$ denotes the sequence of local rules $\theta_i \theta_{i+1} \dots \theta_j$.
A distribution of rules $\theta$ induces a $\nu$-CA $H_{\theta}$ defined by
$$\forall i \in \mathbb{Z}, \forall x \in \mathcal{C}_A, H_{\theta}(x)_i = \theta_i(x_{[i-r_i,i+r_i]})$$
where $r_i$ is the radius of the rule $\theta_i$.
\[finite\_number\_rules\] If $\mathcal{R}$ is finite, then for all distribution $\theta$ on $\mathcal{R}$, $H_{\theta}$ is a r$\nu$-CA.
Let denote $r = \max \{ n \in \mathbb{N} : f \in \mathcal{R} \cap \mathcal{R}_{A,n}\}$ the greatest radius of a rule in $\mathcal{R}$. Then for a rule $f \in \mathcal{R}$ of radius $r_f$, we define a rule $\tilde{f}$ of radius $r$ by $$\tilde{f}(x) = f(x_{[r-r_f, r + r_f]}) \enspace.$$
Let $\theta$ be a distribution of $\mathcal{R}$ and $H_{\theta}$ the $\nu$-CA induced by $\theta$, $$\forall i \in \mathbb{Z}, \forall x \in \mathcal{C}_A, H_{\theta}(x)_i = \theta_i(x_{[i-r_i,i+r_i]} = \tilde{\theta_i}(x_{[i-r,i+r]})$$
where $r_i$ is the radius of the rule $\theta_i$. Then $H_{\theta}$ is r$\nu$-CA of radius $r$.
In this paper, we will consider distributions on a finite set $\mathcal{R}$ of local rules. The proof of the Proposition \[finite\_number\_rules\] shows we can always assume that $\mathcal{R}$ is a subset of $\mathcal{R}_{A,r}$ for an integer $r$. Moreover, each finite distribution $\psi$ (finite sequence of rules in $\mathcal{R}$) of $n$ rules defines a function $h_{\psi} : A^{n+2r} \rightarrow A^n$ by $$\forall x \in A^{n+2r}, \forall i \in \{0, \dots, n-1\}, h_{\psi}(x)_i = \psi_i(x_{[i,i+2r]}) \enspace.$$ These functions are called partial transition functions since they express the behavior of a $\nu$-CA on a finite set of sites : if $\theta$ is a distribution and $i \leq j$ are integers, then $$\forall x \in \mathcal{C}_A, H_{\theta}(x)_{[i,j]} = h_{\theta_{[i,j]}(x_{[i-r,j+r]})} \enspace.$$
Topological dynamics
--------------------
The topological properties for $\nu$-CA are studied according to the Cantor distance $d$. For two configurations $x$ and $y$, the Cantor distance is defined by $$d(x,y) =
\left\{
\begin{array}{cl}
0 & \text{if $x = y$} \\
2^{-k} & \text{ where $k = \min \{ i \in \mathbb{N} : x_{[-i,i]} \neq y_{[-i,i]}\}$, otherwise} \\
\end{array}
\right.
\enspace.$$
The topology defined by the Cantor distance on $\mathcal{C}_A = A^{\mathbb{Z}}$ coincide with the product topology induced by the discrete topology on $A$. Then the space of configurations $(\mathcal{C}_A, d)$ is a Cantor space : it is a perfect, compact, totally disconnected metric space.
Cellular automata and non-uniform cellular automata can be characterized by this topology.
Let $H : \mathcal{C}_A \rightarrow \mathcal{C}_A$ be a function, then
1. $H$ is a CA if and only if $H$ is continuous and commutes with the shift automaton, i.e. $H \circ \sigma = \sigma \circ H$ [@Hedlund1969].
2. $H$ is a $\nu$-CA if and only if $H$ is continuous.
3. If $H$ is a r$\nu-CA$ then $H$ is Lipschitz continuous.
However, the following example shows that there exists $\nu$-CA which are Lipschitz continuous but are not r$\nu$-CA.
The $\nu$-CA $H$ defined on an alphabet $A$ as $$\forall i \in \mathbb{Z}, \forall x \in \mathcal{C}_A, H(x)_i = x_{-i}$$ is Lipschitz continuous (with a Lipschitz constant equal to 1) but it is not a r$\nu$-CA.
In Section \[eq\_sens\_add\], we will study some topological properties of a subclass of $\nu$-CA, namely equicontinuity and sensitivity to initial conditions.
A configuration $x \in \mathcal{C}_A$ is said to be an equicontinuity point of the function $H : \mathcal{C}_A \rightarrow \mathcal{C}_A$ if and only if $$\forall \epsilon > 0, \exists \delta > 0, \forall y \in \mathcal{C}_A, d(x,y) < \delta \Rightarrow \forall n \in \mathbb{N}, d(H^n(x),H^n(y)) < \epsilon \enspace.$$
A function $H : \mathcal{C}_A \rightarrow \mathcal{C}_A$ is said to be equicontinuous if and only if all the configurations are equicontinuous points.
A function $H : \mathcal{C}_A \rightarrow \mathcal{C}_A$ is said to be sensitive to initial conditions (or just sensitive) if and only if $$\exists \epsilon > 0, \forall x \in \mathcal{C}_A, \forall \delta > 0, \exists y \in \mathcal{C}_A, d(x,y) < \delta \text{ and } \exists n \in \mathbb{N}, d(H^n(x), H^n(y)) > \epsilon \enspace.$$
Equicontinuity is a property of stability of the system, while sensitivity to initial conditions is more related to chaotic behavior.
Surjectivity and injectivity
============================
Let $\mathcal{R}$ be a finite set of rules. We assume, without loss of generality, that all rules of $\mathcal{R}$ have same radius $r$. We want to determine which are the distributions of $\Theta_{\mathcal{R}}$ inducing surjective (resp. injective) r$\nu$-CA.
Surjectivity
------------
Let $Surj(\mathcal{R}) = \{ \theta \in \Theta_{\mathcal{R}} : H_{\theta} \text{ is surjective}\}$ denote the set of all distributions inducing surjective r$\nu$-CA. We are going to prove that $Surj(\mathcal{R})$ is a sofic subshift.
Recall that a subset $U$ of distributions is a subshift if it is (topologically) closed and shift invariant, i.e. $\sigma(U) = U$. Equivalently, a subshift $U$ can be defined by a set of forbidden patterns. Then a distribution $\theta$ is in $U$ if and only if no finite pattern of $\theta$ is forbidden. A subshift is said to be sofic if it can be defined by a set of forbidden patterns which is recognizable by a finite automaton [@Lind1995].
The proof consists in three steps. First we show that a r$\nu$-CA induced by a distribution is surjective if and only if all its partial transition functions are surjective. Second we prove that $Surj(\mathcal{R})$ is the subshift that avoids the set of all non-surjective partial transition functions on $\mathcal{R}$. Finally, we show that this set is recognizable by a finite automaton, and hence the subshift is sofic.
Let $\theta \in \Theta_{\mathcal{R}}$, $H_{\theta}$ is surjective if and only if for all $i \leq j$, $h_{\theta_{[i,j]}}$ is surjective.
Assume $H_{\theta}$ is surjective, let $i \leq j$ be two integers and $w \in A^{j-i+1}$. Choose a configuration $x$ such that $x_{[i,j]} = w$. Since $H_{\theta}$ is surjective, there exists $y$ such that $H_{\theta}(y) = x$. $h_{\theta_{[i,j]}}(y_{[i-r,j+r]}) = H_{\theta}(y)_{[i,j]} = x_{[i,j]} = w$. For all $i \leq j$, $h_{\theta_{[i,j]}}$ is surjective.
Assume for all $i \leq j$, $h_{\theta_{[i,j]}}$ is surjective. Let $x$ be a configuration and, for all integer $n \geq 0$, let $Y_n$ be the set $\{ y \in \mathcal{C}_A : H_{\theta}(y)_{[-n,n]} = x_{[-n,n]}\}$.
For all $n \geq 0$, $h_{\theta_{[-n,n]}}$ is surjective, then there exists $w \in A^{2(n+r) + 1}$ such that $h_{\theta_{[-n,n]}}(w) = x_{[-n,n]}$. Then all configuration $y$ such that $y_{[-n-r,n+r]} = w$ are in $Y_n$, and $Y_n$ is not empty.
Let $n \geq 0$ be an integer and $y \in Y_{n+1}$, then $H_{\theta}(y)_{[-n-1,n+1]} = x_{[-n-1,n+1]}$ and $y \in Y_n$.
For all $n$, $Y_n \neq \emptyset$ and $Y_n \subseteq Y_{n+1}$, by compacity there exists $y \in \cap_{n \geq 0} Y_n$. Such an $y$ verifies $H_{\theta}(y) = x$ and therefore $H_{\theta}$ is surjective.
Let $\mathcal{F}_{\mathcal{R}} = \{ \psi \in \mathcal{R}^* : h_{\psi} \text{ is not surjective} \}$ denotes the set of all finite distributions which define non-surjective partial transition functions. Then a distribution $\theta \in \Theta_{\mathcal{R}}$ defines a surjective r$\nu$-CA if and only if $\theta$ avoids the patterns of $\mathcal{F}_{\mathcal{R}}$. $Surj(\mathcal{R})$ is the subshift of $\theta_{\mathcal{R}}$ that avoids $\mathcal{F}_{\mathcal{R}}$.
Recall that the decidability of the surjectivity for classical cellular automata has been studied thanks to DeBruijn graphs [@Durand1998]. We will present here a variant of those graphs for distributions on a finite set of rules. The study of this graph will allow us to show that $\mathcal{F}_{\mathcal{R}}$ is recognizable.
The DeBruijn graph associated to the set of rules $\mathcal{R}$ (we assume the radius of the rules is at least 1) is the graph $\mathcal{G}_{\mathcal{R}}$ which contains $|A|^{2r}$ nodes, each of them is labeled by a different word of $A^{2r}$. For every states $a$ and $b$ of $A$, for every word $w$ of $A^{2r-1}$, for every rule $f$ of $\mathcal{R}$, there exists an edge from the node $aw$ to the node $wb$ (nodes are assimilated to their label). This edge is labeled by $(f,f(awb))$.
Let $A = \{0,1\}$ and $\mathcal{R} = \{\oplus, id\}$ where $$\begin{array}{rccl}
\oplus : & A^3 & \rightarrow & A \\
& (x,y,z) & \rightarrow & x + z \bmod 2\\
\\
id : & A^3 & \rightarrow & A \\
& (x,y,z) & \rightarrow & y\\
\end{array}
\enspace.$$ The DeBruijn graph $\mathcal{G}_{\mathcal{R}}$ associated to $\mathcal{R}$ is the graph
where multiple edges from same origin to same target have been collapsed and labels concatenated.
Consider this graph as a finite automaton where all states are both initial and final and let $\mathcal{L}$ be the language recognized by this automaton.
\[language\] $\mathcal{L} = \{(\psi,u) \in (\mathcal{R} \times A)^* : h_\psi^{-1}(u) \neq \emptyset \}$.
Let $(\psi,u) \in (\mathcal{R} \times A)^*$ be such that $h_\psi^{-1}(u) \neq \emptyset$ and $n = |\psi| = |u|$. There exists $w \in A^{n+2r}$ such that $h_\psi(w) = u$. Indeed, $$w_{[0,2r-1]} \xrightarrow{(\psi_0,u_0)} w_{[1,2r]} \xrightarrow{(\psi_1,u_1)} w_{[2,2r+1]} \xrightarrow{(\psi_2,u_2)} \dots \xrightarrow{(\psi_{n-1},u_{n-1})} w_{[n,n+2r-1]}$$ is a path of $\mathcal{G}_{\mathcal{R}}$ and then $(\psi,u) \in \mathcal{L}$.
Let $(\psi,u)$ be in $\mathcal{L}$, $(\psi,u)$ is the label of a path in $\mathcal{G}_{\mathcal{R}}$ of the form $$a_0 w_0 \xrightarrow{(\psi_0,u_0)} w_0b_1 = a_1 w_1 \xrightarrow{(\psi_1,u_1)} w_1b_2 = a_2 w_2 \xrightarrow{(\psi_2,u_2)} \dots \xrightarrow{(\psi_{n-1},u_{n-1})} w_{n-1}b_n = a_n w_n$$ where for all $i$, $a_i$ and $b_i$ in $A$ and $w_i$ in $A^{2r-1}$. Then by definition of $\mathcal{G}_{\mathcal{R}}$, $h_{\psi}(a_0\dots a_n w_n) = u$ and $h_{\psi}^{-1}(u) \neq \emptyset$.
$Surj(\mathcal{R})$ is a sofic subshift.
We have a finite automaton that recognizes $\mathcal{L} = \{(\psi,u) \in (\mathcal{R} \times A)^* : h_\psi^{-1}(u) \neq \emptyset \}$ (Lemma \[language\]). Then we can construct an automaton $\mathcal{A}$ which recognizes $\mathcal{L}^c = \{(\psi,u) \in (\mathcal{R} \times A)^* : h_\psi^{-1}(u) = \emptyset \}$. We construct a new automaton $\tilde{\mathcal{A}}$ from $\mathcal{A}$ by deleting all the second components of edge labels. A word $\psi \in \mathcal{R}^*$ is recognized by $\tilde{\mathcal{A}}$ if and only if there exists $u \in A^*$ such that $(\psi,u) \in \mathcal{L}^c$, i.e. $h_\psi^{-1}(u) = \emptyset$. Finally, a word $\psi \in \mathcal{R}^*$ is recognized by $\tilde{\mathcal{A}}$ if and only if $h_{\psi}$ is not surjective and the language recognized by $\tilde{\mathcal{A}}$ is $\mathcal{F}_{\mathcal{R}}$. $Surj(\mathcal{R})$ is a sofic subshift.
Injectivity
-----------
We have seen that paths in the DeBruijn graph $\mathcal{G}_{\mathcal{R}}$ associated to a set of rules $\mathcal{R}$ allow to define a word $w$ (from the sequence of visited nodes) and a finite distribution $\psi$ and another word $u$ (from the sequence of visited edges) such that $h_{\psi}(w) = u$. If we consider now bi-infinite paths, we defined in the same way two configurations $x$ and $y$ and a distribution $\theta$ such that $H_{\theta}(x) = y$. The product graph of $\mathcal{R}$ is an extension of its DeBruijn graph which allows to describe simultaneously two configurations which have the same image.
The product graph $\mathcal{P}_{\mathcal{R}}$ is the graph which contains $|A|^{4r}$ nodes, each of them is labeled by a different couple of word of $A^{2r}$. There is an edge from $(u,u')$ to $(v,v')$ labeled by $(f,a) \in \mathcal{R} \times A$ if and only if there is an edge from $u$ to $v$ (resp. from $u'$ to $v'$) labeled by $(f,a)$.
Then a bi-infinite path in $\mathcal{P}_{\mathcal{R}}$ corresponds to two bi-infinite paths in $\mathcal{G}_{\mathcal{R}}$ and the sequence of edge labels used by these two paths is the same. More formally, if $$\dots \xrightarrow{(\theta_{i-1},z_{i-1})} (u_{i},u'_{i}) \xrightarrow{(\theta_i,z_i)} (u_{i+1},u'_{i+1}) \xrightarrow{(\theta_{i+1},z_{i+1})} (u_{i+2},u'_{i+2}) \xrightarrow{(\theta_{i+2},z_{i+2})} \dots$$ is a path in $\mathcal{P}_{\mathcal{R}}$, define two configurations $x$ and $y$ such as for all integer $i$, $x_i$ is the $(r + 1)^{th}$ letter of $u_i$ and $y_i$ is the $(r + 1)^{th}$ letter of $u'_i$. Then $H_{\theta}(x) = H_{\theta}(y) = z$.
On the other hand, if $\theta$ is a distribution and $x$ and $y$ are two configurations such that $H_{\theta}(x) = H_{\theta}(y)$ = z, then $$\dots \xrightarrow{(\theta_{i-1},z_{i-1})} (x_{[i-r,i+r-1]},y_{[i-r,i+r-1]}) \xrightarrow{(\theta_i,z_i)} (x_{[i-r+1,i+r]},y_{[i-r+1,i+r]}) \xrightarrow{(\theta_{i+1},z_{i+1})} \dots$$ is a path in $\mathcal{P}_{\mathcal{R}}$.
As for the surjectivity, denote $\tilde{\mathcal{P}}_{\mathcal{R}}$ the finite automaton obtained from $\mathcal{P}_{\mathcal{R}}$ by deleting all the second components of edge labels. A bi-infinite path defines two configurations $x$ and $y$ (from the sequence of visited nodes) and a configuration $\theta$ (from the sequence of visited edges) such that $H_{\theta}(x) = H_{\theta}(y)$. The converse is also trivially true.
Then $\theta$ leads to a non-injective r$\nu$-CA $H_ {\theta}$ if and only there exists two configurations $x \neq y$ such that $H_{\theta}(x) = H_{\theta}(y)$ if and only if there exists a bi-infinite path in $\tilde{\mathcal{P}}_{\mathcal{R}}$ labeled by $\theta$ which visits a node $(u,u')$ where $u \neq u'$.
Recall that a $\zeta$-rational language is a set of bi-infinite words recognized by a finite automaton, that is to say the set of all labels of successful bi-infinite paths in the automaton. A path in such automaton is successful if and only it crosses infinitely many times initial states for negative indexes and infinitely many times final states for positive indexes (Büchi acceptance condition).
$Inj(\mathcal{R}) = \{ \theta \in \Theta_{\mathcal{R}} : H_{\theta} \text{ is injective}\}$, i.e. the set of all distributions that induce injective r$\nu$-CA, is a $\zeta$-rational language.
Consider now $\tilde{\mathcal{P}}_{\mathcal{R}}$ as a finite automaton where all the states are initial and the final states are the states of the form $(u,u')$ with $u \neq u'$. A bi-infinite path is successful in this graph if and only it crosses infinitely many times initial states for negative indexes (automatically because all states are initial) and at least one final state. Then the language recognized (under this acceptance condition) is exactly $\Theta_{\mathcal{R}} \smallsetminus Inj(\mathcal{R})$.
But if a language is recognized by an automaton with this new acceptance condition, it is recognized by an automaton with the Büchi acceptance condition and then is $\zeta$-rational. It is known that $\zeta$-rational are closed under complement, since $\Theta_{\mathcal{R}} \smallsetminus Inj(\mathcal{R})$ is $\zeta$-rational then $Inj(\mathcal{R})$ is $\zeta$-rational.
Number conserving
=================
In physics, a lot of transformations are conservative : a certain quantity remains invariant during a whole experiment. Think to conservation laws of mass and energy for example. As $CA$ and $\nu$-CA are used to represent phenomena from physics, such a property of conservation has been introduced. We will generalize some existing results of the uniform case.
In this section, the alphabet $A$ we consider is a “numerical” alphabet of the form $\{0, 1, \dots, s-1\}$ where $s$ is the cardinal of $A$. A configuration $x \in \mathcal{C}_A$ is said to be *finite* if and only if the support of $x$, i.e. the set $\{i \in \mathbb{Z}, x_i \neq 0\}$, is finite. Let $\mathcal{C}_A^F$ denotes the set of all finite configurations on $A$ and $\underline{0}$ be the configuration with empty support, i.e. such that for all integer $i$, $\underline{0}_i = 0$.
For every configuration $x \in \mathcal{C}_A$, define the *partial charge* of $x$ between the index $-n$ and $n$ by $$\mu_n(x) = \sum_{i = -n}^n x_i$$ and the *global charge* of $x$ by $$\mu(x) = \lim_{n \rightarrow \infty} \mu_n \enspace.$$ Then, if $x$ is not a finite configuration, $\mu(x) = \infty$.
A $\nu$-CA $H$ is *number-conserving on finite configuration* (FNC) if and only if for all $x \in \mathcal{C}_A^F$, $\mu(x) = \mu(H(x))$.
A $\nu$-CA $H$ is FNC if and only if it preserves the charge of finite configurations. Consequently, for all finite configuration $x$, $H(x)$ is a finite configuration and $H(\underline{0}) = \underline{0}$.
In the case of non-finite configurations, the conservation of the charge is expressed as conservation of the average charge. The average charge of a configuration $x$ on a window of size $n$ is the quantity $\frac{\mu_n(x)}{2n+1}$. Then a $\nu$-CA $H$ will be number-conserving if the average charge of a configuration and the average charge of its image are asymptotically the same $$\forall x \in \mathcal{C}_A, \frac{\mu_n(H(x))}{2n+1} \sim \frac{\mu_n(x)}{2n+1} \enspace.$$
Then if $\mu_n(x) \neq 0$ the quantity $\frac{\mu_n(H(x))}{\mu_n(x)}$ represents the relative gain/loss of charge on a window of size $2n+1$.
Let $H$ be a $\nu$-CA and $x$ be a configuration. If $x \neq \underline{0}$ then there exists $n_0$ such that for all $n$ greater than or equals to $n_0$, $\mu_n(x) \neq 0$. Then the sequence $\frac{\mu_n(H(x))}{\mu_n(x)}$ is defined for $n$ greater than or equals to $n_0$. Let $m(x) = \liminf_{n \rightarrow \infty} \frac{\mu_n(H(x))}{\mu_n(x)}$ and $M(x) = \limsup_{n \rightarrow \infty} \frac{\mu_n(H(x))}{\mu_n(x)}$.
$H$ is said to be number-conserving (NC) if and only if
1. $H(\underline{0}) = \underline{0}$
2. $\forall x \in \mathcal{C}_A \smallsetminus \{\underline{0}\}, m(x) = M(x) = 1$ (the sequence $\frac{\mu_n(H(x))}{\mu_n(x)}$ converges on 1).
This definition of number-conserving $\nu$-CA does not depend from the origin chosen for the lattice. In fact, the sequences $\frac{\mu_n(H(x))}{\mu_n(x)}$ and $\frac{\mu_n(H(\sigma(x)))}{\mu_n(\sigma(x))}$ have the same adherence values for all configuration $x$. Then $$\forall x \in \mathcal{C}_A, m(x) = m(\sigma(x)) \text{ and } M(x) = M(\sigma(x)) \enspace.$$
Let $H$ be a r$\nu$-CA of radius $r$, $H$ is NC if and only if $H$ is $NFC$.
Assume that $H$ is NC. Since $H(\underline{0}) = \underline{0}$, all images of finite configurations are finite configurations. Let $x$ be in $\mathcal{C}_A^F \smallsetminus \{\underline{0}\}$, the sequences $\mu_n(x)$ and $\mu_n(H(x))$ are stationary and converge respectively on $\mu(x)$ and $\mu(H(x))$. Then $\lim_{n \rightarrow \infty} \frac{\mu_n(H(x))}{\mu_n(x)} = \frac{\mu(H(x))}{\mu(x)} = 1$. $\mu(x) = \mu(H(x))$ and $H$ is FNC.
Assume that $H$ is not NC. If $H(\underline{0}) \neq \underline{0}$ then $H$ is not FNC, else there exists a configuration $x$ such that $m(x) \neq 1$ or $M(x) \neq 1$. If $x$ is a finite configuration then $H$ is not FNC.
We assume now that $x$ is not a finite configuration ($\mu(x) = \infty$). We will now be interested in the case $M(x) > 1$ (the other case $m(x) < 1$ has a similar proof). By definition of upper limit, for all $\epsilon > 0$, there exists an infinite number of indexes $n$ such that
$$\frac{\mu_n(H(x))}{\mu_n(x)} \geq M(X) - \epsilon \enspace.$$
Let $\epsilon > 0$ and $k > 0$, there exists infinitely many indexes $n$ such that $$\frac{\mu_n(H(x))}{\mu_n(x)} \geq M(X) - \frac{\epsilon}{2}$$ which is equivalent to say that there exists infinitely many indexes $n$ such that $$\label{eq_k}
\mu_n(H(x)) \geq (M(X) - \epsilon) \mu_n(x) + \frac{\epsilon}{2} \mu_n(x)\enspace.$$
$\lim_{n \rightarrow \infty} \mu_n(x) = \infty$ then there exists $n_0$ such that for all integer $n > n_0$, $\mu_n(x) \geq \frac{2k}{\epsilon}$. But as there are infinitely many indexes that verify inequality (\[eq\_k\]), one of them is greater than $n_0$ and for this index $n$ we have $$\mu_n(H(x)) \geq (M(X) - \epsilon) \mu_n(x) + k \enspace.$$
We have proved that $$\forall \epsilon > 0, \forall k > 0, \exists n, \mu_n(H(x)) \geq (M(X) - \epsilon) \mu_n(x) + k \enspace.$$
Choose $\epsilon$ such that $M(x) - \epsilon > 1$ and $k = 2r(s-1)$, then there exists an integer $n$ such that $$\mu_n(H(x)) \geq (M(X) - \epsilon) \mu_n(x) + 2r(s-1) \enspace.$$
Let $y$ be the finite configuration such that for all integer $i$, $y_i = x_i$ if $|i| \leq n$, 0 otherwise.
$$\begin{array}{rcl}
\mu(H(y)) & = & \sum_{i \in \mathbb{Z}} H(y)_i\\
& = & \sum_{i = -n-r}^{n+r} H(y)_i \\
& \geq & \sum_{i = -n+r}^{n-r} H(x)_i \\
& \geq & \sum_{i = -n}^n H(x)_i - 2r(s-1) \\
& \geq & (M(x)-\epsilon)\sum_{i = -n}^n x_i \\
& \geq & (M(x)-\epsilon)\sum_{i \in \mathbb{Z}} y_i\\
& > & \mu(y) \enspace.
\end{array}$$ Hence $H$ is not FNC.
#### Application to r$\nu$-CA defined on a finite set of rules.
Let $\mathcal{R}$ be as usual a finite set of local rules of radius $r> 0$. Let $NC(\mathcal{R}) = \{ \theta \in \Theta_{\mathcal{R}} : H_{\theta} \text{ is NC}\}$ be the set of all distributions that induce number-conserving r$\nu$-CA. We will prove that $NC(\mathcal{R})$ is a subshift of finite type.
Let $\theta \in \Theta_{\mathcal{R}}$. Then $\theta \in NC(\mathcal{R})$ if and only if $\forall j \in \mathbb{Z}, \theta_{[j-2r,j]} \notin \mathcal{F}_{\mathcal{R}}$ where $$\mathcal{F}_{\mathcal{R}} = \{\psi \in \mathcal{R}^{2r+1} : \exists u \in A^{2r+1}, \psi_{2r}(u) \neq u_0 + \sum_{i=0}^{2r-1} \psi_{i+1}(0^{2r-i}u_{[1,i+1]}) - \psi_{i}(0^{2r-i}u_{[0,i]})\} \enspace.$$
Assume that $\theta$ is in $NC(\mathcal{R})$, let $j \in \mathbb{Z}$ and $u \in A^{2r+1}$. $H_{\theta}$ is NC then $H_{\theta}(\underline{0}) = \underline{0}$ and for all integer $i$, $\theta_i(0^{2r+1}) = 0$.
Let $x$ be the finite configuration such that $x_{[j-r,j+r]} = u$ and $x_i = 0$ elsewhere, let $y$ be the finite configuration such that $y_{[j-r,j+r]} = 0u_{[1,2r]}$ and $y_i = 0$ elsewhere.
$H_{\theta}$ is NC (and FNC) then $\mu(H(x)) = \mu(x)$, i.e.
$$\label{eq_u}
\sum_{i=0}^{2r} \theta_{j+i-2r}(0^{2r-i}u_{[0,i]}) + \sum_{i=1}^{2r} \theta_{j+i}(u_{[i,2r]}0^i) = \sum_{i = 0}^{2r} u_i \enspace.$$
As same $\mu(H(y)) = \mu(y)$, i.e.
$$\label{eq_0u}
\sum_{i=1}^{2r} \theta_{j+i-2r}(0^{2r-i+1}u_{[1,i]}) + \sum_{i=1}^{2r} \theta_{j+i}(u_{[i,2r]}0^i) = \sum_{i = 1}^{2r} u_i \enspace.$$
Subtracting (\[eq\_0u\]) to (\[eq\_u\]), we obtain
$$\theta_j(u) = u_0 + \sum_{i=1}^{2r} \theta_{j+i-2r}(0^{2r-i+1}u_{[1,i]}) - \sum_{i=0}^{2r-1} \theta_{j+i-2r}(0^{2r-i}u_{[0,i]})$$
which can be rewritten
$$\theta_j(u) = u_0 + \sum_{i=0}^{2r-1} \theta_{j+i+1-2r}(0^{2r-i}u_{[1,i+1]}) - \theta_{j+i-2r}(0^{2r-i}u_{[0,i]}) \enspace.$$
That is true for all word $u$ then $\theta_{[j-2r,j]} \notin \mathcal{F}_{\mathcal{R}}$.
Assume that for all integer $j$, $\theta_{[j-2r,j]} \notin \mathcal{F}_{\mathcal{R}}$. Let $j$ be an integer, $\theta_{[j,j+2r]} \notin \mathcal{F}_{\mathcal{R}}$ then, taking $u = 0^{2r+1}$, we have $$\theta_{j + 2r}(0^{2r+1}) = 0 + \sum_{i=0}^{2r-1} \theta_{j + i +1}(0^{2r+1}) - \theta_{j + i}(0^{2r+1})$$ which leads to $\theta_j(0^{2r+1}) = 0$. This will justify that all following sums have in fact a finite support and are well-defined.
Let $x$ be a finite configuration, $$\sum_{j \in \mathbb{Z}} H_{\theta}(x)_j = \sum_{j \in \mathbb{Z}} \theta_j(x_{[j-r,j+r]})$$ $$= \sum_{j \in \mathbb{Z}} \left( x_j + \sum_{i=0}^{2r-1} \theta_{j+i+1-2r}(0^{2r-i}x_{[j-r+1,j-r+i+1]}) - \theta_{j+i-2r}(0^{2r-i}x_{[j-r,j-r+i]}) \right)$$ $$= \sum_{j \in \mathbb{Z}} x_j + \sum_{i=0}^{2r-1} \left( \sum_{j \in \mathbb{Z}} \theta_{j+i+1-2r}(0^{2r-i}x_{[j-r+1,j-r+i+1]}) - \sum_{j \in \mathbb{Z}} \theta_{j+i-2r}(0^{2r-i}x_{[j-r,j-r+i]}) \right)$$ but $$\sum_{j \in \mathbb{Z}} \theta_{j+i+1-2r}(0^{2r-i}x_{[j-r+1,j-r+i+1]}) = \sum_{j \in \mathbb{Z}} \theta_{j+i-2r}(0^{2r-i}x_{[j-r,j-r+i]})$$ then $$\mu(H_{\theta}(x)) = \sum_{j \in \mathbb{Z}} H_{\theta}(x)_j = \sum_{j \in \mathbb{Z}} x_j = \mu(x)$$ and $H_{\theta}$ is FNC then NC because it is a r$\nu$-CA.
$NC(\mathcal{R})$ is a subshift of finite type.
$NC(\mathcal{R})$ is the set of distributions which avoid the pattern of $\mathcal{F}_{\mathcal{R}}$ which is finite as a subset of $\mathcal{R}^{2r+1}$.
Equicontinuity and sensitivity for linear r$\nu$-CA {#eq_sens_add}
===================================================
In this part, we will have a look on dynamical properties on linear r$\nu$-CA. All along this section, $(A,+,.)$ denotes a finite commutative ring, $0$ and $1$ denote the neutral elements of $(A,+)$ and $(A,.)$, respectively.
Then for all integer $n$, $(A^n, +, .)$ defines an A-algebra by
1. $\forall u,v \in A^n, u + v = (u_0 + v_0, \dots, u_{n-1} + v_{n-1})$
2. $\forall u,v \in A^n, uv = (u_0v_0, \dots, u_{n-1}v_{n-1})$
3. $\forall \lambda \in A, \forall u \in A^n, \lambda u = (\lambda u_0, \dots, \lambda u_{n-1})$
Similarly $(\mathcal{C}_A, +, .)$ defines an A-algebra by
1. $\forall x,y \in \mathcal{C}_A, \forall i \in \mathbb{Z}, (x + y)_i = x_i + y_i$
2. $\forall x,y \in \mathcal{C}_A, \forall i \in \mathbb{Z}, (x y)_i = x_i y_i$
3. $\forall \lambda \in A, \forall x \in \mathcal{C}_A, \forall i \in \mathbb{Z}, (\lambda x)_i = \lambda x_i$
A $\nu$-CA $H : \mathcal{C}_A \rightarrow \mathcal{C}_A$ is said to be *linear* if and only if for all configurations $x$ and $y$, $H(x+y) = H(x) + H(y)$. Similarly a local rule $f$ of radius $r$ is said to be linear if and only if for all words $u$ and $v$ in $A^{2r+1}$, $f(u+v) = f(u) + f(v)$.
A local rule $f$ of radius $r$ is linear if and only if there exists a word $\lambda$ in $A^{2r+1}$ such that $$\forall u \in A^{2r+1}, f(u) = \lambda \bullet u := \sum_{i=0}^{2r} \lambda_i u_i$$
A $\nu$-CA $H : \mathcal{C}_A \rightarrow \mathcal{C}_A$ is linear if and only if $$\forall i \in \mathbb{Z}, \exists r \in \mathbb{N}, \exists f \in \mathcal{R}_{A,r}, \forall x \in \mathcal{C}_A, H(x)_i = f(x_{[i-r,i+r]})$$ and $f$ is linear for all integer $i$.
We will be interested in equicontinuity and sensitivity for linear $\nu$-CA. In the general case, a $\nu$-CA is not sensitive if and only if it admits an equicontinuous point. In the case of additive $\nu$-CA a stronger property holds.
Let $H$ be a linear $\nu$-CA then $H$ is either sensitive or equicontinuous.
$H$ is linear, then for all integer $n$, $H^n$ is linear. For all integers $n$ and $i$, there exists an integer $r \geq 0$ and $\lambda \in A^{2r+1}$ such that for all configuration $x$, $H^{n}(x)_i = \lambda \bullet x_{[i-r,i+r]}$. Let $r_i^n$ denotes $\max (\{i > 0 : \lambda_{r - i} \neq 0 \text{ or } \lambda_{r + i} \neq 0\} \cup \{0\})$. It is easy to see that $r_i^n$ is well-defined whatever the choice of $r$ and $\lambda$ is done.
Assume there exists $i$ such that the sequence $(r_i^n)_{n \in \mathbb{N}}$ is not bounded. Let $x$ be a configuration, let $\delta = 2^{-m} > 0$, there exists an integer $n$ such that $r_i^n > 2|i| + 1 + m$, let $y$ be the configuration such that for all integer $j$, $y_j = 0$ if $|j| \neq r_i^n$ ; 1 otherwise. Then $d(x,x+y) = d(0,y) < \delta$ and $d(H^n(x), H^n(x+y)) = d(0, H^n(y)) > 2^{-i}$. Then $H$ is sensitive with sensitivity constant $2^{-i}$.
Assume at opposite that for all integer $i$ the sequence $(r_i^n)_{n \in \mathbb{N}}$ is bounded by the integer $M_i > 0$. Let $x$ be a configuration, let $\epsilon = 2^{-m} > 0$, let $\delta = 2^{-(m + M)}$ where $M = \max \{M_i : -m \leq i \leq m\}$, let $y$ be a configuration such that $d(x,y) < \delta$, then for all integer $n$, $d(H^n(x) = H^n(y)) < \epsilon$ because $x_{[-m-M, m + M]} = y_{[-m-M, m + M]} \rightarrow H^n(x)_{[-m,m]} = H^n(y)_{[-m,m]}$. Then $H$ is equicontinuous.
Then $H$ is either sensitive or equicontinuous.
From now on $\mathcal{R}$ is a finite set of linear local rules of radius $r$.
Let $\psi \in \mathcal{R}^*$ of size $n \geq r$, then $\psi$ is a right-wall if and only if all the sequences $(u_k)_{k \in \mathbb{N}}$ defined by $$\begin{array}{rcl}
u_0 & = & 0^n \\
u_1 & = & h_{\psi}(0^ru_0v) \text{ where $v \in A^r$} \\
u_{k+1} & = & h_{\psi}(0^ru_k0^r) \text{ for $k > 1$}
\end{array}$$ verify ${u_k}_{[0,r-1]} = 0^r$. Left-wall are defined similarly.
Let $\theta \in \Theta_{\mathcal{R}}$. Then, $H_{\theta}$ is sensitive if and only if there exist two integers $k^-$ and $k^+$ such that
1. for all integer $i < k^-$, for all integer $n \geq r - 1$, $\theta_{[i - n,i]}$ is not a left-wall
2. for all integer $i > k^+$, for all integer $n \geq r - 1$, $\theta_{[i,i+n]}$ is not a right-wall
If all the rules of $\mathcal{R}$ have radius 1 then the language $\{ \theta \in \Theta_{\mathcal{R}} : H_\theta \text{ is sensitive}\}$ is a $\zeta$-rational language.
|
---
abstract: 'A general and systematic regularization is developed for the exact solitonic form factors of exponential operators in the (1+1)-dimensional sine-Gordon model by analytical continuation of their integral representations. The procedure is implemented in Mathematica. Test results are shown for four- and six-soliton form factors.'
author:
- |
T. Pálmai[^1]\
\
[*Department of Theoretical Physics,*]{}\
[*Budapest University of Technology and Economics,*]{}\
[*H-1111 Budafoki út 8, Hungary*]{}
bibliography:
- 'sgff.bib'
title: 'Regularization of multi-soliton form factors in sine-Gordon model'
---
Introduction
============
Form factors (matrix elements of local operators) are important quantities in quantum field theories. It is a remarkable feature of certain two-dimensional field theories (integrable models), that their S-matrices can be obtained exactly in the framework of factorized scattering theory [@Zamolodchikov:1978; @Mussardo:1992]. Furthermore, in integrable models there is a rather restrictive set of equations satisfied by the form factors (that is the form factor axioms [@Smirnov:1992; @Mussardo:1992]), which makes it possible in many cases to obtain them exactly as well. For instance, in the sine-Gordon model all form factors of exponential operators are known [@Smirnov:1992; @Lukyanov:1997; @Babujian:1998]. The spectrum of the sine-Gordon model consists of a soliton-antisoliton doublet and their bound states, called “breathers”. While the breather-breather form factors can be given explicitly (see e.g. [@Lukyanov:1997]), the solitonic ones, in general, are only known in terms of some highly non-trivial integral representations. In addition, the integrals converge in a limited domain of the parameters. In this paper we give a regularization procedure to calculate the solitonic form factors in the sine-Gordon model for arbitrary choice of the parameters. The regularized multi-soliton form factors could them be used to obtain correlation functions of direct physical interest, e.g. in condensed matter physics [@Essler:2004].
The outline of the paper is as follows. In Section 2 the sine-Gordon model along with its exact form factors are reviewed and based on [@Lukyanov:1997] integral representations for the form factors of exponential operators are given. Section 3 is devoted to the analysis of a certain function which appears in the integral representations. Giving this function’s asymptotic series and identifying its poles make it possible to analytically continue the integral representations. In Section 4 explicit formulae are provided for the four-soliton form factors. Section 5 is devoted to discussion of test results, while Section 6 is left for conclusions and outlook.
Form factors in the sine-Gordon model
=====================================
Definitions, S-matrix and the form factor axioms
------------------------------------------------
The sine-Gordon model is defined by the classical Lagrangian $$\label{Lagr}
L=\int_{-\infty}^{\infty}dx\left[\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi+\frac{m_{0}^{2}}{\beta^{2}}\cos\left(\beta\varphi\right)\right].$$ Define the parameter $$\label{xi}
\xi=\frac{\beta^{2}}{1-\beta^{2}}$$ which is relevant in the low-energy description of the theory. The spectrum of the quantum theory contains the soliton-antisoliton doublet and their bound states, the “breathers”. The number of breather states ($B_1$, $B_2$, ..., $B_{\bar{N}}$) is bounded, there are $\bar{N}=\left\lfloor \frac{1}{\xi}\right\rfloor $ of them. For our purposes it is enough to consider only the solitonic particles of the spectrum, indexed in the following with $\pm$ (soliton-antisoliton).
### S-matrix
The S-matrix for the soliton-antisoliton doublet reads $$\left(\begin{array}{cccc}
S\\
& S_{T} & S_{R}\\
& S_{R} & S_{T}\\
& & & S
\end{array}\right),$$ with the non-zero elements [@Zamolodchikov:1978] $$\begin{aligned}
S_{++}^{++}\left(\Theta\right) & = & S_{--}^{--}\left(\Theta\right)=S\left(\Theta\right),\\
S_{+-}^{+-}\left(\Theta\right) & = & S_{-+}^{-+}\left(\Theta\right)=S_{T}\left(\Theta\right)=S\left(\Theta\right)\frac{\sinh\frac{\Theta}{\xi}}{\sinh\frac{i\pi-\Theta}{\xi}},\\
S_{+-}^{-+}\left(\Theta\right) & = & S_{-+}^{+-}\left(\Theta\right)=S_{R}\left(\Theta\right)=S\left(\Theta\right)\frac{\sinh\frac{i\pi}{\xi}}{\sinh\frac{i\pi-\Theta}{\xi}},\end{aligned}$$ where $$\begin{gathered}
S(\Theta)=-(-1)^N\prod_{k=1}^N\frac{ik\pi\xi+\Theta}{ik\pi\xi-\Theta}\\
\times
\exp\left[-i\int_0^\infty\frac{dt}{t}\sin(\Theta t)\frac{2\sinh\frac{\pi(1-\xi)t}{2}e^{-N\pi\xi t}+(e^{-N\pi\xi t}-1)\left(e^{\frac{\pi(\xi-1)t}{2}}+e^{-\frac{\pi(\xi+1)t}{2}}\right)}{2\sinh\frac{\pi\xi t}{2}\cosh\frac{\pi t}{2}}\right],\end{gathered}$$ which is independent of the integer $N$, however the integral converges in a larger domain of $\mathbb{C}$ for $N>0$.
### Form factors
Consider the matrix elements $$\mathcal{F}_{a_{1}\ldots a_{n}}^{b_{1}\ldots b_{m}}\left(\Theta_{1}',\ldots\Theta_{m}'|\Theta_{1},\ldots\Theta_{n}\right)=\,_{out}\left\langle A^{b_{m}}\left(\Theta_{m}'\right)\ldots A^{b_{1}}\left(\Theta_{1}'\right)|O|A_{a_{1}}\left(\Theta_{1}\right)\ldots A_{a_{n}}\left(\Theta_{n}\right)\right\rangle _{in}$$ of the local, hermitian operator $O$ between asymptotic states. The form factors are defined by $$\mathcal{F}_{a_{1}\ldots a_{n}}\left(\Theta_{1},\ldots\Theta_{n}\right)=\langle0|O|A_{a_{1}}\left(\Theta_{1}\right)\ldots A_{a_{n}}\left(\Theta_{n}\right)\rangle_{in},$$ as the matrix elements of the operator between the vacuum and an $n$-particle state. Crossing symmetry implies $$\mathcal{F}_{a_{1}\ldots a_{n}}^{b_{1}\ldots b_{m}}\left(\Theta_{1}',\ldots\Theta_{m}'|\Theta_{1},\ldots\Theta_{n}\right)=\mathcal{F}_{a_{1}\ldots a_{n}(-b_{1})\ldots(-b_{m})}\left(\Theta_{1},\ldots\Theta_{n},\Theta_{1}'+i\pi,\ldots\Theta_{m}'+i\pi\right),$$ which is understood as an analytic continuation. The form factors can be reconstructed based on the following axioms.
1. Analyticity and the physical matrix elements. $\mathcal{F}_{a_{1}\ldots a_{n}}\left(\Theta_{1},\ldots\Theta_{n}\right)$ is analytic in the variables $\Theta_{i}-\Theta_{j}$ inside the physical strip $0<{\rm Im}\,\Theta<2\pi$ except for simple poles. It is the physical matrix element when all $\Theta_{i}$ are real and ordered as $\Theta_{1}<\Theta_{2}<\ldots<\Theta_{n}$.
2. Relativistic invariance. The form factors satisfy $$\mathcal{F}_{a_{1}\ldots a_{n}}\left(\Theta_{1}+z,\ldots\Theta_{n}+z\right)=e^{zS\left(O\right)}\mathcal{F}_{a_{1}\ldots a_{n}}\left(\Theta_{1},\ldots\Theta_{n}\right),$$ where $S\left(O\right)$ is the spin of the operator $O$.
3. Watson’s theorem. The following symmetry properties are satisfied $$\mathcal{F}_{a_{1}\ldots a_{j}a_{j+1}\ldots a_{n}}\left(\Theta_{1},\ldots\Theta_{j},\Theta_{j+1},\ldots\Theta_{n}\right)=S_{a_{j+1}a_{j}}^{c_{j}c_{j+1}}\left(\Theta_{j+1}-\Theta_{j}\right)\mathcal{F}_{a_{1}\ldots c_{j}c_{j+1}\ldots a_{n}}\left(\Theta_{1},\ldots\Theta_{j+1},\Theta_{j},\ldots\Theta_{n}\right),$$ $$\mathcal{F}_{a_{1}\ldots a_{n}}\left(\Theta_{1},\ldots\Theta_{n}+2\pi i\right)=e^{2\pi i\omega\left(O,\Psi\right)}\mathcal{F}_{a_{n}a_{1}\ldots a_{n-1}}\left(\Theta_{n},\Theta_{1},\ldots\Theta_{n-1}\right),$$ where the latter is understood as an analytic continuation and $\omega\left(O,\Psi\right)$ is the mutual non-locality index of the operator $O$ and $\Psi$, the “elementary” field, upon which the whole operator product algebra can be constructed.
4. Kinematical poles. $\mathcal{F}_{a_{1}\ldots a_{n}}\left(\Theta_{1},\ldots\Theta_{n}\right)$ has simple poles at $\Theta_{n}=\Theta_{j}+i\pi$ with residues $$\begin{aligned}
{1}
-i\mathcal{F}_{a'_{1}\ldots\hat{a}'_{j}\ldots a'_{n}}\left(\Theta_{1},\ldots\hat{\Theta}_{j},\ldots\Theta_{n-1}\right)&\left\{ \delta_{a_{1}}^{a'_{1}}\ldots\delta_{a_{j-1}}^{a'_{j-1}}S_{a_{n-1}c_{1}}^{a'_{n-1}\left(-a_{n}\right)}\left(\Theta_{n-1}-\Theta_{j}\right)S_{a_{n-2}c_{2}}^{a'_{n-2}c_{1}}\left(\Theta_{n-2}-\Theta_{j}\right)\ldots\right.\nonumber \\
& \hspace{17em}\times S_{a_{j+1}a_{j}}^{a'_{j+1}c_{n-j-2}}\left(\Theta_{j+1}-\Theta_{j}\right)\nonumber \\
& \qquad-e^{2\pi i\omega\left(O,\Psi\right)}S_{c_{1}a_{1}}^{\left(-a_{n}\right)a'_{1}}\left(\Theta_{j}-\Theta_{1}\right)\ldots S_{c_{j-2}a_{j-2}}^{c_{j-3}a'_{j-2}}\left(\Theta_{j}-\Theta_{j-2}\right)\nonumber \\
& \hspace{12em}\left.\times S_{a_{j}a_{j-1}}^{c_{j-2}a'_{j-1}}\left(\Theta_{j}-\Theta_{j-1}\right)\delta_{a_{j+1}}^{a'_{j+1}}\ldots\delta_{a_{n-1}}^{a'_{n-1}}\right\} .\end{aligned}$$ In the absence of bound states these are the only singularities of $\mathcal{F}_{a_{1}\ldots a_{n}}\left(\Theta_{1},\ldots\Theta_{n}\right)$ in the strip $0<{\rm Im}\,\Theta_j<2\pi$ for real $\{\Theta_i\}_{i\neq j}$.
Integral representations of multi-soliton form factors
------------------------------------------------------
In [@Lukyanov:1997] it is proposed that the $2n$-particle form factors of the exponential operator $e^{ia\varphi}$ in the sine-Gordon model can be represented by $$\begin{gathered}
\mathcal{F}_{\sigma_1\ldots\sigma_{2n}}(\Theta_1,\ldots,\Theta_{2n})=\langle0|e^{ia\varphi}|A_{\sigma_{2n}}\left(\Theta_{2n}\right)\ldots A_{\sigma_{1}}\left(\Theta_{1}\right)\rangle\\=\mathcal{G}_{a}\langle\langle Z_{\sigma_{2n}}\left(\Theta_{2n}\right)\ldots Z_{\sigma_{1}}\left(\Theta_{1}\right)\rangle\rangle\equiv\mathcal{G}_{a}F_{\sigma_1\ldots\sigma_{2n}}(\Theta_1,\ldots,\Theta_{2n})\label{eq:FF}\end{gathered}$$ where $\sum_{i=1}^{2n}\sigma_{i}=0$ because of charge conservation and $\mathcal{G}_{a}$ is the vacuum expectation value of the exponential operator [@Lukyanov:1997; @Lukyanov:1996]. The operators $Z_{\pm}\left(\Theta\right)$ are defined by $$Z_{+}\left(\Theta\right)=\sqrt{i\frac{\mathcal{C}_{2}}{4\mathcal{C}_{1}}}e^{\frac{a\Theta}{\beta}}e^{i\phi\left(\Theta\right)},$$ $$\begin{gathered}
Z_{-}\left(\Theta\right)=\sqrt{i\frac{\mathcal{C}_{2}}{4\mathcal{C}_{1}}}e^{-\frac{a\Theta}{\beta}} \left\{ e^{\frac{i\pi}{2\beta^{2}}}\int_{C_{+}}\frac{d\gamma}{2\pi}e^{\left(1-\frac{2a}{\beta}-\frac{1}{\beta^{2}}\right)\left(\gamma-\Theta\right)}e^{-i\bar{\phi}\left(\gamma\right)}e^{i\phi\left(\Theta\right)}\right. \\ \left.-e^{-\frac{i\pi}{2\beta^{2}}}\int_{C_{-}}\frac{d\gamma}{2\pi}e^{\left(1-\frac{2a}{\beta}-\frac{1}{\beta^{2}}\right)\left(\gamma-\Theta\right)}e^{i\phi\left(\Theta\right)}e^{-i\bar{\phi}\left(\gamma\right)}\right\} .\end{gathered}$$ Since $\phi(\Theta)$ and $\bar\phi(\gamma)$ are free fields the averaging $\langle\langle\ldots\rangle\rangle$ is performed by the multiplicative Wick’s theorem, using $$\begin{aligned}
\langle\langle e^{i\phi\left(\Theta_{2}\right)}e^{i\phi\left(\Theta_{1}\right)}\rangle\rangle&=G\left(\Theta_{1}-\Theta_{2}\right),
\\
\langle\langle e^{i\phi\left(\Theta_{2}\right)}e^{-i\bar{\phi}\left(\Theta_{1}\right)}\rangle\rangle&=W\left(\Theta_{1}-\Theta_{2}\right)=\frac{1}{G\left(\Theta_{1}-\Theta_{2}-\frac{i\pi}{2}\right)G\left(\Theta_{1}-\Theta_{2}+\frac{i\pi}{2}\right)},
\\
\langle\langle e^{-i\bar{\phi}\left(\Theta_{2}\right)}e^{-i\bar{\phi}\left(\Theta_{1}\right)}\rangle\rangle&=\bar{G}\left(\Theta_{1}-\Theta_{2}\right)=\frac{1}{W\left(\Theta_{1}-\Theta_{2}-\frac{i\pi}{2}\right)W\left(\Theta_{1}-\Theta_{2}+\frac{i\pi}{2}\right)}.\end{aligned}$$ The appearing functions and constants are as follows. $$\begin{aligned}
G(\Theta)&=i\mathcal{C}_1\sinh\left(\frac{\Theta}{2}\right)\exp\left\{\int_0^\infty \frac{dt}{t} \frac{\sinh^2t\left(1-\frac{i\Theta}{\pi}\right)\sinh t(\xi-1)}{\sinh(2t)\cosh(t)\sinh(\xi t)}\right\},
\\
\label{Wfun}
W\left(\Theta\right)&=-\frac{2}{\cosh\left(\Theta\right)}\exp\left\{ -2\int_{0}^{\infty}\frac{dt}{t}\frac{\sinh^{2}t\left(1-\frac{i\Theta}{\pi}\right)\sinh t\left(\xi-1\right)}{\sinh2t\sinh\xi t}\right\},
\\
\bar G (\Theta)&=-\frac{\mathcal{C}_2}{4}\xi \sinh\left(\frac{\Theta+i\pi}{\xi}\right)\sinh(\Theta),\end{aligned}$$ $$\begin{aligned}
\mathcal{C}_1&=\exp\left\{-\int_0^\infty\frac{dt}{t} \frac{\sinh^2\left(\frac{t}{2}\right)\sinh t(\xi-1)}{\sinh(2t)\cosh(t)\sinh(\xi t)}\right\}=G(-i\pi),
\\
\mathcal{C}_2&=\exp\left\{4\int_0^\infty\frac{dt}{t} \frac{\sinh^2\left(\frac{t}{2}\right)\sinh t(\xi-1)}{\sinh(2t)\sinh(\xi t)}\right\}=\frac{4}{\left[W\left(\frac{i\pi}{2}\right)\xi\sin\left(\frac{\pi}{\xi}\right)\right]^2}.\end{aligned}$$
The integration contours appearing in the consequent expressions for the form factors are such that the “principal poles” of the $W$-functions are always between the contour and the real line. (We define the “principal pole” of $W\left(\Theta\right)$ as the pole located at $\Theta=-\frac{i\pi}{2}$).
For the two-particle form factors it is only necessary to evaluate Eq. (\[eq:FF\]) for two $Z$ operators. Let $A=-\left(\frac{1}{\xi}+\frac{2a}{\beta}\right),$ then the result is
$$\begin{aligned}
{1}
\langle\langle Z_{+}\left(\Theta_{2}\right) & Z_{-}\left(\Theta_{1}\right)\rangle\rangle=\frac{i\mathcal{C}_{2}}{4\mathcal{C}_{1}}e^{\frac{a}{\beta}\left(\Theta_{2}-\Theta_{1}\right)}G(\Theta_{1}-\Theta_{2})e^{-A\Theta_{1}}\nonumber \\
& \qquad\times\left\{ e^{\frac{i\pi}{2\beta^{2}}}\int\frac{d\gamma}{2\pi}e^{A\gamma}W\left(\gamma-\Theta_{2}\right)W\left(\Theta_{1}-\gamma\right)-e^{-\frac{i\pi}{2\beta^{2}}}\int\frac{d\gamma}{2\pi}e^{A\gamma}W\left(\gamma-\Theta_{2}\right)W\left(\gamma-\Theta_{1}\right)\right\} ,\end{aligned}$$
$$\begin{aligned}
{1}
\langle\langle Z_{-}\left(\Theta_{2}\right) & Z_{+}\left(\Theta_{1}\right)\rangle\rangle=\frac{i\mathcal{C}_{2}}{4\mathcal{C}_{1}}e^{\frac{a}{\beta}\left(\Theta_{1}-\Theta_{2}\right)}G(\Theta_{1}-\Theta_{2})e^{-A\Theta_{2}}\nonumber \\
& \times\left\{ e^{\frac{i\pi}{2\beta^{2}}}\int\frac{d\gamma}{2\pi}e^{A\gamma}W\left(\Theta_{2}-\gamma\right)W\left(\Theta_{1}-\gamma\right)-e^{-\frac{i\pi}{2\beta^{2}}}\int\frac{d\gamma}{2\pi}e^{A\gamma}W\left(\gamma-\Theta_{2}\right)W\left(\Theta_{1}-\gamma\right)\right\} ,\end{aligned}$$
The four-particle form factors can also be obtained through evaluating Eq. (\[eq:FF\]) with the result
$$\begin{aligned}
\langle\langle Z_{\sigma_{4}}\left(\Theta_{4}\right)Z_{\sigma_{3}}\left(\Theta_{3}\right)Z_{\sigma_{2}}\left(\Theta_{2}\right)Z_{\sigma_{1}}\left(\Theta_{1}\right)\rangle\rangle & = & \frac{\xi\mathcal{C}_{2}^{3}}{1024\pi^{2}\mathcal{C}_{1}^{2}}G\left(\Theta_{3}-\Theta_{4}\right)G\left(\Theta_{2}-\Theta_{4}\right)G\left(\Theta_{1}-\Theta_{4}\right)\nonumber \\
& & \times G\left(\Theta_{2}-\Theta_{3}\right)G\left(\Theta_{1}-\Theta_{3}\right)G\left(\Theta_{1}-\Theta_{2}\right)J_{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}},\label{eq:4sol}\end{aligned}$$
where
$$J_{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}=e^{\frac{a}{\beta}\sum_{i=1}^{4}\sigma_{i}\Theta_{i}}e^{-A\sum_{\sigma_{i}=-1}\Theta_{i}}I_{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}$$
and $I_{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}$’s are given by
$$\begin{aligned}
I_{--++} & = & e^{\frac{i\pi}{\beta^{2}}}\boldsymbol{P}\left(I_{22},I_{31}\right)-\boldsymbol{P}\left(I_{22},I_{40}\right)-\boldsymbol{P}\left(I_{31},I_{31}\right)+e^{-\frac{i\pi}{\beta^{2}}}\boldsymbol{P}\left(I_{31},I_{40}\right),\\
I_{-+-+} & = & e^{\frac{i\pi}{\beta^{2}}}\boldsymbol{P}\left(I_{13},I_{31}\right)-\boldsymbol{P}\left(I_{22},I_{31}\right)-\boldsymbol{P}\left(I_{13},I_{40}\right)+e^{-\frac{i\pi}{\beta^{2}}}\boldsymbol{P}\left(I_{22},I_{40}\right),\\
I_{-++-} & = & e^{\frac{i\pi}{\beta^{2}}}\boldsymbol{P}\left(I_{04},I_{31}\right)-\boldsymbol{P}\left(I_{13},I_{31}\right)-\boldsymbol{P}\left(I_{04},I_{40}\right)+e^{-\frac{i\pi}{\beta^{2}}}\boldsymbol{P}\left(I_{13},I_{40}\right),\\
I_{++--} & = & e^{\frac{i\pi}{\beta^{2}}}\boldsymbol{P}\left(I_{04},I_{13}\right)-\boldsymbol{P}\left(I_{13},I_{13}\right)-\boldsymbol{P}\left(I_{04},I_{22}\right)+e^{-\frac{i\pi}{\beta^{2}}}\boldsymbol{P}\left(I_{13},I_{22}\right),\\
I_{+-+-} & = & e^{\frac{i\pi}{\beta^{2}}}\boldsymbol{P}\left(I_{04},I_{22}\right)-\boldsymbol{P}\left(I_{13},I_{22}\right)-\boldsymbol{P}\left(I_{04},I_{31}\right)+e^{-\frac{i\pi}{\beta^{2}}}\boldsymbol{P}\left(I_{13},I_{31}\right),\\
I_{+--+} & = & e^{\frac{i\pi}{\beta^{2}}}\boldsymbol{P}\left(I_{13},I_{22}\right)-\boldsymbol{P}\left(I_{22},I_{22}\right)-\boldsymbol{P}\left(I_{13},I_{31}\right)+e^{-\frac{i\pi}{\beta^{2}}}\boldsymbol{P}\left(I_{22},I_{31}\right).\end{aligned}$$
The integrals $I_{ij}$ have four components, $I_{ij,k}$ $k=1,\ldots,4$ and the operation $\boldsymbol{P}$ is defined by $$\boldsymbol{P}\left(a,b\right)=e^{\frac{i\pi}{\xi}}\left(a_{1}b_{1}-a_{2}b_{2}\right)-e^{-\frac{i\pi}{\xi}}\left(a_{3}b_{3}-a_{4}b_{4}\right).$$ $I_{ij,k}$’s read $$I_{04,k}=\int e^{\left(A+\alpha_{k}\right)x}W\left(\Theta_{4}-x\right)W\left(\Theta_{3}-x\right)W\left(\Theta_{2}-x\right)W\left(\Theta_{1}-x\right)dx,$$ $$I_{13,k}=\int e^{\left(A+\alpha_{k}\right)x}W\left(x-\Theta_{4}\right)W\left(\Theta_{3}-x\right)W\left(\Theta_{2}-x\right)W\left(\Theta_{1}-x\right)dx,$$
$$I_{22,k}=\int e^{\left(A+\alpha_{k}\right)x}W\left(x-\Theta_{4}\right)W\left(x-\Theta_{3}\right)W\left(\Theta_{2}-x\right)W\left(\Theta_{1}-x\right)dx,$$
$$I_{31,k}=\int e^{\left(A+\alpha_{k}\right)x}W\left(x-\Theta_{4}\right)W\left(x-\Theta_{3}\right)W\left(x-\Theta_{2}\right)W\left(\Theta_{1}-x\right)dx,$$
$$I_{40,k}=\int e^{\left(A+\alpha_{k}\right)x}W\left(x-\Theta_{4}\right)W\left(x-\Theta_{3}\right)W\left(x-\Theta_{2}\right)W\left(x-\Theta_{1}\right)dx$$
with $\alpha_{1}=-1-\frac{1}{\xi}$, $\alpha_{2}=1-\frac{1}{\xi}$, $\alpha_{3}=-1+\frac{1}{\xi}$, $\alpha_{4}=+1+\frac{1}{\xi}$ coming from writing $\bar{G}(x)$ as the sum of four exponentials; the contours are as before.
In general, the $2n$-particle form factor is realized as $$\begin{aligned}
{1}
\langle\langle\prod_{i=1}^{n}Z_{+}\left(\Theta_{i+n}\right)\prod_{i=1}^{n}Z_{-}\left(\Theta_{i}\right)\rangle\rangle= & \left(\frac{i\mathcal{C}_{2}}{8\pi\mathcal{C}_{1}}\right)^{n}e^{\frac{a}{\beta}\sum_{i=1}^{n}\left(\Theta_{i+n}-\Theta_{i}\right)}e^{-A\sum_{i=1}^{n}\Theta_{i}}\prod_{j>i}G\left(\Theta_{i}-\Theta_{j}\right)\nonumber \\
& \times\int\left\{ \prod_{{\substack{i=1\\ \phantom{j\neq i} }}}^{n}d\gamma_{i}e^{A\gamma_{i}}\left(e^{\frac{i\pi}{2\beta}}W\left(\Theta_{i}-\gamma_{i}\right)-e^{-\frac{i\pi}{2\beta}}W\left(\gamma_{i}-\Theta_{i}\right)\right)\right.\nonumber \\
& \qquad\qquad\times\left.\prod_{j=1}^{n}W\left(\gamma_{i}-\Theta_{j+n}\right)\prod_{{\substack{j=1\\ j\neq i}}}^n W\left(\text{sign}\left(j-i\right)\left(\gamma_{i}-\Theta_{j}\right)\right)\right\} \cdot\prod_{j>i}\bar{G}\left(\gamma_{j}-\gamma_{i}\right)\end{aligned}$$ The last product gives the numerical factor $\left(-\frac{\mathcal{C}_{2}\xi}{16}\right)^{\frac{n(n-1)}{2}}$ and the sum of $4^{\frac{n(n-1)}{2}}$ exponentials containing $\gamma_{i}$’s. All in all, we have $\left(n+1\right)$ combinations of the $W$-functions, which must be integrated over with some exponential factors. Note that the exponential factors do not alter the structure (e.g. the poles) of the integrands. The other kinds of $2n$-particle form factors can be obtained e.g. through the symmetry properties of form factors (Watson’s theorem).
The problem with such integrals is that they diverge for either
$${\rm Re}\, a>\frac{\beta}{2}$$
or
$${\rm Re}\, a<-\frac{1}{\beta}+\frac{\beta}{2}.$$
For such choices of $a$ the integrands have essential singularities at ${\rm Re}\,x\to\pm\infty$. In the next section we prove that $W\left(x\right)$ has an asymptotic series in exponentials of $x$, therefore the divergent integrals can always be analytically continued to obtain a finite result. Our strategy is to first deform the integration contours to the real line, then extract the divergent terms of the integrands in the form of exponentials and give their contributions exactly by the analytic continuation rules $$\begin{aligned}
\int_{-\infty}^{0}\exp\left(\alpha x\right)dx & \equiv & +\frac{1}{\alpha},\qquad\alpha\in\mathbb{C},\\
\int_{0}^{+\infty}\exp\left(\alpha x\right)dx & \equiv & -\frac{1}{\alpha},\qquad\alpha\in\mathbb{C}.\end{aligned}$$ Then if the integral is expected to be analytic and $\int_{0}^{\infty}f\left(x\right)dx$ exists we have $$\int_{-\infty}^{\infty}f\left(x\right)dx\equiv\sum_{i}\frac{a_{i}}{\alpha_{i}}+\int_{-\infty}^{0}\left(f\left(x\right)-[f]\left(x\right)\right)dx+\int_{0}^{\infty}f(x)dx\label{eq:analc}$$ for some $f\left(t\right)$ admitting an asymptotic expansion in exponentials, $$f\left(x\right)=\sum_{\alpha_{i}<0}a_{i}e^{\alpha_{i}x}+O\left(e^{\alpha_{+}x}\right)\equiv [f](x)+O\left(e^{\alpha_{+}x}\right) ,\qquad x\to-\infty,\quad\alpha_{+}>0.$$ The previous equation defines the function $[f](x)$. The case when $\int_{0}^{\infty}f\left(x\right)dx=\infty$ is similar.
It should be noted that for some combination of the parameters, the analytic continuation may still produce an infinite result, that is in the case $\alpha_i=0$ for some $i$. This happens e.g. for the integral $I_{22,1}$ when $\frac{a}{\beta}=\frac{1}{2}$. These infinities, however, must and indeed do cancel out from our end results, the form factors, therefore the $\alpha_{i}=0$ terms in the asymptotic series should be omitted before making the analytic continuation prescribed in (\[eq:analc\]).
Analysis of the $W$-function
============================
Asymptotic series
-----------------
The function $W\left(x\right)$ is given by (\[Wfun\]). The asymptotic series of $\cosh(x)^{-1}$ reads as $$\cosh(x)^{-1}=2e^{-sx}\left(1-e^{-2sx}+e^{-4sx}+\ldots\right),\qquad{\rm Re}\, x\to s\cdot\infty,$$ where $\text{s\ensuremath{\equiv}sign}{\rm Re}\, x$ was introduced for convenience. The exponent of the remaining part of $W(x)$ can be rewritten as $$\int_{0}^{\infty}\frac{dt}{t}\frac{\sinh\left(\xi-1\right)t}{\sinh\left(\xi t\right)\sinh\left(2t\right)}\left(1-\cosh\left(2t\right)\cos\left(\frac{2tx}{\pi}\right)+i\sinh\left(2t\right)\sin\left(\frac{2tx}{\pi}\right)\right).$$ Differentiate the previous formula with respect to $x$ and obtain $$\begin{aligned}
\frac{2}{\pi}\int_{0}^{\infty}dt\frac{\sinh\left(\xi-1\right)t}{\sinh\left(\xi t\right)\sinh\left(2t\right)}\left(\cosh\left(2t\right)\sin\left(\frac{2tx}{\pi}\right)+i\sinh\left(2t\right)\cos\left(\frac{2tx}{\pi}\right)\right) & =\label{eq:olverhez}\\
\frac{i}{\pi}\left(\int_{-\infty}^\infty \frac{\sinh\left(\xi-1\right)t}{\sinh\left(\xi t\right)}(1-\coth(2t))e^{it\frac{2x}{\pi}}dt\right).\end{aligned}$$ The asymptotic series of the Fourier integrals was first obtained by [@Olver:1974], where an analog to Watson’s lemma for Laplace transforms was discussed. Given a function $q\left(t\right)$ with the asymptotic series near $t=0$: $$q\left(t\right)=\sum_{n=0}^{\infty}b_{n}t^{n+\lambda-1}$$ with some $0<\lambda\leq1$, the asymptotic series of $$F\left(x\right)=\int_{0}^{\infty}q\left(t\right)e^{\pm itx}dt$$ as $x\to\infty$ is given by $$F\left(x\right)=\sum_{n=0}^{\infty}b_{n}e^{\pm\frac{i\pi}{2}\left(n+\lambda\right)}\Gamma\left(n+\lambda\right)x^{-n-\lambda}+O\left(e^{-\mu x}\right),\qquad x\to\infty,$$ where $O\left(e^{-\mu x}\right)$ denotes corrections “beyond all orders”, i.e. exponentially small terms ($\mu>0$). Applying this construction to the integrals occurring in Eq. (\[eq:olverhez\]) we get $$\frac{2}{\pi}\int_{0}^{\infty}dt\frac{\sinh\left(\xi-1\right)t}{\sinh\left(\xi t\right)}\coth\left(2t\right)\sin\left(\frac{2tx}{\pi}\right)=\frac{2}{\pi}\frac{\xi-1}{2\xi}{\rm Im}\,\lim_{\lambda\to0}e^{\frac{i\pi}{2}\lambda}\Gamma\left(\lambda\right)+O\left(e^{-\mu_{1}x}\right)=\frac{\xi-1}{2\xi}+O\left(e^{-\mu_{1}x}\right)$$ and $$\frac{2}{\pi}\int_{0}^{\infty}dt\frac{\sinh\left(\xi-1\right)t}{\sinh\left(\xi t\right)}\cos\left(\frac{2tx}{\pi}\right)=O\left(e^{-\mu_{2}x}\right)$$ considering that $q\left(t\right)$ is odd in the first integral and even in the second which implies that all but the first term of the first integral disappears because of the ${\rm Im}$/${\rm Re}$ operation, respectively. For simplicity it was assumed, that $x$ is real. It is easy to check our statements remain true if this condition is relaxed.
To find the exponentially small terms the integral is evaluated by the residue theorem which yields only the residues times $2\pi i$ since the integrands are of small enough order on the half circles $C_{R}^{\pm}=\left\{ z\,:\,\pm{\rm Im}\, z>0,\,|z|=R\right\} $ as $R\to\infty$ (where $C_{R}^{+}$ is associated with ${\rm Re}\, x>0$ while $C_{R}^{-}$ with ${\rm Re}\, x<0$). The result is $$\begin{aligned}
{1}
s\left[\frac{\xi-1}{2\xi}+\sum_{k=1}^{\infty}\left(-1\right)^{k+1}\right. & \cot\left(\frac{\pi\xi\left(2k+1\right)}{2}\right)e^{-\left(2k+1\right)sx}+\sum_{k=1}^{\infty}\left(-1\right)^{k}e^{-2ksx}\nonumber \\
& \left.-\frac{2}{\xi}\sum_{k=1}^{\infty}\sin\left(\frac{\pi k}{\xi}\right)\cot\left(\frac{2\pi k}{\xi}\right)e^{-\frac{2k}{\xi}sx}\right]+\frac{2i}{\xi}\sum_{k=1}^{\infty}\left(-1\right)^{k+1}\sin\left(\frac{\left(\xi-1\right)\pi k}{\xi}\right)e^{-\frac{2k}{\xi}sx}.\end{aligned}$$ When integrated with respect to $x$ one gets the exponent of the $W$-function:
$$\frac{\xi-1}{2\xi}sx+\sum_{k=1}^{\infty}\frac{1}{k}\sin\left(\frac{\pi k}{\xi}\right)\left[\cot\left(\frac{2\pi k}{\xi}\right)-is\right]e^{-\frac{2k}{\xi}sx}-\sum_{k=1}^{\infty}\frac{1}{k}\left. \begin{cases}
i^{k+1}\cot\left(\frac{\pi\xi k}{2}\right),&\text{ for odd $k$}\\
i^{k},&\text{ for even $k$}
\end{cases}\right\} e^{-ksx}+C_{s}.$$ The integration constant, $C_{s}$ is determined from the relation
$$\bar{G}(x)=\frac{1}{W\left(x+\frac{i\pi}{2}\right)W\left(x-\frac{i\pi}{2}\right)}$$
and the explicit form of the function $\bar{G}(x)$, that is
$$\bar{G}\left(x\right)=-\frac{\mathcal{C}_{2}}{4}\xi\sinh\left(\frac{\Theta+i\pi}{\xi}\right)\sinh\left(\Theta\right),$$
with the result
$$-4e^{C_{s}}=\frac{4i}{\sqrt{\mathcal{C}_{2}\xi}}e^{-\frac{is\pi}{2\xi}}.$$
Now we are ready to give the asymptotic expansion of $W\left(x\right)$:
$$\label{Was}
W\left(x\right)=\frac{4i}{\sqrt{\mathcal{C}_{2}\xi}}e^{-\frac{is\pi}{2\xi}}e^{-\frac{\xi+1}{2\xi}sx}\left(\sum_{l=0}^{\infty}\left(-1\right)^{l}e^{-2lsx}\right)\prod_{k=1}^{\infty}\left(\left[\sum_{l=0}^{\infty}\frac{a_{k}^{l}}{l!}e^{-\frac{2kl}{\xi}sx}\right]\left[\sum_{l=0}^{\infty}\frac{b_{k}^{l}}{l!}e^{-klsx}\right]\right),\quad {\rm Re}\,x\to s\cdot\infty,$$
where the coefficients depend only on $\xi$ and $\text{s\ensuremath{\equiv}sign}\left({\rm Re}\,x\right)$ and are expressed as
$$a_{k}=\frac{1}{k}\left[\frac{\cos\left(\frac{2\pi k}{\xi}\right)}{2\cos\left(\frac{\pi k}{\xi}\right)}-is\sin\left(\frac{\pi k}{\xi}\right)\right]$$
and $$b_{k}=-\frac{1}{k} \begin{cases}
i^{k+1}\cot\left(\frac{\pi\xi k}{2}\right)&\text{ for odd $k$,}\\
i^{k}&\text{ for even $k$.}
\end{cases}$$
Note, that, strictly speaking, our expansion is limited to the case of irrational $\xi$ parameters since otherwise the coefficients $a_{k}$ and $b_{k}$ always become infinite for some $k$. However, such infinities can be shown to cancel out.
Let $\xi=\frac{n_{1}}{n_{2}}$, where $n_{1}$ and $n_{2}$ are relative primes. We have singular $a_{k}$’s whenever $\frac{2kn_{2}}{n_{1}}=\text{odd}$, which immediately implies $n_{1}=\text{even}$ and $n_{2}=\text{odd}$ and the $N$th singular $a$-coefficient is indexed by $\frac{Nn_{1}}{2}$, where $N$ is necessarily odd. On the other hand $b_{l}$ is singular if $\frac{ln_{1}}{n_{2}}=\text{even}$ while $l$ is odd, implying again $n_{1}=\text{even}$ and $n_{2}=\text{odd}$ and the $N$th singular $b$-coefficient is indexed by $Nn_{2}$. Now in both cases the $N$th singular term contribute terms of order $e^{-Nn_{2}x}$ to the exponent of $W\left(x\right)$. All that remains is to show that the $N$th diverging coefficients cancel each other. To see this let $\xi=\frac{n_{1}}{n_{2}}\left(1+\varepsilon\right)$ or equivalently
$$\begin{aligned}
n_{1} & \to & n_{1}\left(1+\varepsilon\right),\\
n_{2} & \to & n_{2}\left(1-\varepsilon\right),\end{aligned}$$
resulting in $$\begin{aligned}
{\rm Re}\, a_{Nn_{1}/2}=\frac{1}{Nn_{1}}\frac{\cos\left(\pi Nn_{2}\right)}{\cos\left(\pi Nn_{2}/2\right)} & \longrightarrow & -\frac{1}{Nn_{1}}\frac{1}{\cos\left(\pi Nn_{2}\left(1-\varepsilon\right)/2\right)},\\
b_{Nn_{2}}=-\frac{1}{Nn_{2}}i^{Nn_{2}+1}\frac{\cos\left(\pi Nn_{1}/2\right)}{\sin\left(\pi Nn_{1}/2\right)} & \longrightarrow & -\frac{1}{Nn_{2}}\frac{\left(-1\right)^{\frac{N\left(n_{1}+n_{2}\right)+1}{2}}}{\sin\left(\pi Nn_{1}\left(1+\varepsilon\right)/2\right)}\end{aligned}$$ which if expanded in $\varepsilon$ yield $${\rm Re}\, a_{Nn_{1}/2}=b_{Nn_{2}}+O\left(\varepsilon\right)=\left(-1\right)^{\frac{Nn_{2}+3}{2}}\frac{4}{\pi N^{2}n_{1}n_{2}}\frac{1}{\varepsilon}+O\left(\varepsilon\right),$$ which agrees for $\varepsilon\to0$. Because of the cancellation the following rules can be formulated for rational $\xi$’s:
$$a_{k}=-\frac{is}{k}\text{ for }\frac{2k}{\xi}=\text{odd},$$
$$b_{k}=0\text{ for }k=\text{odd and }k\xi=\text{even}.$$
Poles
-----
With the asymptotic expansion at hand the divergences of the integral representations can be readily remedied. However there is another issue with the integrals containing $W$-functions. In fact, $W(x)$ has a number of poles on the line ${\rm Re}\, x=0$. When the integration contour is fixed (which is the desired scenario), poles can cross it and one needs to analytically continue the result by adding the residue contributions of the crossing poles. In the followings we determine the poles of $W(x)$. The poles of $W(x)$ are easily extracted from the identity ([@Takacspc], but also follows from a similar representation of $G(x)$, given in [@FeherTakacs]) $$\begin{aligned}
{2}\label{Wreg}
W\left(x\right)=-\frac{2}{\cosh x} & \prod_{k=1}^{N}\frac{\Gamma\left(1+\frac{2k-\frac{5}{2}+\frac{ix}{\pi}}{\xi}\right)\Gamma\left(1+\frac{2k-\frac{1}{2}-\frac{ix}{\pi}}{\xi}\right)\Gamma\left(\frac{2k-\frac{1}{2}}{\xi}\right)^{2}}{\Gamma\left(1+\frac{2k-\frac{3}{2}}{\xi}\right)^{2}\Gamma\left(\frac{2k+\frac{1}{2}-\frac{ix}{\pi}}{\xi}\right)\Gamma\left(\frac{2k-\frac{3}{2}+\frac{ix}{\pi}}{\xi}\right)}\nonumber \\
& \hspace{3cm}\times\exp\left\{ -2\int_{0}^{\infty}\frac{dt}{t}\frac{e^{-4Nt}\sinh^{2}t\left(1-\frac{i x}{\pi}\right)\sinh t\left(\xi-1\right)}{\sinh2t\sinh\xi t}\right\} .\end{aligned}$$ They originate from the poles of the gamma functions and the roots of $\cosh x$. It is apparent that ${\rm Re}\, x=0$ for every pole. We are interested in the poles of $W\left(x-x_{0}\right)$ which cross the real line (or the original integration contour) when $\xi$ is decreased. The previous equation yields two infinite series of poles given by $$\begin{aligned}
\xi+2k-\frac{5}{2}+\frac{ix_{1,k,n}}{\pi} & = & -n\xi,\label{xipeq}\\
\xi+2k-\frac{1}{2}-\frac{ix_{2,k,n}}{\pi} & = & -n\xi,\end{aligned}$$ $n$ being a non-negative integer and $k$ being a positive number. We have the following estimates for the series of poles: $$\begin{aligned}
{\rm Im}\, x_{1,k,n} & > & \frac{\left(4k-5\right)\pi}{2},\\
{\rm Im}\, x_{2,k,n} & < & \frac{\left(1-4k\right)\pi}{2}.\end{aligned}$$ For $|{\rm Im}\, x_{0}|\leq\frac{3}{2}\pi$ the only poles that can cross the real line are $$x_{n}\equiv x_{1,1,n}=i\pi\left(n\xi-\frac{1}{2}\right),\qquad n=1,2,\ldots.\label{eq:xipoles}$$ and no poles can cross the original contour, which intersects the ${\rm Im}\, x=0$ line at $x-x_{0}=-\frac{i\pi}{2}-\varepsilon$, $\varepsilon\to0$. With reference to the form factors, note that because of Watson’s theorem it is enough to give a calculation method when all the rapidities satisfy $|{\rm Im}\,\Theta_{i}|\leq\pi$. Thus it is not necessary to analyze further the $\xi$-dependent poles of $W\left(x-x_{0}\right)$, that is covering the case $|{\rm Im}\, x_{0}|>\pi$.
Implementation of the four-soliton form factor formula
======================================================
As an example of the machinery outlined in Section 2 in this section the implementation of the four-soliton form factors is discussed. Implementing Eq. (\[eq:4sol\]) is non-trivial only in the calculation of the integrals $I_{ij,k}$.
First, by Cauchy’s theorem we deform the integration contour to the real line. For this we need to identify the poles between the real line and the original contour, which consists of the the principal poles and (possibly) several $\xi$-dependent poles (given by Eq. (\[eq:xipoles\])) of $W$-functions. The principal poles give the following contributions to $I_{22,k}$:
$$\begin{aligned}
P_{1,k} & = & -\frac{4}{\pi\sqrt{\mathcal{C}_{2}}}e^{\left(A+\alpha_{k}\right)\left(\Theta_{1}+\frac{i\pi}{2}\right)}W\left(\Theta_{1}-\Theta_{4}+\frac{i\pi}{2}\right)W\left(\Theta_{1}-\Theta_{3}+\frac{i\pi}{2}\right)W\left(\Theta_{2}-\Theta_{1}-\frac{i\pi}{2}\right),\\
P_{2,k} & = & -\frac{4}{\pi\sqrt{\mathcal{C}_{2}}}e^{\left(A+\alpha_{k}\right)\left(\Theta_{2}+\frac{i\pi}{2}\right)}W\left(\Theta_{2}-\Theta_{4}+\frac{i\pi}{2}\right)W\left(\Theta_{2}-\Theta_{3}+\frac{i\pi}{2}\right)W\left(\Theta_{1}-\Theta_{2}-\frac{i\pi}{2}\right),\\
P_{3,k} & = & -\frac{4}{\pi\sqrt{\mathcal{C}_{2}}}e^{\left(A+\alpha_{k}\right)\left(\Theta_{3}-\frac{i\pi}{2}\right)}W\left(\Theta_{3}-\Theta_{4}-\frac{i\pi}{2}\right)W\left(\Theta_{2}-\Theta_{3}+\frac{i\pi}{2}\right)W\left(\Theta_{1}-\Theta_{3}+\frac{i\pi}{2}\right),\\
P_{4,k} & = & -\frac{4}{\pi\sqrt{\mathcal{C}_{2}}}e^{\left(A+\alpha_{k}\right)\left(\Theta_{4}-\frac{i\pi}{2}\right)}W\left(\Theta_{4}-\Theta_{3}-\frac{i\pi}{2}\right)W\left(\Theta_{2}-\Theta_{4}+\frac{i\pi}{2}\right)W\left(\Theta_{1}-\Theta_{4}+\frac{i\pi}{2}\right),\end{aligned}$$
if ${\rm Im}\,\Theta_{1}>-\frac{\pi}{2}$, ${\rm Im}\,\Theta_{2}>-\frac{\pi}{2}$, ${\rm Im}\,\Theta_{3}<+\frac{\pi}{2}$, ${\rm Im}\,\Theta_{4}<+\frac{\pi}{2}$, respectively. The $\xi$-dependent poles yield
$$\begin{aligned}
X_{1,k} & = & \sum_{n=1}^{N_{1}}2\pi ir_{n}e^{\left(A+\alpha_{k}\right)\left(\Theta_{1}-x_{n}\right)}W\left(\Theta_{1}-\Theta_{4}-x_{n}\right)W\left(\Theta_{1}-\Theta_{3}-x_{n}\right)W\left(\Theta_{2}-\Theta_{1}+x_{n}\right)\\
X_{2,k} & = & \sum_{n=1}^{N_{2}}2\pi ir_{n}e^{\left(A+\alpha_{k}\right)\left(\Theta_{2}-x_{n}\right)}W\left(\Theta_{2}-\Theta_{4}-x_{n}\right)W\left(\Theta_{2}-\Theta_{3}-x_{n}\right)W\left(\Theta_{1}-\Theta_{2}+x_{n}\right)\\
X_{3,k} & = & \sum_{n=1}^{N_{3}}2\pi ir_{n}e^{\left(A+\alpha_{k}\right)\left(\Theta_{3}+x_{n}\right)}W\left(\Theta_{3}-\Theta_{4}+x_{n}\right)W\left(\Theta_{2}-\Theta_{3}-x_{n}\right)W\left(\Theta_{1}-\Theta_{3}-x_{n}\right)\\
X_{4,k} & = & \sum_{n=1}^{N_{4}}2\pi ir_{n}e^{\left(A+\alpha_{k}\right)\left(\Theta_{4}+x_{n}\right)}W\left(\Theta_{4}-\Theta_{3}+x_{n}\right)W\left(\Theta_{2}-\Theta_{4}-x_{n}\right)W\left(\Theta_{1}-\Theta_{4}-x_{n}\right)\end{aligned}$$
where $N_{1,2}=\left\lfloor \left({\rm Im}\,\Theta_{1,2}/\pi+1/2\right)/\xi\right\rfloor $ and $N_{3,4}=\left\lfloor \left(-{\rm Im}\,\Theta_{3,4}/\pi+1/2\right)/\xi\right\rfloor $, $x_{n}$ are defined by Eq. (\[eq:xipoles\]) and $r_{n}$ is the residue of $W\left(x\right)$ at $x_{n}$, calculated numerically by the definition:
$$\label{resid}
r_{n}=\lim_{x\to x_{n}}\left(x-x_{n}\right)W\left(x\right).$$
Second, the integrals, $I_{ij,k}$ with the deformed contours are evaluated by the analytic continuation formula (\[eq:analc\]). In conclusion one gets $$I_{22,k}=\sum_i\frac{a_i}{\alpha_i} +\int_{-\infty}^{0}\left(A\left(x\right)-[A](x)\right)dx+\int_{0}^{\infty}A(x)dx+\sum_{n=1}^{4}\left(P_{n,k}+X_{n,k}\right),\label{eq:I11k}$$ with $$\begin{aligned}
A\left(x\right)&=\exp\left[\left(A+\alpha_{k}\right)x\right]W\left(x-\Theta_{4}\right)W\left(x-\Theta_{3}\right)W\left(\Theta_{2}-x\right)W\left(\Theta_{1}-x\right)\\
&=[A](x)+O(e^{\alpha_+ x})=\sum_{\alpha_i<0} a_i e^{\alpha_ix}+O(e^{\alpha_+ x}),\qquad {\rm Re}\,x\to-\infty.\end{aligned}$$
Upon generalization to the $2n$-soliton form factors the only non-trivial component of this procedure is the determination of the residues picked up when deforming the contour. For an integrand $$A\left(x\right)=e^{Bx}\prod_{i=1}^{N}W\left(s_{i}\left(x-\Theta_{i}\right)\right),\qquad s_i=\pm1$$ we have $$\int_{C}A(x)dx=\int_{-\infty}^\infty A(x)dx+P+X,$$ with the residue contributions $$P=-\frac{4}{\pi\sqrt{\mathcal{C}_{2}}}\sum_{i=1}^{N}\Theta\left[-s_{i}{\rm Im}\,\Theta_{i}+\frac{\pi}{2}\right]e^{B\left(\Theta_{i}-s_{i}\frac{i\pi}{2}\right)}\prod_{j\neq i}W\left(s_{j}\left[\Theta_{i}-\Theta_{j}-s_{i}\frac{i\pi}{2}\right]\right),$$ and $$X=\sum_{i=1}^{N}\sum_{n=1}^{N_{i}}2\pi ir_{n}e^{B\left(\Theta_{i}+s_{i}r_{n}\right)}\prod_{j\neq i}W\left(s_{j}\left[\Theta_{i}-\Theta_{j}+s_{i}r_{n}\right]\right),\qquad N_{i}=\left\lfloor \frac{\pi-2s_{i}{\rm Im}\,\Theta_{i}}{2\pi\xi}\right\rfloor .$$
To conclude this section, we give some details of the Mathematica [@Mathematica] package [SGFF.M]{}. After the above, only one element remains that is not straightforward in the implementation: the calculation of the asymptotic series of the product of several $W$-functions. Simple products of the series quickly produce an intractable number of terms, most of which are inaccurate (higher order terms, to which further orders in the series of the constituent functions would contribute). Our solution makes use of 2-by-$n$ matrices containing the coefficients and the exponents of the terms in the asymptotic series. We have the following key procedures in the package [SGFF.M]{}.
- $\mathtt{PRO}\left[\mathfrak{A},\mathfrak{B}\right]$ calculates the 2-by-$m$ matrix corresponding to the product of asymptotic series $\mathfrak{A}$, $\mathfrak{B}$ including orders only with accurate coefficients.
- $\mathtt{SHI}\left[\mathfrak{A},a\right]$ generates the 2-by-$n$ matrix corresponding to the asymptotic series of $f\left(x+a\right)$ from that of $f\left(x\right)$ (i.e. $\mathfrak{A}$).
- $\mathtt{AInt}\left[\mathfrak{A}\right]$ gives the integral of the asymptotic series corresponding to $\mathfrak{A}$ on the negative half-line.
- $\mathtt{Asyfun}\left[\mathfrak{A},x\right]$ yields the value of the asymptotic series corresponding to $\mathfrak{A}$ at $x$.
In terms of these procedures $I_{22,k}$ is calculated as $$I_{22,k}=\mathtt{AInt}\left[\mathfrak{A}\right]+\int_{-\infty}^{0}\left(A\left(x\right)-\mathtt{Asyfun}\left[\mathfrak{A},x\right]\right)dx+\int_{0}^{\infty}A(x)dx+\sum_{n=1}^{4}\left(P_{n,k}+X_{n,k}\right),\label{eq:I11k}$$ with $A(x)$ as before, and $$\mathfrak{A}=\mathtt{PRO}\left[\mathtt{PRO}\left(\mathtt{PRO}\left[\mathtt{SHI}\left[\mathfrak{W}^{*},-\Theta_{4}\right],\mathtt{SHI}\left[\mathfrak{W}^{*},-\Theta_{3}\right]\right],\mathtt{SHI}\left(\mathfrak{W},-\Theta_{2}\right]\right],\mathtt{SHI}\left[\mathfrak{W},-\Theta_{1}\right]\right],$$ $\mathfrak{W}$ being the asymptotic series of $W(x)$ for ${\rm Re}\,x\geq0$. The remaining integrals are performed by the routine $\mathtt{NIntegrate}$.
From a practical point of view one should note, that the second term in (\[eq:I11k\]) can be numerically unstable. On the other hand, provided that the truncated asymptotic series is a good enough approximation of $A(x)$ for $x<0$ the contribution of this term can be neglected altogether. Therefore, we omitted this term from our code and supposed that the input rapidities are big enough for this to cause no harm. This can be assumed safely, since Lorentz invariance implies that the rapidities can be shifted by an arbitrary real number.
Considering now the general case of $2n$-particle form factors, we give the asymptotic series of $$\label{integrand}
A\left(x\right)=e^{Bx}\prod_{i=1}^{N}W\left(s_{i}\left(x-\Theta_{i}\right)\right),$$ diverging for $x\to-\infty$, as $$\mathfrak{A}=\mathtt{PRO}_{i=1}^{N}\left[\mathtt{SHI}\left[\mathtt{Co}_{s_{i}}\left(\mathfrak{W}\right),-\Theta_{i}\right]\right],\qquad\mathtt{Co}_{s_{i}}\left(\mathfrak{W}\right)=\left\{ \begin{array}{c}
\mathfrak{W}\phantom{^{^{*}}},\quad s_{i}=+1\\
\mathfrak{W}^{*},\quad s_{i}=-1
\end{array}\right.$$
The main functions available in [SGFF.M]{} are to calculate the two-, four- and six-soliton form factors.
- $\mathtt{FF2}[\Theta_1,\Theta_2]$ gives the two-particle form factors $$\{F_{+-}(\Theta_{1,i}-\Theta_{2,i}),F_{-+}(\Theta_{1,i}-\Theta_{2,i})\}$$ where $\Theta_{1}$ and $\Theta_{2}$ are arrays of the same length with elements $\Theta_{a,i}$ ($a=1,2$, $i=1,2,\ldots N$) rapidities where the two-soliton form factors are to be evaluated.
- $\mathtt{FF4}[\Theta_1,\Theta_2,\Theta_3,\Theta_4]$ gives the four-particle form factors $$\{F_{--++}(\Theta_{1,i},\Theta_{2,i},\Theta_{3,i},\Theta_{4,i}),F_{-+-+}(\ldots),F_{+--+}(\ldots),F_{-++-}(\ldots),F_{+-+-}(\ldots),F_{++--}(\ldots)\}.$$
- $\mathtt{FF6}[\Theta_1,\Theta_2,\Theta_3,\Theta_4,\Theta_5,\Theta_6]$ gives the six-particle form factors $$\begin{aligned}
\{F_{+++---}&(\Theta_{1,i},\Theta_{2,i},\Theta_{3,i},\Theta_{4,i},\Theta_{5,i},\Theta_{6,i}),\,
F_{++-+--}(\ldots),\,F_{++--+-}(\ldots),\,F_{++---+}(\ldots),\\
&F_{+-+-+-}(\ldots),\,F_{+-++--}(\ldots),\,F_{+-+--+}(\ldots),\,F_{+--++-}(\ldots),\,F_{+--+-+}(\ldots),\\
&F_{+---++}(\ldots),\,F_{-+++--}(\ldots),\,F_{-++-+-}(\ldots),\,F_{-++--+}(\ldots),\,F_{-+-++-}(\ldots),\\
&F_{-+--++}(\ldots),\,F_{-+-+-+}(\ldots),\,F_{--+++-}(\ldots),\,F_{--++-+}(\ldots),\,F_{--+-++}(\ldots),\,F_{---+++}(\ldots)
\}.\end{aligned}$$
- $\mathtt{FF2p}[\Theta_1,\Theta_2]$ gives the two-particle form factors for physical rapidities, i.e. ones with imaginary parts of $\pm \pi$.
- $\mathtt{FF4p}[\Theta_1,\Theta_2,\Theta_3,\Theta_4]$ gives the four-particle form factors for physical rapidities.
- $\mathtt{FF6p}[\Theta_1,\Theta_2,\Theta_3,\Theta_4,\Theta_5,\Theta_6]$ gives the six-particle form factors for physical rapidities.
Note that accurate results can only be expected when all rapidities have big enough positive real parts and imaginary part in the interval $[-\pi,\pi]$. The functions calculating form factors only at physical rapidities are considerably faster compared to the general ones if the form factors are needed in more than one points; they calculate the necessary $W$-function values for the integrals only once as part of the initialization.
The parameters that can be specified in $\tt SGFF.M$ are the following, which can be edited e.g. in Mathematica before loading the package.
- $\mathtt{\xi}$ is the IR parameter (\[xi\]).
- $\mathtt{aover\beta}$ is the ratio of the parameter $a$ appearing in the operator $O=e^{i a\varphi}$ and the UV parameter $\beta$ of the Lagrangian.
- $\mathtt{NN}$ is the regularization parameter for the $G-$ and $W-$functions denoted by $N$ in the formula (\[Wreg\]).
- $\mathtt{Na}$ is the maximum number of terms treated in the individual asymptotic series in the formula (\[Was\]).
- $\mathtt{Ni}$ is the number of interpolation points used to calculate the integrands of type (\[integrand\]). When evaluating the form factors at general rapidities, mainly $\mathtt{Ni}$ determines the time of evaluation. However, it is this parameter that influences the accuracy the most, as well. A safe choice is $\mathtt{Ni}=2000$.
- $\mathtt{\varepsilon}$ is a technical parameter for the calculation of the residues (\[resid\]), $\varepsilon=x-x_n$.
- $\mathtt{aa}$ and $\mathtt{bb}$ are the lower and upper bounds of the integrals of the type $\int_0^\infty A(x)dx$ in (\[eq:I11k\])
Tests
=====
The four-particle form factor $F_{--++}$ was checked against the free fermion point result (omitting the vacuum expectation value) $$\begin{aligned}
{1}
\langle\langle Z_{+}\left(\Theta_{4}\right)Z_{+}\left(\Theta_{3}\right)Z_{-}\left(\Theta_{2}\right)Z_{-}\left(\Theta_{1}\right)\rangle\rangle= & \sin^{2}\left(\sqrt{2}\pi a\right)e^{\sqrt{2}a\left(\Theta_{4}+\Theta_{3}-\Theta_{2}-\Theta_{1}\right)}\\
& \times\frac{\sinh\left(\frac{\Theta_{1}-\Theta_{2}}{2}\right)\sinh\left(\frac{\Theta_{3}-\Theta_{4}}{2}\right)}{\cosh\left(\frac{\Theta_{3}-\Theta_{1}}{2}\right)\cosh\left(\frac{\Theta_{3}-\Theta_{2}}{2}\right)\cosh\left(\frac{\Theta_{4}-\Theta_{1}}{2}\right)\cosh\left(\frac{\Theta_{4}-\Theta_{2}}{2}\right)}.\nonumber \end{aligned}$$ We do not show test results for this formula since our calculations agreed with the exact results to the machine precision (of 15 digits).
Also, we investigated whether the numerically obtained form factors satisfy the form factor axioms. In the four-particle case the equation
$$\label{W4_0}
F_{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}\left(\Theta_{1}+z,\Theta_{2}+z,\Theta_{3}+z,\Theta_{4}+z\right)=F_{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}\left(\Theta_{1},\Theta_{2},\Theta_{3},\Theta_{4}\right)$$
must hold, which was checked. Watson’s theorem is another axiom, e.g. in the form
$$\label{W4_1}
F_{--++}\left(\Theta_{1},\Theta_{2},\Theta_{3},\Theta_{4}\right)=S_{+-}^{-+}\left(\Theta_{3}-\Theta_{2}\right)F_{--++}\left(\Theta_{1},\Theta_{3},\Theta_{2},\Theta_{4}\right)+S_{+-}^{+-}\left(\Theta_{3}-\Theta_{2}\right)F_{-+-+}\left(\Theta_{1},\Theta_{3},\Theta_{2},\Theta_{4}\right),$$
and $$\label{W4_2}
F_{--++}\left(\Theta_{1},\Theta_{2},\Theta_{3},\Theta_{4}+2\pi i\right)=e^{2\pi i\omega}F_{+--+}\left(\Theta_{4},\Theta_{1},\Theta_{2},\Theta_{3}\right),$$ $\omega=\frac{a}{\beta}$ being the mutual non-locality index.
The residues of the kinematic poles of the four-particle form factors were also checked by: $$\label{kin4}
i\lim_{\Theta_{4}\to\Theta_{2}+i\pi}\left(\Theta_{4}-\Theta_{2}-i\pi\right)F_{--++}\left(\Theta_{1},\Theta_{2},\Theta_{3},\Theta_{4}\right)=F_{-+}\left(\Theta_{1}-\Theta_{3}\right)\left[S_{+-}^{+-}\left(\Theta_{3}-\Theta_{2}\right)-e^{2\pi i\omega}S_{--}^{--}\left(\Theta_{2}-\Theta_{1}\right)\right].$$ Testing the kinematical poles is especially important: for the cases when the two-soliton form factor is known explicitly (e.g. for half-integer $\frac{a}{\beta}$), equation (\[kin4\]) gives the only check that is independent of numerical integrals and their analytic continuations. E.g. for $\frac{a}{\beta}=1$ the two-particle form factors are known to be [@Lukyanov:1997] $$F_{\mp\pm}^\beta(\Theta)=\frac{G(\Theta)}{G(-i\pi)}\cot\left(\frac{\pi\xi}{2}\right)\frac{4i\cosh\left(\frac{\Theta}{2}\right)e^{\mp\frac{\Theta+i\pi}{2\xi}}}{\xi\sinh\left(\frac{\Theta+i\pi}{\xi}\right)}.$$
In Tables 1 and 2 we listed test results for the four-particle form factors. One can see that magnitude of the error varies greatly for different scenarios. This is because we work in fixed precision (double precision) and the integrals appearing in the formulas can assume values of very different magnitudes and rounding errors can get magnified.
LHS RHS
------------------------- --------------------------------------- ---------------------------------------
(\[W4\_0\]), $\xi=2.23$ $0.45330-1.4092i$ $0.45336-1.4093i$
(\[W4\_0\]), $\xi=0.34$ $0.00089-0.051i$ $0.00091-0.049i$
(\[W4\_1\]), $\xi=2.23$ $0.453360-1.4093198i$ $0.453358-1.4093196i$
(\[W4\_1\]), $\xi=0.34$ $0.0009063-0.04937438i$ $0.0009065-0.04937441i$
(\[W4\_2\]), $\xi=2.23$ $-0.04255089122137+0.03246926430660i$ $-0.04255089122139+0.03246926430663i$
(\[W4\_2\]), $\xi=0.34$ $-0.043292833089+0.00219194033i$ $-0.043292833083+0.00219194037i$
: Comparison of the LHS’s and RHS’s of the form factor axioms (\[W4\_0\]) (where $z=1$ was taken), (\[W4\_1\]), (\[W4\_2\]) in the four-soliton case. The rapidities were chosen to be $\Theta_{1}=7.6,\,\Theta_{2}=7,\,\Theta_{3}=7.2,$ and $\Theta_{4}=6-i\pi$. In all the tests $\frac{a}{\beta}=\frac{5}{4}$ was set.
\[tab4\]
LHS RHS
------------------------ ----------------------------------- -----------------------------------
(\[kin4\]), $\xi=2.23$ $\phantom{-}0.8211182+0.7147548i$ $\phantom{-}0.8211175+0.7147545i$
(\[kin4\]), $\xi=1.17$ $-0.2812726+0.0213804i$ $-0.2812724+0.0213801i$
(\[kin4\]), $\xi=0.34$ $-0.4726029- 0.6620907i$ $-0.4726070- 0.6620917i$
: Comparison of residues of four-particle form factors with exact results (\[kin4\]). We took $\frac{a}{\beta}=1$ and for the rapidities $\Theta_{1}=7.6,\,\Theta_{2}=7,\,\Theta_{3}=7.2,$ and $\Theta_4=7+10^{-8}+i\pi$.
\[tab41\]
We also implemented the 6-particle form factors. The numerical results agreed with the exact results for the free fermion point, where the 6-particle form factor $F_{---+++}$ reads $$\begin{aligned}
{1}
\frac{-i\sin^{3}\left(\sqrt{2}\pi a\right)e^{\sqrt{2}a\left(\Theta_{6}+\Theta_{5}+\Theta_{4}-\Theta_{3}-\Theta_{2}-\Theta_{1}\right)}\sinh\left(\frac{\Theta_{1}-\Theta_{2}}{2}\right)\sinh\left(\frac{\Theta_{4}-\Theta_{5}}{2}\right)}{\cosh\left(\frac{\Theta_{4}-\Theta_{1}}{2}\right)\cosh\left(\frac{\Theta_{4}-\Theta_{2}}{2}\right)\cosh\left(\frac{\Theta_{4}-\Theta_{3}}{2}\right)\cosh\left(\frac{\Theta_{5}-\Theta_{1}}{2}\right)\cosh\left(\frac{\Theta_{5}-\Theta_{2}}{2}\right)}\nonumber \\
\times\frac{\sinh\left(\frac{\Theta_{1}-\Theta_{3}}{2}\right)\sinh\left(\frac{\Theta_{4}-\Theta_{6}}{2}\right)\sinh\left(\frac{\Theta_{2}-\Theta_{3}}{2}\right)\sinh\left(\frac{\Theta_{5}-\Theta_{6}}{2}\right)}{\cosh\left(\frac{\Theta_{5}-\Theta_{3}}{2}\right)\cosh\left(\frac{\Theta_{6}-\Theta_{1}}{2}\right)\cosh\left(\frac{\Theta_{6}-\Theta_{2}}{2}\right)\cosh\left(\frac{\Theta_{6}-\Theta_{3}}{2}\right)}\end{aligned}$$ In addition, the following kinematic pole equation was tested: $$\begin{aligned}
{1}
i\lim_{\Theta_{6}\to\Theta_{3}+i\pi} & \left(\Theta_{6}-\Theta_{3}-i\pi\right)F_{---+++}\left(\Theta_{1},\Theta_{2},\Theta_{3},\Theta_{4},\Theta_{5},\Theta_{6}\right)=\nonumber \\
& F_{--++}\left(\Theta_{1},\Theta_{2},\Theta_{4},\Theta_{5}\right)\left[S_{+-}^{+-}\left(\Theta_{5}-\Theta_{3}\right)S_{+-}^{+-}\left(\Theta_{4}-\Theta_{3}\right)-e^{2\pi i\omega}S_{--}^{--}\left(\Theta_{3}-\Theta_{1}\right)S_{--}^{--}\left(\Theta_{3}-\Theta_{2}\right)\right].\label{kin6}\end{aligned}$$ Watson’s theorem requires e.g. $$\begin{aligned}
F_{---+++}\left(\Theta_{1},\Theta_{2},\Theta_{3},\Theta_{4},\Theta_{5},\Theta_{6}\right) & = & S_{+-}^{+-}\left(\Theta_{4}-\Theta_{3}\right)F_{--+-++}\left(\Theta_{1},\Theta_{2},\Theta_{4},\Theta_{3},\Theta_{5},\Theta_{6}\right)\nonumber \\
& & \qquad+S_{+-}^{-+}\left(\Theta_{4}-\Theta_{3}\right)F_{---+++}\left(\Theta_{1},\Theta_{2},\Theta_{4},\Theta_{3},\Theta_{5},\Theta_{6}\right)\label{W6_1}.\end{aligned}$$ Test results for the six-particle form factors are listed in Table 3.
\[tab6\]
LHS RHS
------------------------- ---------------------------- ---------------------------
(\[W6\_1\]), $\xi=1.17$ $-0.50782662-2.333030973i$ $-0.50782660-2.33030977i$
(\[W6\_1\]), $\xi=0.34$ $-0.3945330-0.3095434i$ $-0.3945333-0.3095431i$
(\[kin6\]), $\xi=1.17$ $-2.84279-1.63925i$ $-2.84263-1.63902i$
(\[kin6\]), $\xi=0.34$ $-0.0151096-0.147483i$ $-0.0151107-0.147442i$
: Comparison of the LHS’s and RHS’s of the form factor axioms in the six-soliton case. For (\[W6\_1\]) the rapidities were chosen to be $\Theta_{1}=2.1,\,\Theta_{2}=1.9,\,\Theta_{3}=6,\,\Theta_{4}=5.9,\,\Theta_{5}=1.2,\,\Theta_{6}=5.5+i\pi$, while for (\[kin6\]), $\Theta_{1}=2.1,\,\Theta_{2}=1.9,\,\Theta_{3}=5.9,\,\Theta_{4}=1.2,\,\Theta_{5}=6,\,\Theta_{6}=5.90001+i\pi$ was taken. In all the tests $\frac{a}{\beta}=\frac{5}{4}$ was set.
It interesting to note, that the use of multi-soliton form factors extends to the calculation of soliton-breather and breather-breather form factors by virtue of the bound state pole axiom [@Smirnov:1992]. In case of higher breather-breather form factors, using soliton-antisoliton form factors can be preferable: e.g. to calculate the $B_n$–$B_m$ form factor one needs to evaluate either the $(n+m)$–$B_1$- or the four-soliton form factors.
Conclusions and outlook
=======================
We established a method to obtain the multi-soliton form factors numerically in the (1+1)-dimensional sine-Gordon model. The form factors are known in terms of integral representations, whose domains of convergence were extended by analytical continuation. In order to do this we needed the asymptotic series of the $W$-function. Detailed formulae were only shown for the four-soliton form factors, however the number of treatable particles is not limited by the procedure. Test results obtained by the code provided for the four- and six-soliton form factors were shown.
Based on the formalism developed for finite volume form factors in [@PT1:2007; @PT2:2008; @Pmu:2008], a program to investigate the sine-Gordon form factors is currently underway [@FeherTakacs], which can now be extended to multi-soliton states [@FPT:2011]. The present formalism is also expected to be relevant to boundary form factors [@BPT:2006], for which finite size corrections have been developed in [@KT:2007] and applied to sine-Gordon theory in [@LT:2011]. Future applications will also include the calculation of finite temperature correlation functions based on the formalism developed in [@Essler:2007; @Essler:2009; @Pozsgay].
Acknowledgements {#acknowledgements .unnumbered}
================
The author is indebted to Gábor Takács for a number of valuable discussions and for carefully reading the manuscript.
[^1]: E-mail address: palmai@phy.bme.hu
|
---
abstract: |
In this paper, we derive *a priori* estimates for the gradient and second order derivatives of solutions to a class of Hessian type fully nonlinear parabolic equations with the first initial-boundary value problem on Riemannian manifolds. These *a priori* estimates are derived under conditions which are nearly optimal. Especially, there are no geometric restrictions on the boundary of the Riemannian manifolds. And as an application, the existence of smooth solutions to the first initial-boundary value problem even for infinity time is obtained.
*Mathematical Subject Classification (2010):* 35B45, 35R01, 35K20, 35K96.
*Keywords:* Fully nonlinear parabolic equations; Riemannian manifolds; *a priori* estimates; Hessian; First initial-boundary value problem.
address:
- 'Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China'
- 'Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China'
author:
- 'Ge-jun Bao'
- 'Wei-song Dong\*'
title: Estimates for a class of hessian type fully nonlinear parabolic equations on Riemannian manifolds
---
Introduction
============
Let $(M^n, g)$ be a compact Riemannian manifold of dimension $n \geq 2$ with smooth boundary $\partial M$ and ${\bar{M}}:= M \cup \partial M$. We study the Hessian type fully nonlinear parabolic equation $$\label{eqn}
f (\lambda[\nabla^{2}u + \chi], - u_t) = \psi (x, t)$$ in $M_T = M \times (0,T] \subset M \times \mathbb{R}$, where $f$ is a symmetric smooth function of $n + 1$ variables, $\nabla^2 u$ denotes the Hessian of $u (x, t)$ with respect to the space $x \in M$, $u_t$ is the derivative with respect to the time $t \in (0, T]$, $\chi$ is a smooth (0, 2) tensor on ${\bar{M}}$ and $\lambda [\nabla^{2} u + \chi] = (\lambda_1 ,\ldots,\lambda_n)$ denotes the eigenvalues of $\nabla^{2} u + \chi$ with respect to the metric $g$. While the first initial-boundary value problem requires: $$\label{eqn-b}
u = \varphi, \; \mbox{on $\mathcal{P}M_T$},$$ where $\mathcal{P} M_T = B M_T \cup S M_T$ and $B M_T = M \times \{0\}$, $S M_T = \partial M \times [0,T]$. We assume $\psi \in C^{4, 1} (\overline {M_T}) $, $\varphi \in C^{4, 1}(\mathcal{P}M_T)$.
As in [@CNS], we assume $f \in C^\infty (\Gamma) \cap C^0 (\overline{\Gamma})$ to be defined on an open, convex, symmetric proper subcone $\Gamma \subset \mathbb{R}^{n + 1}$ with vertex at the origin and $$\Gamma^{+} \equiv \{\lambda \in \mathbb{R}^{n + 1}:\mbox{ each component }
\lambda_{\ell} > 0, \; 1 \leq \ell \leq n+1\} \subseteq \Gamma.$$ In this work we assume only a few conditions on $f$, which are almost optimal, but the followings are essential as the structure conditions. We assume that $f$ satisfies: $$\label{f1}
f_{\ell} \equiv \frac{\partial f}{\partial \lambda_{\ell}} > 0 \mbox{ in } \Gamma,\ \ 1\leq \ell \leq n + 1,$$ $$\label{f2}
f\mbox{ is concave in }\Gamma,$$ and $$\label{f5}
f > 0 \mbox{ in } \Gamma, \ \ f = 0 \mbox{ on } \partial \Gamma, \ \ \inf_{M_T}\psi > 0.$$
In [@G] Guan has developed some methods to derive *a priori* second order estimates under nearly optimal conditions for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds. More recently, Guan and Jiao [@GJ] further developed the methods to cover more general elliptic equations. In this paper we prove the mechanism in [@G] to derive second order estimates is also valid for a wild class of Hessian type fully nonlinear parabolic equations on Reimannian manifolds. Following [@GJ] we assume: $$\label{f3}
T_{\lambda} \cap \partial \Gamma^{a} \; \mbox{is a
nonempty compact set, for any $\lambda \in \Gamma$ and $0 < a < f (\lambda)$},$$ where $\partial \Gamma^{\sigma} = \{\lambda \in \Gamma: f(\lambda) = \sigma\}$ is the boundary of $\Gamma^{\sigma} = \{\lambda \in \Gamma: f (\lambda) > \sigma\}$ and $T_{\lambda}$ denotes the tangent plane at $\lambda$ of $\partial \Gamma^{f (\lambda)}$, for $\sigma \in {\hbox{\bbbld R}}^+$ and $\lambda \in \Gamma$. By assumptions (\[f1\]) and (\[f2\]), $\partial \Gamma^{\sigma}$ is smooth and convex.
Since we need no geometric boundary conditions, we have to assume, and which is more convenience in application, that there exists an admissible function (see Section 2) $\underline{u} \in C^{2, 1} (\overline {M_T})$ satisfying $$\label{sub}
f(\lambda[\nabla^{2} \underline{u} + \chi], - \underline{u}_{t}) \geq \psi(x,t) \mbox{ in } M_T$$ with ${\underline}u = \varphi$ on $\partial M \times (0, T]$ and ${\underline}u \leq \varphi$ on ${\bar{M}}\times \{0\}$, which we call a *subsolution*. If the inequality holds strictly, then we call ${\underline}u$ a *strict* subsolution. In [@L], Lieberman proved that there exists a strict subsolution under conditions that for any compact subset $K$ of $\overline {M_T} \times \Gamma$, there exists a positive constant $R(K)$ such that $ f(R\lambda) > \psi (x, t)$ for any $R \geq R(K)$, $(x, t, \lambda) \in K$, and that there is a positive constant $R_1$ such that $(\kappa, R_1) \in \Gamma$, where $\kappa = (\kappa_0, \ldots, \kappa_{n - 1})$ is the space-time curvatures of $SM_T$(see [@L]).
Without loss of generality, we assume the compatibility condition, that is for all $x \in {\bar{M}}$, $(\lambda[\nabla^{2} \varphi(x,0) + \chi], -\varphi_t(x, 0)) \in \Gamma$, and $$\label{comp}
f (\lambda [\nabla^2 \varphi (x, 0) + \chi(x)], - \varphi_t (x, 0)) = \psi (x, 0).$$ We remark that this condition is actually ensured by the short time existence of solution to equation and . Now we can give out our main result as below.
\[bd-th1\] Let $u \in C^{4, 1} (M_T) \cap C^{2, 1}(\overline {M_T})$ be an admissible solution of (\[eqn\]) in $M_T$ with $u = \varphi \; \mbox{on} \; \mathcal{P} M_T$. Suppose $f$ satisfies - , the following and hold: $$\label{c-290}
\sum_{\ell =1}^{n+1} f_\ell \lambda_\ell \geq - K_0 (1 + \sum_{\ell = 1}^{n + 1} f_\ell), \; \; \forall \lambda \in \Gamma.$$ Then we have $$\label{gsui-1}
\max_{\overline {M_T}} (|\nabla^2 u| + |u_t| ) \leq C,$$ where $C > 0$ depends on $|u|_{C^1_x (\overline {M_T})}$, $|\underline{u}|_{C^{2,1} (\overline {M_T})}$, $|\psi|_{C^{2, 1}(\overline {M_T}))}$, $|\varphi|_{C^1_t(\overline {M_T}))}$ and other known data.
We remark that in Theorem \[bd-th1\], is only needed when deriving second order boundary estimates, and there the norms are defined as below: $$\begin{aligned}|u|_{C^1_x (\overline {M_T})} = & |u|_{C^0 (\overline {M_T})} + |\nabla u|_{C^0 (\overline {M_T})},\;
|\varphi|_{C^1_t (\overline {M_T})} = |\varphi_t|_{C^0 (\overline {M_T})},\\
|\psi|_{C^{2, 1} (\overline {M_T})} = & |\psi|_{C^0 (\overline {M_T})}
+ |\nabla \psi|_{C^0 (\overline {M_T})} + |\nabla^2 \psi|_{C^0 (\overline {M_T})} + |\psi_t|_{C^0 (\overline {M_T})}.
\end{aligned}$$
For the gradient estimates, firstly from $\Gamma \subset \{ \lambda \in \mathbb{R}^{n + 1}: \sum_{\ell =1}^{n + 1} \lambda_\ell \geq 0\}$, we see that $u$ is a subsolution of $$\label{ups}
\left\{\begin{aligned} - h_t + \triangle h + \;& \sum \chi_{ii} (x, t) = 0, \; & \mbox{in} \; M_T,\\
h = \;& \varphi, \; & \mbox{on} \; \mathcal{P} M_T.
\end{aligned}\right.$$ If we assume $h$ is the solution of the above linear equation, we may easily get $u \leq h$ on $\overline {M_T}$ by the comparison principle. On the other hand, since ${\underline}u$ is a subsolution of and , we have ${\underline}u \leq u$. Therefore, we have the $C^0$ estimates that $$\label{C0}
\sup_{M_T} |u| + \sup_{S M_T} |\nabla u| \leq C.$$ While it is evident that on $B M_T$, we have $\nabla u = \nabla \varphi$. So the following theorem completes the main work of this paper.
\[bd-th2\] Suppose $f$ satisfies - . Let $u \in C^{4, 1} (M_T) \cap C^{2, 1}(\overline {M_T})$ be an admissible solution of (\[eqn\]) in $M_T$. Then $$\label{gsui-2}
\sup_{M_T}|\nabla u| \leq C ( 1 + \sup_{\mathcal{P} M_T}|\nabla u|),$$ where $C$ depends on $|\psi|_{C^1_x (\overline {M_T})}$, $|u|_{C^0 (\overline {M_T})}$ and other known data, under either of the following additional assumptions: $(\mathbf{i})$ $f$ satisfies and $$\label{f6}
f_\jmath (\lambda) \geq \nu_1 ( 1 + \sum_{\ell =1}^{n+1} f_\ell (\lambda) ) \mbox{ for any }
\lambda \in \Gamma \mbox{ with } \lambda_\jmath < 0,$$ where $1 \leq \jmath \leq n + 1$, and $\nu_1$ is a uniform positive constant; $(\mathbf{ii})$ and hold, as well as that $(M, g)$ has nonnegative sectional curvature.
Based on the above *a priori* estimates and , equation becomes uniformly parabolic equation. Then by Evans-Krylov Theorem [@Evans; @K], we can obtain the *a priori* $C^{2 + \alpha, 1 + \alpha/2}$ estimates. Therefore it is possible to apply the theory of linear uniformly parabolic equations (see [@L] for more) to get higher order estimates. We remark that, the *a priori* estimates in Theorem \[bd-th1\], Theorem \[bd-th2\] and do not depend on the time $t$ explicitly, and as a byproduct of these estimates, we have the following (long time, i.e. $T = \infty$) existence results. We note that a function in $C^{\infty} (\overline {M_T})$ means that it is sufficiently smooth about $(x, t) \in \overline {M_T}$, and note $M_\infty = M \times \{t > 0\}$.
\[jsui-th4\] Let $\psi \in C^{\infty} (\overline {M_T}) $ and $\varphi \in C^{\infty} (\mathcal{P} M_T)$, $0 < T \leq \infty$. Suppose $f$ satisfies - , and holds. In addition that either and or $(M^n, g)$ has nonnegative sectional curvature holds. Then there exists a unique admissible solution $u \in C^{\infty} (\overline {M_T})$ to the first initial-boundary value problem and .
The above theorem is a direct result of the short time existence and the uniform estimates in Theorem \[bd-th1\] and Theorem \[bd-th2\], because at each beginning time we can take $\varphi = u$, which enables us to assume the compatibility condition, for more one can see Theorem 15.9 in [@L].
Here we give some typical examples of our equation, for example $f = \sigma^{1/k}_k$ for $k \geq 2$ (the reason why $k$ cannot equal to $1$ is due to condition , see [@G]) and $f = (\sigma_k / \sigma_l)^{1/(k - l)}$, $1 \leq l < k \leq n + 1$, both of which are defined on the cone $\Gamma_{k} = \{\lambda \in \mathbb{R}^{n+1}: \sigma_{j} (\lambda) > 0, j = 1, \ldots, k\}$, where $\sigma_{k} (\lambda)$ are the elementary symmetric functions $\sigma_{k} (\lambda) = \sum_ {i_{1} < \ldots < i_{k}}
\lambda_{i_{1}} \ldots \lambda_{i_{k}}$. Another interesting example is given by $f = \log P_k$, where $$P_k (\lambda) = \prod_{i_1 < \cdots < i_k} (\lambda_{i_1} + \cdots + \lambda_{i_k}), \ \ 1 \leq k \leq n + 1,$$ defined in the cone $\mathcal{P}_k := \{\lambda \in \mathbb{R}^{n + 1}: \lambda_{i_1} + \cdots + \lambda_{i_k} > 0\}$.
Krylov in [@K1] introduced three parabolic type equations analogous to Monge-Ampère equation in $\mathbb{R}^n$. One type which is studied on Riemannian manifolds by Jiao and Sui in [@JS] recently is $$\label{eqn-js}
f (\lambda[\nabla^{2}u + \chi]) - u_t = \psi (x, t),$$ under assumptions that $\inf_{\mathcal{P} M_T} (\varphi_t + \psi) = \nu_0 > 0$, and $\psi (x, t)$ is concave with respect to $x \in M$. This type equation in $\mathbb{R}^n$ when $f = \sigma_n^{1/n}$ with $\chi \equiv 0$ was firstly considered by Ivochkina and Ladyzhenskaya in [@IL1] and [@IL2]. Another type is $$\label{MA-p}
- u_t \det(\nabla^2 u) = \psi^{n + 1},$$ which is a typical form of our case in $\mathbb{R}^{n + 1}$ with $\chi \equiv 0$. Some other cases can be fined in Chou and Wang [@CW2001] or Wang [@WX2].
At the end of the introduction, we describe the outline of our paper. In Section 2, we state some preliminaries and introduce our main tool (Theorem \[3I-th3\]) to establish the $C^2$ *a priori* estimates, and two propositions which are needed when deriving the second order estimates on boundary. In Section 3, we establish the estimates for $|u_t|$ that do not depend on $T$ explicitly, after which we have a bound for the constant $C(\epsilon, |u_t|, K_0, \sup_{M_T} \psi)$ in Proposition \[par-2\]. Then the mechanism in [@G] is valid for the second order boundary estimates, and the global and boundary estimates for second order derivatives are derived in Section 4 and Section 5 respectively. In Section 6, we establish the interior gradient estimates as the end.
Preliminaries {#gj-P}
=============
From now on, we stipulate that the Latin alphabet $i, j, k, \cdots $ are valued between $1$ and $n$ when there is no other statement. Firstly, we give some notations and formulas on Riemannian manifolds, throughout the paper $\nabla$ denotes the Levi-Civita connection of $(M^n, g)$. Let $e_1, \ldots, e_n$ be a local frame on $M^n$. We denote $g_{ij} = g (e_i, e_j)$, $\{g^{ij}\} = \{g_{ij}\}^{-1}$. Define the Christoffel symbols $\Gamma_{ij}^k$ by $\nabla_{e_i} e_j = \Gamma_{ij}^k e_k$ and the curvature coefficients $$R_{ijkl} = g( R (e_k, e_l) e_j, e_i), \;\; R^i_{jkl} = g^{im} R_{mjkl}.$$ We shall use the notation $\nabla_i = \nabla_{e_i}$, $\nabla_{ij} = \nabla_i \nabla_j - \Gamma_{ij}^k \nabla_k $, etc. Finally we recall the following formula on Riemannian manifolds $$\label{hess-A70}
\nabla_{ijk} v - \nabla_{jik} v = R^l_{kij} \nabla_l v,$$ which will be frequently used in following sections.
Let $u$ be an admissible solution of equation . For simplicity we define $U \equiv \nabla^2 u + \chi$, ${\underline}U \equiv \nabla^2 {\underline}u + \chi$ and under an orthonormal local frame $e_1, \ldots, e_{n}$, we write $U_{ij} \equiv U (e_i, e_j) = \nabla_{ij} u + \chi_{ij}$. Direct calculating yields that $$\begin{aligned}
\nabla_k U_{ij} \equiv & \nabla U (e_k, e_i, e_j)
= \nabla_k \chi_{ij} + \nabla_{kij} u, \\
\nabla_{kl} U_{ij} \equiv & \nabla^2 U (e_k, e_l, e_i, e_j )
= \nabla_k \nabla_l U_{ij} - \Gamma_{kl}^m \nabla_m U_{ij}.
\end{aligned}$$ For the convenience, sometimes we denote $- u_t$ by $U_{n+1 n+1}$, i.e. $U_{n+1 n+1} = -u_t$, and $U_{i n+1} = U_{n+1 i} = 0$ where $1 \leq i \leq n$.
Let $F$ be the function defined by $F (A) = f (\lambda [A])$ for $A \in \mathbb{S}^{n + 1}$ with $\lambda [A] \in \Gamma$, where $\mathbb{S}^{n + 1}$ is the set of $(n + 1) \times (n + 1)$ symmetric matrices. We call a function $u \in C^{2, 1}(M_T)$ *admissible* if $(\lambda[\nabla^{2} u + \chi], - u_t) \in \Gamma$ in $M_T$. It is shown in [@CNS] that ensures that equation is parabolic (i.e. $\{\partial F (A) / \partial A_{ij}\}$ is positive definite) with respect to admissible solutions, while implies that the function $F$ is concave. For an admissible solution $u \in C^{2, 1}(M_T)$ denote $$\hat{U} \equiv [U_{ij}, - u_t] \equiv
\left( \begin{array}{cc} U_{ij} & 0\\ 0 & -u_t\\ \end{array}
\right),$$ under an orthonormal local frame $e_1, \ldots, e_n$. Therefore equation can be locally written as $$\label{eqn'}
F (\hat{U}) = \psi (x, t).$$ We denote $$F^{ij} = \frac{\partial F}{\partial A_{ij}} (\hat{U}), \;\;
F^{ij, kl} = \frac{\partial^2 F}{\partial A_{ij} \partial A_{kl}} (\hat{U}), \;\;
F^{\tau} = \frac{\partial F}{\partial A_{n + 1, n + 1}} (\hat{U}) \equiv f_\tau.$$ The matrix $[\{F^{ij}\}, F^\tau]$ has eigenvalues $f_1, \ldots, f_n, f_\tau$ and is positive definite by assumption . Moreover, when $[\{U_{ij}\}, - u_t]$ is diagonal so is $[\{F^{ij}\}, F^\tau]$, and as in [@G] the following identities hold $$F^{ij} U_{ij} = \sum f_i \lambda_i, \;\; F^{ij} U_{ik} U_{kj} = \sum f_i \lambda_i^2,$$ where $\lambda (\{U_{ij}\}) = (\lambda_1, \ldots, \lambda_{n})$.
The following theorem proved in [@GJ] is the keystone in deriving *a priori* $C^2$ estimates in our paper.
\[3I-th3\] Suppose $f$ satisfies , and . Let $\Im $ be a compact set of $\Gamma$ and $\sup_{\partial \Gamma} f < a \leq b < \sup_{\Gamma} f$. There exist positive constants $\theta = \theta (\Im, [a, b])$ and $R = R (\Im, [a, b])$ such that for any $\lambda \in \Gamma^{[a, b]} = \bar \Gamma^a \backslash \Gamma^b$, when $|\lambda| \geq R$, $$\label{3I-100}
\sum_{\ell=1}^{n + 1} f_\ell (\lambda) (\mu_{\ell} - \lambda_\ell) \geq
\theta + \theta \sum_{\ell=1}^{n + 1} f_\ell (\lambda) + f (\mu) - f (\lambda), \; \forall \mu \in \Im.$$
For $\forall \, v \in C^{2, 1} (\overline {M_T})$, we define the linear operator $\mathcal{L}$ by $$\mathcal{L} v = F^{ij} \nabla_{ij} v - F^\tau v_t .$$ Choose a smooth orthonormal local frame $e_1, \ldots, e_n$ about $(x, t)$ such that $\{U_{ij}(x,t)\}$ is diagonal. From Lemma 6.2 in [@CNS] and Theorem \[3I-th3\], it is easily to prove that there exist positive constants $\theta$, $R$ depending only on ${\underline}u$ and $\psi$ such that when $|\lambda| = |\lambda [\hat{U}]| \geq R$, $$\label{gj}
\mathcal{L} ({\underline}u - u) = F^{ii} ( {\underline}{U_{ii}} - U_{ii} ) + F^{\tau} (-{\underline}u_t + u_t) \geq \theta (1+ \sum F^{ii} + F^{\tau}).$$
\[rmk-0\] If ${\underline}u$ is a strict subsolution. Note that $\{\lambda ({\underline}{\hat{U}}) : (x, t) \in \overline {M_T}\}$ is contained in a compact subset of $\Gamma$, here ${\underline}{\hat{U}} = [{\underline}U, - {\underline}u_t]$. We see that there exist constants $\varepsilon, \delta_0 > 0$ such that $\lambda
[{\underline}U (x, t) - \varepsilon g, -{\underline}{u}_t - \varepsilon] \in \Gamma$ for all $(x, t) \in \overline {M_T}$ and $F ( [{\underline}U - \varepsilon g, - {\underline}u_t - \varepsilon]) \geq \psi + \delta_0$. By the concavity of $F$, we have $$\begin{aligned}
F^{ij} ( {\underline}{U}_{ij} - U_{ij} ) - F^{\tau} ( {\underline}{u}_t - u_t )
\geq \varepsilon (\sum F^{ii} + F^\tau) + \delta_0.
\end{aligned}$$ That means is valid in the whole $\overline {M_T}$ if ${\underline}u$ is a strict subsolution.
The following two propositions play the key role in the second order boundary estimates, which are the generalized counterpart results in [@G].
\[par-1\] Let $F(\hat U) = f (\lambda (U), - u_t)$. There is an index $1 \leq r \leq n$ such that $$\label{par-11}
\sum_{l<n} F^{ij} U_{il} U_{lj} \geq \frac{1}{2} \sum_{i \neq r} f_i \lambda_i^2.$$
This can be proved by exactly the same method as the prove of Proposition 2.7 in [@G]. So we omit the proof.
One more result we need is the following which actually is a combination of generalized Lemma 2.8 and Corollary 2.9 in [@G]. The method of this proof is from [@GSS].
\[par-2\] Suppose $f = f (\lambda (U), - u_t)$ satisfies , and . Then for any index $1 \leq r \leq n$ and $\epsilon > 0$, $$\label{par-21}
\sum f_i |\lambda_i| \leq \epsilon \sum_{i \neq r} f_i \lambda_i^2 +
C(\epsilon, |u_t|, K_0, \sup_{M_T} \psi) (1 + \sum f_i + f_\tau).$$
Firstly, if $\lambda_r \leq 0$, by , we have $$\begin{aligned}
\sum f_i |\lambda_i|
= & 2 \sum_{\lambda_i > 0} f_i \lambda_i - \sum f_i \lambda_i\\
\leq & \epsilon \sum_{\lambda_i > 0} f_i \lambda_i^2 +
\frac{1}{\epsilon} \sum_{\lambda_i > 0} f_i + f_\tau (- u_t) + K_0 (1 + \sum f_i + f_\tau)\\
\leq & \epsilon \sum_{i \neq r} f_i \lambda_i^2 +
C (\epsilon, K_0) \sum f_i + \max \{|u_t|, K_0\} f_\tau + K_0.
\end{aligned}$$
Secondly, if $\lambda_r \geq 0$, then by , we have $$\begin{aligned}
\sum f_i |\lambda_i| = & \sum f_i \lambda_i - 2 \sum_{\lambda_i < 0} f_i \lambda_i\\
\leq & \epsilon \sum_{\lambda_i < 0} f_i \lambda_i^2 + \frac{1}{\epsilon}
\sum_{\lambda_i < 0} f_i + \sum f_i + f_\tau + f_\tau u_t
+ \psi - f (\mathbf{1})\\
\leq & \epsilon \sum_{\lambda_i < 0} f_i \lambda_i^2
+ C (\epsilon, |u_t|, \sup_{M_T} \psi) (1 + \sum f_i + f_\tau)
\end{aligned}$$ since $f (\mathbf{1}) > 0$ and where $\mathbf{1} = (1, \cdots, 1) \in \mathbb{R}^{n + 1}$. This proves .
estimates for $u_t$
====================
The assumption is crucial for the estimates of $u_t$. In a forthcoming paper we will consider the estimates of $u_t$ without this restriction. By the compatibility condition , on $BM_T$, we have $u_t = \varphi_t$, which is also valid apparently on $ \partial M \times [0, T]$. Hence we have $\sup_{\mathcal{P}M_T} |u_t| \leq C$. Now by differentiating equation with respect to $t$ we see that $\mathcal{L} u_t = \psi_t$. Let $a$ be a positive constant to be determined, by , if $|u_t|$ is sufficiently large, by Theorem \[3I-th3\], we have $$\mathcal{L}(- u_t + a ({\underline}u - u)) \geq - C + a \theta (1 + \sum F^{ii} + F^{\tau})
\geq 0$$ when $a \geq \frac{C}{\theta}$. Similarly we can prove the same result holds for $u_t$. Thus by maximum principle we have $$\sup_{M_T} |u_t| \leq C .$$ here $C$ depends on $|\psi|_{C^1_t}$, $|u|_{C^0}$, $|\varphi|_{C^1_t}$and other known data.
Therefore, the estimate for $|u_t|$ implies that the constants $C(\epsilon, |u_t|, K_0, \sup_{M_T} \psi)$ in Proposition \[par-2\] are bounded, which enables us to apply the mechanism in [@G] to derive the second order boundary estimates.
$C^{2}$ global estimates
========================
In this section, we derive *a priori* global estimates for the second order derivatives. We set $$W = \max_{(x,t) \in \overline {M_T}} \max_{\xi \in T_x M, |\xi| = 1}
(\nabla_{\xi\xi} u + \chi_{\xi \xi})
\exp (\frac{\delta}{2} |\nabla u|^{2} + a (\underline{u} - u)),$$ where $a \gg 1 \gg \delta$ are positive constants to be determined later. It suffices to estimate $W$. We may assume $W$ is achieved at $(x_{0}, t_{0}) \in \overline {M_T} - \mathcal{P} M_T$ for some unit vector $\xi \in T_{x_0} M^n$. Choose a smooth orthonormal local frame $e_{1}, \ldots, e_{n}$ about $x_{0}$ such that $e_1 (x_0) = \xi$, $\nabla_{e_i} e_j = 0$, and $\{U_{ij} (x_0, t_0)\}$ is diagonal. We may also assume $U_{11} \geq \ldots \geq U_{n n}$, $U_{11} \geq \sup_{M_T} |u_t|$. Therefore $W = U_{11} (x_0, t_0) e^{\phi (x_0, t_0)}$, where $\phi = \frac{\delta}{2} |\nabla u|^{2} + a (\underline{u} - u)$, and $|U_{ii}| \leq n |U_{11}|$ which derived from $- u_t + U_{11} + \cdots + U_{nn} > 0$.
At the point $(x_{0}, t_{0})$ where the function $\log (U_{11}) + \phi$ attains its maximum, we have $$\label{gs3}
\frac{\nabla_i U_{11}}{U_{11}} + \nabla_i \phi = 0
\mbox{ for each } i = 1, \ldots, n,$$ $$\label{gs4}
\frac{(\nabla_{11} u)_t}{U_{11}} + \phi_t \geq 0,$$ and $$\label{gs5}
0 \geq \sum_{i = 1}^{n} F^{ii} \Big\{ \frac{\nabla_{ii} U_{11}}{U_{11}}
- \frac{(\nabla_i U_{11})^{2}}{U^{2}_{11}} + \nabla_{ii} \phi \Big\}.$$ Differentiating equation (\[eqn\]) twice, we obtain $$\label{gs1}
F^{ij} \nabla_{k} U_{ij} - F^{\tau} \nabla_k u_t = \nabla_k \psi \;\; \mbox{ for } \; 1 \leq k \leq n,$$ and $$\begin{aligned}
\label{gs2}
F^{ij} \nabla_{11} U_{ij} + & \, F^{ij,kl} \nabla_1 U_{ij} \nabla_1 U_{kl}
+ F^{\tau\tau} (\nabla_1 u_t)^2 \\
-& \, 2 F^{ij, \tau} \nabla_1 U_{ij} \nabla_1 u_t - F^\tau \nabla_{11} u_t
= \nabla_{11} \psi .
\end{aligned}$$
Hence, combining , , and , and noting $ \nabla_{ii}U_{11} \geq \nabla_{11} U_{ii} - C U_{11}$ (see [@G]), we have $$\label{gs11}
\mathcal{L} \phi \leq E + C\Big(1 + \sum F^{ii} \Big)$$ when $U_{11}$ is sufficiently large, where $$\begin{aligned}
E = \, & \frac{F^{ii} (\nabla_{i} U_{11})^{2}}{{U^{2}_{11}}}
+ \frac{ F^{ij,kl} \nabla_1 U_{ij} \nabla_1 U_{kl}
- 2 F^{ij, \tau} \nabla_1 U_{ij} \nabla_1 u_t
+ F^{\tau\tau} (\nabla_1 u_t)^2 }{U_{11}}.
\end{aligned}$$ By some straightforward calculation, we have, at $(x_0, t_0)$, $$\label{test1}
\nabla_i \phi = \delta \nabla_j u \nabla _{ij} u + a \nabla_i ({\underline}u - u), \;\;
\phi_t = \delta \nabla_j u (\nabla_j u)_t + a ({\underline}u - u)_t,$$ $$\label{test2}
\begin{aligned}
\nabla_{ii} \phi = \,& \delta (\nabla_{ij} u \nabla_{ij} u
+ \nabla_j u \nabla_{iij} u) + a \nabla_{ii} ({\underline}u - u)\\
\geq \,& \frac{\delta}{2} U_{ii}^2 - C \delta
+ \delta \nabla_{j} u \nabla_{iij} u + a \nabla_{ii} ({\underline}u - u).
\end{aligned}$$ Thus, by and we have, $$\label{test4}
\begin{aligned}
F^{ii} \nabla_{ii} \phi \geq \,& \frac{\delta}{2} F^{ii} U^2_{ii}
+ \delta \nabla_j u F^{ii} (\nabla_{jii} u + R^l_{iij} \nabla_l u)\\
& + a F^{ii} \nabla_{ii} ({\underline}u - u) - C \delta \sum F^{ii}\\
\geq \,& \frac{\delta}{2} F^{ii} U^2_{ii} + \delta F^{\tau} \nabla_{j} u \nabla_j u_t + a F^{ii} \nabla_{ii} ({\underline}u - u) \\
\, & - C \delta (1 + \sum F^{ii}).
\end{aligned}$$ Therefore, by , and we obtain $$\label{gs12}
\begin{aligned}
a \mathcal{L} ({\underline}{u} - u) \,& \leq E - \frac{\delta}{2} F^{ii} U^2_{ii}
+ C ( 1+ \sum F^{ii} ).
\end{aligned}$$
For fixed $0 < s \leq 1/3$, let $$J = \{i: U_{ii} \leq - s U_{11}, 1 < i \leq n\}, \;\;
K = \{i: U_{ii} > - s U_{11}, 1 \leq i \leq n\}.$$ Using a result of Andrews [@A] and Gerhardt [@GC] (see [@U] also), and noting that $U_{n+1 j} = 0$ for all $j = 1, 2, \cdots, n$, we have, $$\label{gj-S130}
\begin{aligned}
- F^{ij, kl} \nabla_1 U_{ij} \nabla_1 U_{kl} + \,& 2 F^{ij, \tau} \nabla_1 U_{ij} \nabla_1 u_t - F^{\tau\tau} \nabla_1 u_t \nabla_1 u_t\\
\geq \,& \sum_{1 \leq i \neq j \leq n + 1} \frac{F^{ii} - F^{jj}}{U_{jj} - U_{ii}}
(\nabla_1 U_{ij})^2 \\
\geq \,& 2 \sum_{2 \leq i \leq n} \frac{F^{ii} - F^{11}}{U_{11} - U_{ii}}
(\nabla_1 U_{i 1})^2 \\
\geq \,& \frac{2 (1-s)}{(1+s) U_{11}} \sum_{i \in K} (F^{ii} - F^{11})
((\nabla_i U_{11})^2 - C /s),
\end{aligned}$$ where in the last inequality we used the following result which can be readily proved with , that for any $ s \in (0, 1)$ $$\label{chu1}
\begin{aligned}
(1 - s)(\nabla_i U_{11})^2 \leq (\nabla_1 U_{1i})^2 + C (1 - s)/s .
\end{aligned}$$ From , combining and that $\nabla_i \phi \leq \delta \nabla_i u U_{ii} + C a$ at $(x_0, t_0)$, we get $$\label{gj-S140}
\begin{aligned}
E
\leq \,& \sum_{i \in J} F^{ii} (\nabla_i \phi)^2
+ C \sum_{i \in K} F^{ii} + C F^{11} \sum_{i \in K} (\nabla_i \phi)^2\\
\leq \,& C a^2 \sum_{i \in J} F^{ii} + C \delta^2 F^{ii} U_{ii}^2
+ C \sum_{i \in K} F^{ii} + C (\delta^2 U_{11}^2 + a^2) F^{11}.
\end{aligned}$$ Therefore by and , we finally obtain $$\label{gs7}
\begin{aligned}
0 \geq \,& (\frac{\delta}{2} - C \delta^2 ) F^{ii} U_{ii}^2 - C a^2 \sum_{i \in J} F^{ii}
- C (\delta^2 U_{11}^2 + a^2) F^{11} \\
& + a \mathcal{L}({\underline}u - u ) - C (1 + \sum F^{ii}).
\end{aligned}$$ Observe that $$F^{ii} U_{ii}^2 \geq F^{11} U_{11}^2 + \sum_{i \in J} F^{ii} U_{ii} \geq F^{11} U_{11}^2 + s^2 U_{11}^2 \sum_{i \in J} F^{ii}.$$ We may firstly choose $\delta$ small sufficiently such that $\frac{\delta}{2} - C \delta^2 > c_0 > 0$ . Then we assume $U_{11} > R$, where $R$ is the positive constant such that holds and fix $a$ large enough so that $a \mathcal{L}({\underline}u - u ) - C (1 + \sum F^{ii}) \geq 0 $ holds, then we would get a contradiction provided $U_{11}$ is sufficiently large from . Thus we get an upper bound for $U_{11}$.
$C^{2}$ boundary estimates
==========================
Throughout this section we assume the function $\varphi \in C^{4,1} (\mathcal{P} M_T)$ is extended to a $C^{4,1}$ function on $\overline {M_T}$, which is still denoted by $\varphi$.
Fix a point $(x_{0}, t_{0}) \in S M_T$. We shall choose a smooth orthonormal local frame $e_1, \ldots, e_n$ around $x_0$ such that when restricted to $\partial M$, $e_n$ is normal to $\partial M$. Since $u - {\underline}{u} = 0$ on $S M_T$, we have $$\label{hess-a200}
\nabla_{\alpha \beta} (u - {\underline}{u})
= - \nabla_n (u - {\underline}{u}) \varPi (e_{\alpha}, e_{\beta}), \;\;
\forall \; 1 \leq \alpha, \beta < n \;\;
\mbox{on $SM_T$},$$ where $\varPi$ denotes the second fundamental form of $\partial M$. Therefore, $$|\nabla_{\alpha \beta} u| \leq C, \;\; \forall \; 1 \leq \alpha, \beta < n \;\; \mbox{on} \;\; S M_T.$$
Let $\rho (x)$ denote the distance from $x \in M$ to $x_{0}$, and set $$M_T^{\delta} = \{X = (x, t) \in M \times (0,T]:
\rho (x) < \delta, \; t \leq t_{0} + \delta\}.$$ Since $\partial M$ is smooth, we may also assume the distance function $d (x, t) \equiv d (x)$ to the boundary $SM_T$ is smooth in $M_T^{\delta}$.
\[lem1\] There exist some uniform positive constants $a, \delta, \varepsilon$ sufficiently small and $N$ sufficiently large such that the function $$v = (u - \underline{u}) + ad - \frac{Nd^{2}}{2}$$ satisfies $$\label{v}
\mathcal{L} v \leq - \varepsilon (1 + \sum F^{ii} + F^\tau) \mbox{ in }M_T^{\delta},
\ v \geq 0\mbox{ on } \mathcal{P} M_T^{\delta}.$$
We note that to ensure $v \geq 0$ in $\mathcal{P} M_{\delta}$ we may require $\delta \leq 2a/N$ after $a, N$ being fixed. It is easy to see that $$\label{bs4}
\mathcal{L} v
\leq \mathcal{L}(u - {\underline}u) + C (a + Nd) \sum F^{ii}
- N F^{ij} \nabla_i d \nabla_j d.$$ Fix $\theta > 0$ small and $R > 0$ large enough such that holds at every point in $\bar{M}_T^{\delta_{0}}$ for some fixed $\delta_{0} > 0$. Let $\lambda = \lambda [U]$ be the eigenvalues of $U$. At a fixed point in $M_{\delta}$ where $\delta < \delta_0$, we consider two cases: (a) $|\lambda| < R$ and (b) $|\lambda| \geq R$.
In case (a), since $|u_t| \leq C$, by , there are uniform bounds $$c_{1} I \leq \{F^{ij}\} \leq C_{1} I, \;\; c_1 \leq F^\tau \leq C_1$$ for some positive constants $c_1, C_1$, and therefore $F^{ij} \nabla_{i} d \nabla_{j} d \geq c_{1}$ since $|\nabla d| \equiv 1$. Since $\mathcal{L}(u - {\underline}u) \leq 0$, we may fix $N$ large enough so that Lemma \[lem1\] holds for any $a, \varepsilon \in (0,1]$, as long as $\delta$ is sufficiently small.
In case (b), since $F^{ij}\nabla_i d \nabla_j d \geq 0$, by and (\[bs4\]) we may further require $a$ and $\delta$ small enough so that Lemma \[lem1\] holds.
With the help of $ \nabla_{ij} (\nabla_{k} u)
= \nabla_{ijk} u + \Gamma_{ik}^l \nabla_{j l} u + \Gamma_{jk}^l \nabla_{ i l} u
+ \nabla_{\nabla_{ij} e_k} u $ and , we obtain $$\label{hess-E170}
\begin{aligned}
|\mathcal{L} \nabla_k (u - \varphi)|
\leq \,& 2 |F^{ij} \Gamma_{ik}^l \nabla_{j l} u| + C \Big(1 + \sum F^{ii} + F^\tau \Big) \\
\leq \,& C \Big(1 + \sum f_i |\lambda_i| + \sum f_i + f_\tau \Big).
\end{aligned}$$ According to we have $$\mathcal{L} |\nabla_{\gamma} (u - \varphi)|^2
\geq F^{ij} U_{i \gamma} U_{j \gamma}
- C \Big(1 + \sum f_i |\lambda_i| + \sum f_i + f_\tau \Big).$$ Let $$\label{hess-E176}
\varPsi
= A_1 v + A_2 \rho^2 - A_3 \sum_{\gamma < n} |\nabla_{\gamma} (u - \varphi)|^2.
$$ For any $K > 0$, since $\nabla_l (u - \varphi) = 0$ on $\partial M$ with $1 \leq l \leq n-1$, when $A_2 \gg A_3 \gg 1$, we have $( A_2 - K ) \rho^2 \geq A_3 \sum_{l < n} |\nabla_l (u - \varphi)|^2$ in $\overline M_T^\delta$. Hence we can choose $A_1 \gg A_2 \gg A_3 \gg 1$ such that $\varPsi \geq K (d + \rho^2)$ in $\overline M_T^\delta$. By Proposition \[par-1\] and Proposition \[par-2\] and Lemma \[lem1\], it follows that in $M_T^{\delta}$, $\mathcal{L} (\varPsi \pm \nabla_{\alpha} (u - \varphi)) \leq 0$, and $\varPsi \pm \nabla_{\alpha} (u - \varphi) \geq 0$ on $\mathcal{P} M_T^{\delta}$ when $A_1 \gg A_2 \gg A_3 \gg 1$. By the maximum principle we derive $\varPsi \pm \nabla_{\alpha} (u - \varphi) \geq 0$ in $M_T^{\delta}$ and therefore $$|\nabla_{n \alpha} u (x_0, t_0)| \leq \nabla_n \varPsi (x_0, t_0) \leq C,
\;\; \forall \; \alpha < n.$$
It remains to derive $\sup_{S M_T} \nabla_{n n} u \leq C$, since $- u_t + \triangle u + \sum \chi_{ii} \geq - C$. For $(x, t) \in SM_T$, let $\tilde U (x, t)$ be the restriction to $T_x \partial M$ of $U (x, t)$, viewed as a bilinear map on the tangent space of $\partial M$ at $x$, and $\lambda ( \tilde U (x, t) )$ be the eigenvalues with respect to the induced metric of $(M^n, g)$ on $\partial M$. Similarly one can define $\tilde {{\underline}U } (x, t)$ and $\lambda (\tilde {{\underline}U } (x, t) )$. On $SM_T$, we define that $$\tilde{F} ([\tilde U, - u_t]) := \lim_{R \rightarrow \infty} f (\lambda (\tilde U), R, -u_t).$$ Due to Trudinger [@T], we need only show that the following quantity $$m := \min_{(x, t) \in S M_T}
\Big( \tilde F ([\tilde U, -u_t])(x, t) - \psi (x,t) \Big)$$ is positive (see [@G]). We can assume that $m$ is finite. Note that $\tilde{F}$ is concave, and it is easily seen that the following holds $$\begin{aligned}
\hbar := \,& \min_{(x, t) \in S M_T} \Big( \tilde{F}( [\tilde {{\underline}U}, - {\underline}u_t]) (x, t) - \psi (x, t) \Big) > 0,
\end{aligned}$$ and $\hbar$ may equal to infinity. Without loss of generality we assume $m < \hbar/2$. Suppose $m$ is achieved at a point $(x_{0}, t_{0}) \in S M_T$. Now we give some notations. Choose a local orthonormal frame $e_1 \ldots, e_n$ around $x_0$ as before, that is $e_n$ is normal to $\partial M$. Therefore locally we have $\tilde U = \{U_{\alpha \beta}\}$, where $1 \leq \alpha, \beta \leq n-1$. We denote that $\tilde{F}^{\alpha \beta}_0$ is the first order derivative of $\tilde F$ with respect to $\tilde U_{\alpha \beta}$ at $(x_0, t_0)$, and $\tilde{F}^{\tau}_0$ is the first order derivative of $\tilde F$ with respect to $- u_t$ at $(x_0, t_0)$.
By we have on $ SM_T $, $$\label{c-220}
U_{\alpha {\beta}} = {\underline}{U}_{\alpha {\beta}}
- \nabla_n (u - {\underline}{u}) \sigma_{\alpha {\beta}}$$ where $\sigma_{\alpha {\beta}} = \langle \nabla_{\alpha} e_{\beta}, e_n \rangle$; note that $\sigma_{\alpha \beta} = \varPi (e_\alpha, e_\beta)$ on $SM_T$. Since on $SM_T$ we have ${\underline}{u}_t = u_t$, it follows that at $(x_0, t_0)$, $$\label{c-225}
\begin{aligned}
\nabla_n (u - {\underline}{u}) {\tilde{F}}^{\alpha {\beta}}_0 \sigma_{\alpha {\beta}}
= \,& {\tilde{F}}^{\alpha {\beta}}_0 ({\underline}{U}_{\alpha \beta} - U_{\alpha {\beta}})
\geq {\tilde{F}}[{\underline}{U}_{\alpha \beta}, -{\underline}{u}_t] - {\tilde{F}}[U_{\alpha \beta}, -u_t] \\
= \,& {\tilde{F}}[{\underline}{U}_{\alpha {\beta}}, - {\underline}u_t] - \psi - m
\geq \hbar - m.
\end{aligned}$$ Setting $\eta = {\tilde{F}}^{\alpha {\beta}}_0 \sigma_{\alpha {\beta}}$ which is well defined in $M_T^\delta$, by we obtain $$\label{c-230}
\eta (x_0) \geq \frac{\hbar}{2 \nabla_n (u - {\underline}{u}) (x_0, t_0)}
\geq \epsilon_1 \hbar > 0$$ for some uniform $\epsilon_1 > 0$.
Let $$Q \equiv {\tilde{F}}^{\alpha {\beta}}_0 \Big( {\underline}U_{\alpha {\beta}} - U_{\alpha {\beta}} (x_0, t_0) \Big)
+ {\tilde{F}}^\tau_0 \Big( u_t(x_0, t_0) - \varphi_t \Big) + \psi (x_0, t_0) - \psi (x, t),$$ and we see that $Q$ is well defined in $M_T^\delta$ for small $\delta$ . Since ${\tilde{F}}$ is concave, we have on $ SM_T$ $$\label{c-210}
\begin{aligned}
{\tilde{F}}^{\alpha {\beta}}_0 \Big(U_{\alpha {\beta}} - U_{\alpha {\beta}} (x_0, t_0)\Big)
& - {\tilde{F}}_0^\tau \Big(u_t - u_t (x_0, t_0)\Big) - F (\hat U ) + F (\hat U (x_0, t_0))\\
& \geq {\tilde{F}}([U_{\alpha {\beta}}, -u_t]) - F (\hat U ) - m \geq 0 .
\end{aligned}$$ We define $$\varPhi = - \eta \nabla_n (u - {\underline}u) + Q ;$$ it follows from and that $\varPhi (x_0, t_0) = 0$ and $\varPhi \geq 0$ on $ SM_T $. For $(x, 0) \in B M_T^\delta$ we have $$\varPhi (x, 0) \geq \varPhi (\hat x, 0) - C d (x) \geq - C d (x),$$ where $C$ depends on $C^1$ norms of $\nabla^2 {\underline}u$, $\psi$, $\varphi_t$, and $\hat x \in \partial M$ such that $d (x) = {dist}_{M} (x, \hat x)$ . Besides with some calculation, we have $$\label{gblq-B360}
\begin{aligned}
\mathcal{L} \varPhi
\leq C \sum f_i + C \sum f_i |\lambda_i| + C f_\tau + C .
\end{aligned}$$ Therefore, applying Proposition \[par-1\], Proposition \[par-2\] and Lemma \[lem1\] again, as well as choosing $A_1 \gg A_2 \gg A_3 \gg 1$, we derive $$\label{cma-106}
\left\{ \begin{aligned}
\mathcal {L} \,& (\varPsi + \varPhi) \leq 0 \;\; \mbox{in $M_T^\delta$}, \\
& \varPsi + \varPhi \geq 0 \;\; \mbox{on $\mathcal{P} M_T^\delta$}.
\end{aligned} \right.$$ By the maximum principle, $\varPsi + \varPhi \geq 0$ in $M_T^\delta$. Thus $\nabla_n \varPhi (x_0, t_0) \geq - \nabla_n \varPsi (x_0, t_0) \geq -C $. While we also have $$\label{c-255}
- C \leq \nabla_n \varPhi (x_0, t_0)
\leq \Big(- \eta (x_0) \Big) \nabla_{nn} u (x_0, t_0) + C.$$ This with yields that $$\nabla_{nn} u (x_0, t_0) \leq \frac{C}{\epsilon_1 \hbar}.$$
By now we have got *a priori* upper bounds for all eigenvalues of $\{U_{ij} (x_0, t_0)\}$ and hence the eigenvalues are contained in a compact subset of $\Gamma$ by . Therefore by we obtain $m > 0$ (for the detailed proof one can see [@GSS]). This completes the proof of Theorem \[bd-th1\].
Gradient estimates
==================
We now deal with the interior estimates of $|\nabla u|$ with conditions appeared in Guan [@G1] and Li [@Li90] respectively which correspond to the two cases in Theorem \[bd-th2\].
Case $(\mathbf{i})$: we set $$W = \sup_{(x, t) \in M_T} |\nabla u| \phi^{-\delta},$$ where $\phi = - u + \sup u + 1$ and $\delta < 1$ a positive constant. It suffices to estimate $W$. We may assume $W$ is achieved at $(x_{0}, t_{0}) \in \overline {M_T} - \mathcal{P} M_T$. Choose a smooth orthonormal local frame $e_1, \ldots, e_n$ about $x_0$ as before such that $\nabla_{e_i} e_j = 0$ at $x_0$. Assume $U (x_0, t_0)$ is diagonal. Differentiating the function $\log \omega - \delta \log \phi$ at $(x_0, t_0)$, where $\omega = |\nabla u|$, we obtain, $$\label{g1}
\frac{\nabla_i \omega}{\omega} - \frac{\delta \nabla_i \phi}{\phi} = 0, \; \mbox{ for every }i = 1, \ldots, n,$$ $$\label{g0}
\frac{\omega_t}{\omega} - \frac{\delta \phi_t}{\phi} \geq 0,$$ and $$\label{g2}
\begin{aligned}
0 \geq \frac{\nabla_{ii}\omega}{\omega} + \frac{(\delta - \delta^2) |\nabla_i \phi|^2}{\phi^2} - \frac{\delta \nabla_{ii} \phi}{\phi}.
\end{aligned}$$ Next, by and , $$\label{w}
\begin{aligned}\omega \nabla_{ii} \omega
= \;& (\nabla_{lii} u + R^k_{iil} \nabla_k u)\nabla_l u + \nabla_{il} u \nabla_{il} u - \frac{\delta^2 \omega^2 (\nabla_i \phi)^2}{\phi^2}\\
\geq \;& \nabla_l U_{ii} \nabla_l u - C|\nabla u|^2
- \frac{\delta^2 \omega^2 (\nabla_i \phi)^2}{\phi^2}.
\end{aligned}$$ Now multiply $F^{ii}$ on both side of and substitute in it. It follows that $$\label{w2}
\begin{aligned}
0 \geq \;& \frac{1}{\omega^2} F^{ii} \nabla_l u \nabla_l U_{ii} - C \sum F^{ii} - \frac{\delta F^{ii} \nabla_{ii} \phi}{\phi}\\
\;& + \frac{(\delta - 2\delta^2) }{\phi^2} F^{ii} |\nabla_i \phi|^2.
\end{aligned}$$ Now compute $F^{ii} \nabla_l u \nabla_l U_{ii}$ using . With we have $$\label{w3}
\begin{aligned}
F^{ii} \nabla_l u \nabla_l U_{ii} = \nabla_l u \psi_{x_l} + F^{\tau} \nabla_l u \nabla_l u_t
\geq - C |\nabla u| + \frac{\delta \omega^2}{\phi} F^{\tau} \phi_t
\end{aligned}$$ Therefore, in view of and , we obtain $$\label{w4}
\begin{aligned}
0 \geq \;& - C (1 + \sum F^{ii})
+\frac{\delta}{\phi} (F^{\tau} \phi_t - F^{ii} \nabla_{ii} \phi) + \frac{(\delta - 2\delta^2) }{\phi^2} \sum F^{ii} |\nabla_i \phi|^2.
\end{aligned}$$ Besides by , we have $$\label{w6}
F^{\tau} \phi_t - F^{ii} \nabla_{ii} \phi = \sum F^{ii}U_{ii} - F^{\tau} u_t - F^{ii} \chi_{ii} \geq - C (1 + \sum F^{ii} + F^\tau).$$
Without loss of generality, we may assume $|\nabla u (x_{0}, t_0)| \leq n \nabla_{1} u (x_{0}, t_0)$. By and noting that $U(x_0, t_0)$ is diagonal, we derive at $(x_0, t_0)$ $$\label{w5}
U_{11} = \sum_{k \geq 2} \frac{\nabla_k u \chi_{1k}}{\nabla_1 u} - \frac{\delta |\nabla u|^2}{2 \phi} + \chi_{11}
\leq C - \frac{\delta |\nabla u|^2}{2 \phi}.$$ Therefore, if $|\nabla u|$ is sufficiently large (otherwise we are done), by , we obtain $$F^{ii} (\nabla_i u)^2 \geq F^{11}|\nabla_1 u|^2 \geq \nu_0/n^2 |\nabla u|^2 (1 + \sum f_i + f_\tau).$$ Choosing $0 < \delta < \frac{1}{2}$ and substituting the above inequality and in , then it follows $|\nabla u(x_0, t_0)| \leq C$, and $C$ depends on $|\psi|_{C^1_x (\overline {M_T})}$, $|u|_{C^0 (\overline {M_T})}$, $K_0$ and other known data.
Case $(\mathbf{ii})$: Since $(M^n, g)$ has nonnegative sectional curvature, under an orthonormal local frame, i.e. $R^k_{iil} \nabla_k u \nabla_l u \geq 0$. In the proof of case $(\mathbf{i})$, we therefore have in place of , $$\label{w7}
\begin{aligned}
\omega \nabla_{ii} \omega \geq (\nabla_{lii} u + R^k_{iil} \nabla_k u)\nabla_l u
\geq \nabla_l u \nabla_l U_{ii} - C |\nabla u|.
\end{aligned}$$ Taking $\phi = u - {\underline}u + \sup |u - {\underline}u| + 1$ and $\delta < 1$, by , , and , we obtain $$\label{w8}
0 \geq \frac{\delta}{\phi} \mathcal{L} ({\underline}u - u) + \frac{(\delta - \delta^2) }{\phi^2} \sum F^{ii} |\nabla_i \phi|^2
- \frac{C}{|\nabla u|} (1 + \sum F^{ii})$$ Now if $\lambda[U(x_0, t_0), - u_t (x_0, t_0)] \geq R$, by Theorem \[3I-th3\], we may derive a bound for $|\nabla u (x_0, t_0)|$. If $\lambda[U(x_0, t_0), - u_t (x_0, t_0)] \leq R$, since $|u_t| \leq C$ where $C$ is independent of $|\nabla u|$, by , there exists a positive constant $C_0$ such that $\frac{1}{C_0} I \leq \{F^{ij}\} \leq C_0 I$, where $I$ is the unit matrix in the set of $n \times n$ symmetric matrices $\mathbb{S}^n$. It follows from that $$0 \geq \frac{(\delta - \delta^2) }{C_0 \phi^2 } |\nabla ({\underline}u- u)|^2 - \frac{C}{|\nabla u|} (1 + n C_0).$$ This proves $|\nabla u (x_0, t_0)| \leq C$ for $\delta < 1$. Thus Theorem \[bd-th2\] is completed.
**Acknowledgement:** The authors wish to thank Doctor Heming Jiao for many insightful suggestions and comments.
[99]{} B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differ. Eqns. 2 (1994), 151-171. G. Bao, W. Dong, H. Jiao, Regularity for an obstacle problem of Hessian equations on Riemannian manifolds, to appear in Journal of Differential Equations. L. A. Caffarelli, L. Nirenberg, J. Spruck, Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), 261-301. K. S. Chou, X.-J. Wang, Variational theory for Hessian equations, Comm. Pure Appl. Math. 54 (2001), 1029-1064. L. C. Evans, Classical solutions of fully nonlinear, convex, second order elliptic equations, Comm. Pure Appl. Math. 25 (1982), 333-363. C. Gerhardt, Closed Weingarten hypersurfaces in Riemannian manifolds, J. Differ. Geom. 43 (1996), 612-641. B. Guan, The Dirichlet problem for Hessian equations on Riemannian manifolds, Calc. Var. Partial Differ. Eqns. 8 (1999), 45-69. B. Guan, Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Math. J. 163 (2014), 1491-1524. B. Guan, H. Jiao, Estimates for a class of fully nonlinear elliptic equations on Riemannian manifolds, preprint. B. Guan, S. Shi, Z. Sui, On estimates for fully nonlinear parabolic equations on Riemannian manifolds, preprint. N. M. Ivochkina, O. A. Ladyzhenskaya, On parabolic equations generated by symmetric functions of the principal curvatures of the evolving surface or of the eigenvalues of the Hessian. Part I: Monge-Amp$\grave{e}$re equations, St. Petersburg Math. J. 6 (1995), 575-594. N. M. Ivochkina, O. A. Ladyzhenskaya, Flows generated by symmetric functions of the eigenvalues of the Hessian, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 221 (1995), 127-144, 258 (Russian). English transl. in J. Math. Sci. 87 (1997), 3353-3365. H. Jiao, Z. Sui, The first initial-boundary value problem for a class of fully nonlinear parabolic equations on Riemannian manifolds, Int. Math. Res. Not. DOI: 10.1093/imrn/rnu014. N. V. Krylov, Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation, Sibirsk. Math. Zh. 17 (1976), 290-303 (Russian). English transl. in Siberian Math. J. 17 (1976), 226-236. N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 1, 75-108. English transl. in Math. USSR-Izv 22 (1984), no. 1, 67-98. Y.-Y. Li, Some existence results of fully nonlinear elliptic equations of Monge-Ampère type, Comm. Pure Appl. Math. 43 (1990), 233-271. G. M. Lieberman, Second order parabolic differential equations, World Scientific Publ., Singapore, 1996. N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995), 151-164. J. I. E. Urbas, Hessian equations on compact Riemannian manifolds, Nonlinear Problems in Mathematical Physics and Related Topics, II, Kluwer/Plenum, New York, 2002, 367-377. X.-J. Wang, The $k$-Hessian equation, Geometric Analysis and PDEs. Lecture Notes in Math. Dordrecht: Springer, 2009: 177-252.
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---
abstract: 'The form of collinear gauge invariance for power suppressed operators in the soft-collinear effective theory is discussed. Using a field redefinition we show that it is possible to make any power suppressed ultrasoft-collinear operators invariant under the original leading order gauge transformations. Our manipulations avoid gauge fixing. The Lagrangians to ${\cal O}(\lambda^2)$ are given in terms of these new fields. We then give a simple procedure for constructing power suppressed soft-collinear operators in [SCET$_{\rm II}$ ]{}by using an intermediate theory [SCET$_{\rm I}$ ]{}.'
author:
- 'Christian W. Bauer'
- Dan Pirjol
- 'Iain W. Stewart'
title: On Power Suppressed Operators and Gauge Invariance in SCET
---
Introduction
============
The soft-collinear effective theory (SCET) has been proposed as a systematic approach for separating hard and soft scales in processes with energetic quarks and gluons [@bfl; @bfps; @cbis; @bps2]. The infrared physics is described in the effective theory in terms of collinear, soft, and ultrasoft fields with well defined momentum scaling. These fields are used to construct operators such as Lagrangians and currents that describe long distance effects, while hard corrections are contained in Wilson coefficients. This formalism builds on and extends earlier techniques used for discussing factorization [@reviews].
The degrees of freedom in SCET include collinear quarks $\xi_n$ and gluons $A_n^\mu$ with momentum scaling $p_c^\mu = (n\cdot p, {{\bar n}}\cdot p, p_\perp) \sim
Q(\lambda^2, 1, \lambda)$, soft modes $q_s, A_s^\mu$ with momenta $p_s^\mu \sim
Q\lambda$, and ultrasoft (usoft) modes $q_{us}, A_{us}^\mu$ with momenta $p_{\rm
us}^\mu \sim Q\lambda^2$. Here $Q$ is the hard scale, $\lambda\ll 1$ is the expansion parameter, and $n_\mu, {{\bar n}}_\mu$ are two light-cone unit vectors satisfying $n^2 = {{\bar n}}^2 = 0$ and $n\cdot {{\bar n}}= 2$. The explicit set of required fields may differ depending on the relevant scales in a given process. For instance, in Drell-Yan it is useful to have collinear fields for two light-like directions and for multijet-production more than two directions are required [@bfprs; @jets].
In many exclusive heavy meson decays to energetic light hadrons there are important effects at the scales $Q^2$, $Q\Lambda$, and $\Lambda^2$, where $\Lambda\sim 0.5\,{\rm GeV}$ is a hadronic scale. To correctly account for these effects, a sequence of two effective theories, SCET$_{\rm I}$ and SCET$_{\rm
II}$, can be used [@bps4].[^1] One thus distinguishes between $$\begin{aligned}
\label{SCET12}
&& \mbox{ SCET$_{\rm I}$}: \qquad\ \mbox{collinear fields with
$(p^+_c,p^-_c,p^\perp_c)\sim Q(\lambda^2,1,\lambda)$ and usoft fields}{\nonumber}\\
&& \mbox{\hspace{2.6cm} with
$p^\mu_{us} \sim Q\lambda^2$ where $\lambda \sim \sqrt{\Lambda/Q}$} {\nonumber}\\
&& \mbox{ SCET$_{\rm II}$}: \qquad \mbox{collinear fields with
$(p^+_c,p^-_c,p^\perp_c)\sim Q(\eta^2,1,\eta)$ and soft fields} {\nonumber}\\
&&\mbox{\hspace{2.6cm} with $p^\mu_s \sim Q\eta$ where
$\eta \sim \Lambda/Q$ \,. } {\nonumber}\end{aligned}$$ For clarity the power counting parameter $\eta$ is used for SCET$_{\rm II}$ rather than $\lambda$. In exclusive processes the energetic/soft hadrons are described by collinear/soft fields in SCET$_{\rm II}$. Both fields have $p_\perp\sim \Lambda$ which is appropriate for describing the constituents of hadrons of size $r_\perp \sim 1/\Lambda$. For exclusive processes the theory SCET$_{\rm I}$ plays an intermediate role by describing in a local way the fluctuations with $p^2\sim Q\Lambda$ that are involved in interactions between soft and collinear fields in SCET$_{\rm II}$. In contrast, SCET$_{\rm I}$ suffices for describing factorization in inclusive processes like $B\to
X_s\gamma$, as well as some exclusive processes like $B\to \gamma
e\nu$ [@bps2; @bgamenu]. Interactions in SCET$_{\rm II}$ are discussed in Refs. [@bps2; @HN] and power corrections in SCET$_{\rm I}$ were studied in Refs. [@chay; @chay2; @bpspc; @bcdf; @bps4; @ps1; @bf2; @ira]. Quark masses were considered in Ref. [@adam].
The symmetries of the effective theory provide an important guiding principle for constraining the form of operators, especially at the level of power corrections. The SCET has a rich symmetry structure, reflecting the interplay between the different length scales it describes. The constraints include power counting, collinear/soft/ultrasoft gauge invariance, reductions in spin structures, and a reparameterization invariance [@bfl; @bfps; @cbis; @bps2; @chay; @rpi; @bpspc] (see Ref. [@rev2] for a brief review of the symmetries). At a given order in $\lambda$ the most general set of operators for a given process can be constructed using
- [*Power counting:*]{} Restricts the type of fields and derivatives allowed in the operator
- [*Gauge invariance:*]{} Requires operators to be built out of gauge invariant building blocks.
- [*Reparameterization invariance:*]{} Corresponds to the restoration of Lorentz invariance order by order in $\lambda$.
- [*Locality:*]{} The theory [SCET$_{\rm I}$ ]{}is only non-local in ${\cal O}(Q)$ momenta. Only inverse powers of the large label momentum are allowed and collinear Wilson lines have to be built out of ${\cal O}(1)$ gluons.
Note that [SCET$_{\rm I}$ ]{}is constructed in a local manner, but after doing this it is useful to consider a field redefinition $\xi_n \to Y\xi_n$ which introduces non-locality at the usoft scale. The locality restriction does not apply to [SCET$_{\rm II}$ ]{}. Integrating out $p^2\sim Q\Lambda$ modes immediately results in operators involving the soft Wilson line $S$ [@bps2], and it contains inverse powers of $1/\Lambda$ momenta. In the following we will focus on gauge invariance and discuss subtleties which arise in constructing invariant operators at subleading order.
---------------------------------------------------------------------------------------------------------------------------------------------------
Object Collinear ${\cal U}_c$ Usoft $U_{us}$
--------- --------------- ------------------------------------------------------------------- ----------------------------------------------- -- --
$\xi_{n}$ ${\cal U}_c\ \xi_{n}$ $U_{us}\,\xi_{n}$
$gA_{n}^\mu$ ${\cal U}_c\: gA^\mu_{n}\: {\cal U}_c^{\dagger} + $U_{us}\, gA_{n}^\mu\, U_{us}^{\dagger}$
{\cal U}_c \big[i{{\cal D}}^\mu , {\cal U}_c^{\dagger} \big]$
$W$ ${\cal U}_c\, W$ $U_{us}\, W\, U_{us}^{\dagger}$
\[3pt\] $q_{us}$ $q_{us}$ $U_{us}\, q_{us}$
$gA_{us}^\mu$ $gA_{us}^\mu$ $U_{us} gA_{us}^\mu U^\dagger_{us} +
U_{us} [i\partial^\mu , U_{us}^{\dagger}]$
$Y$ $Y$ $U_{us}\, Y$
---------------------------------------------------------------------------------------------------------------------------------------------------
: Gauge transformations for the collinear and usoft fields from Ref. [@bps2], where $i{{\cal D}}^\mu \equiv \frac{n^\mu}{2}\,
{\bar {\cal P}}+ {\cal P}_\perp^\mu + \frac{{{\bar n}}^\mu}{2}\, i\, n{\!\cdot\!}D_{\rm us} $. The collinear fields and transformations are understood to have momentum labels and involve convolutions, but for simplicity these indices are suppressed. The usoft transformations do not change the momentum labels of collinear fields. []{data-label="table_gt"}
------------------------------------------------------------------------------------------------------------------------------------------------
Objects Collinear ${\cal U}_c$ Soft $U_s$
--------- --------------- ------------------------------------------------------------------ --------------------------------------------- -- --
$\xi_{n}$ ${\cal U}_c\ \xi_{n}$ $\xi_{n}$
$gA_{n}^\mu$ ${\cal U}_c\: gA^\mu_{n}\: {\cal U}_c^{\dagger} + $ gA_{n}^\mu $
{\cal U}_c \big[ i\partial_c^\mu {\cal U}_c^{\dagger} \big]$
$W$ ${\cal U}_c\, W$ $W$
\[3pt\] $q_{s}$ $q_{s}$ $U_s\, q_{s}$
$g A_{s}^\mu$ $g A_{s}^\mu$ $U_{s}\, g A_{s}^\mu U_s^{\dagger} +
U_s [ i \partial_s^\mu, U_s^{\dagger}]$
$S$ $S$ $U_s\, S$
------------------------------------------------------------------------------------------------------------------------------------------------
: Gauge transformations for collinear and soft fields in ${\rm SCET}_{\rm II}$ from Ref. [@bps2]. Momentum labels are suppressed, and $\partial_c^\mu$ and $\partial_s^\mu$ are defined to only pick out collinear and soft momenta respectively. Here $i\partial_c^\mu\ne i {\cal D}^\mu$ since usoft fields are not included in ${\rm SCET}_{\rm II}$.[]{data-label="table_gt2"}
The gauge transformations for the SCET fields were derived in [@bps2] and are summarized in Tables \[table\_gt\] and \[table\_gt2\]. Here $\partial_c^\mu {\cal U}_c \sim Q(\lambda^2,1,\lambda)$, $\partial_s^\mu U_s
\sim Q\lambda$, and $\partial^\mu U_{us} \sim Q\lambda^2$ distinguish the collinear, soft and usoft gauge transformations respectively. Partial derivatives without a subscript are usoft, so $i\partial_\mu\sim Q\lambda^2$. In Table I we have used $$\begin{aligned}
\label{Dc}
i{{\cal D}}^\mu \equiv \frac{n^\mu}{2}\, {\bar {\cal P}}+ {\cal P}_\perp^\mu +
\frac{{{\bar n}}^\mu}{2}\, i\, n{\!\cdot\!}D_{\rm us}\end{aligned}$$ in the fundamental representation. Note that only the $n {\!\cdot\!}A_{us}$ component of the usoft gauge field appears here and that the components of ${{\cal D}}^\mu$ have the same scaling in $\lambda$ as the collinear gluon field, so all transformations are homogeneous. Thus, power counting strongly constrains the leading usoft-collinear interactions. It also forces us to have a multipole expansion so that only the $n{\!\cdot\!}k$ momenta of collinear particles can be changed by interactions with usoft gluons. In Refs. [@bfl; @bfps; @cbis; @bps2] this expansion is done in momentum space while in Refs. [@bcdf; @HN; @bf2] it is done in position space. This leads to formulations of SCET whose operators appear slightly different, but whose final predictions for physical observables have to be the same.
In this paper we discuss how gauge invariance is realized for power suppressed operators in both ${\rm SCET}_{\rm I}$ and ${\rm SCET}_{\rm II}$. ${\rm
SCET}_{\rm I}$ is studied in Section \[sect1\] where we clarify the nature of collinear gauge invariance in power suppressed operators with ultrasoft derivatives. This is done by showing that it is possible to arrange these power suppressed operators such that only the original [*leading order*]{} gauge transformations are needed at any order in the power expansion. This was also the goal of a recent study by Beneke and Feldmann [@bf2] and a comparison is given with their results. The form of our transformed fields is different than theirs, reflecting a freedom in choice of viable field redefinitions. We found that it was not necessary to do any gauge fixing in our manipulations.
In SCET$_{\rm II}$ the soft and collinear gauge invariance alone allow a large number of operators, reflecting the more non-local nature of this theory. In particular, gauge invariance does not uniquely fix the path of the Wilson lines. However, since SCET$_{\rm II}$ is matched on from SCET$_{\rm I}$ and not from full QCD, one can obtain information about the operators relevant for a given process from the structure of operators in SCET$_{\rm I}$. We illustrate the SCET$_{\rm I}$$\to$ SCET$_{\rm II}$ matching by several examples in Section \[sect2\].
Gauge Invariance in SCET$_{\rm I}$ {#sect1}
==================================
At leading order the SCET$_{\rm I}$ Lagrangian for collinear quarks is [@bfps; @cbis] $$\begin{aligned}
\label{L0}
{\cal L}_{\xi\xi}^{(0)}
&=& \bar \xi_n \left[ i n{\!\cdot\!}D
+ i \Dslash^\perp_c W \frac{1}{{\bar {\cal P}}} W^\dagger
i \Dslash^\perp_c\right] \frac{\bnslash}{2} \xi_n \,,\end{aligned}$$ where the collinear covariant derivatives are $iD_c^\mu = {\cal P}^\mu +
gA_n^\mu$ with label operators ${{\cal P}}^\mu$, the full derivative $in{\!\cdot\!}D=in{\!\cdot\!}\partial + g n{\!\cdot\!}A_{us} + gn{\!\cdot\!}A_n$, and the Wilson line $W$ is built out of ${{\bar n}}{\!\cdot\!}A_n$ fields where $f(i{{\bar n}}{\!\cdot\!}D_c) = W f({\bar {\cal P}})
W^\dagger$ $$\begin{aligned}
\label{W}
W = \Big[ \sum_{\rm perms} \exp\Big( -\frac{g}{{\bar {\cal P}}}\:
{{\bar n}}{\!\cdot\!}A_{n,q}(x) \ \Big) \Big] \,.\end{aligned}$$ Under the gauge transformations in Table \[table\_gt\] covariant derivatives acting in the fundamental representation transform under collinear and usoft transformations as $$\begin{aligned}
\label{Dtrans1}
{\cal U}_c &:& \quad in{\!\cdot\!}D \to {\cal U}_c\, in{\!\cdot\!}D \, {\cal
U}_c^{\dagger}\,,
\phantom{{}_c}\qquad
iD_c^\perp \to {\cal U}_c\, i D_c^\perp {\cal U}_c^{\dagger}\,,\qquad
i{{\bar n}}{\!\cdot\!}D_c \to {\cal U}_c\, i{{\bar n}}{\!\cdot\!}D_c \, {\cal U}_c^{\dagger}\,, \\
U_{us} &:& \quad in{\!\cdot\!}D \to U_{us}\, in{\!\cdot\!}D \, U_{us}^{\dagger}\,,\qquad
iD_c^\perp \to U_{us}\, i D_c^\perp U_{us}^{\dagger}\,,\qquad
i{{\bar n}}{\!\cdot\!}D_c \to U_{us}\, i{{\bar n}}{\!\cdot\!}D_c \, U_{us}^{\dagger}\,.{\nonumber}\end{aligned}$$ It is straightforward to verify that all factors of ${\cal U}_c$ or $U_{us}$ drop out of ${\cal L}_{\xi\xi}^{(0)}$, which has been shown to be the most general possible operator consistent with gauge invariance, power counting, and reparameterization invariance [@bps2; @rpi]. The same is true of the leading order collinear gluon action $$\begin{aligned}
\label{Lcg}
{\cal L}_{cg}^{(0)} &=& \frac{1}{2 g^2}\, {\rm tr}\ \bigg\{
\Big[i{{\cal D}}^\mu +g A_{n,q}^\mu \,, i{{\cal D}}^\nu + g A_{n,q'}^\nu \Big] \bigg\}^2 \\
&& +2\, {\rm tr} \Big\{ \, \bar c_{n,p'}\, \Big[ i{{\cal D}}_\mu , \Big[ i{{\cal D}}^\mu +
g A_{n,q}^\mu \,, c_{n,p}\,\Big]\Big] \Big\}
+ \frac{1}{\alpha}\, {\rm tr}\ \Big\{ [i{{\cal D}}_\mu\,, A_{n,q}^\mu]\Big\}^2\,. {\nonumber}\end{aligned}$$ The terms on the second line are the gauge fixing terms for a general covariant gauge, where $c_n$ are adjoint ghost fields.
Beyond leading order the form of the subleading Lagrangians can be determined by matching calculations and use of the SCET symmetries. There is a reparameterization invariance [@LM] (RPI), which in SCET is due to the freedom in choosing the basis vectors $n$ and ${{\bar n}}$, and in decomposing the momenta ${{\bar n}}{\!\cdot\!}(p+k)$ and $(p_\perp^\mu+k_\perp^\mu)$ into collinear $p$ and usoft $k$ components [@chay; @rpi]. This RPI connects collinear and usoft derivatives, $$\begin{aligned}
\label{rpid}
{\bar {\cal P}}+ i{{\bar n}}{\!\cdot\!}\partial\,,\qquad {{\cal P}}_\perp^\mu+i\partial_\perp^\mu\,,\end{aligned}$$ and also relates the Wilson coefficients of leading and subleading operators [@chay; @rpi; @bcdf; @ps1].
To turn the derivatives in Eq. (\[rpid\]) into covariant derivatives we make use of gauge symmetry. This forces the label operator to be replaced by the collinear covariant derivative $iD_c^\mu$, but as we shall see it allows some freedom in the usoft term [@chay2]. In Refs. [@chay; @rpi] the usoft derivative was made covariant with the choice $iD_{us}^\mu$, so the RPI combinations in Eq. (\[rpid\]) become $$\begin{aligned}
\label{c1}
&& \mbox{choice i)}\qquad\qquad
i{{\bar n}}{\!\cdot\!}D= i{{\bar n}}{\!\cdot\!}D_c + i{{\bar n}}{\!\cdot\!}D_{us}\,,\qquad
i D_\perp^\mu = i D_{c,\perp}^\mu + iD_{us,\perp}^\mu \,.\qquad\qquad\qquad\end{aligned}$$ For the purpose of gauge transformations this corresponds to promoting the ultrasoft field to a full background field of a quantum collinear gauge field so that $$\begin{aligned}
\label{gauge1}
gA_n^\mu \to {\cal U}_c\, gA_n^\mu \, {\cal U}_c^{\dagger}
+ {\cal U}_c [{\cal P}^\mu + iD^\mu_{\rm us}\,, {\cal U}_c^{\dagger}]\,,\end{aligned}$$ and the combined field $A^\mu = A_n^\mu + A_{\rm us}^\mu$ transforms as $$\begin{aligned}
gA^\mu \to {\cal U}_c gA^\mu {\cal U}_c^{\dagger} + {\cal U}_c [{\cal P}^\mu
+ i\partial_{\rm us}^\mu\,, {\cal U}_c^{\dagger}] \,.\end{aligned}$$ With this choice one still has homogeneous gauge transformations in Table \[table\_gt\] at leading order, which we will call $G^{(0)}$, however one also induces subleading collinear transformations for $A_n^\perp$ and ${{\bar n}}{\!\cdot\!}A_n$ suppressed by $\lambda$ and $\lambda^2$ respectively $$\begin{aligned}
G^{(1)}:\qquad A^\mu_{n,\perp} \to {\cal U}_c [iD_{\perp,us}^\mu ,
{\cal U}_c^\dagger ] \,,
\qquad {{\bar n}}{\!\cdot\!}A_n \to {\cal U}_c [i{{\bar n}}{\!\cdot\!}D_{us} , {\cal
U}_c^\dagger ] \,.\end{aligned}$$ Thus, much like the reparameterization invariance, there are gauge transformations that connect the leading and subleading terms. This observation was first made in Ref. [@chay2]. For example, using the gauge completion given in Eq. (\[c1\]) the $O(\lambda)$ Lagrangian is $$\begin{aligned}
\label{L1}
{\cal L}_{\xi\xi}^{(1)} &=& \bar \xi_n \Big[ i \Dslash^\perp_{\rm us}
\frac{1}{{{\bar n}}{\!\cdot\!}iD_c} i \Dslash^\perp_c
+ i \Dslash^{\perp}_c \frac{1}{{{\bar n}}{\!\cdot\!}iD_c} i \Dslash^\perp_{us}\Big]
\frac{\bnslash}{2} \xi_n\,.\end{aligned}$$ Under a collinear gauge transformation $G^{(0)}$ from Table \[table\_gt\] one finds $$\begin{aligned}
{\cal L}^{(1)} \to {\cal L}^{(1)} - \bar \xi_n \left[
[i\Dslash^\perp_{\rm us}, {\cal U}_c^\dagger] {\cal U}_c
\frac{1}{{{\bar n}}{\!\cdot\!}iD_c}
i \Dslash^\perp_c + i \Dslash^{\perp}_c \frac{1}{{{\bar n}}{\!\cdot\!}iD_c}
{\cal U}_c^\dagger [i\Dslash^\perp_{us}, {\cal U}_c] \right]
\frac{\bnslash}{2} \xi_n\,. \end{aligned}$$ The second term cancels against the $G^{(1)}$ variation of the leading order Lagrangian ${\cal L}_{\xi\xi}^{(0)}$, implying that the effective Lagrangian is invariant up to this order. The other subleading actions with usoft fields are [@bcdf; @bps4; @feldmann; @ps1] $$\begin{aligned}
\label{subLs}
{\cal L}^{(2a )}_{\xi q} &=& \bar\xi_n \frac{1}{i{{\bar n}}{\!\cdot\!}D_c}\:
ig \big\{ \Mslash_\perp + \frac{\bnslash}{2} n{\!\cdot\!}M \big\}
\, W \, q_{us} \mbox{ + h.c.} \,, {\nonumber}\\
{\cal L}_{cg}^{(1)} &=& \frac{2}{g^2}\: {\rm tr}
\Big\{ \big[i {D}_0^\mu , iD_c^{\perp\nu} \big]
\big[i {D}_{0\mu} , iD_{us\,\nu}^\perp \big] \Big\}\,,\\
{\cal L}_{cg}^{(2)} &=& \frac{1}{g^2}\: {\rm tr}
\Big\{ \big[i {D}_0^\mu , iD_{us}^{\perp\nu} \big]
\big[i {D}_{0\mu} , iD_{us\,\nu}^\perp \big] \Big\}
+ \frac{1}{g^2}\: {\rm tr}
\Big\{ \big[i D_{us}^{\perp\mu} , iD_{us}^{\perp\nu} \big]
\big[i {D}_{c\mu}^\perp , i{D}_{c\nu}^\perp \big] \Big\}{\nonumber}\\
&+&\frac{1}{g^2}\: {\rm tr}
\Big\{ \big[i {D}_0^\mu , i n{\!\cdot\!}D \big]
\big[i {D}_{0\mu} , i {{\bar n}}{\!\cdot\!}D_{us} \big] \Big\}
+ \frac{1}{g^2}\: {\rm tr}
\Big\{ \big[i D_{us}^{\perp\mu} , iD_{c}^{\perp\nu} \big]
\big[i {D}_{c\mu}^\perp , i{D}_{us\nu}^\perp \big] \Big\}{\nonumber}\,,\end{aligned}$$ where $i g M^\mu = [i{{\bar n}}{\!\cdot\!}D_c, i D_{us}^\mu + {{\bar n}}^\mu gn{\!\cdot\!}A_n /2 ]$ and $iD_0^\mu = iD_c^\mu + i {{\bar n}}^\mu n{\!\cdot\!}D_{us}/2$. (The terms ${\cal
L}_{\xi q}^{(1)}$ and ${\cal L}_{\xi q}^{(2b)}$ do not depend on ultrasoft covariant derivatives and are shown below in Eq. (\[Lxiq\]).) Similar manipulations show that the results in Eq. (\[subLs\]) are invariant with terms canceled by the $G^{(1)}$ transformation of ${\cal L}_{\xi q}^{(1)}$ and ${\cal L}_{cg}^{(0,1)}$.
Although operators with usoft fields are gauge invariant, the presence of $G^{(1)}$ requires transformations of operators at different powers in $\lambda$ to cancel one another. This is unsatisfactory since constraining operators at any particular order requires transforming lower order operators. Furthermore this would mean we would only be able to assign an unambiguous meaning to the sum of leading and subleading matrix elements. Instead, we would like to use fields with no $G^{(1)}$ transformation, so that operators are manifestly invariant under $G^{(0)}$ at each order in $\lambda$. In other words the terms at a given order are invariant without needing the transformation of lower order terms. To this end, consider the field redefinitions $$\begin{aligned}
\label{redef}
g {{\bar n}}{\!\cdot\!}\hat A_n = g{{\bar n}}{\!\cdot\!}A_n -
{\cal W}[i{{\bar n}}{\!\cdot\!}D_{us}, {\cal W}^\dagger ]\,,\qquad
g \hat A_n^\perp = g A_n^\perp -
{\cal W}[i D_{us}^\perp, {\cal W}^\dagger ]\,,\qquad\end{aligned}$$ where $g n{\!\cdot\!}\hat A_n = gn{\!\cdot\!}A_n$, and $\hat A_n^\mu$ are new collinear gluon fields. Here ${\cal W}$ is the product of Wilson lines defined in Ref. [@bcdf] which in position space is $$\begin{aligned}
\label{cW}
{\cal W}(x) &=& P \exp\Big( ig \int_{-\infty}^0\!\!\!\!\! ds\,
{{\bar n}}{\!\cdot\!}(A_n \!+\!A_{us})({{\bar n}}s\!+\!x) \Big)
\bigg[ P \exp\Big( i g \int_{-\infty}^0 \!\!\!\!\! ds\,
{{\bar n}}{\!\cdot\!}A_{us}({{\bar n}}s\!+\!x) \Big)\bigg]^\dagger .\end{aligned}$$ In Eq. (\[cW\]) the collinear fields $A_n^\mu(X\!+\!x)$ are the Fourier transforms of $A_{n,p}^\mu(x)$ with $X$ the conjugate variable to $p$. Under collinear gauge transformations ${\cal W}\to U_c {\cal W}$, while under usoft gauge transformations ${\cal W}\to U_{us} {\cal W} U_{us}^\dagger$. The presence of ${\cal W}$ in Eq. (\[redef\]) causes $\hat A_n$ to be defined in terms of a non-linear function of $A_n$. Note that our transformation in Eq. (\[redef\]) differs from that in Ref. [@bf2], as we discuss in more detail below. Under a collinear gauge transformation the $\perp$ component of the new collinear gluon field transforms as (suppressing momentum space labels) $$\begin{aligned}
g \hat A^\perp_n &\to & {\cal U}_c\, g A_n^\perp {\cal U}_c^\dagger
+ {\cal U}_c [{{\cal P}}_\perp + i D_{us}^\perp , {\cal U}_c^\dagger ]
- {\cal U}_c {\cal W} [ i D^\perp_{us}, {\cal W}^\dagger {\cal U}_c^\dagger]
\nonumber\\
&=& {\cal U}_c\, g A_n^\perp {\cal U}_c^\dagger + {\cal U}_c {{\cal P}}^\perp
{\cal U}_c^\dagger
+ {\cal U}_c\, iD^\perp_{us} {\cal U}_c^\dagger
- {\cal U}_c\, {\cal W} iD^\perp_{us} {\cal W}^\dagger {\cal U}_c^\dagger {\nonumber}\\
&=& {\cal U}_c g\hat A_n^\perp {\cal U}_c^\dagger + {\cal U}_c {{\cal P}}_\perp
{\cal U}_c^\dagger \,.\end{aligned}$$ Only hatted fields appear in the final result. With a similar set of steps we find $g {{\bar n}}{\!\cdot\!}\hat A_n \to {\cal U}_c\, g {{\bar n}}{\!\cdot\!}\hat A_n {\cal
U}_c^\dagger + {\cal U}_c {\bar {\cal P}}\, {\cal U}_c^\dagger$. Therefore $$\begin{aligned}
\label{gthat}
g\hat A_n^\mu \to {\cal U}_c\: g\hat A_n^\mu\, {\cal U}_c^\dagger
+ {\cal U}_c\, [i{\cal D}^\mu\,, {\cal U}_c^\dagger]\,,\end{aligned}$$ just like in Table \[table\_gt\]. Thus, in terms of the hatted fields, transformations that involve suppressed terms like $G^{(1)}$ never appear. This is the desired result.
To express the Lagrangians in terms of hatted fields it is useful to have the inverse transformation to Eq. (\[redef\]). This is complicated by the factors of ${\cal W}={\cal W}[{{\bar n}}{\!\cdot\!}A_n, {{\bar n}}{\!\cdot\!}A_{us}]$ given in Eq. (\[redef\]), which depend non-linearly on the gluon fields. Now, we know that $$\begin{aligned}
i{{\bar n}}{\!\cdot\!}D\: {\cal W}
= {\cal W} g {{\bar n}}{\!\cdot\!}A_{us} \,,\end{aligned}$$ which implies that in terms of the hatted fields ${\cal W}={\cal W}[{{\bar n}}{\!\cdot\!}\hat A_n, {{\bar n}}{\!\cdot\!}A_{us}]$ satisfies the equation $$\begin{aligned}
&& 0 = ( i {{\bar n}}{\!\cdot\!}\hat D_c + {\cal W} i{{\bar n}}{\!\cdot\!}D_{us} {\cal W}^\dagger )
{\cal W} - {\cal W} g{{\bar n}}{\!\cdot\!}A_{us}
= i{{\bar n}}{\!\cdot\!}\hat D_c\: {\cal W}\,. \end{aligned}$$ However, this equation has a unique solution $\hat W$. Switching to momentum labels and residual coordinates $x$ [@cbis], this $\hat W$ is just the standard Wilson line in Eq. (\[W\]) expressed in terms of the ${{\bar n}}{\!\cdot\!}\hat
A_n$ collinear field (since they are defined by the same equation). This gives the remarkable result that after the field redefinition we have to all orders in $\lambda$ $$\begin{aligned}
{\cal W} = \hat W = \Big[ \sum_{\rm perms} \exp\Big( -\frac{g}{{\bar {\cal P}}}\:
{{\bar n}}{\!\cdot\!}\hat A_{n,q}(x) \ \Big) \Big] \,,\end{aligned}$$ which is independent of the usoft gauge field. Under the gauge transformations $\hat W\to U_c\hat W$ and $\hat W\to U_{us} \hat W U_{us}^\dagger$ just like we had for $W$. Thus, the inverse transformation to Eq. (\[redef\]) can be written $$\begin{aligned}
\label{redef2}
g {{\bar n}}{\!\cdot\!}A_n = g{{\bar n}}{\!\cdot\!}\hat A_n +
\hat W [i{{\bar n}}{\!\cdot\!}D_{us}, {\hat W}^\dagger ]\,,\qquad
g A_n^\perp = g \hat A_n^\perp +
{\hat W}[i D_{us}^\perp, {\hat W}^\dagger ]\,,\qquad\end{aligned}$$ This corresponds to gauging the RPI combinations in Eq. (\[rpid\]) to $$\begin{aligned}
\label{c2}
&& \mbox{choice ii)}\qquad\quad
i{{\bar n}}{\!\cdot\!}\hat D = i{{\bar n}}{\!\cdot\!}\hat D_c + {\hat W} i{{\bar n}}{\!\cdot\!}D_{us}
{\hat W}^\dagger \,,\qquad
i \hat D_\perp^\mu = i \hat D_{c,\perp}^\mu + {\hat W} iD_{us,\perp}^\mu
{\hat W}^\dagger \,,\qquad\qquad\end{aligned}$$ rather than using choice i) in Eq. (\[c1\]). Under a collinear and usoft gauge transformations these derivatives transform exactly as in Eq. (\[Dtrans1\]) $$\begin{aligned}
\label{Dtrans2}
{\cal U}_c :&& in{\!\cdot\!}\hat D \to {\cal U}_c\, in{\!\cdot\!}\hat D \,
{\cal U}_c^\dagger\,, \phantom{{}_c}\qquad
i\hat D_c^\perp \to {\cal U}_c\, i \hat D_c^\perp {\cal U}_c^\dagger\,,\qquad
i{{\bar n}}{\!\cdot\!}\hat D_c \to {\cal U}_c\, i{{\bar n}}{\!\cdot\!}\hat D_c\, {\cal U}_c^\dagger\,,
\\
U_{us} :&& in{\!\cdot\!}\hat D \to U_{us}\, in{\!\cdot\!}\hat D\, U_{us}^\dagger\,,\qquad
i\hat D_c^\perp \to U_{us}\, i \hat D_c^\perp U_{us}^\dagger\,,\qquad
i{{\bar n}}{\!\cdot\!}\hat D_c \to U_{us}\, i{{\bar n}}{\!\cdot\!}\hat D_c\, U_{us}^\dagger\,.{\nonumber}\end{aligned}$$
In Ref. [@bf2] transformations were also made with the aim of determining fields that could be used in power suppressed operators while avoiding gauge transformations that mix different orders in $\lambda$. Similar to the construction here their initial fields transform as in Eq. (\[gauge1\]) and the desired final collinear transformations are identical to the form in Ref. [@bps2], shown in our Table \[table\_gt\]. In Ref. [@bf2] the new collinear quark and gluon fields were defined as $$\begin{aligned}
\label{BF}
\xi_n &=& R W_c^\dagger \hat \xi_n \,, \\
g A_{\perp c} &=& R \Big( W_c^\dagger i \hat D_{\perp c} W_c^\dagger -
i\partial_c^\perp \Big) R^\dagger\,,\nonumber\\
gn{\!\cdot\!}A_{c} &=& R \Big( W_c^\dagger i n{\!\cdot\!}\hat D W_c -
i n{\!\cdot\!}D_{us}({{\bar n}}{\!\cdot\!}x n/2)\Big) R^\dagger \,,\nonumber \end{aligned}$$ where the fields on the left-hand side are understood to be in a light-like axial gauge with ${{\bar n}}{\!\cdot\!}A_c=1$. The matrix $R$ is defined as $R(x) = P\exp( ig \int_C dz_\mu A_{\rm us}^\mu(z))$ with the path $C$ a straight line connecting $\frac12 {{\bar n}}_\mu n{\!\cdot\!}x$ to $x$. In Ref. [@bf2] the collinear fields were constructed entirely in position space, and a multipole expansion was performed on the usoft fields $\phi_{us}(x)
= \phi_{us}(x_-) + (x_\perp \cdot i\partial_\perp) \phi_{us}(x_-) + \ldots$. The transformation with the matrix $R$ was then necessary to connect collinear and usoft fields which are at different space-time points. After inserting these fields into the effective Lagrangian, operators involving the matrix $R$ were expanded using the Fock-Schwinger gauge for the ultrasoft gluon field.
The results in Eq. (\[BF\]) differ from our field transformation in Eq. (\[redef2\]) in several respects. First, we did not need to redefine the collinear quark field $\xi_{n,p}(x)$ since our labeled collinear fields carry residual ultrasoft momentum through their $x$ dependence. For the gluons our transformation changes ${{\bar n}}\cdot A_n$ but not the $n\cdot A_n$ field, whereas Eq. (\[BF\]) does the exact opposite. For the $A_n^\perp$ field our hatted field is not surrounded by W’s, and we have a covariant usoft derivative while Eq. (\[BF\]) has a normal derivative. The fact that both our usoft and collinear fields are local in the coordinate $x$ representing residual momenta $k^\mu\sim Q\lambda^2$ means that we did not need to consider a matrix like $R$. Also, note that in our procedure for transforming the fields we did not require any gauge fixing at intermediate steps. Finally, we comment that the form of our field redefinition leads to an interesting result for ${\cal W}$ in terms of the new fields, namely ${\cal W}=\hat W$ with no higher order terms in $\lambda$.
The use of position and momentum space makes a more direct comparison difficult. However, any field redefinitions that lead to the desired result are equally valid and both Eq. (\[BF\]) and Eq. (\[redef2\]) satisfy this criteria. In general one knows that field redefinitions should only affect the form of operators and the result for Green’s functions, but should not affect S-matrix elements. Thus, equivalent effective theories are often realized with different fields. We expect that there should be a field redefinition which would relate our fields $\hat A_n$ to the fields $\hat A_n$ in Ref. [@bf2], although we have not constructed it in closed form.
\
Having established collinear gauge fields whose transformations never mix orders in $\lambda$, we now rewrite all subleading Lagrangians to order $\lambda^2$ using Eq. (\[redef2\]). For simplicity we omit the hats in the following equations, however all collinear gauge fields should be understood to be the hatted ones. For the collinear quark Lagrangian we find $$\begin{aligned}
\label{Lxxnew}
{\cal L}_{\xi\xi}^{(1)}
&=& \big(\bar \xi_n W\big)\, i\Dslash^\perp_{us} \frac{1}{{\bar {\cal P}}} \big(W^\dagger
i \Dslash^\perp_c \frac{\bnslash}{2} \xi_n \big)
+ \big( \bar \xi_n i \Dslash^{\perp}_c W\big) \frac{1}{{\bar {\cal P}}} i\Dslash^\perp_{us}
\big( W^\dagger \frac{\bnslash}{2} \xi_n \big){\nonumber}\\
{\cal L}_{\xi\xi}^{(2)} &=& \big( \bar \xi_n W\big)
i\Dslash_{us}^\perp \frac{1}{{\bar {\cal P}}} i\Dslash_{us}^\perp \frac{\bnslash}{2}
\big( W^\dagger \xi_n\big) +
\big( \bar \xi_n i\Dslash^\perp_c W\big) \frac{1}{{\bar {\cal P}}^2} i{{\bar n}}{\!\cdot\!}D_{us}
\frac{\bnslash}{2}\big( W^\dagger i\Dslash^\perp_c \xi_n\big) \,, $$ where we have used the fact that $$\begin{aligned}
\frac{1}{i{{\bar n}}{\!\cdot\!}D} = \frac{1}{i{{\bar n}}{\!\cdot\!}D_c}
- W \frac{1}{{\bar {\cal P}}^2} i{{\bar n}}{\!\cdot\!}D_{us} W^\dagger + \ldots \,.\end{aligned}$$ It is easy to see that the results in Eq. (\[Lxxnew\]) are invariant under the transformations in Table \[table\_gt\]. For the mixed collinear-usoft quark interactions we find the invariant results $$\begin{aligned}
\label{Lxiq}
{\cal L}^{(1)}_{\xi q} &=& \bar\xi_n \: \frac{1}{i{{\bar n}}{\!\cdot\!}D_c}\:
ig\, \Bslash_\perp^c W q_{us} \mbox{ + h.c.}\,,{\nonumber}\\
{\cal L}^{(2a )}_{\xi q} &=& \bar\xi_n \frac{\bnslash}{2}
\frac{1}{i{{\bar n}}{\!\cdot\!}D_c}\:
ig\, n{\!\cdot\!}M \, W \, q_{us} \mbox{ + h.c.} \,, {\nonumber}\\
{\cal L}^{(2b)}_{\xi q} &=& \bar\xi_n \frac{\bnslash}{2}
i\Dslash_\perp^{\,c} \frac{1}{(i{{\bar n}}{\!\cdot\!}D_c)^2}\: ig\, \Bslash_\perp^c W
\: q_{us} \mbox{ + h.c.}\,, $$ where $ig \Bslash_\perp^c =[i{{\bar n}}{\!\cdot\!}D^c,i\Dslash_\perp^{c}]$ and we have used the fact that the transformation of ${\cal L}_{\xi q}^{(1)}$ makes $$\begin{aligned}
ig M_\perp^\mu &=& [i{{\bar n}}{\!\cdot\!}D_c , W iD_{us\perp}^\mu W^\dagger ]
= [ W {\bar {\cal P}}W^\dagger, W i D_{us\perp}^\mu W^\dagger ]
= W [{\bar {\cal P}}, i D_{us\perp}^\mu ] W^\dagger = 0 \,.\end{aligned}$$ Finally, for the subleading terms in the mixed usoft-collinear gluon action we find $$\begin{aligned}
{\cal L}_{cg}^{(1)} &=& \frac{2}{g^2}\: {\rm tr}
\Big\{ \big[i {D}_0^\mu , iD_c^{\perp\nu} \big]
\big[i {D}_{0\mu} , W iD_{us\,\nu}^\perp W^\dagger \big] \Big\}\,,\\
{\cal L}_{cg}^{(2)} &=& \frac{1}{g^2}\: {\rm tr}
\Big\{ \big[i {D}_0^\mu , W iD_{us}^{\perp\nu} W^\dagger \big]
\big[i {D}_{0\mu} , W iD_{us\,\nu}^\perp W^\dagger \big] \Big\} {\nonumber}\\
&&\!\!\!\!\!\!\!\!\!\!\!
+\frac{1}{g^2}\: {\rm tr}
\Big\{ W \big[ i D_{us}^{\perp\mu} , iD_{us}^{\perp\nu} \big] W^\dagger
\big[i {D}_{c\mu}^\perp , i{D}_{c\nu}^\perp \big] \Big\}
+\frac{1}{g^2}\: {\rm tr}
\Big\{ \big[i {D}_0^\mu , i n{\!\cdot\!}D \big]
\big[i {D}_{0\mu} , W i {{\bar n}}{\!\cdot\!}D_{us} W^\dagger \big] \Big\} {\nonumber}\\
&&\!\!\!\!\!\!\!\!\!\!\!
+ \frac{1}{g^2}\: {\rm tr}
\Big\{ \big[W i D_{us}^{\perp\mu} W^\dagger , iD_{c}^{\perp\nu} \big]
\big[i {D}_{c\mu}^\perp , W i{D}_{us\nu}^\perp W^\dagger \big]
\Big\}{\nonumber}\,,
$$ where $i D_0^\mu = i{\cal D}^\mu + g A_{n}^\mu$.
Power Suppressed Soft-Collinear Operators {#sect2}
=========================================
In SCET$_{\rm II}$ the structure of operators with soft and collinear fields is still constrained by properties such as power counting, gauge invariance, and reparameterization invariance. However the non-local nature of the theory makes it more difficult to simply write down the most general operators in an arbitrary case. To see this we consider a simple example, namely a heavy-to-light current. In the full theory we have $\bar q \Gamma b$ and in the effective theory $$\begin{aligned}
\label{heavylight}
C({\bar {\cal P}})\: \bar \xi_n W \Gamma S^\dagger h_v. \end{aligned}$$ The Wilson lines $W$ and $S$ are required to ensure collinear and soft gauge invariance respectively. However, neither gauge invariance nor power counting determines the exact path of $S$ from $x$ to $\infty$, since all $A_s^\mu$ fields scale the same way. Thus, additional input is needed to constrain these operators. From direct matching calculations, which integrate out fluctuations with $p^2\sim Q\Lambda$, it is straightforward to determine that $S$ is a straight Wilson line along the $n$ direction built out of $n{\!\cdot\!}A_s$ fields [@bps2]. An alternative procedure is [@bps4] $$\begin{aligned}
&& \mbox{ i)\phantom{ii}\quad Match QCD onto SCET$_{\rm I}$ at a scale
$\mu^2\sim Q^2$ (with $p_c^2\sim Q\Lambda$)}{\nonumber}\\
&& \mbox{ ii)\phantom{i}\quad Factorize the usoft-collinear interactions
with the field redefinitions, } {\nonumber}\\
&& \mbox{\hspace{2cm}
$\xi_n = Y\xi_n^{(0)}$ and $A_n^\mu = Y A_n^{(0)\mu} Y^\dagger $.}{\nonumber}\\
&& \mbox{ iii)\quad Match SCET$_{\rm I}$ onto SCET$_{\rm II}$ at a scale
$\mu^2\sim Q\Lambda$ (with $p_c^2\sim \Lambda^2$) } \,. {\nonumber}\qquad\qquad\qquad\end{aligned}$$ For the heavy-to-light case we have i) $\bar q \Gamma b \to C({\bar {\cal P}})\: \bar \xi_n
W \Gamma\: h_v^{us}$, and then ii) $C({\bar {\cal P}})\: \bar \xi_n W \Gamma\: h_v^{us} =
C({\bar {\cal P}})\: \bar \xi_n^{(0)} W^{(0)} \Gamma\: Y^\dagger h_v^{us}$. For the final step we rename the usoft fields as soft fields $Y^\dagger h_v^{us}=S^\dagger
h_v^{s}$, and then lower the offshellness of the collinear fields. Since the leading collinear Lagrangians in [SCET$_{\rm I}$ ]{}and [SCET$_{\rm II}$ ]{}are the same all possible time-ordered products agree exactly and we can simply replace $C({\bar {\cal P}})\: \bar
\xi_n^{(0)} W^{(0)}\to C({\bar {\cal P}})\: \bar \xi_n^{\rm II} W^{\rm II}$. The final result is identical to Eq. (\[heavylight\]) but the steps are simpler than those carried out in the appendix of Ref. [@bps2]. From the two-step approach it is also clear why the Wilson coefficient does not pick up any dependence on the soft momentum in this example.
The two-stage matching procedure becomes even more useful in cases where [SCET$_{\rm I}$ ]{}contains time-ordered products, since these can induce non-trivial jet functions involving $p^2\sim Q\Lambda$ fluctuations. SCET$_{\rm I}$ gives a well defined set of Feynman rules for computing these jet functions at tree level and in loops, and does so in a manner independent from the computation of Wilson coefficients at the hard scale $p^2\sim Q^2$. Since the operator in [SCET$_{\rm I}$ ]{}is a time-ordered product we are guaranteed that the running to the scale $\mu^2=Q\Lambda$ is determined by that of the product of the hard Wilson coefficients. A final benefit is that power counting in [SCET$_{\rm I}$ ]{}constrains the allowed scaling of operators in [SCET$_{\rm II}$ ]{}, and in particular, places a limit on the number of factors of $1/\Lambda$ that can be induced from $1/(Q\Lambda)$ terms as we discuss below. This provides a complementary procedure to constraining the powers of $1/\Lambda$ with reparameterization invariance as first described in Ref. [@HN].
Let us consider a generic matching calculation $$\begin{aligned}
\label{M12}
{\rm SCET}_{\rm I}\ [ p_c^2 \sim Q\Lambda\,, p_{us}^2\sim \Lambda^2]
\ \ \stackrel{\mu^2\sim Q\Lambda}{\longrightarrow} \ \
{\rm SCET}_{\rm II}\ [ p_c^2 \sim \Lambda^2\,, p_s^2\sim\Lambda^2] \,.\end{aligned}$$ First construct all time-ordered products, $T_{\rm I}^j$, of SCET$_{\rm I}$ operators which contribute at a given order in the power counting. To match these onto SCET$_{\rm II}$ operators we take matrix elements, $$\begin{aligned}
\label{melt}
\langle\: \phi_I(p_i^2\sim \Lambda^2)\: | \: T^j_I\:
|\: \phi'_I(p_i^2\sim \Lambda^2)\: \rangle\,.\end{aligned}$$ Here the states have particles with ultrasoft momenta $p_{us}^2\sim \Lambda^2$, but with small collinear momenta $p_c^2\sim \Lambda^2$. These are allowed states in the Hilbert space of SCET$_{\rm I}$, since for example $p_\perp^2$ momenta of this size correspond to having zero label $\perp$ momenta, but non-zero residual $\perp$ momenta. These are also obviously states in SCET$_{\rm II}$. As in any matching calculation, we can use any convenient states, and one usually chooses free particle states. Note that the external collinear particles in (\[melt\]) have reduced offshellness, however this is not in general the case for the internal propagators.
As an additional constraint, the matching in Eq. (\[M12\]) must be carried out in a manner that accounts for the fact that only certain products of collinear fields have [*gauge invariant*]{} label momentum, and that these momentum components are not lowered in matching these products of fields onto collinear fields in [SCET$_{\rm II}$ ]{}. This means that only gauge invariant products of collinear fields should be integrated out in the matching (guaranteeing that gauge invariant products are also left over). This automatically builds in the fact that the low energy operators in [SCET$_{\rm II}$ ]{}must be built out of gauge invariant products $\Phi_1 = W^\dagger \xi_n$, $\Phi_2 = [W^\dagger D_c^\perp W]$, ${\cal
S}_1 = S^\dagger q_s$, etc. This properly matches the theory [SCET$_{\rm I}$ ]{}onto the subset of phase space that is described by fields in [SCET$_{\rm II}$ ]{}. This matching will be perturbative as long as the scale $\sqrt{Q\Lambda}\gg\Lambda $.
A useful benefit of the two-stage procedure is that the power counting is transparent. Thus even though we are integrating out an intermediate scale $p^2\sim Q\Lambda$ that involves factors of the hadronic scale $\Lambda$, we need not worry about missing operators that would be power suppressed but are enhanced by explicit factors of $1/\Lambda$. The power counting for the matching process is $$\begin{aligned}
\label{pc12}
T^I \sim \lambda^{2k} \rightarrow {\cal O}^{II} \sim \eta^{k+E} \,,\end{aligned}$$ where the final scaling is independent of how factors of $\eta$ are partitioned between coefficients and operators in SCET$_{\rm II}$ (we will choose to make Wilson coefficients in SCET$_{\rm II}$ dimensionless and order $\eta^0$). This equation says that T-products which are order $\lambda^{2k}$ in SCET$_{\rm I}$ will match onto operators in SCET$_{\rm II}$ that are order $\eta^{k+E}$ with $E\ge 0$. Here the factor $\eta^E$ is the extra factor obtained by lowering the offshellness of the external collinear fields and thereby changing their power counting. For example $(\xi_n^{\rm I} \sim\lambda=\sqrt{\eta}) \to (\xi_n^{\rm
II}\sim \eta)$, which agrees with the formula having $E=1/2$. In general $E=1/2$ for external $\xi_n$ or $A_n^\perp$, $E=0$ for external ${{\bar n}}{\!\cdot\!}A_n$ or $W$, and $E=1$ for external $n{\!\cdot\!}A_n$.
To illustrate these points we consider several examples. First consider the example of factorization in $B\to D\pi$ [@bps], but using the two-stage procedure. Matching the two $(\bar c b)_{V-A} (\bar d u)_{V-A}$ electroweak four quark operators onto operators in SCET$_{\rm I}$ gives $$\begin{aligned}
{Q}_{\bf 0}^{\rm I} &=& \big[{\bar h_{v'}^{us}} \, \,\Gamma_h
{h_v^{us}}\big] \big[{ \bar\xi_{n,p'} W}
{C_{\bf 0}({\bar {\cal P}}_+)} \Gamma_l { W^\dagger \xi_{n,p} }\big] \,,\\
{Q}_{\bf 8}^{\rm I} &=& \big[{ \bar h_{v'}^{us} } \, \Gamma_h T^A {
h_v^{us}}\big] \big[{ \bar\xi_{n,p'} W} { C_{\bf 8}({\bar {\cal P}}_+)}\Gamma_l
T^A { W^\dagger \xi_{n,p} }\big] \nonumber \,,\end{aligned}$$ where ${\bar {\cal P}}_+ = {\bar {\cal P}}^\dagger + {\bar {\cal P}}$ and the Wilson coefficients $C_{\bf 0,8}$ contain the hard $p^2\sim Q^2$ effects. Next decouple the usoft interactions from the leading collinear Lagrangian with the field redefinitions $\xi_n =
Y\xi_n^{(0)}$ and $A_n^\mu= Y A_n^{(0)\mu} Y^\dagger$ [@bps2]. This leaves $$\begin{aligned}
{Q}_{\bf 0}^{\rm I} &=& \big[{\bar h_{v'}^{us}} \,\Gamma_h \,
{h_v^{us}}\big] \big[{ \bar\xi_{n,p'}^{(0)} W^{(0)} }
{C_{\bf 0}({\bar {\cal P}}_+)}\Gamma_l { W^{(0)\dagger} \xi_{n,p}^{(0)} }\big] \,, \\
{Q}_{\bf 8}^{\rm I} &=& \big[{ \bar h_{v'}^{us} } \,\Gamma_h Y T^A Y^\dagger {
h_v^{us}}\big] \big[{ \bar\xi_{n,p'}^{(0)} W^{(0)} } { C_{\bf 8}({\bar {\cal P}}_+)}
\Gamma_l
T^A { W^{(0)\dagger} \xi_{n,p}^{(0)} }\big] \nonumber \,.\end{aligned}$$ In this result the ultrasoft and collinear fields are completely factorized. The collinear fields still have large offshellness $p^2\sim Q\Lambda$, so we need step iii). Taken with leading order Lagrangian insertions this example is just like the heavy-to-light current, so we match directly onto the [SCET$_{\rm II}$ ]{}operators $$\begin{aligned}
{Q}_{\bf 0}^{\rm II} &=& \big[{\bar h_{v'}^{s}} \, \Gamma_h \,
{h_v^{s}}\big] \big[{ \bar\xi_{n,p'} W}
{C_{\bf 0}({\bar {\cal P}}_+)}\Gamma_l { W^\dagger \xi_{n,p} }\big] \,,\\
{Q}_{\bf 8}^{\rm II} &=& \big[{ \bar h_{v'}^{s} } \, \Gamma_h S T^A S^\dagger {
h_v^{s}}\big] \big[{ \bar\xi_{n,p'} W} { C_{\bf 8}({\bar {\cal P}}_+)}\Gamma_l
T^A { W^\dagger \xi_{n,p} }\big] \nonumber \,.\end{aligned}$$ This is the same as the result originally derived in Ref. [@bps]. It is easy to see that no other [SCET$_{\rm II}$ ]{}operators are possible at this order.
This algebra was quite simple, however we have not yet seen the full power of the intermediate theory with the above example. The procedure becomes useful once we consider time-ordered products in [SCET$_{\rm I}$ ]{}, since then one can obtain non-trivial jet functions $J$ in [SCET$_{\rm II}$ ]{}which lead to Wilson coefficients $C(z_i)\: J(z_i,x_j,y_k)$ for the SCET$_{\rm II}$ operators. This jet function has convolutions with variables $z_i$ that correspond to the $p^-$ momentum dependence in the hard coefficient $C$. It also can have dependence on the $x_j$ momentum fractions of collinear fields in the [SCET$_{\rm II}$ ]{}operators we match onto. Finally, since collinear fields in [SCET$_{\rm I}$ ]{}are affected by the $k^+$ usoft momenta (through the $in{\!\cdot\!}\partial$ term in their action) the jet $J$ can depend on the momentum fractions $y_k$ which correspond to the soft $+$-momenta of gauge invariant products of soft fields in [SCET$_{\rm II}$ ]{}.
An example of a more involved matching calculation was given for the case of heavy-to-light form factors in Refs. [@bps4; @ps1] and we will not repeat this example here. To illustrate this case of matching further consider the toy example of light-light soft-collinear currents. In Ref. [@HN] these currents were derived by direct matching from QCD, so we contrast this procedure with the matching onto [SCET$_{\rm II}$ ]{}operators by using [SCET$_{\rm I}$ ]{}. Such operators are matched from contributions in [SCET$_{\rm I}$ ]{}which provide mixing between collinear and usoft quarks. Consider $$\begin{aligned}
\label{Ts}
T_0^{(3)} &=& \int\! d^4 x\, T[ J_{\xi\xi}^{(2)}(0) , i {\cal L}_{\xi
q}^{(1)}(x) ]
{\nonumber}\\
J_{\xi q}^{(4)} &=& \bar\xi_n W \Gamma q_{us}
\,,\end{aligned}$$ where $J_{\xi\xi}^{(2)} = \bar\xi_n W \Gamma W^\dagger \xi_n$ and ${\cal L}_{\xi
q}^{(1)}(x)$ is given in Eq. (\[Lxiq\]) (hard coefficients are suppressed since they are not crucial to our discussion). The order in $\lambda$ is denoted by the exponent in brackets. To match these operators onto [SCET$_{\rm II}$ ]{}we use the procedure explained above. For the local operator $J_{\xi q}^{(4)}$ this matching is simple. We first perform the field redefinition $\xi_n =
Y\xi_n^{(0)}$ and $A_n^\mu = Y A_n^{(0)\mu} Y^\dagger$ to write $$\begin{aligned}
J_{\xi q}^{(4)} &=& \left[ \bar\xi^{(0)} _n W^{(0)} \right] \Gamma
\left[Y^\dagger q_{us}\right]\end{aligned}$$ where we have indicated the gauge invariant blocks of fields by the square brackets. The final step is to identify the usoft fields with soft fields and to lower the off-shellness of the collinear fields. At tree level this leads to the operator $$\begin{aligned}
\label{O1}
O_1 = [\bar\xi_n W] \Gamma [ S^\dagger q_s ]\end{aligned}$$ in [SCET$_{\rm II}$ ]{}which is order $\eta^{5/2}$. This follows from Eq. (\[pc12\]) with $k=2$ and $E=1/2$.
For the time-ordered product $T_0^{(3)}$ we follow similar steps. After the field redefinition $$\begin{aligned}
T_0^{(3)} \!\!&=&\!\! \int\!\! d^4x\, T\Big\{ \left[\bar\xi_n^{(0)}
W^{(0)}\right] \Gamma
\left[W^{(0)\dagger} \xi_n^{(0)}\right](0) , \left[\bar\xi_n^{(0)} W^{(0)}
\right] \left[ W^{(0)\dagger} i\Dslash_\perp^c W^{(0)} \right] \left[Y^\dagger
q_{us} \right](x) \Big\}.\ \ \\end{aligned}$$ Consider the matrix element between a collinear fermion, a $\perp$ collinear gluon and a soft fermion. To match onto [SCET$_{\rm II}$ ]{}we contract the $[W^{(0)\dagger}
\xi_n^{(0)}][\bar\xi_n^{(0)} W^{(0)}]$ product, lower the off-shellness of the remaining $[\bar\xi_n^{(0)} W^{(0)}]$ and $[ W^{(0)\dagger} i\Dslash_\perp^c
W^{(0)}]$ and rename the $[Y^\dagger q_{us}]$ to $[S^\dagger q_{s}]$. At tree the two collinear fermion fields get contracted giving a propagator as shown in the first diagram of Fig. \[fig\_match\]. This gives the operator $$\begin{aligned}
\label{O2}
O_2 = [\bar\xi_n W] \Gamma \frac{\nslash}{2}
[W^\dagger i\Dslash_\perp^c W ] \frac{1}{n{\!\cdot\!}{{\cal P}}} [S^\dagger q_s]\end{aligned}$$ in [SCET$_{\rm II}$ ]{}which is the same operator as Ref. [@HN]. Note that while in [SCET$_{\rm I}$ ]{}$T_{0}^{(3)}$ was larger by one power of $\lambda$ than $J_{\xi
q}^{(4)}$, the resulting two operators are the same order in $\eta$. This is because in lowering the off-shellness of $[ W^{(0)\dagger} i\Dslash_\perp^c
W^{(0)}]$ the power counting of the $\perp$ gluon is reduced from $\lambda$ to $\eta=\lambda^2$. This agrees with Eq. (\[pc12\]) with $E=1/2$, so $O_2\sim
\eta^{5/2}$ just like $O_1$.[^2]
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There are additional contributions in [SCET$_{\rm I}$ ]{}that one can write down at order $\lambda^4$, such as $T[ J_{\xi\xi}^{(2)}(0), i {\cal L}_{\xi q}^{(2)}(x) ]$, $T[ J_{\xi\xi}^{(2)}(0), i {\cal L}_{\xi q}^{(1)}(x),i {\cal L}_{\xi q}^{(1)}(y)
] $, and $T[ J_{\xi\xi}^{(3)}(0) , i {\cal L}_{\xi q}^{(1)}(x) ]$. At tree level all these contributions contain factors of $D^c$, which receive an additional suppression factor when matching onto [SCET$_{\rm II}$ ]{}. However, at higher orders in perturbation theory these operators can contribute since more collinear fields are contracted. For the operators displayed in Eqs. (\[O1\],\[O2\]) they give rise to non-trivial jet functions. Consider for example the time-ordered product $$\begin{aligned}
T_0^{(4)} = \int\!\! d^4x\ T[ J_{\xi\xi}^{(3)}(0), i {\cal L}_{\xi q}^{(1)}(x) ]\end{aligned}$$ where $J_{\xi\xi}^{(3)}= (\bar\xi_n W) \Gamma (1/{\bar {\cal P}}) (W^\dagger
i\Dslash_\perp^{\:c} W) (\bnslash/2) (W^\dagger \xi_n)$. Operators like $T_0^{(4)}$ appear for example in the matching of QCD onto [SCET$_{\rm I}$ ]{}for the electromagnetic current of light quarks (see the second reference in [@bgamenu]). Gauge invariant blocks of collinear fields in the time-ordered product are contracted when matching onto [SCET$_{\rm II}$ ]{}. An example is illustrated in the second diagram in Fig. \[fig\_match\] where the factors of fields containing $D_\perp^c$ derivatives are contracted. Such a graph does not exhibit the additional suppression factor, as there is no collinear covariant perpendicular derivative left over. Thus, this operator can contribute to the operator $O_1$ and induce a non-trivial Wilson coefficient $J$. Therefore, the operators ${\cal O}_{1,2}$ in [SCET$_{\rm II}$ ]{}contributing to light-light soft-collinear current at any order in the matching from [SCET$_{\rm I}$ ]{}have the form $$\begin{aligned}
\label{O12s}
O_1 &=& J_1(\omega,y)\, (\bar\xi_n W)_\omega \Gamma (S^\dagger q_s)_y \,,{\nonumber}\\
O_2 &=& J_2(\omega_i,y)\, (\bar\xi_n W)_{\omega_1} \Gamma \frac{\nslash}{2}
[W^\dagger i\Dslash_\perp^c W ]_{\omega_2} \frac{1}{n{\!\cdot\!}{{\cal P}}}
(S^\dagger q_s)_y \,,\end{aligned}$$ where $(\bar\xi_n W)_\omega = [\bar\xi_n W \delta(\omega-{\bar {\cal P}}^\dagger)]$ and $(S^\dagger q_s)_y = [\delta(y-n{\!\cdot\!}P) S^\dagger q_s]$.
Finally, this procedure can also be used to match onto the Lagrangian for mixed soft-collinear interactions in SCET$_{\rm II}$. After making the field redefinition in step ii) there are no usoft-collinear Lagrangian interactions at order $\lambda^0$ in [SCET$_{\rm I}$ ]{}. Therefore from Eq. (\[pc12\]) it follows that it is not possible to construct a gauge invariant order $\eta^0$ soft-collinear Lagrangian. This is true for both quarks and gluons. This very simple power counting argument clarifies the original argument based on gauge invariance and power counting in Ref. [@bps2] and supplements the direct matching calculations in Ref. [@HN]. In the language of the power counting formulae in Ref. [@bpspc] the power counting for soft-collinear Lagrangian terms in [SCET$_{\rm II}$ ]{}corresponds to an index factor $(k-3) V_{SC}^k$ in the equation for $\delta$ which gives the power counting for an arbitrary time-ordered product. Here $V_{SC}^k$ counts the number of insertions of soft-collinear Lagrangian operators that are order $\eta^k$. The factor of $(k-3)$ agrees with the phase space argument in Ref. [@HN].
At order $\lambda^2$ we have a time-ordered product $\int d^4x T\{ {\cal L}_{\xi
q}^{(1)}(0), i{\cal L}_{\xi q}^{(1)}(x) \}$, which can induce suppressed operators in the [SCET$_{\rm II}$ ]{}Lagrangian. Contracting the collinear quarks in a $W^\dagger \xi_n(0) \bar\xi_n W(x)$ factor this gives an operator whose form agrees with Eq. (17) of Ref. [@HN]. At tree level in the matching we find $$\begin{aligned}
\label{LS1}
{\cal L}_{qqBB}^{(1)} &=& (\bar q_s S) \Big( W^\dagger i g \Bslash_\perp^c W
\frac{1}{{\bar {\cal P}}^\dagger} \Big) \frac{\nslash}{2} \Big( \frac{1}{{\bar {\cal P}}} W^\dagger
i g \Bslash_\perp^c W \Big) \frac{1}{n{\!\cdot\!}{{\cal P}}} ( S^\dagger q_s ) \,.\end{aligned}$$ Here the factor $\nslash/(2 n{\!\cdot\!}{{\cal P}})$ is again from the collinear quark propagator, and from Eq. (\[pc12\]) we count $E=1$ since two $\perp$ gluons are external and have their power counting changed in passing to [SCET$_{\rm II}$ ]{}. The superscript $(1)$ indicates that this operator contributes at order $\eta$ in [SCET$_{\rm II}$ ]{}. The factor of $\eta$ is derived by noting that the operator in Eq. (\[LS1\]) is $\sim \eta^4$ and so counts as $V_{SC}^4=1$. Thus subtracting three we see that it contributes an $\eta$ to the $\delta$ power counting formula.
Conclusion
==========
In this paper we discussed a few issues related to the gauge invariance of the soft-collinear effective theory beyond leading order. Together with power counting and reparameterization invariance, gauge invariance constrains the form of the allowed effective theory operators. However, there is some freedom in splitting the QCD gluon field into collinear and ultrasoft fields in the effective theory. In Sec. II we showed that the choice which gives $$\begin{aligned}
\label{final}
&& \qquad \qquad\quad
i{{\bar n}}{\!\cdot\!}\hat D = i{{\bar n}}{\!\cdot\!}\hat D_c + {\hat W} i{{\bar n}}{\!\cdot\!}D_{us}
{\hat W}^\dagger \,,\qquad
i \hat D_\perp^\mu = i \hat D_{c,\perp}^\mu + {\hat W} iD_{us,\perp}^\mu
{\hat W}^\dagger \,,\qquad\qquad\end{aligned}$$ corresponds to collinear and usoft fields which transform in a homogeneous way under the gauge transformations at any order in $\lambda$. This result uniquely fixes how power suppressed ultrasoft derivatives appear which are related to the collinear derivatives by reparameterization invariance. Using the new fields we then gave results for the subleading collinear and usoft-collinear effective Lagrangians to $O(\lambda^2)$, which by themselves are invariant under the collinear gauge transformations in Table \[table\_gt\].
A related construction was presented in Ref. [@bf2] using a position space multipole expansion. The collinear field redefinition adopted here differs from the one there. Our construction has the benefit of avoiding gauge fixing in the derivation. The explicit form of the transformation relating the fields in Ref. [@bf2] to the fields we have here remains an open and interesting question.
For SCET$_{\rm II}$, power counting, RPI and gauge invariance also give restrictions on allowed operators, which are however not as strict as in SCET$_{\rm I}$. The reason is that [SCET$_{\rm II}$ ]{}is non-local at the scale over which soft particles are propagating, whereas [SCET$_{\rm I}$ ]{}is only non-local at the hard scale $Q$. (This is the case before we decide to induce by hand a non-local $Y$ in [SCET$_{\rm I}$ ]{}by making a field redefinition.) Thus, additional input is needed to construct operators in SCET$_{\rm II}$, and one has to carefully consider which modes are integrated out in arriving at the low energy theory. In Ref. [@bps4] it was proposed that soft-collinear operators in SCET$_{\rm
II}$ could be constructed in an elegant manner by making use of factored ultrasoft-collinear operators in SCET$_{\rm I}$. In Sec. III we presented details of this matching calculation in several examples, and showed how the constraints from power counting and gauge invariance on SCET$_{\rm I}$ restrict the form of the operators induced in matching onto SCET$_{\rm II}$.
[99]{}
C. W. Bauer, S. Fleming and M. Luke, Phys. Rev. D [**63**]{}, 014006 (2001) \[arXiv:hep-ph/0005275\]. C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D [**63**]{}, 114020 (2001) \[arXiv:hep-ph/0011336\]. C. W. Bauer and I. W. Stewart, Phys. Lett. B [**516**]{}, 134 (2001). \[arXiv:hep-ph/0107001\]. C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D [**65**]{}, 054022 (2002) \[arXiv:hep-ph/0109045\]. For reviews R. L. Jaffe, \[arXiv:hep-ph/9602236\]; G. Sterman, TASI lectures 1995, \[arXiv:hep-ph/9606312\]; J.C. Collins, D.E. Soper, and G. Sterman in [*Perturbative Quantum Chromodynamics*]{}, Ed. by A. H. Mueller, World Scientific Publ., 1989, p. 1-93. S. J. Brodsky and G. P. Lepage, in [*Perturbative Quantum Chromodynamics*]{}, pp. 93-240; S. J. Brodsky, SLAC report SLAC-PUB-9281, arXiv:hep-ph/0208158; J. C. Collins and D. E. Soper, Ann. Rev. Nucl. Part. Sci. [**37**]{}, 383 (1987). C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein, I. W. Stewart, Phys. Rev. D [**66**]{}, 014017 (2002) \[arXiv:hep-ph/0202088\]. C. W. Bauer, A. V. Manohar and M. B. Wise, arXiv:hep-ph/0212255. C. W. Bauer, D. Pirjol and I. W. Stewart, arXiv:hep-ph/0211069. S. Descotes-Genon and C. T. Sachrajda, Nucl. Phys. B [**650**]{}, 356 (2003) \[arXiv:hep-ph/0209216\]; E. Lunghi, D. Pirjol and D. Wyler, Nucl. Phys. B [**649**]{}, 349 (2003) \[arXiv:hep-ph/0210091\]; see also S. W. Bosch, R. J. Hill, B. O. Lange and M. Neubert, arXiv:hep-ph/0301123. R. J. Hill and M. Neubert, arXiv:hep-ph/0211018. J. Chay and C. Kim, Phys. Rev. D [**65**]{}, 114016 (2002) \[arXiv:hep-ph/0201197\]. J. Chay and C. Kim, arXiv:hep-ph/0205117. C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D [**66**]{}, 054005 (2002) \[arXiv:hep-ph/0205289\]. M. Beneke, A. Chapovsky, M. Diehl, T. Feldmann, Nucl. Phys. B [**643**]{}, 431 (2002) \[arXiv:hep-ph/0206152\]. M. Beneke and T. Feldmann, Phys. Lett. B [**553**]{}, 267 (2003) \[arXiv:hep-ph/0211358\]. D. Pirjol and I. W. Stewart, arXiv:hep-ph/0211251. I. Z. Rothstein, arXiv:hep-ph/0301240. A. K. Leibovich, Z. Ligeti and M. B. Wise, arXiv:hep-ph/0303099. A. V. Manohar, T. Mehen, D. Pirjol, I. W. Stewart, Phys. Lett. B [**539**]{}, 59 (2002) \[arXiv:hep-ph/0204229\]. I. W. Stewart, Nucl. Phys. B (Proc. Suppl.) [**115**]{}, 107 (2003) \[arXiv:hep-ph/0209159\]. M. E. Luke and A. V. Manohar, Phys. Lett. B [**286**]{}, 348 (1992) \[arXiv:hep-ph/9205228\]. T. Feldmann, arXiv:hep-ph/0209239. C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. Lett. [**87**]{}, 201806 (2001) \[arXiv:hep-ph/0107002\].
[^1]: In Ref. [@bps2] a version of SCET was constructed that simultaneously involves collinear, soft, and usoft fields. While it is possible that some physical process may simultaneously require these degrees of freedom, here we restrict ourselves to the degrees of freedom of [SCET$_{\rm I}$ ]{}-[SCET$_{\rm II}$ ]{}which suffice for most applications.
[^2]: Note that in matching we always expand the upper theory in a series of terms to match it onto the lower theory. Therefore, it is not unusual that operators in [SCET$_{\rm I}$ ]{}match onto operators of different orders in [SCET$_{\rm II}$ ]{}.
|
---
abstract: |
We study the spectral densities of (pseudo)scalar currents at finite temperature in general case when mass of two quarks are different. Such spectral densities are necessary for the phenomenological investigation of hadronic parameters. We use quark propagator at finite temperature and show that an additional branch cut arises in spectral density, which corresponds to particle absorption from the medium. The obtained results at $T\rightarrow0$ limit are in good agreement with the vacuum results.
PACS: 11.10.Wx, 11.55.Hx, 12.38.Bx
author:
- |
Elşen Veli Veliev $^{*1}$, Kazem Azizi $^{\dag2}$, Hayriye Sundu $^{*3}$, Gülşah Kaya $^{*4}$\
$^{*}$ Physics Department, Kocaeli University, Umuttepe Yerleşkesi\
41380 Izmit, Turkey\
$^{\dag}$ Physics Division, Faculty of Arts and Sciences, Doǧuş University,\
Ac[i]{}badem-Kad[i]{}köy\
$^1$ e-mail:elsen@kocaeli.edu.tr\
$^2$e-mail:kazizi@dogus.edu.tr\
$^3$email:hayriye.sundu@kocaeli.edu.tr\
$^4$email:gulsahbozkir@kocaeli.edu.tr
title: ' **Spectral Density of (Pseudo)Scalar Currents at Finite Temperature**'
---
Introduction
============
Heavy ion collision experiments provide an opportunity to investigate particle properties in the medium. Inspired by these experiments, there is an increasing interest to investigate the properties of hadronic matter under extreme condition [@1],[@2]. In general, the media created by these collisions consist different mesons and baryons. Investigation of hadronic properties at finite temperature and density directly using the fundamental thermal QCD Lagrangian is highly desirable. However, such interactions occur in a region very far from the perturbative regime, where the quark-gluon coupling constant becomes large and the perturbative methods is not suitable for calculation of these properties. Therefore, we need nonperturbative approaches.
Some nonperturbative approaches are Lattice QCD, Heavy Quark Effective Theory (HQET), QCD Sum Rules, etc. Among these approaches, the QCD Sum Rules method [@3] and its extension to finite temperature [@4] has been widely used as an efficient and applicable tool to investigate the hadronic properties [@5]. The main elements in QCD sum rules approach are correlation function and dispersion relation. The correlation function is an object with dual nature. At large negative $q^2$, it can be evaluated by perturbative methods, whereas at positive $q^2$, it must be represented in terms of hadronic observables. Dispersion relation allows us to link the thermal correlator for positive values of $q^2$ to the one for negative values. Spectral functions in different cases were studied in the literature [@6]-[@13].
In the present work, we investigate the two-point thermal correlator using the real time formulation of the thermal field theory [@14]. We calculate the spectral densities of (pseudo)scalar currents at finite temperature, which are necessary for the phenomenological investigation of (pseudo)scalar mesons parameters. We use quark propagator at finite temperature and show how an additional branch cut corresponding to particle absorption from the medium arises in spectral density. We also compare our results with predictions obtained at zero temperature.
Thermal Spectral Densities of (Pseudo) Scalar Currents
=======================================================
We begin by considering the thermal correlation function, $$\label{eqn1}
\Pi(q, T)=i \int d^{4}x e^{iq\cdot x} \Big{\langle} {\cal T} \Big{(}J(x)J^{+}(0)\Big{)}\Big{\rangle}, \\$$ where $J(x)=:\bar{q}_{1}(x)\Gamma q_{2}(x):$ is the interpolating current that carries the quantum numbers of the state concerned and ${\cal T}$ indicates the time ordered product. Here $\Gamma=I$ or $i\gamma_5$ for scalar and pseudoscalar particles, respectively. The thermal average of the operator, $A={\cal T}(J(x)J^{+}(0))$, appearing in the above thermal correlator is expressed as $$\label{eqn2}
\langle A\rangle=Tr e^{-\beta H}A/Tr e^{-\beta H}, \\$$ where $H$ is the QCD Hamiltonian, and $\beta=1/T$ stands for the inverse of the temperature. Traces are carried out over any complete set of states. In the real time version, thermal correlator has the form of a $2\times2$ matrix. However, this matrix may be diagonalized, when it is expressed by a single analytic function, which determines completely the dynamics of the corresponding two-point function [@15]. As this function is simply related to any one, say the $11$-component of the matrix, we need to calculate only this component of the correlation function. The $11$-component of the thermal quark propagator is a sum of vacuum quark propagator expression and a term depending on the Fermi distribution function, $$\label{eqn3}
S(q)=(\gamma^{\mu}q_{\mu}+m)\Big{(}\frac{1}{q^{2}-m^{2}+i\varepsilon}+2\pi
i n(|q_0|)\delta(q^{2}-m^{2})\Big{)}, \\$$ where $n(x)$ is the Fermi distribution function, $n(x)=[\exp(\beta
x)+1]^{-1}$. Now, we proceed to obtain the temperature-dependent dispersion relation. The time ordering product in Eq. (1) can be expressed as $$\label{eqn4}
\langle T(J(x)J^{+}(x'))\rangle=\theta(x_{0}-x'_{0})\langle J(x) J^{+}(x')\rangle+\theta(x'_{0}-x_{0})\langle J^{+}(x')J(x)\rangle, \\$$ where $\theta(x)$ is step function.
Using Kubo-Martin-Schwinger relation, $\langle J(x_{0})
J^{+}(x'_{0})\rangle=\langle J^{+}(x'_{0})J(x_{0}+i\beta)\rangle$ for thermal expectation and making Fourier and some other transformations, we get the following expression for the thermal correlation function in momentum space [@14]: $$\label{eqn5}
\Pi (|\textbf{q}|, q_{0})=\frac{1}{2\pi}\int_{-\infty}^{\infty}dq'_{0}M(|\textbf{q}|, q'_{0})\Big{(}\frac{1}{q_{0}-q'_{0}+i\varepsilon}-\frac{\exp(-\beta q_{0})}{q_{0}-q'_{0}-i\varepsilon}\Big{)} , \\$$ where $$\label{eqn6}
M(|\textbf{q}|, q_{0})=\int d^{4}x e^{i q\cdot x}\langle J(x) J^{+}(0)\rangle . \\$$ In the above transformations, the following standard integral representation for the $\theta$- step function is used: $$\label{eqn7}
\theta(x_{0}-x'_{0})=\frac{1}{2 i \pi}\int_{-\infty}^{\infty}dk_{0}\frac{\exp[ik_{0}(x_{0}-x'_{0})]}{k_{0}-i\varepsilon} . \\$$ The imaginary part of the correlation function can be simply evaluated using the formula $\frac{i}{x+i \varepsilon}=\pi
\delta(x)+i P (\frac{1}{x})$, which leads to [@16]: $$\label{eqn8}
\Pi (q,T)=\int_{0}^{\infty}ds \frac{\rho (s)}{s+Q_{0}^{2}} , \\$$ where $\rho (q,T)=\frac{1}{\pi} Im \Pi(q,T) \tanh \frac{\beta
q_{0}}{2}$ and $Q_{0}^{2}=-q_{0}^{2}$. In some cases, the correlation function has ultraviolet divergent. If the spectral density does not vanish at $s\rightarrow\infty$, the dispersion integral in Eq. (8) diverges. A standard way to overcome this problem is to subtract first few terms of its Taylor expansion at $q^{2}=0$ from $\Pi(q,T)$. The thermal correlation function in momentum space can be written as $$\label{eqn9}
\Pi(q,T)=-i \int \frac{d^{4}k}{(2\pi)^{4}}Tr (\Gamma S(k) \Gamma S(k-q)) , \\$$ where $\Gamma=I$ and $i\gamma_{5}$ for scalar and pseudoscalar particles, respectively. Inserting propagators from Eq. (3) in Eq. (9) and carrying out the $k_{0}$ integration, we obtain the imaginary part of $\Pi(q,T)$ in the following form: $$\begin{aligned}
\label{eqn10}
&&Im\Pi(q,T)=-N_{c}\int\frac{d\textbf{k}}{8\pi^{2}}\frac{1}{\omega_{1}\omega_{2}}\Big{[}(\omega_{1}^{2}-\textbf{k}^{2}+\textbf{k}\cdot
\textbf{q}-\omega_{1}q_{0}\pm m_{1}m_{2}) \nonumber\\&& \times
[(1-n_{1}-n_{2}+2n_{1}n_{2})
\delta(q_{0}-\omega_{1}-\omega_{2})-(n_{1}+n_{2}-2n_{1}n_{2})\delta(q_{0}-\omega_{1}+\omega_{2})]
\nonumber\\&&+(\omega_{1}^{2}-\textbf{k}^{2}+\textbf{k}\cdot
\textbf{q}+\omega_{1}q_{0}\pm m_{1}m_{2}) \nonumber\\&& \times
[(1-n_{1}-n_{2}+2n_{1}n_{2})
\delta(q_{0}+\omega_{1}+\omega_{2})-(n_{1}+n_{2}-2n_{1}n_{2})\delta(q_{0}+\omega_{1}-\omega_{2})]\Big{]}
,\end{aligned}$$ where $m_{1}$ and $m_{2}$ are quark masses, $\omega_{1}=\sqrt{\textbf{k}^{2}+m_{1}^{2}}$ , $\omega_{2}=\sqrt{(\textbf{k-q})^{2}+m_{2}^{2}}$ , $n_{1}=n(\omega_{1})$, $n_{2}=n(\omega_{2})$ and the plus and minus signs in front of $m_{1}$, $m_{2}$, correspond to scalar and pseudoscalar particles, respectively. The term, which does not include the Fermi distribution functions, show the vacuum contribution. Terms including the Fermi distributions depict medium contributions. The delta-functions in the different terms of Eq. (10) control the regions of non-vanishing imaginary parts of $\Pi(q,T)$ , which define the position of the branch cuts. As seen the term including $\delta(q_{0}-\omega_{1}-\omega_{2})$ gives contribution when $q_{0}=\omega_{1}+\omega_{2}$ . Using Cauchy-Schwarz inequality, $( \sum_{i=1}^{{n}}
a_{i}^{2})(\sum_{i=1}^{n}b_{i}^{2})\geq(\sum_{i=1}^{n}a_{i}
b_{i})^{2}$ we see that, $$\label{eqn11}
\omega_{1}\omega_{2}=\sqrt{\textbf{k}^{2}+m_{1}^{2}}\sqrt{(\textbf{k-q})^{2}+m_{2}^{2}}\geq|\textbf{k}| |\textbf{k-q}|+m_{1} m_{2} ,\\$$ and for $q_{0}=\omega_{1}+\omega_{2}$, we get, $$\label{eqn12}
q_{0}^{2}=m_{1}^{2}+\textbf{k}^{2}+m_{2}^{2}+(\textbf{k-q})^{2}+2\omega_{1}\omega_{2}\geq (m_{1}+m_{2})^{2}+\textbf{q}^{2} .\\$$ Therefore, we obtain the first branch cut, $q^{2}\geq
(m_{1}+m_{2})^{2}$, which coincides with zero temperature cut describing the standard threshold for particle decays. This term survives at zero temperature and it is called the annihilation term. On the other hand, the term including $\delta(q_{0}-\omega_{1}+\omega_{2})$ gives contribution when $q_{0}=\omega_{1}-\omega_{2}$. Similarly to the above expression, we obtain, $$\label{eqn13}
q_{0}^{2}=m_{1}^{2}+\textbf{k}^{2}+m_{2}^{2}+(\textbf{k-q})^{2}-2\omega_{1}\omega_{2}\leq (m_{1}-m_{2})^{2}+\textbf{q}^{2} ,\\$$ and therefore an additional branch cut arises at finite temperature, $q^{2}\leq (m_{1}-m_{2})^{2} $, which corresponds to particle absorption from the medium. It is called scattering term and vanishes at $T=0$.
In the following, we restrict our calculations with $|\textbf{q}|=0$, when there is no angular dependence. Note that, with $|\textbf{q}|=0$, the value of $|\textbf{k}|$, fixed by the $\delta$-functions in Eq. (10), is the magnitude of three momentum of quark or antiquark in the center-of-mass of the quark-antiquark system: $$\label{eqn14}
\textbf{k}^{2}=\frac{\Big{(}q_{0}^{2}-(m_{1}+m_{2})^{2}\Big{)}\Big{(}q_{0}^{2}-(m_{1}-m_{2})^{2}\Big{)}}{4 q_{0}^{2}} .\\$$ In the $|\textbf{q}|=0$ case, as it is seen, the terms including $\delta(q_{0}-\omega_{1}-\omega_{2})$ and $\delta(q_{0}+\omega_{1}+\omega_{2})$ functions in Eq. (10), give contributions at the regions, $q_{0}\geq (m_{1}+m_{2})$ and $q_{0}\leq -(m_{1}+m_{2})$, respectively giving the vacuum cuts. Similarly, the terms including $\delta(q_{0}-\omega_{1}+\omega_{2})$ and $\delta(q_{0}+\omega_{1}-\omega_{2})$ functions in Eq. (10), give contributions at in the regions, $0\leq q_{0}\leq
(m_{1}-m_{2})$ and $-(m_{1}-m_{2})\leq q_{0}\leq 0 $, respectively giving the Landau cuts. After straightforward calculations, we find the vacuum part of the $Im \Pi (q,T)$ as: $$\label{eqn15}
Im \Pi (q_{0},T=0)=\frac{N_{c}}{8 \pi q_{0}^{2}}\sqrt{(q_{0}^{2}-m_{1}^{2}-m_{2}^{2})^{2}-4m_{1}^{2} m_{2}^{2}}\Big{(}q_{0}^{2}-(m_{1}-m_{2})^{2}\Big{)} .\\$$ Taking into account both branch cuts after some transformations, the annihilation and scattering parts of spectral density is found as: $$\label{eqn16}
\rho_{a,pert}(s,T)=\rho_{0}(s)\Big{[}1-n\Big{(}\frac{\sqrt{s}}{2}\Big{(}1+\frac{m_{1}^{2}-m_{2}^{2}}{s}\Big{)}\Big{)}-n\Big{(}\frac{\sqrt{s}}{2}\Big{(}1-
\frac{m_{1}^{2}-m_{2}^{2}}{s}\Big{)}\Big{)}\Big{]} ,\\$$ for $(m_{1}+m_{2})^{2}\leq s\leq\ \infty $, $$\label{eqn17}
\rho_{s,pert}(s,T)=\rho_{0}(s)\Big{[}n\Big{(}\frac{\sqrt{s}}{2}\Big{(}1+\frac{m_{1}^{2}-m_{2}^{2}}{s}\Big{)}\Big{)}-n\Big{(}-\frac{\sqrt{s}}{2}\Big{(}1-
\frac{m_{1}^{2}-m_{2}^{2}}{s}\Big{)}\Big{)}\Big{]} ,\\$$ for $0\leq s \leq (m_{1}-m_{2})^{2}$, with $m_{1}\geq m_{2}$. Here, $\rho _{0}(s)$ is the spectral density in the lowest order of perturbation theory at zero temperature and it is given by $$\label{eqn18}
\rho_{0}(s)=\frac{3}{8\pi^{2}s}q^{2}(s)v^{n}(s),\\$$ where $q(s)=s-(m_{1}-m_{2})^{2}$ and $v(s)=\Big{(}1-4m_{1}
m_{2}/q(s)\Big{)}^{1/2}$. Here $n=3$ and $n=1$ for scalar and pseudoscalar particles, respectively. As it is seen, at $T\rightarrow0$ limit these expressions are in good consistency with the vacuum expressions. Moreover, the obtained results are well consistent with the existing results in $m_{1}=m_{2}$ case [@17]-[@19] for the scalar and pseudoscalar particles.
As an example, we present the dependence of the annihilation and scattering parts of the spectral density for $K^{\pm}$ and $D^{\pm}$ particles in Figs. 1 and 2. In numerical analysis, we use the values $m_{s}=0,13$ GeV and $m_{c}=1,46 $ GeV for the quark masses. As it is clear, in the region of the standard threshold for particle decays, the $\rho_{0}(s)$ is replaced by the annihilation term. In the case of light mesons, the values of $\rho_{a,pert}(s,T)$ considerably differ from those of the $\rho_{0}(s)$. However, in the case of heavy mesons, the $\rho_{a,pert}(s,T)$ and $\rho_{0}(s)$ values are very close to each other. From Fig. 1, we also see that the in light $K^{\pm}$ cases, the medium contributions play important role and consist higher percentage of the total value.
Our concluding result is that the thermal contributions contribute significantly to the spectral function.
Acknowledgement
===============
The authors would like to thank T. M. Aliev for his useful discussions. This work are supported in part by the Scientific and Technological Research Council of Turkey (TUBITAK) under the research project No. 110T284 and in part by Kocaeli University under the research project No. 2010/32.
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abstract: 'The dynamics of tilted vortex lines in Josephson-coupled layered superconductors is considered within the time-dependent Ginzburg-Landau theory. The frequency and angular dependences of the complex-valued vortex mobility $\mu$ are studied. The components of the viscosity and inertial mass tensors are found to increase essentially for magnetic field orientations close to the layers. For superconducting/normal metal multilayers the frequency ($\omega$) range is shown to exist where the $\mu^{-1}$ value depends logarithmically on $\omega$.'
address: |
Institute for Physics of Microstructures, Russian Academy of Sciences\
603600, Nizhny Novgorod, GSP-105, Russia
author:
- 'A. S. Mel’nikov'
title: Inertial Mass and Viscosity of Tilted Vortex Lines in Layered Superconductors
---
The inertial mass $M$ and viscosity $\eta$ are known to be important quantities for characterizing vortex dynamic response in type-II superconductors. Without taking account of pinning effects, the vortex line bending and traction by the superflow the simple phenomenological description of the vortex dynamics in isotropic superconductors (see, e.g. Ref. [@gitt]) is based on the following equation for the flux line velocity ${\bf V}$: ${M\dot{\bf V} +\eta{\bf V}=\phi_0{\bf j}_{tr}\times{\bf n}/c}$, where ${{\bf j}_{tr}={\bf j}_0exp(i\omega t)}$ is the transport current density, ${\bf n}$ is the unit vector which points in the magnetic field direction, $\phi_{0}$ is the flux quantum. In the context of many applications (including high-frequency phenomena such as microwave or infrared response) it is convenient to introduce also the complex-valued dynamic mobility $\mu=(i\omega M +\eta)^{-1}$ which may be extended to include both pinning and flux-creep effects [@mobil; @clem2]. For $T$ close to $T_{c}$ the quantity $\eta$ may be estimated within the Bardeen-Stephen model or calculated more exactly using the time-dependent Ginzburg-Landau (TDGL) theory [@gorkov]. The inertial effects may play an essential role not only in high-frequency classical dynamics but also in quantum tunneling phenomena at extremely low temperatures [@review]. There exist several contributions to the vortex inertial mass [@review; @suhl; @kupr; @duan2; @duan; @sim1; @sim2; @coffey], i.e., one from the electronic states within the normal core (the electronic contribution $M_c$), one connected with the electric field energy ($M_{em}$) and one arising from mechanical stress and strain effects. Note that for isotropic compounds the term $M_{em}$ is much less than the electronic contribution ${M_c\sim \hbar^2 N_f}$ ($N_f$ is the density of states at the Fermi level).
Recently a great deal of attention has been devoted to the peculiarities of the vortex dynamics in extremely anisotropic high-$T_{c}$ superconductors which pertain to a larger class of materials including quasi-two dimensional (2D) organic superconductors, chalcogenides and artificial superconducting multilayers. A common feature of these systems is the weak interlayer Josephson coupling which results in a short effective coherence length $\xi_{c}$ for the order parameter spatial variation along the $c$ direction (perpendicular to the layers). In a broad temperature range the $\xi_{c}$ value may be much smaller than the interlayer distance $D$. This fact results in the quasi-2D character of static and dynamic magnetic properties in these compounds. As a consequence, the description of the dissipative and inertial effects in the vortex motion within the anisotropic TDGL theory employing an effective mass tensor [@asm; @hao; @hao2; @ivlev1] appears to be adequate only over a limited temperature range (${\xi_c(T)>D}$). In this paper we restrict ourselves to the case of weak magnetic fields when the so-called effective cores of neighbouring flux lines do not overlap (see Refs. [@feinberg; @lnb3]) and study the motion of an isolated vortex line which forms the angle $\gamma$ with the $z$ axis ($c$ direction) and lies, e.g., in the $(xz)$ plane. It is the purpose of this Letter to report on a calculation of the vortex dynamic mobility for layered superconducting structures in the temperature range $\xi_c(T)\ll D$. Let us consider a stack of thin superconducting (S) films of thickness $d$ separated by insulating (I) or normal metal (N) layers of thickness ${D\gg d}$. The inertial effects and a viscous drag force acting on a moving flux line are determined by the complicated structure of the effective vortex core [@feinberg; @lnb3]. For ${tan\gamma < \xi/D}$ the effective core size is of the order of the coherence length $\xi$ in the plane $(xy)$. In the angular domain ${Dtan\gamma > \xi}$ the effective core consists of 2D core regions connected by the Josephson-like vortex cores which appear for ${Dtan\gamma > L_{j}}$, where $
L_{j}=(D\hbar c^{2}/
(8\pi e \lambda_{ab}^{2}J_{c}))^{1/2}
$ is the Josephson length, $J_c$ is the interlayer Josephson critical current density, $\lambda_{ab}$ is the London penetration depth for currents along layers. The characteristic dimension of the 2D core region in the plane $(xy)$ is $a_d=min[Dtan\gamma,L_j]$. The dimensions of the Josephson-like core along $y$ and $z$ are $L_j$ and $D$, respectively.
Before presenting the derivation of the vortex dynamic equation, we briefly discuss the main mechanisms responsible for the inertial and dissipative effects in the vortex motion. One may expect that the contributions arising from the regions of 2D pancake normal cores do not change essentially and may be calculated in a manner analogous to the treatment in Refs.[@suhl; @kupr; @duan2; @duan; @sim1]. The novel specific terms in $M$ and $\eta$ come from the 2D and Josephson-like core regions. In particular, the simple qualitative arguments show that with an increase in $\gamma$ the term $M_{em}$ should increase due to the capacitive effects. This term is determined by the spatial distribution of the electric field generated by a moving flux line. Far away from the normal core regions and for rather large distances between the neighbouring 2D pancake vortices (${Dtan\gamma > \xi}$) the averaged $z$ component of the electric field $E_{zn}$ between layers $n$ and ${n+1}$ may be found using the Josephson relation (we take here the gauge ${A_z=0}$): ${E_{zn}=-\hbar {\bf V}\nabla (\theta_n-\theta_{n+1})/(2eD)}$, where ${\theta_n ({\bf r}-{\bf R}(t))}$ is the phase of the complex order parameter in the layer $n$, ${\dot{\bf R}={\bf V}}$ is the vortex velocity. At large distances $\tilde R_n$ from the 2D vortex center (in the 2D core region) ${E_{zn}}$ decreases as $\tilde R_n^{-1}$ to the distances ${\tilde R_n\sim a_d}$. Such an extremely slow decay of ${E_{zn}}$ results in the logarithmic divergence of the kinetic energy of the electric field, which is cut off at a large length scale $a_d$. Thus the contribution proportional to the value ${ln\,(a_d/\xi)}$ should appear in the expression for the inertial mass. This enhancement of the $M_{em}$ value may be significant if the dielectric constant of insulating layers is large. The analogous logarithmic term from the 2D core region contributes to the viscosity coefficient due to the dissipation produced by the interlayer normal currents (see also Ref. [@mel]). Surely, the influence of these interlayer currents on the scalar electrodynamic potential distribution (and, hence, on the complex-valued dynamic mobility) is most significant for S/N multilayers and temperatures close to $T_c$. For large angles ${tan\gamma > L_j/D}$ the formation of Josephson-like cores results in the additional contributions to the vortex viscosity and mass, which should be linear in the length $Dtan\gamma$. These terms may play an essential role in the vortex dynamics as it follows from the results obtained in Refs. [@clem2; @clem1] for the particular case ${\gamma=\pi/2}$. Obviously, the Josephson strings parallel to the $x$ axis contribute to the vortex mobility only for a nonzero velocity component $V_y$. As a consequence, for large tilting angles the vortex mass and viscosity coefficient should depend on the transport current orientation in the plane $(xy)$.
We now continue with the calculation of the viscosity coefficient and the vortex mass within the phenomenological Lawrence-Doniach model [@ld] generalized for the description of the dynamic phenomena (for $T$ close to $T_{c}$). Such a generalization may be written in analogy to the TDGL theory in 3D homogeneous superconductors. In the following treatment we will use the diffusion type TDGL equations which are strictly valid only for gapless superconductors. Nevertheless, we believe the final results to be qualitatively correct for a more general case. Note also that in the TDGL equations given below we neglect the terms connected with the particle-hole asymmetry and responsible for the vortex traction by the superflow [@dorsey; @kopnin]. If we choose the gauge $A_z=0$, the equation for the order parameter $\psi_n({\bf r})={f_{n}exp(i \theta_{n})}$ and the current density (averaged over the periodicity length $D$) have the form: $$\begin{aligned}
\tau\left(\frac{\partial}{\partial t}+
\frac{2ie\varphi_n}{\hbar}
\right)\psi_n=
\xi^{2}\left(\nabla-\frac{2ie}{\hbar c}{\bf A}_n \right)^{2}
\psi_{n}+\psi_{n}\nonumber\\
\label{main1}
-|\psi_{n}|^{2}\psi_n
+\frac{\xi^2}{L_j^{2}}(\psi_{n+1}+\psi_{n-1}-2\psi_{n})
\\
\label{main2}
{\bf j}_{n\parallel}=
\frac{\hbar c^{2}f_{n}^{2}}{8\pi e \lambda_{ab}^{2}}
(\nabla\theta_{n}-
\frac{2e}{\hbar c} {\bf A}_{n})
-\sigma_{ab}
(\nabla \varphi_n+
\frac{1}{c}\dot{\bf A}_n),\end{aligned}$$ Here $\varphi_{n}$ is the electrochemical potential in the $n$-th layer, $\sigma_{ab}$ is the normal state conductivity in the layer direction, ${\tau=\pi\hbar/(8(T_{c}-T))}$. The interlayer current density is given by the expression: $$\label{jz}
(j_{z})_{n,n+1}=J_c f_n f_{n+1}
sin(\tilde\theta_{n+1,n})-
\sigma_c D^{-1}\tilde\varphi_{n+1,n}$$ where $\sigma_c$ is the interlayer normal state conductivity, $\tilde\theta_{n+1,n}= \theta_{n+1}-\theta_n$, $\tilde\varphi_{n+1,n}=\varphi_{n+1}-\varphi_n$. Employing the expressions (\[main2\]),(\[jz\]) for S/N multilayers we assume that all the relevant length scales in the plane $(xy)$ are larger than the periodicity length $D$. As a consequence, our approach is strictly valid only in the limit $\xi\gg D$ [@mel]. We consider the dynamics of a tilted vortex line in the presence of an applied transport current ${\bf J}_{n}(t)$ flowing along the plane $(xy)$. For simplicity we suggest that the total stack thickness is much smaller than the effective skin depth and the current density ${{\bf J}_{n}={\bf J}_{tr}}$ does not depend on the layer number $n$. Considering only straight tilted vortices, we also leave aside the problem of possible vortex line flexures immediately under the surface. According to Ref. [@brandt], for the particular case $Dtan\gamma\ll L_j$ these flexures occur in the boundary layer of the width ${L_s\sim min[\lambda_{ab},a] D/L_j\ll\lambda_{ab}}$, where $a$ is the intervortex spacing. The generalization of the technique discussed below, taking account of the vortex line bending is straightforward (see, e.g., Ref. [@gorkov]).
To obtain the equation of motion we use the procedure similar to the one discussed in Refs. [@gorkov; @mel; @dorsey] and search for the solution of Eq.(\[main1\]) in the form: $$\label{pert}
{\psi_{n}=G_n+g_n=G({\bf r}-{\bf R}-{\bf r}_{n})+
g({\bf r}-{\bf R}-{\bf r}_{n},t)},$$ where ${{\bf r}_n=(nDtg\gamma,0)}$, $G({\bf r}-{\bf r}_{n})$ is the order parameter distribution for a static vortex line and $g$ is the small correction which is of the order of ${{\bf V}=\dot{\bf R}}$. The arguments analogous to the ones used for isotropic type-II superconductors [@gorkov] show that in the limit $\lambda_{ab}\gg D$ the magnetic field generated by a moving vortex line may be neglected within the effective core region (which provides the main contribution to the inertial mass and viscosity). The condition $\lambda_{ab}\gg D$ may be easily met in real layered structures, at least, for temperatures close to $T_c$. As a consequence, in Eqs. (\[main1\]),(\[main2\]) we may take into account only the vector potential ${{\bf A}_n=-4\pi\lambda_{ab}^2c^{-1}{\bf j}_{s}}$ generated by the transport supercurrent density ${{\bf j}_{s}}$ averaged over the structure period. Substituting the expansion (\[pert\]) into Eq. (\[main1\]) and neglecting the higher order terms in $V$ one obtains the equation for the correction $g_n$. We next multiply this equation by ${G_{pn}={\bf p}\nabla G_n}$ (${\bf p}$ is an arbitrary translation vector in the plane $(xy)$) and integrate over the area $S$ of the circle ${|{\bf r}-{\bf r}_n-{\bf R}|\leq R_{1}}$ where $R_{1}$ meets the condition ${R_{1}\gg Dtan\gamma}$. After simple transformations the real part of the resulting equation reads: $$\label{equ1}
\frac{\phi_{0}}{c}{\bf p} ({\bf j}_{s}\times{\bf z}_0)=
\frac{\eta_0}{\pi}Re\int\limits_{S}
G_{pn}^* (G_{vn} -
\frac{2ie}{\hbar}\varphi_n G_n-
\dot g_n) d^2 r,$$ where $\eta_{0}=0.5\sigma_{ab} u\phi_{0} H_{c2}c^{-2}$, $H_{c2}$ is the upper critical field for ${\gamma = 0}$, ${u=\hbar c^{2}/(32\lambda_{ab}^{2}\sigma_{ab}(T_{c}-T))}$, and ${G_{vn}={\bf V}\nabla G_n}$. The numerical factor $u$ is determined by the pair-breaking mechanism (see Refs. [@watts; @eliash] for details). The term in the left-hand side (l.h.s.) of Eq. (\[equ1\]) corresponds to the Lorentz force acting on a 2D pancake vortex while the right-hand side (r.h.s.) contains the terms responsible for the viscosity and inertial mass. We do not consider the extrinsic forces due to pinning and the interaction with other vortex lines, which add to the l.h.s. of Eq. (\[equ1\]). Let us examine at first the contribution to the integral in the r.h.s. of Eq. (\[equ1\]) which comes from the domain ${|{\bf r}-{\bf R}-{\bf r}_{n}|\gg\xi}$ (outside the normal cores). In this case one can put ${|\psi_{n}|\simeq 1}$. The continuity equation for the layered system reads: $$\begin{aligned}
\label{cont1}
\frac{\partial \rho_n}{\partial t} + div{\bf j}_n
+\frac{(j_z)_{n,n+1}-(j_z)_{n-1,n}}{D} = 0,\\
\label{cont2}
\rho_n = \frac{\varepsilon}{4\pi D^2}
(2\varphi_n - \varphi_{n+1}-\varphi_{n-1}),\end{aligned}$$ where $\rho_n$ is the averaged charge density in the layer $n$ and $\varepsilon$ is the high-frequency dielectric constant. We neglect here the difference between the electrochemical and scalar electrodynamic potentials, i.e., assume the Thomas-Fermi screening length to be less than all the relevant length scales. Using Eqs. (\[main1\]),(\[cont1\]),(\[cont2\]) one obtains: $$\begin{aligned}
\label{phase}
\frac{2e\tau}{\hbar}\Phi_n=
\xi^2\Delta\theta_n + \frac{\xi^2}{L_j^2}
(sin\tilde\theta_{n+1,n}-sin\tilde\theta_{n,n-1})\\
\label{poten}
\Phi_n=\left(s+\frac{\varepsilon s}{4\pi\sigma_c}
\frac{\partial}{\partial t}\right)
(\varphi_{n+1}+\varphi_{n-1}-2\varphi_{n}),\end{aligned}$$ where ${s=\sigma_c\xi^2/(u\sigma_{ab} D^2)}$ and $
{\Phi_n=\varphi_n+\hbar\dot\theta_n/(2e)}
$ is the gauge-invariant scalar potential. Note that we assumed ${u\stackrel{_>}{_\sim}1}$ which is consistent with the microscopic theory results for gapless superconductors [@watts; @eliash]. Let us consider the two-term expansion for the order parameter phase: ${\theta_n=\theta_{nv}+\chi_n
=\theta_{v}({\bf r}-{\bf R}-{\bf r}_n)
+\chi_n}$, where $\theta_v({\bf r}-{\bf r}_n)$ is the phase distribution for a static vortex line and $\chi_n$ is the first-order in $V$ correction. To solve the system (\[phase\]),(\[poten\]) we use here the linear approximation for the interlayer Josephson current density and replace the terms ${sin\tilde\theta_{n+1,n}}$ by ${\tilde\theta_{n+1,n}}$. The validity of such an approximation was discussed in detail in Ref.[@lnb3]. The solution of the system (\[phase\]),(\[poten\]) gives us a possibility to evaluate the integral in the r.h.s. of Eq. (\[equ1\]) far away from the normal cores taking account of the expressions ${G_{pn}\simeq i({\bf p}\nabla\theta_{nv})exp(i\theta_{nv})}$, ${g_n\simeq i\chi_n exp(i\theta_{nv})}$. Note that the contribution which comes from the normal core region (${|{\bf r}-{\bf r}_n-{\bf R}|\stackrel{_<}{_\sim}\xi}$) does not differ essentially from the one obtained previously for a vortex line in isotropic superconductors [@gorkov; @kupr]. Finally one obtains the following equation of vortex line motion in the Fourier representation: $$\label{mot}
\mu_x^{-1} V_x(\omega){\bf x}_0 +
\mu_y^{-1} V_y(\omega){\bf y}_0 =
\frac{\phi_{0}}{c}{\bf J}_{tr}(\omega)\times{\bf z}_0,$$ where ${\bf x}_0$, ${\bf y}_0$, ${\bf z}_0$ are the unit vectors of the coordinate system. We follow here the treatment in Refs. [@gitt; @kupr] and include both the supercurrent and normal current densities to the driving Lorentz force in the r.h.s. of Eq. (\[mot\]). The complex-valued dynamic mobilities $\mu_{x,y}$ appear to depend strongly on the parameter $s$ and tilting angle $\gamma$. The $s$ parameter may be written in the form: $s={\it l}_c^2/D^2$, where the length ${\it l}_c=\xi\sqrt{\sigma_{c}/u\sigma_{ab}}$ is the electric field penetration depth along $z$. For S/I multilayers one has ${\it l}_c\sim \xi_c$ and, as a consequence, $s\ll 1$ in the temperature range considered in this paper. For this limit one obtains: $$\begin{aligned}
\label{small1}
\mu_x^{-1}=[i\omega(M_c+M_{2D})+\eta_c+\eta_{2D}] D^{-1}
\\
\label{small2}
\mu_y^{-1}=\mu_x^{-1}+(i\omega M_j +\eta_j)tan\gamma
\\
\label{small3}
M_{2D}=\frac{\hbar^2 \varepsilon}{8e^2 D}
ln(1+a_d/\xi)\,;\,
\eta_{2D}=\frac{4\pi\sigma_c M_{2D}}{\varepsilon}
\\
\label{small4}
\eta_j\simeq \frac{\sigma_c \phi_0^2}{\pi c^2 DL_j} \,;\,
M_j = \frac{\varepsilon \eta_j}{4\pi\sigma_c}\end{aligned}$$ where $\eta_c=\eta_0\alpha_1 D$, $M_c=\eta_0\tau\alpha_2 D$ and $\alpha_1$, $\alpha_2$ are the constants of the order unity. The terms $\eta_{c}$ and $i\omega M_c$ are connected with the dissipation and inertial effects in normal core domains. The logarithmic terms $\eta_{2D}$, $M_{2D}$ in Eqs. (\[small3\]) may be considered as the contributions to the viscosity and inertial mass of a single 2D pancake vortex. Let us compare the $M_{2D}$ value to the inertial mass $M_c$ of the normal core for high-$T_c$ materials (for the angular domain $tan\gamma\gg \xi/D$). Considering ${Bi-2:2:1:2}$ as an example we take $D\simeq 15 \AA$, ${\varepsilon\simeq 10}$, ${\sigma_{ab}^{-1}(T\sim T_c)\sim 10^{-4}\Omega cm}$, ${\xi(T=0)=\xi_0\simeq 20-40\AA}$, ${T_c\simeq 80-90K}$, ${L_j/D\simeq 300-1000}$ [@muller] and obtain $$\frac{M_{2D}}{M_c}\simeq
\frac{4\varepsilon \xi_0^2 T_c}
{\pi^2 \alpha_2\hbar uD^2\sigma_{ab}}
ln\,\frac{a_d}{\xi} \sim
0.1\, ln\,\frac{a_d}{\xi}$$ Thus, for $tan\gamma > L_j/D$ one has ${M_{2D}\sim M_c}$. This rough estimate suggests that the contribution $M_{2D}$ to the vortex mass of electromagnetic origin may be significant in layered high-$T_c$ superconductors. In the angular domain $Dtan\gamma\gg L_j$ the vortex dynamic mobility depends essentially on the orientation of the current with respect to the in-plane magnetic field component. Such a dependence results from the contribution to $\mu_y^{-1}$, proportional to the length $Dtan\gamma$ of the Josephson string connecting the neighbouring 2D pancake vortices. The terms $\eta_j$ and $M_j$ correspond to the viscosity and inertial mass per unit length of the Josephson vortex. Note that the expressions (\[small4\]) are in good agreement with the ones obtained previously in Refs. [@clem2; @clem1] for vortices parallel to the layers.
The normal currents in nonsuperconducting layers lead to the penetration of the field ${\bf E}$ generated by a 2D vortex (moving in the plane $z=nD$) at a finite length ${\it l}_c$ along $z$. For S/N multilayers the ${\it l}_c$ value may be larger than the structure period $D$. This fact results in an essential decrease of the potential $\varphi_n$ in the 2D core regions ($\xi<{|{\bf r}-{\bf R}-{\bf r}_{n}| < a_d}$). As a consequence, the order parameter phase in these regions satisfies the diffusion type equation (see Eq. (\[phase\])), where $D_{\theta}=\xi^2/\tau$ is the diffusion constant. The characteristic time scale of the phase distortion propagation through the 2D core region is $t_0\sim a_d^2/D_{\theta}$. If the frequency $\omega$ of the applied ac field is higher than the value $\omega_0\sim t_0^{-1}$, then the essential time dispersion comes into play and the logarithmic divergence of the mobility coefficient is cut off at a length scale $L_{\omega}$ determined by the $\omega$ value. The upper cutoff $L_{\omega}$ is of the order of a diffusion length ${L_{\omega}= \sqrt{D_{\theta}/\omega}}$. The validity of these qualitative arguments is proved by direct calculations. As an example we consider here only the limit ${{\it l}_c\gg D}$ (${s\gg 1}$) which is relevant to S/N multilayers, at least, for temperatures close to $T_c$. We also restrict ourselves to the frequency range ${\omega\tau\ll 1}$ (${L_{\omega}\gg\xi}$). The complex-valued dynamic mobility has the form $\mu_{x,y}^{-1}=(1+i\omega \tau/u)\eta_{x,y}$, where the quantities $M_{x,y}=\tau\eta_{x,y}/u$ and $\eta_{x,y}$ may be considered as the components of the effective mass and viscosity tensors, respectively. Evaluating the integral in Eq. (\[equ1\]) to the logarithmic accuracy one obtains: $$\begin{aligned}
\label{sl1}
\eta_{x}\simeq \eta_0 (F+Q/q);\,
\eta_{y}\simeq \eta_0 (F+qQ)
\\
\label{sl2}
q=\sqrt{1+(Dtan\gamma/L_j)^{2}};\,\,
F=\beta_1 +ln(1+L_m/\xi)\\
\label{sl3}
Q=ln\left(1+\frac{min[L_{\omega},a_d\sqrt{s}]}
{\xi+L_m}\right)
+\beta_2,\end{aligned}$$ where ${|\beta_{1,2}|\stackrel{_<}{_\sim}1}$, ${L_m=min[a_d,L_{\omega}]}$. The logarithmic in $\omega$ terms contribute to the dynamic mobility as well as to the effective mass and viscosity in the frequency range ${s^{-1}\xi^2 a_d^{-2}< \omega\tau <1}$. For ${\sqrt{s}Dtan\gamma<\xi}$ the logarithmic terms in Eqs. (\[sl2\]),(\[sl3\]) are small and the effective mass and viscosity are determined by the dynamic processes in the normal core regions. In the opposite limit (${\sqrt{s}Dtan\gamma>\xi}$) these terms are large and the values $\beta_{1}$, $\beta_{2}$ may be neglected in a wide frequency range. The latter conclusion is not valid only for ${Dtan\gamma\gg L_{j}}$ and ${L_{\omega}\ll L_j}$, when the logarithmic term in Eq. (\[sl3\]) vanishes while the value ${\beta_2\sim -iL_{\omega}^2/L_j^2}$ provides an essential imaginary contribution to the coefficient $\eta_y$. As a consequence, the $M_y$ and $\eta_y$ values introduced above become complex-valued and, hence, in this case it is not adequate to define these quantities as the effective mass and viscosity (otherwise these definitions may be used). For higher frequences ${\omega\tau>1}$ the vortex mobility is completely determined by the order parameter dynamics in the normal core region, which was analysed, e.g., in Ref. [@kupr].
In summary, we have obtained the equation of tilted vortex motion, which may be applicable to various layered systems, including high-$T_c$ copper oxides. The angular dependences of the viscosity coefficient and the inertial mass have been investigated. For S/N mutilayers the complicated effective core structure is shown to result in the specific frequency dependence of the complex-valued dynamic mobility even in the frequency range ${\omega\tau<1}$. The approach developed above provides a starting point for the study of high-frequency linear response in the mixed state of layered superconductors. For this purpose the vortex dynamic mobility should be certainly extended to include both pinning and flux-creep effects [@mobil; @clem2].
This work was supported, in part, by a fellowship of the International Center for Fundumental Physics in Moscow, Russian Foundation for Fundumental Research (Grant No. 96-02-16993a) and Russian State Program on Condensed Matter Physics (Grant No.95042).
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abstract: 'Nontrivial collective behavior may emerge from the interactive dynamics of many oscillatory units. Chimera states are chaotic patterns of spatially localized coherent and incoherent oscillations. The recently-introduced notion of a weak chimera gives a rigorously testable characterization of chimera states for finite-dimensional phase oscillator networks. In this paper we give some persistence results for dynamically invariant sets under perturbations and apply them to coupled populations of phase oscillators with generalized coupling. In contrast to the weak chimeras with nonpositive maximal Lyapunov exponents constructed so far, we show that weak chimeras that are chaotic can exist in the limit of vanishing coupling between coupled populations of phase oscillators. We present numerical evidence that positive Lyapunov exponents can persist for a positive measure set of this inter-population coupling strength.'
address: 'Centre for Systems, Dynamics and Control and Department of Mathematics, University of Exeter, Exeter EX4 4QF, UK'
author:
- Christian Bick and Peter Ashwin
bibliography:
- 'citations\_18\_9.bib'
title: Chaotic Weak Chimeras and their Persistence in Coupled Populations of Phase Oscillators
---
Introduction
============
The emergence of collective behavior is a fascinating effect of the interaction of oscillatory units [@Strogatz2000; @Pikovsky2003; @Strogatz2004] and coupled phase oscillators [@Ashwin1992] serve as paradigmatic mathematical models to study these dynamical states in many system of interest ranging from technology to neuroscience [@Acebron2005; @Tchistiakov1996; @Ashwin2015].
In addition to global synchronization, the emergence of locally synchronized coherence-incoherence patterns—commonly known as chimera states [@Kuramoto2002; @Abrams2004]—has received a lot of attention in recent years [@Panaggio2015a]. These are particularly of interest where the patterns have broken symmetry with respect to the networks. Similar states have been shown to exist in real-world experiments [@Tinsley2012; @Hagerstrom2012; @Martens2013] and may be exploited for applications [@Bick2014a]. While chimera states are typically studied at or near the limit of infinitely many oscillators as stationary patterns of the phase density distribution [@Omel'chenko2013], they have generally only been described phenomenologically.
Ashwin and Burylko recently introduced a testable definition of a chimera state in the context of finite networks of indistinguishable phase oscillators—a weak chimera [@Ashwin2014a]. Weak chimeras are defined for oscillators where the phases $\vphi_k\in \Tor=\R/2\pi \Z$ evolve according to $$\frac{\ud\vphi_k}{\ud t} = \omega - \frac{1}{n} \sum_{j=1}^n H_{kj} g(\vphi_k-\vphi_j)
\label{eq:COsc}$$ in terms of partial frequency synchronization on trajectories; here $H_{kj}$ gives the network topology (respecting a permutation symmetry that acts transitively on the indices of the oscillators) and $g(\phi)$ is the generalized coupling (phase interaction) function. Such weak chimeras cannot appear in fully symmetric globally coupled phase oscillator networks or in any system with three or fewer oscillators. For coupling functions with more than one Fourier mode, there are examples of weak chimeras in systems of four oscillators that are relative periodic orbits, relative quasiperiodic orbits for six oscillators, and weak chimeras of heteroclinic type in a system of ten oscillators [@Ashwin2014a]. Note that coupling functions with multiple Fourier modes are not necessary for the occurrence of attracting weak chimeras: they can be found even for Kuramoto–Sakaguchi coupling $g(\phi)=\sin(\phi+\alpha)$ in such a system with only four oscillators [@Panaggio2015b].
However, the weak chimeras in [@Ashwin2014a] fail to capture one important dynamical feature expected of chimera states in higher-dimensional phase oscillator network: namely that they are chaotic. Moreover, the definition assumes existence of limiting frequency differences, which may not be the case for general trajectories even for a chaotic invariant set possessing a natural measure. Numerical investigations indicate that the chimeras in certain nonlocally coupled rings of oscillators may exhibit positive maximal Lyapunov exponents [@Wolfrum2011a]—while for attractive coupling, they may appear only as transients for typical initial conditions [@Wolfrum2011b]. By contrast, weak chimeras constructed in [@Ashwin2014a; @Panaggio2015b] have Lyapunov exponents that are presumably nonpositive, and can be attractors or repellers. A natural question is whether it is possible find “chaotic” weak chimeras in relatively small finite networks of indistinguishable oscillators—are there weak chimeras whose dynamics have positive maximal Lyapunov exponents for typical orbits?
In this paper we construct systems that exhibit such chaotic weak chimera states. It is already known that positive maximal Lyapunov exponents may arise in a fully symmetric phase oscillator system if the coupling function is chosen appropriately [@Bick2011]. We use this to show the existence of weak chimeras for weakly coupled clusters of oscillators with positive Lyapunov exponents in the limit of vanishing coupling. Then we show numerically that the positive Lyapunov exponents persist for nonvanishing coupling. Consequently, the study of chaotic weak chimeras sheds some light on the dynamics of regular chimera states; indeed one might conjecture that “typical” weak chimeras will be chaotic in all but the smallest and simplest systems.
The paper is organized as follows. In Section \[sec:Persistence\] we discuss some results on persistence of invariant sets for general nonautonomously perturbed dynamical systems and their consequences for weakly coupled product systems. In Section \[sec:Prelims\] we review and make a simple generalization of the definition of a weak chimera state to cases where the frequency difference may not exist on certain trajectories, but may only have upper and lower bounds. In Section \[sec:CWC\] we apply the results of Section \[sec:Persistence\] to prove the existence of weak chimeras for systems that are weakly coupled populations of phase oscillators. We numerically investigate the maximal Lyapunov exponents of these weak chimeras in Section \[sec:Numerics\] and observe that the maximal Lyapunov exponent stays positive for a large (presumably positive measure) set of coupling strengths between clusters. Finally, we give some concluding remarks.
Invariant sets and their persistence under perturbations {#sec:Persistence}
========================================================
In this section we give a general result for the persistence of absorbing sets under general nonautonomous bounded perturbations. Applied to weakly coupled systems, this yields a persistence result for dynamically invariant sets. We will use this result in the following section to show the existence of weak chimeras. The assumptions we make are sufficient to prove the results but not necessary as far as we can determine.
Some notation we will use throughout this manuscript. Let $\R$ denote the field of real numbers and $\Rn$ then $\maxdim$-dimensional vector space over $\R$. If $x,y\in\Rn$ then we denote their scalar product by $x\cdot y$. If $F:\Rn\to\R^m$ then let $F'$ denote the (total) derivative and for $x:\R\to\Rn, t\mapsto x(t)$ write $\dot x=\frac{\ud x}{\ud t}$. A (time-dependent) smooth vector field $f:\Rn\times\R\to\Rn$ defines an ordinary differential equation $$\label{eq:ODE}
\dot x = f(x, t).$$ If the system is autonomous, that is, $f(x, t)=f(x)$, then for $x\in\Rn$ the associated $\alpha$ and $\omega$-limit sets are closed and invariant with respect to the dynamics.
Persistence of absorbing regions
--------------------------------
A set $R\subset\Rn$ with compact smooth boundary is *forward absorbing* for if there is a differentiable function $W:\Rn\to\R$ such that $\partial R = \set{x\in\Rn}{W(x)=0}$ and $W'(x)\cdot f(x, t)<0$ for all $x\in\partial R$, $t\in\R$. A set is *backward absorbing* if it is forward absorbing for $-f$.
In other words, an absorbing set has a boundary given by the zeros of some differentiable function and the vector field $f(x, t)$ points either inward or outward everywhere on this boundary. Clearly, if is autonomous and $R$ is a forward or backward absorbing set then $R$ contains an invariant set of the dynamics.
We now show persistence of absorbing regions for an autonomous system $$\begin{aligned}
\label{eq:aut}
\dot{x}&=f(x)\end{aligned}$$ subject to bounded nonautonomous perturbations. More precisely, consider $$\begin{aligned}
\label{eq:nonaut}
\dot{x}&=f(x)+\e g(x,t)\end{aligned}$$ with $\e\geq 0$ and $g:\Rn\times\R\to\Rn$ smooth with $\norm{g(x,t)}<M<\infty$ for all $t>0$ and $x\in\Rn$.
\[lem:AbsorbingRegion\] Suppose that $R$ is forward absorbing for $\e=0$. Then there is an $\e_0>0$ such that whenever $0\leq\e<\e_0$ the set $R$ is forward absorbing for the dynamics of .
We just have to check that $W'(x)\cdot (f(x)+\e g(x,t))<0$ for all $x\in \partial R$ and $t>0$. Since $W'(x)\cdot f(x)$ is negative on the compact surface $\partial R$ it has a lower bound, i.e., there is a $\xi>0$ such that $W'(x)\cdot f(x)<-\xi$ for all $x\in \partial R$. Thus we have for $x\in\partial R$ $$W'(x)\cdot(f(x)+\e g(x,t)) \leq -\xi + \e \max\set{W'(x)\cdot g(x,t)}{x\in \partial R, t\in\R}$$ so if we pick $$\e_0= \frac{\xi}{\max \tset{M\norm{W'(x)}}{x\in \partial R}}$$ then for each $\e<\e_0$ and $t>0$ we have $W'(x)\cdot(f(x)+\e g(x,t))<0$ as required.
Let $A$ be a compact invariant set for the dynamics of .
(a) A nonnegative differentiable function $V:\Rn\to\R$ is a *Lyapunov function for $A$* if $V(x) = 0$ for all $x\in A$ and $\dot{V}(x) = V'(x)\cdot f(x)<0$ for $x\not\in A$.
(b) If $V$ is a Lyapunov function for $A$ then we call $A$ *sufficiently stable*.
(c) If $V$ is a Lyapunov function for $A$ for $-f$ then we call $A$ *sufficiently unstable*.
One can check that sufficient stability of a set implies asymptotic stability, though the converse only holds under additional assumptions; see for example [@Michel2015].
We now show that Lyapunov functions give absorbing regions arbitrarily close to a compact invariant set $A$ of the unperturbed dynamics as long as the perturbation is sufficiently small. We write $B_{\delta}(A)$ to denote a $\delta$-neighborhood of $A$.
\[thm:NonautPerturb\] Suppose that $A$ is a compact invariant sufficiently stable (or unstable) set for the unperturbed dynamics and $\norm{g(x,t)}<M$ for all $x\in\Rn$ and $t>0$. For any $\delta>0$ there is a compact set $R$ with $$A\subset R \subset B_{\delta}(A)$$ and an $\e_0$ such that whenever $0\leq\e<\e_0$ the set $R$ is an absorbing region for the dynamics of the perturbed system .
Sufficient persistence implies that there is a Lyapunov function $V$. For $\eta>0$ define $$R_\eta:= \set{x\in \Rn}{V(x)\leq\eta}.$$ For given $\delta>0$ choose $\eta>0$ such that $A\subset R_{\eta} \subset B_{\delta}(A)$. Since $V$ is a Lyapunov function the set $R=R_{\eta}$ is an absorbing region for the dynamics of . Lemma \[lem:AbsorbingRegion\] now implies that there is an $\e_0$ such that $R$ is also absorbing for the dynamics of .
Persistence of invariant sets in product systems
------------------------------------------------
Nonautonomous perturbations may arise from weak coupling of two dynamical systems.
### Weak forcing respecting an invariant set
Let $x\in\Rno$, $y\in\Rnt$ and consider the product system $$\label{eq:coupledforced}
\begin{aligned}
\dot{x}&=f(x)+ \e g(x,y)\\
\dot{y}&=h(x,y).
\end{aligned}$$ Suppose that $V\subset\Rnt$ is a compact set such that $\Rno\times V$ is dynamically invariant for all $\e\geq 0$.
\[thm:PersistenceFactor\] Let $A\subset\Rno$ be a compact and sufficiently stable (or sufficiently unstable) set for $\dot{x}=f(x)$ and suppose $\norm{g(x,y)}\leq M<\infty$ for all $(x,y)$. For any $\delta>0$ there exists an $\e_0>0$ such that whenever $0\leq\e<\e_0$ there is a dynamically invariant set $A_{\e} \subset B_\delta(A)\times V$ of the perturbed system .
For given $\delta >0$, Proposition \[thm:NonautPerturb\] yields an $\e_0$ and $R\subset B_\delta(A)$ such that $R$ is absorbing for all $0\leq\e<\e_0$.
Suppose that $A$ is sufficiently stable. The $\omega$-limit set $A_{\e}=\omega(x, y)$ of $(x,y)\in R\times V$ is dynamically invariant and we have $D\subset B_\delta(A)\times V$ since $R$ is absorbing. If $A$ is sufficiently unstable take the $\alpha$-limit set instead.
### Weakly coupled product systems
We now present a similar result for weakly coupled systems, $$\label{eq:coupled}
\begin{aligned}
\dot{x}&=f_1(x)+\e g_1(x,y)\\
\dot{y}&=f_2(y)+\e g_2(x,y)
\end{aligned}$$ with $x\in\Rno$, $y\in\Rnt$ and $f_\ell,g_\ell$ smooth such that $\norm{g_\ell}<M<\infty$, $\ell=1, 2$. We refer to $\dot{x}=f_1(x)$ and $\dot{y}=f_2(y)$ as the *uncoupled factors* of .
\[thm:Products\] Suppose that the uncoupled factors of have sufficiently stable attractors $A_1$ and $A_2$. Let $A=A_1\times A_2$. Then for any $\delta>0$ there exists an $\e_0>0$ such that for all $0\leq \e<\e_0$ there is an invariant set $A_\e\subset B_\delta(A)$ for the dynamics of . The same holds if $A_1$ and $A_2$ are sufficiently unstable.
We repeat the same argument as for Theorem \[thm:NonautPerturb\] for the Lyapunov function $V(x,y):=V_1(x)+V_2(y)$ to show that for any $\delta>0$ there is an $\e_0$ and an absorbing region $R$ with $$A_1\times A_2 \subset R \subset B_{\delta}(A_1\times A_2).$$ for the dynamics of for all $0\leq\e<\e_0$.
Since Theorems \[thm:PersistenceFactor\] and \[thm:Products\] are local results, they clearly generalize the case where the assumptions are satisfied on a sufficiently large neighborhood of the set of interest.
In fact, the existence of absorbing regions $R_1\supset A_1$, $R_2\supset A_2$ suffices to show the existence of an invariant set $A_{\e}\subset R_1\times R_2$ for sufficiently small $\e$. By contrast, a Lyapunov function allows one to construct invariant sets arbitrarily close to the product $A_1\times A_2$.
### Dynamics on the invariant sets
Note that the dynamics on $A_\e$ described in Theorems \[thm:PersistenceFactor\] and \[thm:Products\] may be qualitatively very different than the dynamics on $A$. In fact, even if one assumes appropriate contraction and expansion properties such that $A$ is uniformly normally hyperbolic and $A_\e$ is the invariant set that arises when the vector field is perturbed, one cannot expect that the dynamics on $A$ and $A_\e$ are conjugate [@Fenichel1972].
Persistence relates to the upper semicontinuity of attractors. In fact, many attractors of dynamical systems will be persistent in the sense of upper semicontinuity of the attracting set to a wide class of perturbations that may include nonautonomous perturbations. A more precise statement of this requires framing assumptions on the properties of the attractor and of the class of perturbations. For example, [@Afraimovich1998] give some conditions that ensure persistence of attractors to nonautonomous perturbations, with applications to synchronization problems, while semicontinuity of attractors to discretization is clearly an important topic in the study of numerical approximations of dynamical systems [@Stuart1994].
Weak chimeras for coupled phase oscillators {#sec:Prelims}
===========================================
Frequency synchronization and weak chimeras
-------------------------------------------
We now consider the dynamics of $\maxdim\in\N$ phase oscillators where oscillator $k$ is characterized by its state $\vphi_k\in\Tor$. Rather than restricting to systems of the form , we first consider a more general setting. Let $F=(F_1, \dots, F_\maxdim)$ be a smooth vector field on the torus $\Torn$. The evolution of the $k$th oscillator is given by $$\label{eq:Dyn}
\dot\vphi_k = F_k(\vphi_1,\ldots,\vphi_n).$$
### Average angular frequency intervals and weak chimeras
For any $\vphi^0\in\Torn$ and $T>0$ let us define the [*average angular frequency of oscillator $k$ on the time interval*]{} $[0, T]$ by $$\langle F_k \rangle_{T, \vphi^0} := \frac{1}{T}\int_{0}^{T} F_k(\vphi(t))\ud t,$$ where $\vphi(t)$ is the solution of with initial condition $\vphi(0)=\vphi^0$. More generally, for $A\subset \Torn$ a compact and forward invariant set under the flow determined by $F$ we define $$\begin{aligned}
\Oml_k(F, A) &:= \inf_{\vphi\in A}\liminf_{T\to\infty}\langle F_k \rangle_{T, \vphi},\\
\Omu_k(F, A) &:= \sup_{\vphi\in A}\limsup_{T\to\infty}\langle F_k \rangle_{T, \vphi}\end{aligned}$$ to describe the minimal and maximal average frequency on $A$. Similarly, define the angular frequency differences $$\begin{aligned}
\OmlD{k;j}(F, A) & := \inf_{\vphi\in A}\liminf_{T\to\infty} \big(\langle F_k \rangle_{T, \vphi}-\langle F_j \rangle_{T, \vphi}\big),\\
\OmuD{k;j}(F, A) & := \sup_{\vphi\in A}\limsup_{T\to\infty}\big(\langle F_k \rangle_{T, \vphi}-\langle F_j \rangle_{T, \vphi}\big)\end{aligned}$$ where we use the subscript $\text{df}(k;j)$ to indicate the frequency difference between oscillators $k$ and $j$.
If $F$ describes the dynamics of weakly coupled phase oscillators then $\Oml_k(F, A), \Omu_k(F, A)$ converge under fairly weak assumptions on smoothness of the dynamics [@Karabacak2009]. There are various alternative ways to express the interval of angular frequency differences, for example one can use a continuous version of [@Jenkinson2006 Prop. 2.1] to write $$\begin{aligned}
\OmlD{k;j}(F, A) & = \inf_{\mu\in M_A}\int \langle F_k \rangle_{T, \vphi}-\langle F_j \rangle_{T, \vphi}\, d\mu,\\
\OmuD{k;j}(F, A) & = \sup_{\mu\in M_A}\int \langle F_k \rangle_{T, \vphi}-\langle F_j \rangle_{T, \vphi}\, d\mu.\end{aligned}$$ where $M_A$ is the set of ergodic probability measures invariant under the flow generated by $F$ that are supported on $A$.
The *average angular frequency interval* of the $k$th oscillator on $A$ is given by $$\label{eq:Freqs}
\Omk(F, A) := \left[\Oml_k(F, A), \Omu_k(F, A)\right]\subset\R$$ and the *average angular frequency difference interval* between oscillators $k, j$ is given by $$\label{eq:FreqDiffs}
\OmD{k;j}(F, A) := \left[\OmlD{k;j}(F, A), \OmuD{k;j}(F, A)\right]\subset\R.$$ We say the oscillators $k, j$ are *frequency synchronized* on $A$ if $$\OmD{k;j}(F, A) = \sset{0}.$$
Note that even if $\Omk(F, A)$ and $\Omj(F, A)$ are intervals with interior, it is possible that oscillators $k, j$ are frequency synchronized. We now define weak chimeras by frequency synchronization—this is a generalization of the definition in [@Ashwin2014a].
A compact, connected, chain-recurrent forward-invariant set $A$ is a *weak chimera* if for there are distinct oscillators $k, j, l$ such that $$\begin{aligned}
\OmD{k;j}(F, A) &= \sset{0},\\
\Om_l(F, A)\cap\Omk(F, A)&=\emptyset.\end{aligned}$$
As in [@Ashwin2014a] we assume minimal conditions on $A$ that are satisfied if it is an $\omega$-limit set for the dynamics. The definition in [@Ashwin2014a] can be seen a special case of this definition in the case that there is convergence of the average frequency differences: $\OmD{k;j}(F, A)=\OmD{k;j}(F, A)$.
### Average angular frequencies under perturbations
What is the effect of a bounded perturbation on the average frequencies of oscillators with dynamics given by ? More precisely, if $\e>0$, $M \geq 0$, $Y_k:\R\to\R$ is smooth with $\abs{Y_k(t)}\leq M<\infty$ for $k\in\sset{1, \dotsc, \maxdim}$ and all $t>0$, we are interested in the average angular frequencies of the dynamics determined by the vector field $$\begin{aligned}
\label{eq:Feps}
\dot{\vphi}_k = F^{(\e)}_k(\vphi, t) = F_k(\vphi) + \e Y_k(t).\end{aligned}$$
We now give a (very) approximate bound for the average angular frequencies for dynamics with respect to the perturbed vector field $F^{(\e)}$. For $A\subset\Torn$ define the interval $$\label{eq:Bound}
N_k(F, A) := \Big[\inf_{x\in A}F_k(x), \sup_{x\in A}F_k(x)\Big].$$ Note that if $A$ is compact, so is $N_k(F, A)$.
\[lem:CloseCoarseBound\] Let $A$ be a compact dynamically invariant set for with $\e=0$ and let $\delta\geq 0$, $\e_0>0$ be given. Then there is an $\eta\geq 0$ such that for any $0\leq\e<\e_0$, if there is a compact dynamically invariant set $A_\e\subset B_\delta(A)$ for then $$\begin{aligned}
\Omk(F^{(\e)}, A_\e)&\subset B_{\eta}(N_k(F, B_\delta(A))).\end{aligned}$$
The statement follows from explicit integral estimates; we give here the upper bound—the lower bound is obtained analogously.
If $\vphi(t)$ is a solution of with $\vphi(0)=\vphi^0\in A_\e$ set $D=\set{\vphi(t)}{t\geq 0}$. For any $T>0$ we have $$\begin{aligned}
\langle F^{(\e)}\rangle_{T, \vphi^0} &= \frac{1}{T}\int_{0}^{T}F^{(\e)}_k(\vphi(t))\ud t
= \frac{1}{T}\int_{0}^{T}F_k(\vphi(t)) + \e Y_k(t)\ud t\\
& \leq \sup_{\psi\in D} {F_k(\psi)} + \e M \\
& \leq \sup_{\psi\in {B_\delta(A)}} {F_k(\psi)} + \e_0 M.\end{aligned}$$ Thus, for $\eta=\e_0 M$ we have $$\Omu_k(F^{(\e)}, A_\e)=\sup_{\vphi^0\in A_\e}\limsup_{T\to\infty}\langle F^{(\e)}\rangle_{T, \vphi^0}\leq \sup_{\psi\in {B_\delta(A)}} {F_k(\psi)} + \eta$$ which proves the assertion.
In particular, Lemma \[lem:CloseCoarseBound\] implies that $\Omega_k(F,A)\subset N_k(F,A)$. While Lemma \[lem:CloseCoarseBound\] suffices for our purposes, using continuity of $F$ one can prove a stronger statement: for any $\eta>0$ there exists a $\delta>0$ and $\e_0>0$ such that for any compact and invariant $A_\e\subset B_\delta(A)$ for with $0\leq\e<\e_0$ we have $\Omk(F^{(\e)}, A_\e)\subset B_{\eta}(N_k(F, A))$.
Moreover, if one assumes ergodicity and the existence of suitable invariant measures then the time averages may be replaced by spatial averages. If in addition the measure deforms nicely under the perturbation of the vector field then we can obtain better approximation of the frequencies of the perturbed system than the coarse bound given in Lemma \[lem:CloseCoarseBound\].
Networks of globally coupled phase oscillators
----------------------------------------------
The notion of frequency synchronization and weak chimeras applies to systems of identical and diffusively coupled phase oscillators. We define $\Th:\Torn\times\Torn\to\Rn$ by $$\label{eq:Theta}
\Th_k(\vphi, \psi) := \frac{1}{\maxdim}\sum_{j=1}^{\maxdim}g(\vphi_k-\psi_j).$$ where $g:\Tor\to\R$ is the $2\pi$-periodic coupling (phase interaction) function. Now consider the system of $n$ globally coupled identical phase oscillators $$\label{eq:OscVF}
\dot\vphi_k = F_k(\vphi) = \omega+\Th_k(\vphi, \vphi)$$ with $\Th$ defined as in . We may assume $\omega=0$ without loss of generality.
The system is $\Sn\times\Tor$-equivariant [@Ashwin1992]; the group $\Sn$ of permutations acts by permuting the indices of the oscillators and the continuous symmetry $\Tor$ acts by shifting the phase of all oscillators. As a consequence, both the diagonal $$\begin{aligned}
\Dn &= \set{(\vphi_1, \dotsc, \vphi_\maxdim)\in\Torn}{\vphi_1 = \dotsb = \vphi_\maxdim}\\
\intertext{and the open~\emph{canonical invariant region}}
\label{eq:CIR}
\C &:= \set{(\vphi_1, \dotsc, \vphi_\maxdim)}{\vphi_1 < \dotsb < \vphi_n < \vphi_1+2\pi},\end{aligned}$$ are dynamically invariant with respect to the dynamics of . The latter is bounded by codimension one invariant subspaces corresponding to $(n-1)$-cluster states that all intersect at $\Dn$.
The preservation of phase ordering implies that weak chimeras cannot exist for for any $\maxdim$ even for our more general definition of a weak chimera [@Ashwin2014a].
For some specific solutions, the average angular frequencies are easy to calculate:
(a) If $A=\sset{\vphi^*}$ is a fixed point of (relative to the continuous group action) then we have $\Omk(F, A) = \tsset{\frac{1}{\maxdim}\sum_jg(\vphi^*_n-\vphi^*_j)}$.
(b) If $A$ is an arbitrary (relative) periodic orbit $\vphi(t)$ with period $P>0$ then $\Omk(A) = \tsset{\frac{1}{P}\int_0^P F_k(\vphi(t))\ud t}$.
(c) We have $\Omk(F, \Dn) = \sset{g(0)}$ where $\Dn$ is either a continuum of fixed points (if $g(0)=0$) or the periodic orbit $\vphi(t)=(g(0)t, \dotsc, g(0)t)$. In particular, all oscillators are frequency synchronized.
The following two observations become important in the next section; they assert that changing the coupling function $g$ locally at zero affects the angular frequencies on $\Dn$ but not the dynamics of on compact $A\subset\C$.
First, Lemma \[lem:CloseCoarseBound\] gives information about the average angular frequencies of almost fully (phase) synchronized oscillators. Write $Z^{(\e)}=F+\e Y$ for a small perturbation of $F$ as in and set $D=\set{\vphi(t)}{t\geq 0}$ for a solution $\vphi$ for the flow of $Z^{(\e)}$ that stays close to $\Dn$. We have $\Omk(F, \Dn) = N_k(F, \Dn)= \sset{g(0)}$ and for $\e_0>0$ there exists $\eta>0$ such that $$\label{eq:omgdn}
\Omk(Z^{(\e)}, D) \subset [g(0)-\eta, g(0)+\eta]$$ for all $0\leq\e<\e_0$. In particular, if $D \subset \Dn$ then $\eta$ does not depend on $F$.
Second, the following lemma implies that the dynamics on the canonical invariant region are independent of $g$ in a neighborhood of zero. To highlight the dependence of $F$ on $g$, we write $F^{(g)}$ for the remainder of this section.
\[lem:Coupling\] Let $A\subset \C$, with $\C$ as in , be compact for the coupled phase oscillator system with coupling function $g$. Then there exists a closed interval $I\subset (0, 2\pi)$ such $F^{(g)}|_A = F^{(\hat{g})}|_A$ for all coupling functions $\gh$ with $\gh|_{I} = g|_{I}$.
Since $A\subset\C$ note that $\abs{\vphi_j-\vphi_k}>0$ for $k\neq j$ and $(\vphi_1, \dotsc, \vphi_\maxdim)\in A$. Set $J = \bigcup_{j\neq k}\set{\vphi_k-\vphi_j}{\vphi\in A} \subset \Tor\sm\sset{0}$ and we may write $J\subset (0, 2\pi)$. Since $A$ is compact, $I = [\inf J, \sup J]$ is the desired compact interval. Now $F^{(g)}|_A$ only depends on the values $g$ takes on $I$ and the result follows.
Persistence of weak chimeras {#sec:CWC}
============================
We now apply the results of Section \[sec:Persistence\] to networks with weakly coupled populations of phase oscillators, where each population is as introduced in the previous section.
Persistence for weakly symmetrically coupled populations
--------------------------------------------------------
The weakly coupled product of the dynamics with itself which defines a dynamical system with $\vphi = (\vphi_{1}, \vphi_{2})\in\Torn\times\Torn=\Tornn$ where $\vphi_{\ell}=(\vphi_{\ell,1}, \dotsc, \vphi_{\ell,\maxdim})$. More explicitly, the dynamics in the case of weak coupling are given by $$\dot{\vphi}=F^{(\e,g)}(\vphi)$$ where $$\label{eq:DynP}
\begin{aligned}
\dot\vphi_{1,k} &= F^{(\e,g)}_{1,k}(\vphi) := F_k(\vphi_{1}) + \e \Th_k(\vphi_{1},\vphi_{2}) ,\\
\dot\vphi_{2,k} &= F^{(\e,g)}_{2,k}(\vphi) := F_k(\vphi_{2}) + \e \Th_k(\vphi_{2},\vphi_{1}) ,
\end{aligned}$$ for $k=1, \dotsc, \maxdim$ where $F_k$ is given by and $\Theta$ by . If $A_\e\subset\Tornn$ is compact and dynamically invariant for we let $\Omega_{\ell,k}(F^{(\e,g)}, A_\e)$ denote the angular frequency intervals for oscillator $(\ell, k)$ with phase $\vphi_{\ell,k}$, $\ell\in\sset{1, 2}$, $k\in\sset{1, \dotsc, \maxdim}$. Moreover, for $D\subset\Tornn$ write $$N_{\ell, k}(F^{(\e, g)}, D) = \Big[\inf_{\vphi\in D}F^{(\e, g)}_{\ell, k}(\vphi), \sup_{\vphi\in D}F^{(\e, g)}_{\ell, k}(\vphi)\Big]$$ as in and we have $\Omega_{\ell,k}(F^{(\e,g)}, A_\e)\subset N_{\ell, k}(F^{(\e, g)}, A_\e)$ if $A_\e$ is dynamically invariant. Observe that for $\e=0$ the system decouples into two identical groups of $n$ oscillators—both of which with nontrivial dynamics .
Note that, in addition to the continuous $\Tor$ symmetry, is equivariant with respect to the action of a symmetry group $\Sn\wr \Sk{2}$ where $\wr$ is the wreath product [@Dionne1996]. That is, $\Sn\wr\Sk{2}= (\Sn)^2\times_s \Sk{2}$ where the $\Sn$ permute the oscillators within the each group of $n$ oscillators and the $\Sk{2}$ permutes the two groups. Observe that this is only a semidirect product $\times_s$ as the two sets of permutations do not necessarily commute. This group acts transitively on the oscillators: the oscillators are indistinguishable. Both $\Torn\times\Dn\subset\Tornn$ and $\Dn\times\Torn\subset\Tornn$ are dynamically invariant for any $\e\geq 0$ as fixed point subspaces of the action of $\Sn\wr\Sk{2}$.
### Persistence of weak chimeras {#persistence-of-weak-chimeras}
We now state the main result of this section; the notation $\Ai$ suggests that this invariant set corresponds to the cluster of incoherent oscillators for the chimera while the remaining oscillators are coherent in the sense that they are fully phase-synchronized.
\[thm:CWC\] Suppose that $g$ is a coupling function such that $\Ai\subset\C$ is a compact, forward invariant, and sufficiently stable (or unstable) set for the dynamics of . For any $\delta>0$ with $\overline{B_\delta(\Ai)}\subset\C$ there exist a smooth coupling function $\hat{g}$ and an $\e_0>0$ such that the weakly coupled product system for $F^{(\e,\hat{g})}$ has a weak chimera $A_\e$ with $A_\e \subset \Dn\times B_\delta(\Ai)$ for all $0\leq\e<\e_0$.
Let $\delta>0$ be given and let us assume $\Ai$ is sufficiently stable. Compactness and continuity imply that $M := \max_{k\in\sset{1, \dotsc, \maxdim}}\max_{\vphi\in\Tornn}\abs{\Th_k(\vphi)}<\infty$. Define $$N(g) := \bigcup_{k=1}^{\maxdim} N_{2,k}(F^{(0, g)}, \Dn \times B_\delta(\Ai)),$$ assume that $\e_0$ is fixed (we will determine the exact value below) and set $\eta=\e_0M$. If $B_{\eta}(\sset{g(0)})\cap B_{\eta}(N(g)) = \emptyset$ then we choose $\gh=g$, otherwise we choose $\gh$ by Lemma \[lem:Coupling\] such that $F^{(0, \gh)}|_{B_\delta(\Ai)^2}=F^{(0, g)}|_{B_\delta(\Ai)^2}$—implying $N(g)=N(\gh)$—and $\gh(0)$ sufficiently large so that $B_{\eta}(\sset{\gh(0)})\cap B_{\eta}(N(\gh)) = \emptyset$. In particular, for this choice of $\gh$ the set $\Ai$ is still forward invariant, and sufficiently stable in $B_\delta(\Ai)$ for . A similar argument applies if $\Ai$ is sufficiently unstable.
Applying Theorem \[thm:PersistenceFactor\] (note that $\Delta_n$ is compact in the topology of $\Tor^n$) yields an $\e_0>0$—take this as the unspecified value of $\e_0$ above—and compact sets $A_\e\subset\Dn\times B_\delta(\Ai)$ for all $0\leq\e<\e_0$ such that $A_\e$ is dynamically invariant for the flow defined by $F^{(\e, g)}$. Moreover, $A_\e$ can be assumed to be connected and chain-recurrent buy taking a subset if necessary.
It remains to be shown that any $A_\e$ is a weak chimera. For $\e=0$ the dynamics are uncoupled with $\Omega_{1,k}(F^{(0,\gh)}) = N_{1,k}(F^{(0, \gh)}, \Dn \times \Ai) = \sset{\gh(0)}$ for $k\in\sset{1, \dotsc, \maxdim}$. The weak coupling for $\e>0$ in may be seen as a bounded nonautonomous perturbation to each factor, that is, $\dot\vphi_{\ell}=F(\vphi_{\ell})+\e Y(\vphi_{\ell}, t)$ with $\norm{Y}\leq M$. By Lemma \[lem:CloseCoarseBound\] we have (with $\eta$ as above) $$\begin{aligned}
\Om_{1,k}(F^{(\e,\gh)}, A_\e) &\subset
B_\eta(N_{1,k}(F^{(0, \gh)}, \Dn \times B_\delta(\Ai))) = B_\eta(\sset{\gh(0)}),\\
\Om_{2,k}(F^{(\e,\gh)}, A_\e) &\subset B_\eta(N_{2,k}(F^{(0, \gh)}, \Dn \times B_\delta(\Ai))) \subset B_\eta(N(\gh))\end{aligned}$$ for all $k\in\sset{1, \dotsc, \maxdim}$ and any $0\leq \e<\e_0$. For $\vphi\in A_\e$ we have $\vphi_{1, 1}=\dotsb=\vphi_{1, \maxdim}$ and hence $\Om_{1,1}(F^{(\e,g)}, A_\e)=\dotsb=\Om_{1,\maxdim}(F^{(\e,g)}, A_\e)$. Since $B_\eta(\sset{\gh(0)})\cap B_\eta(N(\gh))=\emptyset$ by the choice of $\gh$, we have $\Om_{1,k}(F^{(\e,g)}, A_\e)\cap
\Om_{2,k}(F^{(\e,g)}, A_\e)=\emptyset$ for all $k\in\sset{1, \dotsc, \maxdim}$. Therefore any $A_\e$ is a weak chimera for with the coupling function $\gh$.
We remark that the results of Theorem \[lem:Coupling\] can be generalized in a straightforward way to $m\geq 2$ populations of $n$ coupled phase oscillators with dynamics given by $$\label{eq:DynPnm}
\dot\vphi_{\ell,k} = F_k(\vphi_{\ell}) + \e\sum_{r\neq \ell}\Th_k(\vphi_{\ell}, \vphi_r)$$ for phases $\vphi_{\ell,k}$ with $k=1, \dotsc, \maxdim$ and $\ell=1, \dotsc, m$. This more general system has symmetry $\Sn\wr \Sk{m}$ which acts transitively on the oscillators.
### Stability
The stability of a weak chimeras $A_\e$ of Theorem \[thm:CWC\] as a subset of $\Dn\times\Torn$ depends on the stability properties of $\Ai\subset\Torn$. Introduction of coordinates $\psi_{\ell,k}=\vphi_{\ell,k}-\vphi_{1,k}$ eliminates the phase shift symmetry. In fact $\psi_{1,k} = 0$ and thus the reduced system is a dynamical system on $\Torn$. If $\Ai$ is sufficiently stable so is $A_\e$ in the reduced system; if $\Ai$ is sufficiently unstable in the unperturbed system so is $A_\e$. Of course, if $\Ai$ is sufficiently stable we obtain a sufficiently unstable set by reversing time and vice versa.
Transversal stability of the invariant set $\Dn\times\Torn\subset\Tornn$ is determined by the sign of $g'(0)$ [@Ashwin1992]. More precisely, $\Dn\times\Torn\subset\Tornn$ is asymptotically stable if $g'(0)<0$ and asymptotically unstable if $g'(0)>0$. Through local perturbation of the coupling function, one can get the desired transversal stability properties.
\[cor:StabilitySwap\] Let $A_\e\subset\Dn\times\Torn$ be a weak chimera of with coupling function $g$ such that $A_\e$ is asymptotically stable in $\Dn\times\Torn$. Then there exists a coupling function such that $A_\e$ is asymptotically stable in $\Tornn$.
An application of Lemma \[lem:Coupling\]; choose a coupling function $\gh$ such that $F^{(0,\gh)}|_{\Dn\times\overline{B_\delta(\Ai)}}=F^{(0,g)}|_{\Dn\times\overline{B_\delta(\Ai)}}$ and $\gh'(0)<0$.
Consequently, the weak chimera states constructed above can be asymptotically stable, asymptotically unstable, or of saddle type. They are asymptotically stable (unstable) if $\Ai$ is sufficiently stable (unstable) and $\Dn\times\Torn$ is transversally stable (unstable) and of saddle type if if $\Ai$ is sufficiently stable and $\Dn\times\Torn$ is transversally unstable and vice versa.
### Chaotic weak chimeras: dynamics on the invariant set
Suitable coupling functions may now give rise to chaotic weak chimera in the limit of vanishing coupling. So far, we have not made any assumptions on the dynamics on $\Ai$; they may be chaotic. If $g$ is chosen such that the dynamics of on $\Ai$ has positive Lyapunov exponents then so will the dynamics of on $A_0=\Dn\times\Ai$. Therefore it is reasonable to assume that the positive Lyapunov exponents will persist for a set of $\e>0$ of positive measure, that is $A_\e$ are chaotic weak chimeras for $\e$ sufficiently small.
\[rem:PersistenceChaos\] To rigorously show that positive Lyapunov exponents persist for nonzero $\e>0$ one would have to make further assumptions that guarantee there is a suitable invariant measure for $\e=0$ that deforms nicely for sufficiently small $\e>0$; see also Section \[sec:Persistence\].
We give explicit examples of coupling functions that give rise to chaotic dynamics on $A_0$ and numerical evidence that the persisting weak chimeras $A_\e$ are chaotic for nonzero coupling in Section \[sec:Numerics\].
Persistence for coupling breaking symmetry {#sec:CWCnosym}
------------------------------------------
So far we have considered weak coupling between populations that preserved the symmetry of the system for any choice of $\e$. Using the notation otherwise as for and assume that $Y_1, Y_2: \Tornn\to \Rn$ are Lipshitz continuous. The system $$\label{eq:DynPns}
\begin{aligned}
\dot\vphi_{1,k} &= F^{(\e,g)}_{1,k}(\vphi) = F_k(\vphi_1) + \e Y_{1,k}(\vphi),\\
\dot\vphi_{2,k} &= F^{(\e,g)}_{2,k}(\vphi) = F_k(\vphi_2) + \e Y_{2,k}(\vphi),
\end{aligned}$$ for $k=1, \dotsc, \maxdim$ is a weakly coupled product system. While is $(\Sn\wr\Sk{2})\times\Tor$ equivariant for $\e=0$, for a generic choice of $Y_1, Y_2$ the symmetry is broken whenever $\e>0$[^1]. However, the weak chimera may persist as the next result shows.
\[thm:CWCns\] Suppose that $g$ is a coupling function such that $\Ai\subset\C$ and $\Ac = \Dn\subset\Torn$ are compact, forward invariant and sufficiently stable (or sufficiently unstable) sets for the dynamics of for $F^{(0,g)}$. For any $\delta>0$ there exist a smooth coupling function $\gh$ and $\e_0>0$ such that the weakly coupled product system has a sufficiently stable (unstable) weak chimera $A$ with $A \subset B_\delta(\Ac\times\Ai)$ for $F^{(\e,\gh)}$ and any $0\leq\e<\e_0$.
Persistence of invariant sets $A_\e$ in each factor follows from Theorem \[thm:Products\]. The same argument as in the proof of Theorem \[thm:CWC\] shows that there is a smooth coupling function such that $A_\e$ is a weak chimera since Lemma \[lem:Coupling\] allows us to modify the coupling function in an open neighborhood of zero.
While Theorem \[thm:CWCns\] only gives existence of sufficiently stable (or sufficiently unstable) weak chimeras for systems there may be others of saddle type. Note also that there are coupling functions for which there is a Lyapunov function for the fully synchronized solution [@VanHemmen1993].
Examples of chaotic weak chimeras {#sec:Numerics}
=================================
In this section we explicitly construct coupling functions that give rise to chaotic weak chimeras for with $\e=0$. Moreover, we demonstrate numerically that these chaotic weak chimeras persist for $\e>0$. While we use the methods developed in the previous section to construct the chaotic weak chimeras, we do not check rigorously whether the assumptions are satisfied.
A chaotic weak chimera of saddle type
-------------------------------------
Consider the dynamics of for $\maxdim=4$ oscillators. Suppose that the coupling function is given by the truncated Fourier series $$\label{eq:gchaos}
{g}(\phi) = \sum_{r=0}^{4} c_r \cos (r\phi+\xi_r)$$ with $c_1 = -2$, $c_2 = -2$, $c_3 = -1$, and $c_4 = -0.88$. For $\xi_1 = \eta_1$, $\xi_2 = -\eta_1$, $\xi_3 = \eta_1+\eta_2$, and $\xi_4 = \eta_1+\eta_2$ with $\eta_1=0.11151$, $\eta_2=0.05586$ the function ${g}$ gives rise to a chaotic attractor $\Ai\subset\C$ with positive maximal Lyapunov exponents [@Bick2011].
![\[fig:N4CWCSaddle\] A transient started near a chaotic weak chimera of saddle type for two populations of $\maxdim=4$ oscillators with coupling function for $\eta_1=0.1104$, $\eta_2=0.057511$ and $\e=0.2$. The phase evolution of each oscillator in a co-rotating frame at the speed of oscillator $1$ is shown using a periodic grey scale ($\vphi_{\ell,k}(t)=0$, in black $\vphi_{\ell,k}(t)=\pi$ in white). Middle and bottom panels show the order parameters $R_\ell(t)$ of the populations $\vphi_{\ell}$, $\ell=1,2$, and the instantaneous frequencies $\dot\vphi_{\ell,k}(t)$. Note that for $t\lessapprox 70$ the populations are clearly are not frequency synchronized with chaotic oscillations—evidence of a chaotic weak chimera. After the trajectory leaves the vicinity of the saddle, it converges to a homogeneous chaotic state.](img/gCWCn4Transient.pdf)
![\[fig:N4CWCSaddleScan\] Positive maximal Lyapunov exponents persist for $\e>0$ for two populations of $\maxdim=4$ oscillators with coupling function for $\eta_1=0.1104$, $\eta_2=0.057511$. We approximated the maximal Lyapunov exponent (black dots) by integrating from a random initial condition (sampled uniformly on $\Dn\times\Torn$) for $T=7000$ time units. The shaded regions show the intervals $[\min_{k,t}\dot\vphi_{\ell,k}(t), \max_{k,t}\dot\vphi_{\ell,k}(t)]$ for $\ell=1$ (dark grey) and $\ell=2$ (light grey)—where these do not overlap, there is no frequency synchronization between the two populations and hence a weak chimera. ](img/gLyapScanCinf.pdf)
For this choice of coupling function and suitable $\e$, the product system gives rise to a chaotic weak chimera of saddle type. While these chaotic weak chimeras are attracting in $\Dn\times\Torn$, they are transversally unstable in $\Tornn$ since ${g}'(0)>0$. Figure \[fig:N4CWCSaddle\] shows a trajectory that is initialized slightly off the invariant set. The population order parameter $R_\ell(t) = \tabs{\frac{1}{\maxdim}\sum_{j=1}^\maxdim \exp(i\vphi_{\ell,j})}$, $\ell=1, 2$, characterizes synchronization in the system is equal to one for the synchronized population and fluctuating chaotically for the incoherent population.
Numerical simulations show that these chaotic weak chimeras persist for $\e>0$. We calculated the maximal Lyapunov exponent $\lmax$ by numerically integrating the variational equations for [^2]. As shown in Figure \[fig:N4CWCSaddleScan\], chaotic weak chimeras apparently exist for most values $\e\lessapprox 0.3$.
An attracting chaotic weak chimera
----------------------------------
As indicated in Corollary \[cor:StabilitySwap\] the chaotic weak chimeras of saddle type can be made attracting in $\Tornn$ with a suitable local perturbation to the coupling function $g$. Define $$\Bm(x) := \begin{cases}\exp\fleft(-\frac{1}{1-x^2}\right)&\text{if} -1<x<1,\\
0& \text{otherwise}\end{cases}$$ and let $a\in\R$, $b \in (0, \pi)$, $c\in \Tor$ be parameters. Consider the “bump function” $\Bm_{abc}(\phi) = a \Bm\big(\frac{\phi}{b}-c\big)$ with $\phi$ taken modulo $2\pi$ with values in $(-\pi, \pi]$. Thus, $\Bm_{abc}(\phi)$ is a $2\pi$-periodic $\Cinf$ function. Define $$\label{eq:ghat}
\gh := {g} + \Bm_{abc}.$$ Let $D$ be an open set with $\Ai\subset D\subset\C$. Now choose $a, b, c$ such that $F^{(\e, g)}|_D=F^{(\e, \gh)}|_D$ and $\gh'(0)<0$. The dynamics of with coupling function $\gh$ give rise to attracting chaotic weak chimeras. Figure \[fig:CinftyN4CWC\](<span style="font-variant:small-caps;">a</span>) shows a single trajectory for $\e=0.2$ that is initialized slightly off the invariant set.
\
Attracting chaotic weak chimeras also appear for larger population sizes. With the same coupling function $\gh$ as for $\maxdim=4$ oscillators we also find chaotic weak chimeras for $\maxdim\in\sset{5, 7}$; cf. [@Bick2011]. A trajectory for $\maxdim=7$ oscillator in each population is depicted in Figure \[fig:CinftyN4CWC\](<span style="font-variant:small-caps;">b</span>).
Other coupling functions giving chaotic weak chimeras {#subsec:trigpoly}
-----------------------------------------------------
\
\
The appearance of attracting chaotic weak chimeras is not limited to the perturbed coupling function constructed above; these typically have infinitely many nontrivial coefficients in their Fourier expansion. Attracting chaotic weak chimeras can also be found for trigonometric polynomial coupling functions with only finitely many nontrivial Fourier modes. Consider the dynamics of with two populations of $\maxdim=4$ phase oscillators and coupling function $$g(\phi) = \sum_{r=1}^{L} c_r \cos (r\phi)+ s_r\sin(r\phi)$$ with $L=11$ and Fourier coefficients as specified in Table \[tab:FourierCoeafficients\] in Appendix \[app:TrigPoly\]. This coupling function is analytic and the dynamics show that there is an attracting chaotic weak chimera for a range of $\e$; cf. Figure \[fig:ComegaN4CWC\](<span style="font-variant:small-caps;">a</span>). Our simulations indicate that apart from the chaotic weak chimera there are other stable attractors in the system. Further numerical investigation, using the fixed initial conditions on the chaotic weak chimera in Figure \[fig:ComegaN4CWC\](<span style="font-variant:small-caps;">a</span>), is summarized in Figure \[fig:ComegaN4CWC\](<span style="font-variant:small-caps;">b</span>). Positive maximal Lyapunov exponents can arise for a range of positive coupling values $\e\lessapprox 0.15$. It is possible that adiabatic continuation—that is, using a point on the attractor for one parameter value as an initial condition for a nearby parameter value—may give more detailed insights into how the chaotic weak chimeras develop and bifurcate as $\e$ is varied.
Discussion {#sec:discuss}
==========
In Section \[sec:CWC\] we prove a general existence result for weak chimeras in coupled phase oscillator systems. Chimeras are constructed in “modular” networks that are weakly coupled but the numerical investigations indicate that they exist even beyond the weak coupling limit. As dynamically invariant sets, the weak chimeras are not transient—“persistent” chimera states in systems with generalized coupling were recently observed numerically [@Suda2015]. But the existence of weak chimeras of saddle type induces transient dynamics for initial conditions close to the saddle.
While our results are stated for the symmetric case that each population has the same number of oscillators, it is straightforward to generalize the construction to asymmetric systems with populations of differing sizes, say of $n_1$ and $n_2$ oscillators. If the oscillators belonging to the coherent region are not only frequency synchronized but also phase locked (their phase difference stays constant over time) the minimal number of oscillators to get similar chaotic weak chimera dynamics is $n_1+n_2=5$ since the dynamics are effectively three dimensional; however in this case the oscillators will not longer be indistinguishable.
The weak chimeras of saddle type (i.e., with both stable and unstable directions) are certainly of interest as analogues to the states studied in [@Wolfrum2011b]. Further existence results of such chaotic weak chimeras may be possible with the methods mentioned in Section \[sec:Persistence\]: normally hyperbolic sets persist under small perturbations. Moreover, for suitable choice of parameters, such chimera saddles could have connecting orbits, thus leading to transitions from one chimera state to the next one. Constructions of such connections could be seen as a form of control of spatially localized dynamics similar to chimera control [@Bick2014a].
The relationship between the chaotic weak chimeras we constructed here and “classical” chimera states in multiple populations needs to be clarified further: in [@Abrams2004] the coupling function only contains a single nontrivial Fourier mode and the coupling strength between the populations is almost as strong as within a population. Moreover, “large” populations are considered in the continuum limit of infinitely many oscillators where the dynamics reduce to mean field equations [@Abrams2004; @Panaggio2015a]. By contrast, the chaotic weak chimeras here arise in weakly coupled populations of four to seven oscillators with coupling given by functions with four or more nontrivial Fourier components. Continuation of these solutions in a suitable parameter space may shed some light on whether there these solutions are directly related. It is worth noting that the chaotic weak chimeras constructed here in such small networks also exhibit chaotic fluctuations of the order parameter [@Bick2011] similar what is observed for large ensembles of interacting nonsmooth oscillators [@Pazo2014]. Clearly, our results for small populations extend to more general (limit cycle) oscillators whose phase reduction is given by the coupling functions of our construction; see [@Kori2008] for approaches to design general oscillator systems with a desired phase reduction. In fact, we expect chaotic dynamics to be more common for general oscillator networks that are not necessarily weakly coupled (as long as phases can still be defined, see remarks in [@Bick2015d]) as amplitude variations facilitate chaotic dynamics in globally coupled identical oscillators [@Nakagawa1993].
In summary, the notion of a weak chimera yields a mathematically precise definition of chimera states for finite dimensional systems. Here we show that weak chimeras which mimic the dynamical behavior of regular chimeras can be constructed explicitly. We anticipate that a similar study of weak chimeras may give further insight into how chimeras arise in finite-dimensional coupled phase oscillator networks.
Acknowledgements {#acknowledgements .unnumbered}
================
CB and PA would like to thank Mike Field and Tiago Pereira for helpful discussions. The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013) under REA grant agreement n^o^ 626111 (CB).
A trigonometric polynomial coupling function {#app:TrigPoly}
============================================
Table \[tab:FourierCoeafficients\] gives coefficients for a trigonometric polynomial coupling function that gives an attracting chaotic weak chimera for the system discussed in Section \[subsec:trigpoly\].
$$\begin{aligned}
c_0 &= -0.48239 & s_0 &= 0.0538\\
c_1 &= -0.48239 & s_1 &= -0.05766\\
c_2 &= -0.23244 & s_2 &= 0.03754\\
c_3 &= -0.20325 & s_3 &= 0.0313\\
c_4 &= 0.01322 & s_4 &= -0.00626\\
c_5 &= 0.01261 & s_5 &= -0.0074\\
c_6 &= 0.01191 & s_6 &= -0.00849\\
c_7 &= 0.01111 & s_7 &= -0.00951\\
c_8 &= 0.01023 & s_8 &= -0.01045\\
c_9 &= 0.00927 & s_9 &= -0.01131\\
c_{10} &= 0.00823 & s_{10} &= -0.01209\end{aligned}$$
[^1]: Note that in this case the oscillators may not be indistinguishable anymore.
[^2]: Integration for $T=7000$ time units was carried out in MATLAB using the standard adaptive Runge–Kutta scheme with relative and absolute error tolerances of $10^{-9}$ and $10^{-11}$ respectively.
|
---
abstract: 'We give an easy counter-example to Problem 7.20 from C. Villani’s book on mass transport: in general, the quadratic Wasserstein distance between $n$-fold normalized convolutions of two given measures fails to decrease monotonically.'
address: '[Financial and Actuarial Mathematics](http://www.fam.tuwien.ac.at/), Technical University Vienna, Wiedner Hauptstrasse 8–10, A-1040 Vienna, Austria.'
author:
- 'Walter Schachermayer, Uwe Schmock, and Josef Teichmann'
date: 'October 4, 2006'
title: 'Non-monotone convergence in the quadratic Wasserstein distance'
---
[^1]
We use the terminology and notation from [@vil:03]. For Borel measures $\mu$, $\nu$ on ${{\mathbb R}}^d$ we define the quadratic Wasserstein distance $${{\mathcal T}}(\mu,\nu):=\inf_{(X,Y)}{{\mathbb E}}\bigl[\Vert X-Y\Vert^{2}\bigr]$$ where $\Vert{\mathchoice{\mskip1.5mu}{\mskip1.5mu}{}{}}\cdot{\mathchoice{\mskip1.5mu}{\mskip1.5mu}{}{}}\Vert$ is the Euclidean distance on ${{\mathbb R}}^d$ and the pairs $(X,Y)$ run through all random vectors defined on some common probabilistic space $(\Omega,\mathcal{F},{{\mathbb P}})$, such that $X $ has distribution $\mu$ and $Y $ has distribution $\nu$. By a slight abuse of notation we define ${{\mathcal T}}(U,V):={{\mathcal T}}(\mu,\nu)$ for two random vectors $U$, $V{\mathchoice{\mskip-1.5mu}{\mskip-1.5mu}{}{}}$, such that $U$ has distribution $\mu$ and $V$ has distribution $\nu$. The following theorem (see ) is due to Tanaka [@tan:73].
\[th1\] For $a,b\in{{\mathbb R}}$ and square integrable random vectors $X$, $Y$, $X'$, $Y'$ such that $X$ is independent of $Y$, and $X'$ is independent of $Y'$, and ${{\mathbb E}}[X]={{\mathbb E}}[X']$ or ${{\mathbb E}}[Y]={{\mathbb E}}[Y']$, we have $${{\mathcal T}}(aX+bY,aX'+bY')
\le a^2{{\mathcal T}}(X_,X')+b^2{{\mathcal T}}(Y,Y').$$
For a sequence of i.i.d. random vectors $(X_{i})_{i\in{{\mathbb N}}}$ we define the normalized partial sums $$S_{m}:=\frac{1}{\sqrt{m}} \sum_{i=1}^{m}X_{i},\qquad m\in{{\mathbb N}}.$$ If $\mu$ denotes the law of $X_{1}$, we write $\mu^{(m)}$ for the law of $S_{m}$. Clearly $\mu^{(m)}$ equals, up to the scaling factor $\sqrt
{m}$, the $m$-fold convolution $\mu\ast\mu\ast\dots\ast\mu$ of $\mu$.
We shall always deal with measures $\mu $, $\nu$ with vanishing barycenter. Given two measures $\mu$ and $\nu$ on ${{\mathbb R}}^d$ with finite second moments, we let $(X_{i})_{i\in{{\mathbb N}}}$ and $(X'_{i})_{i\in{{\mathbb N}}}$ be i.i.d. sequences with law $\mu$ and $\nu$, respectively, and denote by $S_{m}$ and $S_{m}
^{\prime}$ the corresponding normalized partial sums. From Theorem \[th1\] we obtain $${{\mathcal T}}\big( \mu^{(2m)},\nu^{(2m)} \big) \le{{\mathcal T}}\big( \mu
^{(m)},\nu^{(m)} \big),\qquad m\in{{\mathbb N}},$$ from which one may quickly deduce a proof of the Central Limit Theorem (compare [@vil:03 Ch. 7.4] and the references given there).
However, we can *not* deduce from Theorem \[th1\] that the inequality $$\label{J3}
{{\mathcal T}}\big(\mu^{(m+1)},\nu^{(m+1)}\big)\le{{\mathcal T}}\big(\mu^{(m)}
,\nu^{(m)}\big)$$ holds true for all $m\in{{\mathbb N}}$. Specializing to the case $m=2$, an estimate, which we can obtain from Tanaka’s Theorem, is $${{\mathcal T}}\big(\mu^{(3)},\nu^{(3)}\big)
\le\frac{1}{3}
\bigl[2{{\mathcal T}}\big(\mu^{(2)},\nu^{(2)}\big)+{{\mathcal T}}(\mu,\nu)\bigr]
\le{{\mathcal T}}(\mu,\nu).$$ This contains some valid information, but does not imply . It was posed as Problem 7.20 of [@vil:03], whether inequality holds true for all probability measures $\mu$, $\nu$ on ${{\mathbb R}}^d$ and all $m\in{{\mathbb N}}$.
The subsequent easy example shows that the answer is no, even for $d=1$ and symmetric measures. We can choose $\mu=\mu_{n}$ and $\nu=\nu_{n}$ for sufficiently large $n\ge2$, as the proposition (see also Remark \[counter-example\_Remark\]) shows.
\[Main\_Prop\] \[prop2\] Denote by [$\mu$]{}$_{n}$ the distribution of $\sum_{i=1}^{2n-1}Z_{i}$, and by $\nu_{n}$ the distribution of $\sum_{i=1}^{2n}Z_{i}$ with $(Z_{i})_{i\in{{\mathbb N}}}$ i.i.d. and ${{\mathbb P}}(Z_{1}=1)={{\mathbb P}}(Z_{1}=-1)=\frac{1}{2}$. Then $$\label{J4}
\lim_{n\rightarrow\infty}\sqrt{n}\,{{\mathcal T}}(\mu_{n}\ast\mu_{n},\nu_{n}\ast
\nu_{n})=\frac{2}{\sqrt{2\pi}},$$ while ${{\mathcal T}}(\mu_{n}\ast\mu_{n}\ast\mu_{n},\nu_{n}\ast\nu_{n}\ast\nu
_{n})\ge1$ for all $ n\in{{\mathbb N}}$.
\[counter-example\_Remark\] If one only wants to find a counter-example to Problem 7.20 of [@vil:03], one does not really need the full strength of Proposition \[Main\_Prop\], i.e. the estimate that ${{\mathcal T}}(\mu_n \ast \mu_n, \nu_n \ast \nu_n) =
\mathcal{O}(1/\sqrt{n})$. In fact, it is sufficient to consider the case $n=2$ in order to contradict the monotonicity of inequality . Indeed, a direct calculation reveals that $${{\mathcal T}}(\mu_{2}\ast\mu_{2},\nu_{2}\ast\nu_{2})=0.625<\frac{2}{3} \leq
{\biggr(\frac{\sqrt{2}}{\sqrt{3}}\biggl)}^2 {{\mathcal T}}(\mu_2 \ast \mu_2 \ast
\mu_2, \nu_2 \ast \nu_2 \ast \nu_2 ).$$
We start with the final assertion, which is easy to show. The $3$-fold convolutions of the measures $\mu_{n}$ and $\nu_{n}$, respectively, are supported on odd and even numbers, respectively. Hence they have disjoint supports with distance $1$ and so the quadratic transportation costs are bounded from below by $1$.
For the proof of , fix $n\in{{\mathbb N}}$, define $\sigma_{n}=\mu_{n}\ast\mu_{n}$ and $\tau_{n}=\nu_{n}\ast\nu_{n}$, and note that $\sigma_{n}$ and $\tau_{n}$ are supported by the even numbers. For $k=-(2n-1),\dots,(2n-1)$ we denote by $p_{n,k}$ the probability of the point $2k$ under $\sigma_{n}$, i.e. $$p_{n,k}=\binom{4n-2}{k+2n-1}\frac{1}{2^{4n-2}}.$$ We define $p_{n,k}=0$ for $ |k| \ge 2n $. We have $\tau_n=\sigma_n\ast\rho$, where $\rho$ is the distribution giving probability $\frac{1}{4}$, $\frac{1}{2}$, $\frac{1}{4}$ to $-2$, $0$, $2$, respectively. We deduce that for $0\le k\le2n-2$, $$\label{netto_equation}
\begin{split}
\tau_{n}(2k+2)
&=\frac{1}{4}p_{n,k}+\frac{1}{4}p_{n,k+2}+\frac{1}{2}p_{n,k+1}\\
&=\frac{1}{4}(p_{n,k}-p_{n,k+1})+\frac{1}{4}(p_{n,k+2}-p_{n,k+1})+\sigma
_{n}(2k+2)\\
&=\frac{1}{4}p_{n,k}\Bigl(1-\frac{p_{n,k+1}}{p_{n,k}}\Bigr)+\frac{1}{4}p_{n,k+1}
\Bigl(\frac{p_{n,k+2}}{p_{n,k+1}}-1 \Bigr) + \sigma_{n}(2k+2).
\end{split}$$ Notice that $p_{n,k}\ge p_{n,k+1}$ for $0\le k\le2n-1$. The term in the first parentheses is therefore non-negative. It can easily be calculated and estimated via $$0\le1-\frac{p_{n,k+1}}{p_{n,k}}
=1-\frac{\binom{4n-2}{k+2n}}{\binom{4n-2}{k+2n-1}}
=1-\frac{2n-k-1}{k+2n}=\frac{2k+1}{2n+k}
\le\frac{2k+1}{2n},$$ for $0\le k\le2n-1$.
Following [@vil:03] we know that the quadratic Wasserstein distance ${{\mathcal T}}$ can be given by a cyclically monotone transport plan $\pi
=\pi_{n}$. We define the transport plan $\pi$ via an intuitive transport map $T$. It is sufficient to define $T$ for $0\le k\le2n-1$, since it acts symmetrically on the negative side. $T$ moves mass $\frac14 p_{n,k}\frac{2k+1}{2n+k} $ from the point $2k$ to $2k+2$ for $k\ge1$. At $k=0$ the transport $T$ moves $\frac{1}{8n}p_{n,0}$ to every side, which is possible, since there is enough mass concentrated at $0$.
By equation we see that the transport $T $ moves $\sigma_{n} $ to $\tau_{n} $, since, for $1\le k\le 2n-2$, the first terms corresponds to the mass, which arrives from the left and is added to $\sigma_{n} $, and the second term to the mass, which is transported away: summing up one obtains $\tau_{n}$. For $k=2n-1$, mass only arrives from the left. At $k = 0 $ mass is only transported away. By the symmetry of the problem around $0$ and by the quadratic nature of the cost function (the distance of the transport is $2$, hence cost $2^{2}$), we finally have $${{\mathcal T}}(\sigma_{n},\tau_{n})
\le 2 \sum_{k=0}^{2n-1}\frac{2^{2}}{4}p_{n,k}\frac{2k+1}{2n+k}
\le \sum_{k=0}^{2n-1}p_{n,k}\frac{2k+1}{n}.$$ By the Central Limit Theorem and uniform integrability of the function $ x
\mapsto x_+
:= \max(0,x) $ with respect to the binomial approximations, we obtain $$\lim_{n\rightarrow\infty}\frac{1}{2\sqrt{n}}\sum_{k=0}^{2n-1}(2k)p_{n,k}=\int_0^{\infty}\frac{x}{\sqrt{2\pi}}e^{-x^2/2}\,dx.$$ Hence $$\limsup_{n\rightarrow\infty}\sqrt{n}\,{{\mathcal T}}(\sigma_{n},\tau_{n})\le\frac
{2}{\sqrt{2\pi}}\approx 0.79788.$$
In order to obtain equality we start from the local monotonicity of the respective transport maps on non-positive and non-negative numbers. It easily follows that the given transport plan is cyclically monotone and hence optimal (see [@vil:03 Ch. 2]). The subsequent equality allows also to consider estimates from below. Rewriting yields $$\tau_{n}(2k+2)=\frac{1}{4}p_{n,k+1}\Bigl (\frac{p_{n,k}}{p_{n,k+1}}-1 \Bigr
)+\frac{1}{4}p_{n,k+2} \Bigl ( 1-\frac{p_{n,k+1}}{p_{n,k+2}} \Bigr )
+\sigma_{n}(2k+2)$$ for $ 0 \le k \le 2n-3 $, and $$\tau_n (2k+2) = \frac{1}{4} p_{n,k+1} \Bigl ( \frac{p_{n,k}}{p_{n,k+1}} -1
\Bigr ) + \sigma_n (2k+2)$$ for $k=2n-2$. Furthermore, $$\frac{p_{n,k}}{p_{n,k+1}}-1
=\frac{\binom{4n-2}{k+2n-1}}{\binom{4n-2}{k+2n}}-1
=\frac{k+2n}{2n-k-1}-1=\frac{2k+1}{2n-k-1}
\ge\frac{2k+1}{2n}$$ for $0\le k\le2n-2$. This yields by a reasoning similar to the above that $${{\mathcal T}}(\sigma_{n},\tau_{n})
\ge\sum_{k=0}^{2n-2}p_{n,k+1} \frac
{2k+1}{n},$$ hence
[10000]{}
[-0.5]{} $$\liminf_{n\rightarrow\infty}\sqrt{n}\,{{\mathcal T}}(\sigma_{n},\tau_{n}) \ge
\frac{2}{\sqrt{2\pi}}.$$
Let $p\ge2$ be an integer. By slight modifications of the proof of Proposition \[Main\_Prop\] we can construct sequences of measures $(\mu_{n})_{n\in{{\mathbb N}}}$ and $(\nu_{n})_{n\in{{\mathbb N}}}$, such that the quadratic Wasserstein distances of $k$-fold convolutions are bounded from below by $1$ for all $k$ which are not multiples of $p$, while $$\lim_{n \to\infty} {{\mathcal T}}(\mu_{n}^{(p)},\nu_{n}^{(p)})= 0.$$
Assume the notations of [@vil:03]. In the previous considerations we can replace the quadratic cost function by any other lower semi-continuous cost function $c:{{\mathbb R}}^2\rightarrow[{\mathchoice{\mskip1.5mu}{\mskip1.5mu}{}{}}0,+\infty]$, which is bounded on parallels to the diagonal and vanishes on the diagonal. For example, if we choose $c(x,y)={|x-y|}^{r}$ for $0<r<\infty$, then we obtain the same asymptotics as in Proposition \[Main\_Prop\] (with a different constant).
We have used in the above proof that $\tau_{n}$ is obtained from $\sigma_{n}$ by convolving with the measure $\rho$. In fact, this theme goes back (at least) as far as L. Bachelier’s famous thesis from 1900 on option pricing [@bac:00 p. 45]. Strictly speaking, L. Bachelier deals with the measure assigning mass $\frac{1}{2}$ to $-1$, $1$ and considers consecutive convolutions, instead of the above $\rho$. Hence convolutions with $\rho$ correspond to Bachelier’s result after two time steps. Bachelier makes the crucial observation that this convolution leads to a *radiation* of probabilities: Each stock price $x$ radiates during a time unit to its neighboring price a quantity of probability proportional to the difference of their probabilities. This was essentially the argument which allowed us to prove . Let us mention that Bachelier uses this argument to derive the fundamental relation between Brownian motion (which he was the first to define and analyse in his thesis) and the heat equation (compare e.g. [@sch:03] for more on this topic).
Having established the above counterexample, it becomes clear how to modify Problem 7.20 from [@vil:03] to give it a chance to hold true. This possible modification was also pointed out to us by C. Villani.
Let $\mu$ be a probability measure on ${{\mathbb R}}^d$ with finite second moment and vanishing barycenter, and $\gamma$ the Gaussian measure with same first and second moments. Does $({{\mathcal T}}(\mu^{(n)},\gamma))_{n\ge1}$ decrease monotonically to zero?
When entropy is considered instead of the quadratic Wasserstein distance the corresponding question on monotonicity was answered affirmatively in the recent paper [@artbalbarnao:04].
One may also formulate a variant of Problem 7.20 as given in (\[J3\]) by replacing the measure $ \nu $ through a log-concave probability distribution. This would again generalize problem 1.
[9]{}
S. Artstein, K. M. Ball, F. Barthe and A. Naor, *Solution of Shannon’s Problem on the Monotonicity of Entropy*, Journal of the AMS **17**(4), 2004, pp. 975–982. L. Bachelier, *Theorie de la Speculation*, Paris, 1900, see also: <http://www.numdam.org/en/>.
W. Schachermayer, *Introduction to the Mathematics of Financial Markets*, LNM 1816 - Lectures on Probability Theory and Statistics, Saint-Flour summer school 2000 (Pierre Bernard, editor), Springer Verlag, Heidelberg (2003), pp. 111–177.
H. Tanaka, *An inequality for a functional of probability distributions and its applications to Kac’s one-dimensional model of a Maxwell gas*, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete **27**, 47–52, 1973
C. Villani, *Topics in Optimal Transportation*, Graduate Studies in Mathematics **58**, American Mathematical Society, Providence Rhode Island, 2003.
[^1]: Financial support from the Austrian Science Fund ([FWF](http://www.fwf.ac.at/)) under grant P 15889, from the Vienna Science and Technology Fund ([WWTF](http://www.wwtf.at/)) under grant MA13, from the European Union under grant HPRN-CT-2002-00281 is gratefully acknowledged. Furthermore this work was financially supported by the Christian Doppler Research Association ([CDG](http://www.cdg.ac.at/)) via [PRisMa Lab](http://www.prismalab.at/). The authors gratefully acknowledge a fruitful collaboration and continued support by Bank Austria Creditanstalt ([BA-CA](http://www.ba-ca.com/)) and the Austrian Federal Financing Agency ([ÖBFA](http://www.oebfa.co.at/)) through CDG
|
---
abstract: 'We study static, spherically symmetric black holes supported by Euler-Heisenberg theory of electrodynamics and coupled to two different modified theories of gravity. Such theories are the quadratic $f(R)$ model and Eddington-inspired Born-Infeld gravity, both formulated in metric-affine spaces, where metric and affine connection are independent fields. We find exact solutions of the corresponding field equations in both cases, characterized by mass, charge, the Euler-Heisenberg coupling parameter and the modified gravity one. For each such family of solutions, we characterize its horizon structure and the modifications in the innermost region, finding that some subclasses are geodesically complete. The singularity regularization is achieved under two different mechanisms: either the boundary of the manifold is pushed to an infinite affine distance, not being able to be reached in finite time by any geodesic, or the presence of a wormhole structure allows for the smooth extension of all geodesics overcoming the maximum of the potential barrier.'
author:
- Merce Guerrero
- 'Diego Rubiera-Garcia'
title: 'Nonsingular black holes in nonlinear gravity coupled to Euler-Heisenberg electrodynamics'
---
Introduction {#sec:I}
============
Black holes are one of the most fascinating objects in Nature. Originally obtained as exact solutions of Einstein’s field equations, their properties were poorly understood for decades until becoming nowadays a full-fledged member of the family of astronomical objects. From a mathematical point of view, they can be formed from a regular distribution of matter in such a way that a trapped surface is developed [@JoshiBook]. From an astrophysical viewpoint, the gravitational collapse out of fuel-exhausted main-sequence stars ($\gtrsim 25M_{\odot}$) provides the physical mechanism for such a generation [@Heger:2002by]. Moreover, no matter the properties and/or symmetries of the original configuration, the outside metric to the end-state of such a collapse will be always the Kerr-Newman solution, described solely by three parameters: mass, charge and angular momentum [@SteBook]. Over the years, we have accumulated plenty of evidence on the reliability of the simpler Kerr solution (since charge can be typically neglected in astrophysical environments [@Zajacek:2019kla]) to describe such objects, as follows from observations of the X-ray radiation emitted from the inner part of their accretion disks [@Bambi:2017iyh; @Bambi:2019xzp], from gravitational wave emission out of binary mergers [@Abbott1; @Abbott2] and from the imaging of the neighborhood ([*i.e.*]{} the shadow) of the central supermassive object of the M87 galaxy [@M87].
Given the remarkable agreement between the theoretical predictions and the astronomical observations regarding black holes, it is worth taking seriously another key feature of their theoretical description, namely, the existence of space-time singularities deep inside them [@WaldBook]. Indeed, as long as General Relativity (GR) holds, the theorems on singularities [@Theorem1; @Theorem2] (for a pedagogical discussion see e.g. [@Senovilla:2014gza]) tell us that the development of a focusing point at a finite affine time for some set of geodesics in the innermost region of a black hole ($r=0$ in the case of a spherically symmetric black hole) is unavoidable provided that standard energy conditions are satisfied by the matter fields. Now, since null and time-like geodesics are associated to the trajectories of light rays and the free-falling of physical observers, respectively, the incompleteness of any of them is an utterly unpleasant feature, being linked to the lack of predictability of our physical theories. Typically, cosmic censorship arguments are developed [@CCC] in order to cover such singularities behind an event horizon, so as not to have observable effects on asymptotic observers. However, it is distressing that one needs to hide under the carpet such an abhorrent feature of an otherwise observationally successful object (outside its event horizon). Therefore, several arguments have been developed in order to overcome this difficulty without jeopardizing the exterior physics to the horizon, the jewel of the crown being the hyphotesis that quantum gravity effects should come to rescue when the growth of curvature approaches the Planck scale [@OritiBook].
The question on how to incorporate such effects has received many different answers along the decades. To play as conservatively as possible, one way to address them is via effective modifications of the gravitational action [@Review1; @Review2], which could be able to provide some hints on the transition from the classical (GR) regime to the quantum non-classical one and, moreover, to yield new phenomenology in astrophysical environments [@Berti]. This way one can still safely use the tools of the differential manifolds paradigm but modifying/reinterpreting some of its building blocks. This is the path followed in the present work, where we shall adhere to the spirit of the Equivalence Principle, guaranteeing the universality of free-fall motion while keeping the minimal coupling of the matter fields to the gravitational sector, but restore the metric and the affine connection to their status as independent entities (Palatini or metric-affine approach [@Olmo:2011uz]). Indeed, GR can be consistently formulated as a metric-affine theory, with the variation of the Einstein-Hilbert action with respect to the independent connection yielding the metric-connection compatibility condition (which is imposed ab initio in the metric formulation), and the predictions of this formulation are exactly the same ones as those of metric-formulated GR [@BeltranJimenez:2019tjy]. However, when more general actions are considered, the dynamics of metric-affine gravities strongly departs from their metric cousins, offering new ways of addressing the issue with space-time singularities.
For the sake of this paper, we shall consider two well known gravitational extensions of GR: quadratic $f(R)$ gravity [@Staro] and Eddington-inspired Born-Infeld (EiBI) gravity [@EiBI]. As the matter sector we shall use an old acquaintance of the singularity-regularization attempts: nonlinear electrodynamics [@AyonBeato:1998ub; @Bronnikov:2000vy; @Dymnikova:2004zc; @Balart:2014cga; @Balart:2014jia; @Dymnikova:2015hka; @Rodrigues:2015ayd; @Nojiri:2017kex; @Chinaglia:2017uqd; @Rodrigues:2018bdc; @Rodrigues:2019xrc]. Indeed, way before the scale where quantum gravity effects are expected to be excited in the innermost region of black holes, the growth of the electric field would induce quantum vacuum polarization effects modifying the classical description of Maxwell electrodynamics. In an effective approach, such effects to one loop and in the slowly-varying approximation can be incorporated by adding a quadratic piece in the electromagnetic field invariants to the Maxwell Lagrangian, yielding the so-called Euler-Heisenberg electrodynamics [@EH; @EffBook]. Within GR such a model has some nice properties, like a finite energy associated to the system of point-like charges and the existence of new gravitational configurations in terms of the structure of horizons [@Yajima:2000kw; @Ruffini:2013hia; @Breton:2019arv]. However, space-time singularities still plague all such configurations, and similar comments apply to any nonlinear electrodynamics satisfying physically reasonable conditions [@Bronnikov:2000yz].
The main aim of this work is to find static, spherically symmetric solutions corresponding to quadratic $f(R)$ and EiBI gravity coupled to Euler-Heisenberg electrodynamics and investigate the existence of nonsingular black holes in both frameworks. We shall find that both of them have a branch of solutions (as given by the combination of the sign of the gravity and matter parameters) allowing for the completeness of all null and time-like geodesics. This restoration of geodesic completeness is achieved via two different mechanisms: in the first one the focusing point is pushed out to an infinite affine distance preventing any set of geodesics to reach it in finite affine time, while in the second one a defocusing sphere is created at some finite affine distance represented by a wormhole throat with a finite area, in such a way that those geodesics able to overcome the potential barrier can be smoothly extended through the throat to another asymptotically flat region of the manifold. We shall furthermore discuss how these particular results fit within general studies aimed to achieve singularity-avoidance without breaking basic mathematical requirements or getting into contradiction with observations.
This work is organized as follows: in Sec. \[sec:II\] we introduce the basic framework in terms of metric-affine gravities and Euler-Heisenberg electrodynamics. In Sec. \[sec:III\] we find spherically symmetric solutions for quadratic $f(R)$ gravity and discuss their properties, with particular emphasis on the horizon and geodesic structure. A similar analysis is carried out for EiBI gravity in Sec. \[sec:IV\]. Finally in Sec. \[sec:V\] we summarize our findings and further discuss our results.
Theoretical framework {#sec:II}
=====================
Metric-affine gravities
-----------------------
For the sake of this work, we shall establish the theoretical framework for the subclass of metric-affine gravities dubbed as Ricci-based gravities (RBGs), defined by the action $$\label{eq:action}
\mathcal{S}_m=\frac{1}{2\kappa^2} \int d^4x \sqrt{-g} \mathcal{L}_G(g_{\mu\nu},R_{\mu\nu}(\Gamma)) + \mathcal{S}_m(g_{\mu\nu},\psi_m) \ ,$$ where $\kappa^2$ is Newton’s constant in suitable units, $g$ is the determinant of the space-time metric $g_{\mu\nu}$, the Ricci tensor (assumed to be symmetric) $R_{\mu\nu}(\Gamma) \equiv {R^\alpha}_{\mu\alpha\nu}(\Gamma)$ is solely built out of the (torsion-free [@Afonso:2017bxr]) affine connection $\Gamma \equiv \Gamma_{\mu\nu}^{\lambda}$ and the matter action $\mathcal{S}_m=\int d^4x \sqrt{-g} \mathcal{L}_m(g_{\mu\nu},\psi_m)$ is assumed to depend only on the space-time metric and on a set of matter fields $\psi_m$ but not on the connection, to ensure the fulfillment of the equivalence principle. These constraints upon the building blocks of the action (\[eq:action\]) guarantee the second-order and ghost-free character of their field equations [@BeltranJimenez:2019acz; @Jimenez:2020dpn]. Moreover, they imply that RBGs do not propagate extra degrees of freedom beyond the two polarizations of the gravitational field of GR and may pass solar system tests provided that the modifications to GR occur in the ultraviolet limit.
From independent variation of the action (\[eq:action\]) with respect to metric and connection, the resulting field equations may be conveniently written in the Einstein-like representation $$\label{eq:eomRBG}
{G^\mu}_{\nu}(q)=\frac{\kappa^2}{\vert \Omega \vert^{1/2}} \left[{T^\mu}_{\nu}-\delta^\mu_\nu \left(\mathcal{L}_G + \frac{T}{2} \right)\right] \ ,$$ where ${G^\mu}_{\nu}(q)$ is the Einstein tensor of a new rank-two tensor $q_{\mu\nu}$ satisfying $\nabla_{\mu}(\sqrt{-q} q^{\alpha\beta})=0$ (so that $\Gamma$ is Levi-Civita of $q$). This tensor is related to the space-time metric as $$\label{eq:Omegadef}
q_{\mu\nu}=g_{\mu\alpha}{\Omega^\alpha}_{\nu} \ ,$$ where the explicit shape of the *deformation matrix* ${\Omega^\mu}_{\nu}$ (vertical bars denoting its determinant) depends on the particular $\mathcal{L}_G$ chosen, but can be written always (and so does $\mathcal{L}_G$ itself) on-shell in terms of the stress-energy tensor $T_{\mu\nu} \equiv \frac{2}{\sqrt{-g}}\frac{\delta \mathcal{S}_m}{\delta g^{\mu\nu}}$ (with $T$ denoting its trace) and possibly the space-time metric as well. Therefore, the right-hand side of the field equations (\[eq:eomRBG\]) can be read off as representing an effective stress-energy tensor [@Afonso:2018bpv].
Euler-Heisenberg electrodynamics
--------------------------------
Nonlinear electrodynamics (NED) are described by a Lagrangian density $$\mathcal{L}_m=\varphi(X,Y) \ ,$$ where $ X= -\frac{1}{2} F_{ \mu \nu}F^{\mu \nu} $ and $Y=-\frac{1}{2}F_{\mu\nu}F^{*\mu\nu}$ are the two electromagnetic field invariants which can be built out of the field strength tensor, $ F_{\mu \nu}= \partial_\mu A_\nu - \partial_\nu A_\mu $, and its dual $F^{*\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$. The corresponding field equations are written as $\nabla_{\mu}(\varphi_X F^{\mu\nu} + \varphi_Y F^{* \mu\nu})=0$, where $\varphi_X \equiv \frac{\partial \varphi}{\partial X}$ and $\varphi_Y \equiv \frac{\partial \varphi}{\partial Y}$. For electrostatic configurations, the only non-zero component is $ F_{tr} \equiv E(r)$, and the field equations in a static, spherically symmetric space-time can be written as $$\label{eq:NEDeom}
X\varphi_X^2=\frac{q}{r^4} \ ,$$ where $X=E^2$ and $q$ is an integration constant identified as the electric charge for a given configuration. The NED stress-energy tensor $${T^\mu}_{\nu}=2(\varphi_X {F^\mu}_{\alpha}{F^\alpha}_{\nu} -\varphi_Y {F^\mu}_{\alpha}{F^{*\alpha}}_{\nu} )-\varphi(X,Y) \ ,$$ for electrostatic configurations can be conveniently split into $2 \times 2$ blocks as $$\begin{aligned}
\label{stress-energy tensor}
{T^\mu}_\nu &=& \dfrac{1}{8\pi}\left(\begin{array}{cc} (\varphi-2X\varphi_X)\, \hat{I}_{2\times 2} & \hat{0\,}_{2\times 2}\\ \hat{0\,}_{2\times 2}& \varphi \, \hat{I}_{2\times 2}\end{array}\right) \ ,\end{aligned}$$ where $ \hat{0}_{2\times 2} $ and $ \hat{I}_{2\times 2} $ are the $2 \times 2$ zero and identity matrices, respectively. From this expression, the trace reads $T=\frac{1}{2\pi}(\varphi-X\varphi_X)$, which is non-vanishing as long as $\varphi \neq X$ (Maxwell electrodynamics).
For the sake of this paper, we shall restrict our considerations to the case of Euler-Heisenberg electrodynamics, which is described by the particular function[^1] $$\label{Eq:Euler-Heisenberg}
\varphi(X) = X\, + \, \beta X^2 \ .$$ For this theory, the NED field equations (\[eq:NEDeom\]) for (electro-) static, spherically symmetric fields read $$E+2\beta E^3=\frac{q}{r^2} \ .$$ As can be easily verified, at $r \rightarrow \infty$ this theory recovers the Coulomb field, $E=q/r^2$, while for $r=0$ we have instead $E=(q/2\beta)^{1/3} r^{-2/3}$. In the latter limit, the contribution to the energy of the EH field behaves as $ \sim \int {T^0_0} r^2 dr \sim r^{1/3} \to 0$, which implies that the total energy associated to electrostatic configurations in EH electrodynamics is finite.
The field equations (\[eq:NEDeom\]) for EH electrodynamics can be solved in exact form as $$\label{Eq:field_invariant}
X(r)= \left[ \sqrt[3]{U + \sqrt{V^3 + U^2}} \,+ \, \sqrt[3]{U - \sqrt{V^3 + U^2}}\; \right]^2 \ ,$$ where $ U = \dfrac{q}{4 \beta r^2} $ and $ V = \dfrac{1}{6 \beta} $. To work with dimensionless variables, let us introduce a new length scale as $$\label{eq:rcdef}
r_{c}^4 =54\pi l^2_\beta \, r_q^2 \ ,$$ with $ l^2_\beta= \beta/\kappa^2 $ the squared NED length and $ r^2_q = \kappa^2 \, q^2/(4\pi) $ the squared charge radius. This way, Eq. can be rewritten in terms of the dimensionless coordinate $ z=r/r_c $ as $$\label{Eq:X_red}
X(z)=\dfrac{1}{6 \beta\, z^{4/3}}\left[ \left( 1+\sqrt{1+z^4}\right)^{1/3}+\left( 1-\sqrt{1+z^4}\right)^{1/3} \right]^2.$$ Moreover, we can get rid of cubic roots via the alternative expression [@Kruglov:2017ymn] $$\label{eq:Xz}
X(z)=\frac{2}{3\beta} \text{Sinh}^2\left[\frac{1}{3} \ln \left(\frac{1}{z^2}\left(1+ \sqrt{z^4+1}\right)\right)\right].$$ In terms of this dimensionless variable the stress-energy tensor (\[stress-energy tensor\]) for EH electrodynamics reads $$\begin{aligned}
\label{final-stress-energy tensor}
{T^\mu}_\nu &=& \dfrac{1}{8\pi} \left(\begin{array}{cc} -X (1+3\beta X)\, \hat{I}_{2\times 2} & \hat{0\,}_{2\times 2}\\ \hat{0\,}_{2\times 2}& X (1+\beta X) \, \hat{I}_{2\times 2}\end{array}\right)\end{aligned}$$ such that its components are found upon substitution of (\[eq:Xz\]). From this form of the stress-energy tensor it can be verified that EH electrodynamics satisfies the weak energy condition. Moreover, when coupled to the Einstein-Hilbert action of GR, the finite character of the electrostatic solutions of this theory manifests in the fact that, besides configurations with two or a single (degenerate) horizons and naked singularities, typical of the Reissner-Nordström solution of GR, there are also configurations with a single non-degenerate horizon (resembling the Schwarzschild black hole). However, in all cases, a singular behaviour is found as follows from the geodesic incompleteness of all such solutions (see however [@Poshteh:2020sgp]).
Quadratic $f(R)$ Gravity {#sec:III}
========================
Derivation of the solution
--------------------------
Our first RBG model to be analyzed is quadratic $ f(R) $ gravity, given by the Lagrangian density $$\label{f(R)}
\mathcal{L}_G= f(R) = R + \alpha R^2 \ ,$$ where $ \alpha $ is a constant with dimensions of length squared. For $f(R)$ gravity, the trace of the RBG field equations provides us with the following relation $ R\, f_R -2f = \kappa^2 T $ (with $f_R \equiv df/dR$), which tells us that the curvature scalar can be removed in favour of the trace of the stress-energy tensor. This fact implies that only NEDs with a non-vanishing trace will yield new dynamics as compared to GR, being the case of the EH electrodynamics considered in this paper. Moreover, for the quadratic Lagrangian (\[f(R)\]) the above equation yields $R=-\kappa^2 T$, which is the same relation as in GR.
The gravitational field equations (\[eq:eomRBG\]) in $ f(R) $ gravity boil down to $$\label{Eq:Ricci_tensor-f(R)}
{R^\mu}_\nu (q) = \dfrac{1}{f_R^2}\left( \dfrac{f}{2} {\delta^\mu}_\nu+ \kappa^2 {T^\mu}_\nu \right) \ ,$$ while the deformation matrix in this case becomes ${\Omega^\mu}_{\nu}=f_R \, \delta^\mu_\nu$. Therefore, from (\[eq:Omegadef\]) the space-time metric $ g_{\mu\nu} $ is conformally related to the Einstein frame metric $ q_{\mu\nu} $ as $$\label{Eq:metrics_fr}
q_{\mu\nu} = f_R \, g_{\mu\nu} \ ,$$ where we recall that $f_R \equiv f_R(T)$. Let us proceed with the resolution of the field equations (\[Eq:Ricci\_tensor-f(R)\]). To work as general as possible, at this stage we shall not impose constraints upon the shape of the function $\varphi(X)$. We begin by considering a static, spherically symmetric line element for the $ q_{\mu\nu} $ geometry as $$\label{Eq:line_element_q}
ds^2_q = -A(x) e^{2\psi(x)} dt^2 + \dfrac{dx^2}{A(x)} +x^2 d\Omega^2,$$ where $\psi(x), \, A(x)$ are the two metric functions, and $d\Omega^2=d\theta^2 + \sin^2 \theta d\phi^2$ is the volume element in the unit two-spheres. Now, using the symmetry in $2\times 2$ blocks of the stress-energy tensor (\[stress-energy tensor\]), the combination $ {R^t}_{t}- {R^x}_{x}=0 $ of the field equations (\[Eq:Ricci\_tensor-f(R)\]) allows to set $\psi(x)=0$ in (\[Eq:line\_element\_q\]) without any loss of generality. Now, defining the usual mass ansatz $$\label{Eq:ansatz}
A(x) = 1- \dfrac{2M(x)}{x} \ ,$$ we plug it into the remaining non-vanishing component of the field equations as $${R^\theta}_{\theta}=\frac{1}{x^2}\left(1-A-xA_x\right)=\frac{2M_x}{x^2} \ .$$ Equaling it to the right-hand side of the field equations (\[Eq:Ricci\_tensor-f(R)\]) we find that the mass function satisfies $$\label{eq:Mx}
M_x=\frac{x^2}{2f_R^2} \left(\frac{f}{2}+\frac{\kappa^2 \varphi}{8\pi}\right) \ .$$ We next need to express this function in terms of the radial coordinate of the space-time metric $ g_{\mu \nu} $, the latter having the line element $$\label{eq:lineg}
ds^2=-C(x)dt^2+\frac{dx^2}{B(x)} +r^2(x)d\Omega^2 \ ,$$ with $C$ and $B$ new functions to be determined. Eq.(\[Eq:metrics\_fr\]) tells us that the relation between the radial coordinates on both frames is given by $$\label{eq:xrfR}
x^2=r^2f_R \ .$$ Taking a derivative here with respect to $r$ and inserting the result in (\[eq:Mx\]) we arrive at $$\label{Eq:diff_mass_fr}
M_r = \dfrac{r^2}{4 \,f_R^{3/2}} \left( f + \dfrac{\kappa^2 \varphi}{4 \pi}\right) \left(f_R +\dfrac{r}{2} f_{R,\,r}\right) \ .$$ Moreover, by using again (\[Eq:metrics\_fr\]) in the temporal and radial sectors, we arrive to the solution of the line element (\[eq:lineg\]) as $$\label{eq:finalsolf(R)}
ds^2=-C(x) dt^2 + \frac{dx^2}{f_R^2 C(x)} +z^2(x)d\Omega^2 \ ,$$ where we have introduced the dimensionless radial function $z(x)$, which is implicitly defined via Eq.(\[eq:xrfR\])[^2], while the function $C(x)$ can also be conveniently written in terms of this radial coordinate function as $$\label{eq:Final-ansatz-f(R)}
C(z) = \frac{1}{f_R} \left(1- \dfrac{1 + \delta_1 G(z)}{\delta_2 \, z \, f_R^{1/2}}\right).$$ In this metric function we have introduced the two main constants characterizing this problem as $$\begin{aligned}
\delta_1&=&\frac{r_c^3}{2\,l_{\beta}^2 \,r_S}=\frac{(54\pi)^{3/4}}{2\,r_S} \sqrt{\frac{r_q^3}{l_{\beta}}} \\
\delta_2&=&\frac{r_c}{r_S} \ ,\end{aligned}$$ where $r_S \equiv 2M_0 $ is Schwarzschild’s radius, while the function $G(z)$ in (\[eq:Final-ansatz-f(R)\]) is obtained in terms of its derivative as $$\label{eq:G(z)}
G_z (z)= \frac{z^2}{f_R^{3/2}}\left(\tilde{f} + \dfrac{\kappa^2 \tilde{\varphi}}{4 \pi} \right)\left(f_R + z\,f_{R, \,z}\right) \ ,$$ and has contributions from the $ f(R) $ sector as $$\begin{aligned}
\tilde{f} &\equiv& l_\beta^2 \, f= \dfrac{2}{9 \pi} \tau^4 (z) \left(1+\dfrac{\tilde{\alpha}}{2}\tau^4(z)\right)\\
f_R &=& 1 + \tilde{\alpha}\, \tau^4(z) \ , \label{eq:fRsol} \label{eq:fRex}\end{aligned}$$ where $ \tilde{\alpha} \equiv 4\alpha/(9 \pi l_\beta^2) $, and from the NED model as $$\begin{aligned}
\tilde{\varphi} &\equiv & l_\beta^2 \, \varphi=\frac{\tau^2(z)}{6\pi}\left(1+\frac{2}{3}\tau(z)\right) \\
\tau(z) &=& \text{Sinh} \ h(z) \label{eq:tau}\\
h(z) &=& \frac{1}{3} \ln \left[\frac{1}{z^2}\left(1+ \sqrt{z^4+1}\right)\right] \label{eq:hz} \ . \label{funcion-zc}\end{aligned}$$ In the last set of equations, we have introduced the EH model in the characterization of the function $\tau(z)$ (which is just the square root of $X(z)$ in Eq.(\[eq:Xz\]) removing the constants). The line element (\[eq:finalsolf(R)\]) with the definitions above is the solution to the problem of electrostatic solutions in quadratic Palatini $f(R)$ gravity coupled to EH electrodynamics, characterized by two integration constants: the mass $M$ and the electric charge $q$; and two new scales: the gravity parameter $\alpha$ and the matter parameter $\beta$, all of such constants encoded in the two parameters $\delta_1,\delta_2$. We also point out that this line element can be alternatively cast in a more Schwarzschild-like fashion by introducing the change of coordinates $ d\tilde{x}^2= f_R^{-2} dx^2$, though we shall not take this path in order not to spoil the simple representation of the function $z(x)$ in Eq.(\[eq:xrfR\]).
In the asymptotic limit, $z \to \infty$, one can verify that $f_R \approx 1+\frac{\tilde{\alpha}}{81z^8}$ while the function $G_z$ in Eq.(\[eq:G(z)\]) boils down to $$G_z \approx \frac{1}{54\pi z^2} - \frac{1}{729\pi z^6} + \mathcal{O}\left(\frac{\alpha}{z^{10}} \right) \ ,$$ which inserted in the metric function (\[eq:Final-ansatz-f(R)\]) and after spelling out the constants $\delta_1,\delta_2$ yields the result $$\label{eqAsymp}
C(r) \underset{r \to \infty}{\approx} 1-\frac{r_S}{r}+\frac{q^2}{r^2}-\frac{\beta q^4}{5r^6} + \mathcal{O}\left(\frac{\alpha}{z^{10}}\right) \ ,$$ where we have taken units such that $\kappa^2=8\pi$. The first three terms in this expression correspond to the Reissner-Nordström black hole of GR, while the next term is the correction from EH electrodynamics. The corrections introduced by $f(R)$ gravity appear only at order tenth, which should not come as a surprise, since the new gravitational dynamics encoded in the theory arise only in the innermost region of the solution, as we shall see next.
![The radial function $z(x)$ in Eq.(\[eq:xrfR\]) for the case $\tilde{\alpha}<0$ and values $\vert \tilde{\alpha} \vert = 0.2 $ (green), $\vert \tilde{\alpha} \vert = 0.5 $ (orange) and $\vert \tilde{\alpha} \vert = 1.0 $ (purple). The wormhole throat is located at $x=0$ ($z=z_c$ as defined in Eq.(\[eq:zcfR\])). As a comparison, we have plotted the GR case, $r^2=x^2$ (black dashed), for which no such a bounce in the radial function is present.[]{data-label="fig:rfrneg"}](fig1.pdf){width="45.00000%"}
Properties of the solution: radial function
-------------------------------------------
We begin now our analysis of the most relevant features of these solutions by considering the behaviour of the radial function $z\, (x)$ in (\[eq:xrfR\]). From (\[eq:fRex\]) and the positivity everywhere of $\tau\,(z)$, one finds that for $\tilde{\alpha}<0$ the function $f_R$ will vanish at a certain $z=z_c$ with $$\label{eq:zcfR}
z_c=\sqrt{\frac{2\,a}{a^2-1}} \ ,$$ where we have introduced the new constant $$a=\exp \left\lbrace 3\,\text{ArcSinh}\left(\vert \tilde{\alpha} \vert^{-1/4}\right)\right\rbrace \ .$$ Unfortunately, Eq.(\[eq:xrfR\]) does not admit a closed expression for $z=z\,(x)$ in its full domain of definition, but it is easy to see that at $z=z_c$ one has $x=0$ and beyond this point the radial function bounces off to another asymptotically flat region of space-time (see Fig.\[fig:rfrneg\]). Therefore, the area of the two-spheres $S=4\pi z^2$ is bounded from below, and the space-time consist of two patches of the radial function $z \in [z_c,\infty)$ or a single one in terms of the radial coordinate $x \in (-\infty,+\infty)$. The natural interpretation for this bouncing behavior and minimum areal function is that of a wormhole structure [@VisserBook], with $z=z_c$ representing its throat. The size of the latter grows with $\vert \tilde{\alpha} \vert$, while it closes in the limit $\vert \tilde{\alpha} \vert \to 0$, corresponding to GR.
In the $ \tilde{\alpha}>0 $ case, things are far less interesting. Indeed, in such a case $f_R$ has not zeros and the radial function $z(x)$ does not yield a bounce, but instead generates two branches of solutions. As depicted in Fig.\[fig:rfrpos\], there is not a smooth transition between these two branches, and the area of the two-spheres can go all the way down to vanishing value. The corresponding solutions are presumably singular and, therefore, we shall no longer consider them here.
![Plot of the dimensionless coordinate $ z $ as a function of $ x $ for the case $ \tilde{\alpha} >0 $. Same notation as in Fig. \[fig:rfrneg\]. []{data-label="fig:rfrpos"}](fig2.pdf){width="45.00000%"}
Properties of the solution: inner behaviour and horizons
--------------------------------------------------------
From now on we shall focus on characterizing the properties of the branch $\tilde{\alpha}<0$, where we have some hope of finding regular black hole solutions. Since the deviations as compared to GR solutions are expected to arise in the innermost region, we need to study the behaviour of the functions $f_R$ and $G(z)$ there. In this sense, a series expansion of the former using (\[eq:fRsol\]) around $z=z_c$, as defined in (\[eq:zcfR\]), yields the result $$\label{Eq:fR_expansion}
f_R \approx f_1
(z-z_c) + \mathcal{O}(z-z_c)^2 \ ,$$ where we have introduced the constant $$f_1(z_c)= \frac{8 \,\text{coth} \ h(z_c)}{3 \,z_c \, \sqrt{z_c^4+1}} \ ,$$ with $h(z)$ defined in Eq.(\[eq:hz\]). As for $G_z$, it turns out to be a tougher nut to crack given its involved functional dependence, but it behaves at $z=z_c$ as $$G_z\approx \frac{C_1(z_c)}{(z-z_c)^{3/2}} + \mathcal{O}(z-z_c)^{-1/2} \ ,$$ where $C_1(z)>0$ is a cumbersome function of the constants of the model. Therefore, $G(z)\approx \frac{-2C(z_c)}{(z-z_c)^{1/2}} + \mathcal{O}(z-z_c)^{1/2}$, which inserted into the expression for the metric function (\[eq:Final-ansatz-f(R)\]) yields $$\label{eq:CzcfR}
C(z) \approx \frac{\tilde{C}_1(z_c) \delta_1}{\delta_2 (z-z_c)^2} + \mathcal{O}(z-z_c)^{-3/2} \ ,$$ where the new constant $\tilde{C}_1(z)>0$ contains all the contributions in $z_c$. Therefore, we see that the metric function $C(z)$ diverges always at $z=z_c$, as a consequence of the poles present in the $f_R$ factor and also in the $G(z)$ function. Moreover, due to the positivity of $\tilde{C}_1(z), \delta_1$ and $\delta_2$ in this expression, one finds that this divergence goes always to $+\infty$ which, together with the asymptotically flat character of the solutions, as given by (\[eqAsymp\]), provides the structure of horizons for these solutions. Indeed, as depicted in Fig.\[fig:met1\], this structure resembles the one of the Reissner-Nordström solution of GR, namely, two-horizon black holes, extreme black holes (with a single degenerated horizon) and naked configurations. However, a systematic classification of the values of $\{\vert \tilde{\alpha} \vert, \delta_1, \delta_2\}$ yielding any such configurations are hard to find, and require instead direct inspection case-by-case. We also see that the single horizon black holes of the EH electrodynamics in GR have been lost, due to the modifications on the geometry caused by the presence of the wormhole throat at a finite distance $z_c$.
![Numerical integration of the metric function $C(z)$ in Eq.(\[eq:Final-ansatz-f(R)\]) for the branch $\tilde{\alpha}<0$ and the choices of $\vert \tilde{\alpha} \vert=5, \delta_2=1/2$ and three values of $\delta_1=50$ (purple), $\delta_1 \approx 85$ (blue) and $\delta_1=120$ (red), corresponding to black holes with two horizons, extreme black holes, and naked configurations, respectively. All solutions are asymptotically flat, $C(z) \underset{z \to \infty}{\approx} 1$. The vertical dotted line represents the wormhole throat $z=z_c$, to which all curves converge.[]{data-label="fig:met1"}](fig3.pdf){width="45.00000%"}
Properties of the solution: geodesic behaviour and regularity
-------------------------------------------------------------
To gain deeper knowledge on the innermost geometry of these solutions let us study their geodesic structure. For any spherically symmetric space-time with line element (\[eq:lineg\]) the geodesic equation may be written as [@OlmoBook] $$\label{Eq:General_geo}
\dfrac{C}{B}\left( \dfrac{dx}{du}\right)^2 = E^2 -V(x) \ ,$$ where we have introduced the effective potential $$\label{Eq:Potential_f(R)}
V(x) = C(x) \left( -k + \dfrac{L^2}{r^2(x)}\right) \ .$$ Here, $u$ is the affine parameter (the proper time for a time-like observer), $k=-1,0$ for time-like and null geodesics, respectively, while $ E $ and $L $ are the total energy and angular momentum per unit of mass for time-like observers, respectively. For spherically symmetric space-times in Palatini $ f(R) $ gravity, from (\[eq:finalsolf(R)\]) the geodesic equation (\[Eq:General\_geo\]) takes the form $$\label{Eq:General_geo_f(R)}
\dfrac{1}{f_R^2 }\left( \dfrac{dx}{du}\right)^2 = E^2 -V(x) \ .$$ It is convenient to rewrite the above equation in terms of the dimensionless radial function $z(x)$ by using Eq. and its derivatives. This allows to write the following form of the geodesic equation $$\label{Eq:Part_geo_f(R)}
\dfrac{d\tilde{u}}{dz} = \pm \dfrac{ 1+\frac{z \, f_R,_z}{2 \, f_R} }{f_R^{1/2}\sqrt{E^2 -C(z) \left( -k + \frac{L^2}{r_c^2 z^2(x)}\right)}} \ ,$$ where we have re-scaled the affine time parameter as $ \tilde{u} = u/r_c $, and the +(-) sign corresponds to ingoing (outgoing) geodesics. For radial null geodesics ($ k=0 $ and $ L=0 $), the above differential equation becomes $$\label{Eq:null_f(R)}
E \,\dfrac{d\tilde{u}}{dz} = \pm \dfrac{ 1+\frac{z \, f_R,_z}{2 \, f_R} }{f_R^{1/2} } \ .$$ At large distances, $ z \gg 1 $, where $f_R \to 1$, this equation can be integrated as $ E \tilde{u}\simeq \pm z $, which is the expected GR behaviour, in agreement with the fact that in this limit the $f(R)$-EH solution boils down to the Reissner-Nordström one. However, departures are expected as the wormhole throat $z=z_c$ is approached. Indeed, using the expansion of $ f_R $ in Eq. we can easily integrate this expression around $z=z_c$ as $$\label{Eq:critical_geo_f(R)}
E \tilde{\lambda}(z)\approx \mp \frac{\sqrt{8/3}\,z_c}{f_1(z_c)} \frac{1}{\sqrt{z-z_c}} \ .$$ From this expression it is readily seen that the affine parameter $\tilde{\lambda}(z)$ diverges to $\pm \infty$ as the wormhole throat $z=z_c$ is approached (see Fig. \[fig:geofr\]). This implies that null radial geodesics require an infinite affine time to get to (or to depart from) the wormhole throat which, consequently, lies on the future (or past) boundary of the space-time[^3]. This way, as opposed to the Reissner-Nordström space-time where null radial geodesics get to $r=0$ in finite affine time without any possibility to further extension beyond this point, they are complete in the geometry explored in this section. We point out that this mechanism for the removal of geodesic incompleteness via the displacement of any potentially pathologically region to the boundary of the space-time has been discussed in detail in Refs.[@Carballo-Rubio:2019fnb; @Carballo-Rubio:2019nel] on very general grounds, and explicitly implemented in other settings within Palatini theories of gravity [@Bambi:2015zch; @Bejarano:2017fgz; @Nascimento:2018sir].
For null geodesics with $L \neq 0$ and for time-like geodesics, the fact that $C(x)$ diverges to $+\infty$ at $z=z_c$, implies that any such geodesics approaching the wormhole throat will see an infinitely repulsive potential barrier, as given by (\[Eq:Potential\_f(R)\]), and will bounce off at a certain radius given by the vanishing of the denominator of (\[Eq:Part\_geo\_f(R)\]), thus not being able to get to the wormhole throat. Consequently, these geodesics are complete in pretty much the same way as in their Reissner-Nordstöm counterparts. The bottom line of this discussion is the null and time-like geodesic completeness of the full spectrum of solutions (in terms of mass and charge) of quadratic $f(R)$ gravity with EH electrodynamics in the $\tilde{\alpha}<0$ branch. Even though one should not worry about the effects of curvature at the wormhole throat, since that region cannot be reached by any observer, a quick computation revels the existence of curvature divergences there of size $K \equiv {K^\alpha}_{\beta\gamma\delta}{K_\alpha}^{\beta\gamma\delta} \sim 1/(z-z_c)^2$, which is nonetheless much weaker than in their GR counterparts $K \sim 1/r^8$.
![The affine parameter $ E \cdot \tilde{u}(z) $ versus the dimensionless radial coordinate $ z $ for ingoing (blue) and outgoing (orange) null radial geodesics. The vertical dashed purple line corresponds to the wormhole throat, $z=z_c$, while the black dashed lines correspond to null radial geodesics in GR.[]{data-label="fig:geofr"}](fig4.pdf){width="45.00000%"}
EiBI gravity {#sec:IV}
============
Derivation of the solution
--------------------------
The action of EiBI gravity can be written as (for a review of this theory see [@BeltranJimenez:2017doy]) $$\mathcal{S}_{EiBI}=\frac{1}{\kappa^2 \epsilon} \int d^4x \left[\sqrt{-\vert g_{\mu\nu} + \epsilon R_{\mu\nu} \vert} - \lambda \sqrt{-g}\right] \ ,$$ where $\epsilon$ is a parameter with dimensions of length squared. In its weak-field limit, $\vert R_{\mu\nu} \vert \ll \epsilon^{-1}$, the theory reduces to GR with an effective cosmological constant, $\Lambda_{eff}=\frac{\lambda-1}{\epsilon}$, while higher order curvature corrections are suppressed by powers of $\epsilon$. For EiBI gravity, the Einstein-frame metric appearing in (\[eq:Omegadef\]) is given by $q_{\mu\nu} =g_{\mu\nu} + \epsilon R_{\mu\nu}$, while the deformation matrix $ {\Omega^\mu}_{\nu} $ can be determined via the algebraic expression $$\label{Eq:def-Omega}
|\hat{\Omega}|^{1/2} ({\Omega^\mu}_{\nu})^{-1} = \lambda \, {\delta^\mu}_{\nu}- \epsilon \kappa^2 {T^\mu}_{\nu} \ .$$ This relation shows that the deformation matrix inherits also in this case the structure in $2 \times 2$ blocks of the stress-energy tensor defined in Eq. . Thus, we are allowed to write an ansatz for $ {\Omega^\mu}_{\nu} $ as $$\label{Eq:Omega}
{\Omega^\mu}_{\nu} = \left(\begin{array}{cc} \Omega_+ \; \hat{I}_{2\times 2}& \hat{0\,}_{2\times 2}\\ \hat{0\,}_{2\times 2}& \Omega_- \; \hat{I}_{2\times 2}\end{array}\right) \ ,$$ where the components of the matrix can be found by substituting them into Eq. and solving the corresponding equations as $$\begin{aligned}
\Omega_+ &=& \lambda-\epsilon \kappa^2 {T^\theta}_{\theta}= \lambda - \dfrac{\epsilon \kappa^2}{8 \pi} \varphi \ , \label{Omplusgen} \\
\Omega_- &=& \lambda-\epsilon \kappa^2 {T^t}_{t}= \lambda - \dfrac{\epsilon \kappa^2}{8 \pi} (\varphi- 2X \varphi_X) \label{Omneggen} \ .\end{aligned}$$ The gravitational field equations in this case are written as $$\label{Eq:Ricci_tensor-EIBI}
{R^\mu}_\nu (q) = \dfrac{1}{\epsilon} \left( \begin{array}{cc} \dfrac{\Omega_+-1}{\Omega_+} \; \hat{I}_{2\times 2}& \hat{0}_{2\times 2}\\ \hat{0}_{2\times 2}& \dfrac{\Omega_- -1 }{\Omega_-}\; \hat{I}_{2\times 2}\end{array} \right).$$ Considering the line element in and following the same steps done in the previous section, besides taking into account the relation (\[eq:Omegadef\]) between metrics, which implies the following relation between radial coordinates $$\label{eq:xrBI}
x^2=z^2 \Omega_-(z) \ ,$$ together with Eq.(\[Eq:def-Omega\]) leads to the expression for the mass function $$\label{Eq:diff_mass_eibi}
M_r= \dfrac{r^2 }{2\, \epsilon }\,(\Omega_--1)\, {\Omega_-}^{1/2}\left( 1 + \dfrac{r \, \Omega_{-,r}}{2 \; \Omega_-^{1/2}} \; \right) \ .$$ Moreover, following a similar procedure and notation as in the $ f(R) $ gravity case, we find the line element for the $ g_{\mu \nu} $ metric as $$\label{eq:finalsolEiBI}
ds^2=-C(x) \, dt^2 + \frac{dx^2}{\Omega_{+}^2 \,C(x)} +z^2(x)\,d\Omega^2 \ ,$$ where the metric function now satisfies $$\label{eq:Final-ansatz-EiBI}
C(z) = \dfrac{1}{\Omega_+}\left( 1- \dfrac{1 + \delta_1 G(z)}{\delta_2 \, z \, \Omega_-^{1/2}}\right) \ .$$ Here, we have introduced the following definitions: the metric is parameterized in terms of two constants defined as $$\begin{aligned}
\delta_1 &=& \dfrac{r_c^3}{r_S \, \epsilon} = \dfrac{3}{2}\left(\dfrac{3}{2 \, \pi}\right) ^{1/4}\frac{1}{ l_\epsilon^2\,r_S} \sqrt{\frac{r_q^3}{l_{\beta}}} \ , \\
\delta_2 &=& \dfrac{r_c}{r_S} \ ,\end{aligned}$$ where we have redefined the EiBI parameter as $ l_{\epsilon}^2= \epsilon\, /(12 \pi \, l_\beta^2) $, which is the analog of $ \tilde{\alpha} $ in the $ f(R) $ case. These two parameters encode the two integration constants, $r_S$ and $r_q^2$, and the two gravity and model parameters, $l_{\epsilon}^2$ and $l_{\beta}^2$, likewise in the $f(R)$ case. As for the $ G(z) $ function, it is obtained via $$\begin{aligned}
\label{eq:GzBI}
G_z(z) &=& z^2 (\Omega_--1)\; \Omega_-^{1/2}\left(1+ \dfrac{z \Omega_{-},{z}}{2 \, \Omega_-} \;\right). \label{Eq:G(z)_EiBI}\end{aligned}$$ and the contributions on the $\Omega_{\pm}$ factors read $$\begin{aligned}
\Omega_{+} &=& \lambda -l_{\epsilon}^2 \,\tau^2(z) \left(1+\frac{ 2\,\tau^2(z)}{3} \right) \ , \label{eq:Op-bi} \\
\Omega_{-} &=& \lambda + l_{\epsilon}^2\,\tau^2(z) \,(1+2\,\tau^2(z)) \ , \label{eq:Om-bi}\end{aligned}$$ where we recall that $\tau(z)$ is defined in Eq.(\[eq:tau\]). The line element written in is the electrostatic solution of EiBI gravity coupled to EH electrodynamics. Like in the $f(R)$ case, one could transform the line element (\[eq:finalsolEiBI\]) into a Schwarzschild-like form via the change of coordinates $d\tilde{x}^2=\frac{dx^2}{\Omega_{+}^2}$, but again we shall not follow that path in order not to spoil the simple representation (\[eq:xrBI\]) of the radial function. From now on, we will consider asymptotically flat solutions, $ \lambda=1 $.
Let us now first analyze the asymptotic limit of the functions in the line element (\[eq:finalsolEiBI\]). For $ z\rightarrow\infty $, one has $z^2 \approx x^2$, and the deformation metric components behave as $$\label{eq:def-matr-inf}
\Omega_{\pm} \approx 1\mp \frac{l_{\epsilon}^2}{9 z^4}+\mathcal{O}\left(\frac{1}{z}\right)^8.$$ Here, the gravitational sector contributes to the line element in a lower power of $ z $ in comparison to the $ f(R) $ case because $ \Omega_{-} $ has a power in $ \t(z) $. As a consequence, the gravitational corrections appear earlier in the metric component $ C(z) $. Expanding the function $ G_z $ in we get $$\label{key}
G_z \approx\frac{ l_{\epsilon}^2}{9 z^2}-\frac{ \epsilon \, (9 \,l_{\epsilon}^2+4)}{486 z^6}+\mathcal{O}\left(\frac{1}{ z}\right)^{10}.$$ Replacing the above expressions into the metric component and reverting back to the original variables leads to $$\label{eqAsymp-EiBI}
C(r) \underset{r \to \infty}{\approx} 1-\frac{r_S}{r}+\frac{q^2}{r^2}+\dfrac{\epsilon \, r_S q^2}{2 \,r^5}-\frac{(\beta +4\epsilon) \,q^4}{5\,r^6} + \mathcal{O}\left(\frac{1}{r}\right)^{10},$$ The first three terms in this expression correspond to the Reissner-Nordström solution, as expected. The fourth one introduces a sort of interaction between mass and charge fueled by the EiBI gravity dynamics, while the last two terms are pure corrections in EH electrodynamics (obviously identical to the one written in Eq.) and in EiBI gravity, respectively.
Properties of the solution: radial function
-------------------------------------------
As in the $f(R)$ case, we now look for the minimum of the radial function $z(x)$ via the relation (\[eq:xrBI\]). Using the expression (\[eq:Om-bi\]) it is clear that the zeros of $\Omega_{-}$ will only occur in the branch $l_{\epsilon}^2<0$ and, therefore, we shall restrict our attention to this branch from now on. The values of $z(x)$ for which the zeros of $\Omega_{-}$ are attained can actually be written in an identical form as Eq. , but now with $$\label{Eq:a-EiBI}
a=\exp\left\lbrace {3\,\text{ArcSinh}\left(\dfrac{1}{2} \sqrt{\sqrt{\dfrac{\vert l_{\epsilon}^2\vert +8}{\vert l_{\epsilon}^2 \vert}}-1}\right)}\right\rbrace.$$
![The dimensionless radial function $ z(x) $ for the case $ l_{\epsilon}^2<0$. The orange curve represents $ \vert l_{\epsilon}^2 \vert = 0.2$, the green $ \vert l_{\epsilon}^2 \vert = 0.01$ and the purple $ \vert l_{\epsilon}^2 \vert = 1$. The blue and black curves represent the case of Maxwell electrodynamics with $ \vert l_{\epsilon}^2 \vert = 0.5$ and $ r_q= 0.5 $, and of GR, respectively.[]{data-label="fig:rEiBIneg"}](fig5.pdf){width="45.00000%"}
As it is depicted in Fig. \[fig:rEiBIneg\], at this point the radial function $z(x)$ takes its minimum value and bounces off, again representing a wormhole structure with $z=z_c$ the location of its throat. For completeness, we have also plotted in Fig. \[fig:rEiBIpos\] the behaviour of $z(x)$ in the branch $ l_\epsilon^2 >0 $, where no wormhole is present and the space-time splits into two disconnected pieces in the $x>0$ and $x<0$ regions. Therefore, these two structures are similar to those found in the $f(R)$ case above, but their effects in the geometry of the corresponding space-times yield large differences, as we shall see next.
Properties of the solution: inner behavior and horizons
-------------------------------------------------------
To study the behavior of the metric functions on the innermost region and, in particular, at the wormhole throat, we begin by expanding the relevant functions around $z \approx z_c$. For the $\Omega_{\pm}$ functions in (\[Omplusgen\]) and (\[Omneggen\]) with the expression (\[final-stress-energy tensor\]) we find $$\begin{aligned}
\Omega_{+} &\approx& \omega_{+}(z_c) +\mathcal{O}(z-z_c)\label{Omega_+_zc} \ , \\
\Omega_{-}&\approx&\ \omega_{-}(z_c) (z-z_c)+ \mathcal{O}(z-z_c)^2 \ , \label{Omega_-_zc}\end{aligned}$$ where we have introduced the constants $$\begin{aligned}
\omega_{+}(z_c)&=&\frac{2}{3} \left(\sech 2\,h(z_c)+2\right) \ , \label{omega_+_zc} \\
\omega_{-}(z_c)&=&\dfrac{4}{3}\,\frac{(\tanh 2 \,h(z_c)+\coth h(z_c))}{ z_c \sqrt{z_c^4+1}} \ , \label{omega_-_zc}\end{aligned}$$ and we recall that $ h(z_c) $ is defined in Eq.. The expansion of the function $G_z$ in Eq.(\[eq:GzBI\]) becomes $$\label{Gz_zc}
G_z \approx \dfrac{ C_2}{\sqrt{z-z_c}}\, +\mathcal{O}(z-z_c)^{1/2} \ ,$$ where the constant $
C_2=z_c^3 \omega_-^{1/2}/2$. Upon integration, this yields the result $$\label{G(z)_zc}
G(z) \approx -\frac{1}{\delta_c} + 2\, C_2\, \sqrt{z-z_c}+\mathcal{O}(z-z_c)^{3/2} \ ,$$ where $\delta_c(z_c)>0$ is a constant needed to match the inner and asymptotic expansions of $G(z)$, and whose explicit dependence on its argument is very cumbersome, though for our analysis only its positivity is relevant. Plugging the expansions (\[Omega\_+\_zc\]), (\[Omega\_-\_zc\]) and (\[G(z)\_zc\]) in the expression (\[eq:Final-ansatz-EiBI\]), we arrive at the behaviour of the metric components: $$\begin{aligned}
\label{C-EiBI-zc}
g_{tt} &\approx& - \frac{3 \left(1+2 \tau^2_c\right) (\delta_1/\delta_c-1)}{2\, z_c \delta_2 \left(3+4 \,\tau^2_c\right) \omega_{-}^{1/2} \sqrt{z-z_c}}\\
&+&\frac{3 \left(\delta_2-\delta_1 z_c^2\right)}{2\, \delta_2 (2+\sinh 2\,h(z_c))}+\mathcal{O}(z-z_c)^{1/2} \ , \nonumber \\
g_{rr}&\approx& \frac{3 \,z_c\,\delta_2 \,\omega_{-}^{1/2}\cosh 2\,h(z_c)}{ 2(\delta_1/\delta_c-1)(3+2 \tau_c)} \sqrt{z-z_c}+\mathcal{O}(z-z_c) \ , \label{grr-EiBI-zc}\end{aligned}$$ where $ \tau_c \equiv \tau (z_c)$.
![The radial function $ z $ for the case $ l_{\epsilon}^2>0 $. Same notation as in Fig. \[fig:rEiBIneg\].[]{data-label="fig:rEiBIpos"}](fig6.pdf){width="45.00000%"}
From these expressions we can proceed to classify the spectrum of solutions in terms of their horizon structure. Indeed, a glance at the expansion (\[C-EiBI-zc\]) shows that such a classification can be performed according to the ratio $\delta_1/\delta_c$, since it controls the sign of the divergence of the metric function $C(z)$ at $z=z_c$. Thus, if $\delta_1/\delta_c<1$ then $C(z_c) \to -\infty$, and the corresponding solutions are Schwarzschild-like black holes with a single horizon. On the contrary, when $\delta_1/\delta_c>1$, then $C(z_c) \to +\infty$ and one finds configurations with the same structure as the one of the Reissner-Nordström solution of GR: black holes with two horizons, extreme black holes, or naked configurations, depending on the value of the constant $\delta_2 $. Moreover, special configurations are found when $\delta_1=\delta_c$ since in such a case, replacing first this constraint in the metric functions of the line element (\[eq:finalsolEiBI\]) and expanding in series of $z_c$ makes the first term in Eq.(\[C-EiBI-zc\]) to go away and only the finite contribution at $z=z_c$ remains. Consequently, the corresponding configurations are Minkowski-like solutions with either a single non-degenerate horizon or none, depending on the value of $\delta_2\gtrless \delta_1 z_c^2$. It should be pointed out that the Schwarzschild/Reissner-Nordström-like structure of horizons resemble the original one of the EH electrodynamics within GR: while in the latter it is the comparison between the total mass of the space-time, $M$, and the total (finite) energy stored in the electrostatic field the one playing the role in classifying such a structure, here is the ratio $\delta_1/\delta_c$ instead. However, the Minkowski-like configurations are a novel feature of these Palatini space-times, having no counterpart in the Einstein-EH system.
![The metric function $C(z)$ in Eq.(\[eq:Final-ansatz-EiBI\]) for $ l_\epsilon^2 =-1$ (for which $\delta_c \approx 3.260$) and the choices of $\{\delta_1=1,\delta_2=1/3\}$ (blue), $\{\delta_1=5,\delta_2=1/3\}$ (violet), $\{\delta_1=5,\delta_2 \approx 0.4675\}$ (green) and $\{\delta_1=5,\delta_2=1\}$ (red), corresponding to Schwarzschild-like black holes with a single horizon, and the three Reissner-Nordstöm like configurations: black holes with two horizons, extreme black holes, and naked solutions, respectively. The two orange lines starting from a finite value of $C(z)$ at $z=z_c$ are Minkowski-like configurations ($\delta_1=\delta_c$) with a single horizon ($\delta_2=0.6$) or none ($\delta_2=1.5$). All solutions are asymptotically flat, $C(z) \underset{z \to \infty}{\approx} 1$. The vertical dotted line represents the wormhole throat $z=z_c$, to which all curves converge.[]{data-label="fig:met2"}](fig7.pdf){width="45.00000%"}
Properties of the solution: geodesic behaviour and regularity
-------------------------------------------------------------
For EiBI gravity with spherically symmetric solutions of the kind studied here, the geodesic equation (\[Eq:General\_geo\]) can be cast, taking into account the line element (\[eq:finalsolEiBI\]), as $$\label{eq:geoBI}
\frac{1}{\Omega_+ ^2} \left(\frac{dx}{du}\right)^2= E^2 -V(x) \ ,$$ with the same notation and conventions as in the $f(R)$ case above. Again, for null radial geodesics it is more useful to write this equation in terms of the radial function $z$. To this end, we take a derivative in Eq.(\[eq:xrBI\]), which allows to cast (\[eq:geoBI\]) in such a case as $$\label{eq:geonull}
\pm E d\tilde{u}=\frac{\Omega_{-}^{1/2}}{\Omega_{+}} \left(1+\frac{z\Omega_{-,z}}{2\Omega_{-}}\right)dz \ ,$$ where again $\pm$ refer to ingoing/outgoing geodesics. It seems not possible to obtain a integration of this equation to find a closed expression for $\tilde{u}(z(x))$ everywhere, but we can resort to series expansions around the wormhole throat $z=z_c$. A glance at Eqs.(\[Omega\_+\_zc\]) and (\[Omega\_-\_zc\]) reveals that $\Omega_+$ is there just a constant that will have no impact in the behaviour of the solutions, while $\Omega_{-}$ contains the key factor in $(z-z_c)$. Thus, a little algebra allows to find the expansion of (\[eq:geonull\]) at $z=z_c$ as $$\pm E d\tilde{u} \approx \frac{\omega_-^{1/2}z_c}{2\omega_+} \frac{1}{\sqrt{z-z_c}} \ .$$ This can be right away integrated as $$\label{eq:affinex}
\pm E (\tilde{u} - \tilde{u}_0) \approx \frac{\omega_-^{1/2}z_c}{\omega_+} \sqrt{z-z_c} \approx \frac{x}{\omega_{+}} + \mathcal{O}(x^2) \ ,$$ where in the last equation we have made use of the fact that, using (\[eq:xrBI\]) and (\[Omega\_-\_zc\]), the radial function can be expanded in series of $x$ as $$z \approx z_c +\frac{x^2}{z_c^2 \omega_{-} } + \mathcal{O}(x^4) \ .$$ Since the domain of definition of the radial coordinate $x$ is the entire real line, nothing prevents the affine parameter in Eq.(\[eq:affinex\]) to cross the wormhole throat and be indefinitely extended to the asymptotic infinity $x = - \infty$. This is shown in Fig. \[fig:geo\], where we numerically integrate the geodesic equation (\[eq:geonull\]) in full range, showing that any such geodesic starting from a certain $\tilde{u}_0$ at $x=+\infty$ departs from the GR behaviour as the wormhole throat, $x=0$, is approached, and continues its path to another asymptotically flat region of space-time, $x = - \infty$. Therefore, null radial geodesics are complete in this geometry.
![The affine parameter $ E \cdot \tilde{u}(x) $ versus the radial coordinate $ x $ for null radial geodesics. The green curve corresponds to $ \vert l_\epsilon^2 \vert=0.01$, the blue curve to $ \vert l_\epsilon^2\vert=1 $, and the orange to $ \vert l_\epsilon^2 \vert=0.2 $. The black dashed line corresponds to the GR behaviour, where these geodesics end at $x=0$ and are therefore incomplete. At the wormhole throat the affine parameter obeys (\[eq:affinex\]) and can be smoothly extended across $x=0$.[]{data-label="fig:geo"}](fig8.pdf){width="40.00000%"}
For time-like geodesics and for null non-radial geodesics, one needs to analyze the behavior of the effective potential according to the expansion of the metric function at $z=z_c$ ($x=0$), as follows from Eq.(\[C-EiBI-zc\]). Using the expansion (\[C-EiBI-zc\]) this reads $$\label{eq:effwth}
V_{eff}\approx -\frac{a}{\vert x \vert} -b + \mathcal{O}(x) \ ,$$ with the constants $$\begin{aligned}
a&=& \frac{3 (1+2 \tau^2_c) (\delta_c-\delta_1)}{2\,\delta_2 \delta_c \left(3+4 \,\tau^2_c\right)}\left(-k+\dfrac{L^2}{r_c^2 z\,_c^2} \right) , \\
b&=&\frac{3 \left(\delta_2-\delta_1 z_c^2\right)}{2\, \delta_2 (2+\sinh 2\,h(z_c))}\left(-k+\dfrac{L^2}{r_c^2 z\,_c^2} \right).\end{aligned}$$ Indeed, likewise the structure of horizons, the fate of any such geodesic depends on the ratio $\delta_1/\delta_c$.
For Schwarzschild-like configurations, $\delta_1<\delta_c$, the potential (\[eq:effwth\]) is infinitely attractive and, therefore, any such geodesic crossing the event horizon of these configurations will unavoidably get to the wormhole throat in finite affine time. At such a point the geodesic equation (\[eq:geoBI\]) behaves as $$\frac{d\tilde{u}}{dx} = \frac{ \vert x \vert^{1/2}}{\omega_{+}a^{1/2}} + \mathcal{O}(x^{3/2}) \to \tilde{u}(x)=\frac{2x \vert x \vert^{3/2}}{3\omega_{+}a^{1/2}} + \mathcal{O}(x^{5/2}) \ .$$ As the coordinate $x$ extends over the whole real axis, it is clear that these geodesics can be naturally extended across $x=0$ and will be therefore complete for any values of the parameters of the model within the constraint $\delta_1<\delta_c$. It should be stressed that, despite the geodesically complete character of these space-times, any extended observer crossing the wormhole throat will find curvature divergences of size $K \sim 1/(z-z_c)^3$ there, which are much weaker than their GR counterparts, $ K \sim 1/r^8$. Therefore, one might wonder what would be the fate of any such observer undergoing arbitrarily large tidal forces as it crosses the wormhole throat. This question has been raised in other geodesically complete space-times similar to the one found here, requiring a detailed analysis upon congruence of geodesics and their effects on extended observers [@Olmo:2015dba], which lies beyond the scope of this paper.
For Reissner-Nordström-like configurations, $\delta_1>\delta_c$, the effective potential (\[eq:effwth\]) flips sign and it is infinitely repulsive at $z=z_c$. Therefore, any of these geodesics will bounce at some $z>z_c$ and will continue its path within the $x>0$ (or $x<0$) region, which is the same behaviour as the one found in the Reissner-Nordström solution of GR.
Finally, for Minkowski-like configurations, $\delta_1=\delta_c$, one has $a=0$ and the potential goes to a constant as $V_{eff}\approx -b + c(z_c)x^2$, where $c(z_c)>0$ is a constant with an involved dependence on $z_c$. Therefore, any particle with energy $E$ above the maximum of this potential will be able to get to the wormhole throat. At that point, its affine parameter will behave as $$\tilde{\lambda}(x) \approx \frac{x}{\sqrt{b+E^2}} +\mathcal{O}(x^3) \ ,$$ and therefore will find no impediment to continue its trip to the $x<0$ region. Moreover, as opposed to the Schwarzschild-like and Reissner-Nordström-like configurations, in this case curvature scalars are all finite at the wormhole throat.
In summary, we have shown that all null and time-like geodesics in these geometries (in the branch $l_\epsilon^2<0$) are complete, no matter the values of mass and charge of the solutions or the value of the EH scale. The mechanism is, however, different from the $f(R)$ case, in that now the wormhole throat is accessible to different sets of geodesics, but all of them can be smoothly extended across the region $x=0$. Therefore, these geometries represent nonsingular space-times.
Conclusion and discussion {#sec:V}
=========================
In this work we have considered two families of gravitational theories extending GR, namely, quadratic $f(R)$ gravity and Eddington-inspired Born-Infeld gravity, both formulated in metric-affine spaces and coupled to Euler-Heisenberg electrodynamics. These two gravity theories have been chosen due to the different way the new dynamics is fed by the matter fields: in the $f(R)$ case the new effects in the gravitational sector are oblivious to anything but to the trace of the stress-energy tensor, while in the EiBI case they are sensible to its full content. The static, spherically symmetric solutions for both settings were found starting from the Einstein-like representation of the field equations. Such solutions suggested that only a branch of them, corresponding to a certain combination of the signs of the gravity and matter parameters, may hope to yield nonsingular solutions. Therefore, we focused on the characterization of such a branch according to the behaviour of the metric functions on the innermost region of the geometries, on the horizon structure, and on the completeness of geodesics.
The main conclusion of this analysis is that both settings do yield null and time-like geodesically complete space-times for all the spectrum of mass and charge of the corresponding solutions, provided that the aforementioned constraint on the signs of the parameters is met. While in both cases the singularity-regularization is possible thanks to the presence of a wormhole structure, the mechanisms for the completeness of geodesics differ. In the $f(R)$ gravity case, which has the same structure of horizons as in the Reissner-Nordström solution of GR, the central region is pushed to an infinite affine distance, so null radial geodesics would take an infinite time to get there, while for time-like geodesics or null non-radial geodesics the presence of an infinitely repulsive potential near the throat prevents them getting near it. Thus, only half of the wormhole (which may be covered by two horizons, a single extreme one, or be naked) is available for travel within the $x>0$ and $x<0$ regions.
As opposed to the $f(R)$ case, for EiBI gravity the throat can be reached in finite affine time by some sets of observers, depending on the ratio $\delta_1/\delta_c$, which classifies the corresponding configurations in terms of horizons as Schwarzschild-like, Reissner-Norström-like, or Minkowski-like. If we focus on Schwarzschild-like configurations, which have a single event horizon, then the wormhole is a one-way structure, pushing out any observer departed from (say) $x>0$ and crossed the event horizon to the other asymptotic region in finite affine time. For Reissner-Nordström-like configurations, no matter their number of horizons, one finds instead that, like in their GR counterparts, any time-like observer could only get as close to the throat as it energy permits (given the existence of the infinite potential barrier), while null radial geodesics would only require a finite affine to get to the throat and cross it. Finally, Minkowski-like configurations (with a single horizon or none) have a finite maximum of its effective potential, thus allowing any observer whose energy is larger than it to cross the wormhole throat. Though curvature divergences generally appear at the throat (except in the Minkowski-like configurations, where curvature scalars are well behaved everywhere), the fact that they are much weaker than their GR counterparts, $\sim (z-z_c)^{-3}$, together with the lessons from previous research in the topic showing that extended observers are not necessarily destroyed in the transit through such regions [@Olmo:2016fuc], raises questions on their true meaning when both the matter fields and the trajectories of idealized observers are well behaved.
The results obtained in this work further support the suitability of some metric-affine theories to get rid of space-time singularities in a variety of settings with conservative modifications of the GR framework. Moreover, these two basic mechanisms for such a singularity avoidance are shared by several other theories, and in agreement with the results of model-independent analysis in spherically symmetric space-times [@Carballo-Rubio:2019nel; @Carballo-Rubio:2019fnb]. There are several challenges following these results, such as its compatibility with the semiclassical calculations of Hawking’s radiation and black hole evaporation, the unsettled issue of topology change raised from the formation of wormholes, or to what extend these results can be sustained when moving to axially symmetric (rotating) scenarios. The latter is of special interest should any effect of metric-affine gravity able to leak to the near-horizon scale, in order to address any of the opportunities offered by multimessenger astronomy. Work along these lines is currently underway.
Acknowledgments {#acknowledgments .unnumbered}
===============
MG is funded by the predoctoral contract 2018-T1/TIC-10431. DRG is funded by the *Atracción de Talento Investigador* programme of the Comunidad de Madrid (Spain) No. 2018-T1/TIC-10431, and acknowledges support from the Fundação para a Ciência e a Tecnologia (FCT, Portugal) research grants Nos. PTDC/FIS-OUT/29048/2017 and PTDC/FIS-PAR/31938/2017, the spanish projects FIS2014-57387-C3-1-P and FIS2017-84440-C2-1-P (MINECO/FEDER, EU), the project SEJI/2017/042 (Generalitat Valenciana), and PRONEX (FAPESQ-PB/CNPQ, Brazil). This article is based upon work from COST Actions CA15117 and CA18108, supported by COST (European Cooperation in Science and Technology).
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[^1]: When considering the effective limit of quantum electrodynamics this Lagrangian picks another term in $Y$, which is vanishing for the electrostatic configurations of this paper. In such a limit, $\beta$ takes the value $\beta=\frac{2\alpha^2}{45m_e^2}$ [@Schwinger], where $m_e$ is electron’s mass and $\alpha$ the fine structure constant. However, for the purposes of this paper we shall take $\beta$ as a free parameter.
[^2]: In an abuse of notation, here we have introduced an implicit factor $r_c$ inside $x$ as $x \to x \,r_c$, so Eq.(\[eq:xrfR\]) reads $x^2=z^2 f_R$.
[^3]: In some sense this means that we have [*half a wormhole*]{}, in that the region $x>0$ ($x<0$) is not accessible to observers living in the region $x<0$ ($x>0$).
|
Thermal suppression of Josephson interlayer coupling by vortex fluctuations [@decoupl] is the key factor, which enhances vortex mobility and limits useful applications of Bi- and Tl-based high T$_{c}$ superconductors (HTSC). Recent discovery of the theoretically predicted [@mish; @bmt] Josephson plasma resonance (JPR) effect in the mixed state of highly anisotropic layered high temperature and organic superconductors [@oph; @mats; @shib; @kadow; @MatsAng; @Bayrakci] provides a new and unique opportunity to probe the Josephson coupling of layers in a wide range of magnetic fields and temperatures. An alternative way to probe Josephson coupling in the vortex state is direct $c$-axis transport measurements [@Transport].
A minimum necessary condition for the existence of a narrow JPR line is short-time/short-range phase coherence between the superconducting layers. In the London regime this coherence is quantitatively characterized by the spatial and temporal behavior of the “local coherence parameter” ${\cal C}_{n}({\bf r},t)=\cos[\varphi
_{n,n+1}({\bf r},t)]$, with $\varphi _{n,n+1}=\varphi _{n+1}-\varphi
_{n}-(2\pi s/\Phi _{0})A_{z}$ being the gauge-invariant phase difference between layers $n$ and $n+1$. Here ${\bf r}=(x,y)$ and $z$ are coordinates in the $ab$ plane and along the $c$ axis, $s$ is the interlayer spacing.
The first important condition is that the typical time of variation of this factor (phase slip time) must be much larger than the period of plasma oscillations. Their ratio was estimated [@kosh] as $\approx 10^4$ at $T=45$ K and magnetic field $B_{z}=0.1$ T. In this case the plasma mode probes a snapshot of the instantaneous phase distribution.
In the decoupled liquid state the cosine factor ${\cal C}_{n}({\bf
r},t=0)$ also rapidly oscillates in space because correlations between pancake positions in neighboring layers are almost absent. The possibility of a sharp resonance in the situation when fluctuations of local Josephson coupling are much stronger than its average is an interesting and nontrivial issue. JPR resonance in such situation occurs because small phase oscillations induced by the external electric field change slowly in space and average out these rapid variations [@bdmb; @LongPaper].
In this Letter, using the high temperature expansion with respect to the Josephson coupling, we establish a relation connecting the plasma frequency with the density correlation function of the pancake liquid. A similar relation was derived in Ref. [@bpm] assuming the Gaussian distribution for the interlayer phase fluctuations. We demonstrate that the pair distribution function of the liquid is unambiguously connected with the dependence of the plasma frequency on the in-plane component of the magnetic field for fixed $c$-axis component. Therefore, this dependence can be used to extract quantitative information about the structure of the liquid phase.
The essential physics of JPR is captured by a simplified equation for small oscillations of the phase difference $\varphi_{n,n+1}^{\prime
}({\bf r},\omega)$ induced by the external microwave electric field with the amplitude ${\cal D}_{z}$ and the frequency $\omega$ (see Ref. ) $$\left[ \frac{\omega ^{2}}{\omega_{0}^{2}}+\lambda
_{J}^{2}\hat{L}\nabla_{{\bf r}} ^{2}-{\cal C}_{n}({\bf r})\right]
\varphi _{n,n+1}^{\prime }=-\frac{\hbar i\omega {\cal D}_{z}}{8\pi
eE_{J}},
\label{DynEq}$$ where $\omega _{0}(T)=c/\sqrt{\epsilon _{c}}\lambda _{c}(T)$ is the zero field plasma frequency, $\epsilon_{c}$ is the high frequency dielectric constant, $\lambda_{c}$ and $\lambda_{ab}$ are the components of the London penetration depth, $E_{J}=E_{0}/\lambda_J^2$ is the Josephson energy per unit area, $E_{0}=s\Phi
_{0}^{2}/16\pi^{3}\lambda _{ab}^{2}$, $\lambda_{J}=\gamma s$ is the Josephson length, and $\gamma=\lambda_{c}/\lambda_{ab}$ is the anisotropy ratio. The inductive matrix $\hat{L}$ is defined as $\hat{L}A_{n}=\sum_{m}L_{nm}A_{m}$ with $ L_{nm}\approx (\lambda
_{ab}/2s)\exp\left( -|n-m|s/\lambda _{ab}\right)$. We neglect in Eq. (\[DynEq\]) time variations of ${\cal C}_{n}({\bf r},t)$ assuming them to be small during the time $1/\omega$. Eq. (\[DynEq\]) also does not take into account charging effects [@koy] and quasiparticle dissipation. These effects don’t influence much the position of the resonance but may modify its lineshape.
Static configurations $\varphi_{n,n+1}({\bf r})$ are mainly determined by thermal fluctuations of pancake vortices and quantitatively characterized by the average value of the cosine factor, ${\cal
C}({\bf B},T)$, $${\cal C}({\bf B},T)=\langle \cos \varphi _{n,n+1}({\bf r})\rangle
\label{AvCos}$$ and its static correlation function $S({\bf r},{\bf B},T)$, $$S({\bf r},{\bf B},T)=\langle \cos [\varphi _{n,n+1}({\bf
r})-\varphi_{n,n+1}(0)]\rangle,
\label{coscorr}$$ where $\langle \ldots \rangle $ denotes the thermal average. In this Letter we consider the pancake liquid state where the correlations between pancake arrangements in different layers are almost absent. In such situation ${\cal C}_{n}({\bf r})$ rapidly oscillates in space, so that $S({\bf r},{\bf B},T)$ drops at distances of the order of the intervortex spacing $a=(\Phi_0/B_z)^{1/2}$, and ${\cal
C}({\bf B},T)\ll 1$.
An important observation is that in spite of rapid variations of ${\cal C}_{n}({\bf r})$ the oscillating phase $\varphi^{\prime }
\equiv \varphi_{n,n+1}^{\prime }({\bf r})$ varies smoothly in space. The typical length scale $L_{\varphi}$ of $\varphi^{\prime }$ variations can be estimated by balancing the typical kinetic energy of supercurrents, $E_{0}(\varphi^{\prime })^{2}/L_{\varphi}^{2}$, with the typical Josephson energy, $E_{J}(\varphi^{\prime })^{2} a/L_{\varphi}$, in the vortex state which is strongly disordered along the $c$ axis. This gives $L_{\varphi}=\lambda_{J}^{2}/a$. Smoothly varying $\varphi^{\prime }$ effectively averages rapid variation of ${\cal
C}_{n}({\bf r})$ and the plasma frequency is simply determined by ${\cal C}({\bf B},T)$ (see Refs. ) $$\omega _{p}^{2}({\bf B},T) =\omega_{0}^{2}(T){\cal C}({\bf B},T)/{\cal
C}(T),
\label{omegap}$$ Here the factor ${\cal C}(T)={\cal C}(0,T)$ takes into account the suppression of zero field plasma frequency by phase fluctuations (see below). Fluctuations of ${\cal C}_{n}({\bf r})$ smoothened over the area $L_{\varphi}^2$, ${\cal
C}(L_{\varphi})=L_{\varphi}^{-2}\int_{r<L_{\varphi}}d{\bf r}{\cal
C}_n({\bf r})$, produce inhomogeneous broadening of the JPR line. Calculating the mean squared fluctuation of ${\cal C}(L_{\varphi})$, $\langle[{\cal C}(L_{\varphi})-{\cal C}]^{2}\rangle\approx
a^{2}/L_{\varphi}^{2}=a^{4}/\lambda_{J}^{4}$, we obtain the estimate for the inhomogeneous line width $$\delta(\omega_p^2)\approx\omega_0^2(T)a^2/\lambda_J^2.
\label{line}$$ Detailed calculations of the intrinsic lineshape due to this mechanism will be published elsewhere [@LongPaper].
We establish now a relation between plasma frequency and the vortex liquid structure. In general, the Josephson coupled system in the London regime, $B_{z}\ll H_{c2}$, may be described in terms of vortex coordinates ${\bf r}_{n\nu}$ (index $\nu$ labels vortices in the layer $n$) and by the regular (spin-wave type) phase difference $\varphi _{n,n+1}^{(r)}({\bf r})$. The free energy functional in terms of these variables is [@bdmb] $$\begin{aligned}
{\cal F}\{{\bf r}_{n\nu },\varphi _{n,n+1}^{(r)}\}&=&\nonumber \\
{\cal F}_{v}({\bf r}_{n\nu })&+&{\cal F}_\varphi \{\varphi
_{n,n+1}^{(r)}\}+
{\cal F}_J \{{\bf r}_{n\nu },\varphi _{n,n+1}^{(r)}\}.
\label{mainf}\end{aligned}$$ Here ${\cal F}_{v}({\bf r}_{n\nu })$ accounts for the two-dimensional energy of pancakes and also their electromagnetic interaction in different layers [@clem]. The second term, $${\cal F}_\varphi\{\varphi _{n,n+1}^{(r)}\}=
\frac{E_0}{2}\sum_n\int d{\bf r}\nabla \varphi _{n,n+1}^{(r)}\hat{L}
\nabla \varphi _{n,n+1}^{(r)} ,
\label{vfunc}$$ is the energy of intralayer currents associated with regular phase fluctuations, and the third term, $$\begin{aligned}
&&{\cal F}_J\{{\bf r}_{n\nu },\varphi _{n,n+1}^{(r)}\}= \nonumber \\
&&E_J\sum_n\int d{\bf r}\left[1-\cos
\left(\varphi _{n,n+1}^{(v)}+\varphi _{n,n+1}^{(r)}-{\frac{2\pi
s}{\Phi_{0}}}B_{x}y\right)\right],
\label{FJ}\end{aligned}$$ is the Josephson energy. We split the phase difference $\varphi_{n,n+1}$ into vortex part, regular spin-wave part, and contribution coming from the in-plane magnetic field $B_{x}$, $$\varphi_{n,n+1}({\bf r})=\varphi _{n,n+1}^{(v)}({\bf r})+\varphi
_{n,n+1}^{(r)}({\bf r})
-(2\pi s/\Phi_{0})B_{x}y.$$ The phase $\varphi_{n,n+1}^{(v)}({\bf r};{\bf r}_{n\nu},{\bf r}_{n+1,\nu})$ is the singular part of the phase difference induced by vortices at positions ${\bf
r}_{n\nu}$, ${\bf r}_{n+1,\nu}$ when Josephson coupling is absent ($E_J=0$): $$\begin{aligned}
&&\varphi_{n,n+1}^{(v)}({\bf r};{\bf r}_{n\nu},{\bf r}_{n+1,\nu})=
\nonumber \\
&&\sum_{\nu}[\phi_v({\bf
r}-{\bf r}_{n\nu})-\phi_v({\bf r}-{\bf r}_{n+1,\nu})],
\label{2D}\end{aligned}$$ where $\phi_v({\bf r})$ is the polar angle of the point ${\bf r}$.
We calculate ${\cal C}({\bf B},T)$ with the functional (\[mainf\]) using high temperature expansion [@kosh] with respect to ${\cal
F}_J$ as: $${\cal C}({\bf B},T)\approx \frac{E_{J}}{2T}\int d{\bf r}S({\bf
r},{\bf B},T)=f({\bf B},T)\frac{E_0B_J}{2TB_z}.
\label{cfun}$$ Here $f({\bf B},T)=a^{-2}\int d{\bf r}S({\bf r},{\bf B},T)$ is a universal function and $B_{J}=\Phi_{0}/\lambda_{J}^{2}$. Eq. (\[cfun\]) is valid until ${\cal C}({\bf B},T)\ll 1$ which corresponds to the field and temperature range $TB_{z}\gg E_0 B_J$. Comparing Eqs. (\[omegap\]) and (\[cfun\]) with Eq. (\[line\]) we obtain an estimate for the relative line width in the liquid state due to the inhomogeneous broadening, $\delta\omega_p/\omega_p\approx
T/E_0\ll 1$.
In the lowest order in $E_{J}$ spin-wave and vortex degrees of freedom do not interact: $$S({\bf r},{\bf B})=S_{v}({\bf r},B_z)S_{r}({\bf r})\cos (2\pi
sB_{x}y/\Phi _{0}),
\label{SvSr}$$ where the correlation function $S_{v}({\bf r})$ is determined by the functional ${\cal F}_v$, while $S_r({\bf r})$ is determined by ${\cal
F}_\varphi$. In the pancake liquid regime, which we consider in this Letter, $B_{x}$ penetrates almost freely into the sample and has no influence on the phase fluctuations $\varphi _{n,n+1}^{(v)}$ and $\varphi_{n,n+1}^{(r)}$. An extra spatial dependence of $\varphi_{n,n+1}$ induced by $B_{x}$ gives possibility to probe phase correlations at given $B_{z}$ by measuring $B_{x}$-dependence of the plasma frequency.
We now express the vortex phase correlation function $S_{v}({\bf r})$ via the density correlation function of pancake vortices induced by $B_{z}$. The difference $\Phi_{v}({\bf
r})=\varphi_{n,n+1}^{(v)}({\bf r})-\varphi_{n,n+1}^{(v)}(0)$ can be connected with pancake densities $\rho _{n}({\bf R})=\sum_{\nu }\delta
({\bf R}-{\bf r}_{n\nu })$ in the layers as $$\Phi _{v}({\bf r})=\int d{\bf R}[\rho _{n}({\bf R})-\rho _{n+1}({\bf R}
)]\beta ({\bf r},{\bf R}). \label{Ph}$$ Here $\beta ({\bf r},{\bf R})=\phi_v({\bf r}/2-{\bf R})-\phi_v(-{\bf
r}/2-{\bf R})$ is the angle at which the segment connecting points $-{\bf r}/2$ and ${\bf r}/2$ is seen from the point ${\bf R}$, $$\cos \beta (r,{\bf R})=(R^{2}-r^{2}/4)[(R^{2}+r^{2}/4)^{2}-({\bf
Rr})^{2}]^{-1/2}.$$ The function $\beta ( {\bf r},{\bf R})$ has a jump of $2\pi $ when point ${\bf R}$ intersects segment $ [-{\bf r}/2,{\bf r}/2]$. Using the Gaussian approximation for $\Phi_{v}({\bf r})$ we obtain for vortex phase correlation function $S_{v}({\bf r})\equiv \langle \cos
[\Phi _{v}({\bf r})]\rangle$ $$S_{v}({\bf r})=\exp [-F_{v}(r)]
\label{SvFv}$$ with $$F_{v}(r)=\langle [\Phi _{v}({\bf r})]^{2}\rangle/2.
\label{appr}$$ Using Eqs. (\[Ph\]) and (\[appr\]) we connect $F_{v}(r)$ with the pair distribution function $h(r)$ defined by the relation $$n_{v}^{2}h(r)=\langle \rho _{n}(0)\rho _{n}({\bf r})\rangle -\langle \rho
_{n}(0)\rho _{n+1}({\bf r})\rangle -n_{v}\delta ({\bf r})$$ as $$\begin{aligned}
&&F_{v}(r)= \label{fvr} \\
&&-\frac{n_{v}^{2}}{2}\int d{\bf R}d{\bf r}^{\prime }h({\bf
r}^{\prime})\left[
\beta\left ({\bf r},{\bf R}+\frac{{\bf r}^{\prime
}}{2}\right)-\beta\left ({\bf r},{\bf R}- \frac{{\bf r}^{\prime
}}{2}\right)\right] ^{2}. \nonumber\end{aligned}$$ For this identity $n_{v}\int d{\bf r}h(r)=-1$ has been used, $n_{v}=\langle \rho _{n}( {\bf r})\rangle=a^{-2}$. Performing integration with respect to ${\bf R}$ and averaging over directions of ${\bf r}^{\prime}$ we finally obtain $$F_{v}(r)=-\pi n_{v}^{2}\int r^{\prime }dr^{\prime }h(r^{\prime
})J(r,r^{\prime }), \label{meq}$$ where the universal function $J(r,r^{\prime })$ is given by $$\begin{aligned}
&&J(r,r^{\prime })=2\pi r^{2}\left[ \ln (r^{\prime }/r)+2\right] ,\ \ \
r^{\prime }>r, \label{Jr} \\
&&J(r,r^{\prime })=4\pi \left[ 2rr^{\prime }-r^{\prime 2}-(r^{\prime
2}/2)\ln (r/r^{\prime })\right] ,\ \ \ r^{\prime }<r. \nonumber\end{aligned}$$ Eqs. (\[SvFv\]), (\[meq\]) and (\[Jr\]) represent general relations connecting phase fluctuations with vortex density fluctuations. The function $F_{v}(r)$ has the following asymptotics $$\begin{aligned}
F_{v}(r) &\approx &-8\pi ^{2}\frac{r}{a}\int_{0}^{\infty
}dxx^{2}h(x),\ \ \ r\gg a, \\
F_{v}(r) &\approx & \frac{\pi r^{2}}{a^{2}} \left( \ln \frac{a}{r}
+2-2\pi\int_{0}^{\infty }dxxh(x)\ln x\right) ,\ \ r\ll a.\end{aligned}$$ Linear increase of $F_{v}(r)$ at large $r$ indicates exponential decay of phase correlations at large distances.
The problem of calculating of $S_v({\bf r})$ is now reduced to calculation of the integral (\[meq\]) with the density correlation function of the liquid. This function is not available analytically. It can be calculated from Monte Carlo simulations or (approximately) using the density functional theory [@Menon]. In the decoupled pancake liquid regime the correlations between two dimensional pancake liquids in different layers are very weak because the energy of the magnetic interlayer interaction of disordered pancakes has an additional small parameter $s^2/\lambda_{ab}^2$ in comparison with the intralayer energy. If these correlations are neglected then the vortex system is equivalent to the one-component two-dimensional Coulomb plasma which was studied extensively [@caillol]. The pair distribution function $h_{2D}(r,B_{z},T,E_{0})$ within this approximation has an exact scaling property $ h_{2D}(r,B_{z},T,E_{0})=h_{2D}(r/a,T/E_{0})$, i.e., it depends on magnetic field only through the spatial scale. As follows from Eq. (\[meq\]) this scaling property is also transferred to the function $F_{v}(r)$, $
F_{v}(r)=F_{v}(r/a,T/E_{0})$. If, in addition, fluctuations of the regular phase are neglected then at $B_{x}=0$ the function $
f(T/E_{0})$ in the expression (\[cfun\]) for ${\cal C}({\bf B},T)$ is field independent. Therefore, the scaling property ${\cal
C}(B_{z},T)\propto 1/B_{z}$ is a consequence of the approximation of completely decoupled pancake liquids in pinning-free layers when spin-wave phase fluctuations are negligible.
We generated the pair distribution functions $h(x)$ with $x=r/a$ for the two-dimensional pancake liquid using Langevin dynamics simulations. The model and the algorithm have been described in Ref. . Fig. 1 shows an example of the pair distribution function $h(x)$ for temperature $T=0.012\pi E_0$ ($\approx 1.7\ T_{m}$) and the corresponding function $F_v(x)$ obtained by numerical integration of Eq. (\[meq\]). To show a weak waving of $F_v(x)$ due to liquid correlations we also plot the derivative of $F_v(x)$.
As temperature approaches $T_{c}$ the role of fluctuations of the regular phase progressively increases. The spin-wave phase correlations decay algebraically $$S_{r}({\bf r})=(\xi_{ab}/r)^{2\alpha },\ \ \alpha =T/2\pi E_{0}(T),
\label{scff}$$ where $\xi_{ab}(T)$ is the superconducting correlation length \[$1/\xi_{ab}$ determines the upper cut-off for momenta characterizing spatial variations of $\varphi_{n,n+1}^{(r)}({\bf r})$ in Eq. (\[vfunc\])\]. As follows from Eq. (\[SvSr\]) the contribution from regular phase fluctuations becomes essential if $S_{r}(r\sim a)\ll 1 $. This gives the condition $T>E_{0}(T)\ln(H_{c2}/B)$.
To relate $\omega_{p}(B,T)$ with the plasma frequency at zero field and the Josephson coupling $E_{J}\propto
\lambda_{c}^{-2}$ we have to take into account that these quantities are also renormalized by the phase fluctuations. Their suppression is determined by the cosine factor at $B=0$, $C(T)$, which was estimated in Ref. as $$C(T)=(\xi_{ab}/\lambda_{J})^{\alpha}.
\label{CT}$$
Finally using Eqs. (\[omegap\]), (\[SvSr\]), (\[SvFv\]), (\[scff\]), and (\[CT\]) we obtain an expression for plasma frequency which incorporates effects of vortices and of regular phase fluctuations at all temperatures : $$\begin{aligned}
&&\frac{\omega_{p}^{2}(B,T)}{\omega_{0}^{2}(T)}=\frac{E_{0}(T)}{2T }
\left(\frac{B_{J}}{B_{z}}\right)^{1-\alpha }
f_{s}\left( \frac{2\pi sB_{x}}{ \sqrt{\Phi
_{0}B_{z}}}\right) , \label{result} \\
&&f_{s}(b)=2\pi \int_{0}^{\infty }dxx^{1-2\alpha }\exp
[-F_{v}(x)]J_{0}(bx),
\label{int}\end{aligned}$$ where $J_{0}(x)$ is the Bessel function. Note that regular phase fluctuations reduce the power index in the $B_{z}$ dependence of ${\cal C}(B_{z},T)$ at $B_{x}=0$ and make it temperature dependent. Using the typical parameters for optimally doped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8-x}$ \[$\lambda_{ab}(0)=2000$ Å, $T_{c}=90$ K\] we obtain that the index $\alpha$ increases from $\alpha\approx 0.05$ at $T=50$ K to $\alpha\approx 0.18$ at $T=$75 K.
Measurements of $B_{x} $-dependence of the resonance frequency $\omega
_{p}$ at fixed $B_{z}$ and $T$ allow one to obtain, in principle, the pair distribution function $h(x,T)$, i.e., to extract quantitative information about correlations in the vortex liquid. Using Eq. (\[result\]) the function $f(b)$ can be found from the dependence $\omega _{p}(B_{x},B_{z})$ as: $$f_{s}\left( \frac{2\pi sB_{x}}{\sqrt{\Phi _{0}B_{z}}}\right)
=\frac{\omega
_{p}^{2}(B_{x},B_{z})}{\omega _{p}^{2}(0,B_{z})}f_{s}(0).$$ With $f_s(b)$ known, the function $F_{v}(x)$ may be found by the reverse Fourier transform from Eq. (\[int\]), $$F_{v}(x)=-\ln \left[ x^{2\alpha }\int_{0}^{\infty
}bdbf_{s}(b)J_{0}(bx)\right] .$$ Finally, the pair distribution function $h(x)$ can be expressed through $F_{v}(x)$ using Eqs. (\[meq\]) and (\[Jr\]): $$h(x)=-\frac{1}{8\pi x^{3}}\frac{d}{dx}\left\{ x^{3}\frac{d}{dx}\left[
x\frac{ d^{3}F_{v}(x)}{dx^{3}}\right] \right\} . \label{de}$$ Unfortunately, high order differentiations in the last equation complicate practical realization of the procedure. As it is shown in Fig. 1, oscillating behavior of liquid correlations is hardly seen in $F_v(x)$ but the first derivative, $F_v'(x)$, may expose it quite clearly. The experimental angular dependence of the plasma frequency [@MatsAng; @Bayrakci] was found to be in a good qualitative agreement with Eq. (\[result\]). However an accuracy of existing measurements is not sufficient to perform the described quantitative analyses.
In conclusion, we obtain a general expression connecting the plasma frequency with the density correlation function of pancake liquid. We demonstrate that the dependence of the plasma frequency on the magnetic field component parallel to the layers $B_x$ at fixed perpendicular component $B_z$ contains full structural information about the vortex liquid at given $B_z$ and temperature.
This work was supported by the National Science Foundation Office of the Science and Technology Center under contract No. DMR-91-20000 and by the U. S. Department of Energy, BES-Materials Sciences, under contract No. W-31- 109-ENG-38. Work in Los Alamos is supported by the U.S. DOE.
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=3.4in
|
---
abstract: 'SPICA, the cryogenic infrared space telescope recently pre-selected for a “Phase A” concept study as one of the three remaining candidates for ESA’s fifth medium class (M5) mission, is foreseen to include a far-infrared polarimetric imager (SPICA-POL, now called ), which would offer a unique opportunity to resolve major issues in our understanding of the nearby, cold magnetized Universe. This paper presents an overview of the main science drivers for , including high dynamic range polarimetric imaging of the cold interstellar medium (ISM) in both our Milky Way and nearby galaxies. Thanks to a cooled telescope, will deliver wide-field 100–350$\, \mu$m images of linearly polarized dust emission in Stokes Q and U with a resolution, signal-to-noise ratio, and both intensity and spatial dynamic ranges comparable to those achieved by [*Herschel*]{} images of the cold ISM in total intensity (Stokes I). The 200$\, \mu$m images will also have a factor $\sim \,$30 higher resolution than [*Planck*]{} polarization data. This will make a unique tool for characterizing the statistical properties of the magnetized interstellar medium and probing the role of magnetic fields in the formation and evolution of the interstellar web of dusty molecular filaments giving birth to most stars in our Galaxy. will also be a powerful instrument for studying the magnetism of nearby galaxies and testing galactic dynamo models, constraining the physics of dust grain alignment, informing the problem of the interaction of cosmic rays with molecular clouds, tracing magnetic fields in the inner layers of protoplanetary disks, and monitoring accretion bursts in embedded protostars.'
author:
- 'Ph. André$^1$, A. Hughes$^{2}$, V. Guillet$^{3,4}$, F. Boulanger$^{3,5}$, A. Bracco$^{1,5,6}$, E. Ntormousi$^{1,7}$, D. Arzoumanian$^{1,8}$, A.J. Maury$^{1,9}$, J.-Ph. Bernard$^{2}$, S. Bontemps$^{10}$, I. Ristorcelli$^{2}$, J.M. Girart$^{11}$, F. Motte$^{12,1}$, K. Tassis$^{7}$, E. Pantin$^1$, T. Montmerle$^{13}$, D. Johnstone$^{14}$, S. Gabici$^{15}$, A. Efstathiou$^{16}$, S. Basu$^{17}$, M. Béthermin$^{18}$, H. Beuther$^{19}$, J. Braine$^9$, J. Di Francesco$^{14}$, E. Falgarone$^{5}$, K. Ferrière$^{2}$, A. Fletcher$^{20}$, M. Galametz$^{1}$, M. Giard$^{2}$, P. Hennebelle$^{1}$, A. Jones$^{3}$, A.A. Kepley$^{21}$, J.Kwon$^{22}$, G. Lagache$^{18}$, P. Lesaffre$^{5}$, F. Levrier$^{5}$, D. Li$^{23}$, Z.-Y. Li$^{24}$, S.A. Mao$^{25}$, T. Nakagawa$^{22}$, T. Onaka$^{26}$, R. Paladino$^{27}$, N. Peretto$^{28}$ A. Poglitsch$^{1,29}$, V. Revéret$^1$, L. Rodriguez$^1$, M. Sauvage$^1$, J.D.Soler$^{1,19}$, L. Spinoglio$^{30}$, F. Tabatabaei$^{31}$, A. Tritsis$^{32}$, F. van der Tak$^{33}$, D. Ward-Thompson$^{34}$, H. Wiesemeyer$^{25}$, N. Ysard$^{3}$, H. Zhang$^{35}$'
bibliography:
- 'spica-pol.bib'
- 'Refs\_FB.bib'
title: 'Probing the cold magnetized Universe with SPICA-POL (B-BOP)'
---
observations: submillimeter – space missions – interstellar medium: structure – stars: formation – magnetic fields
[**Preface**]{}
The following set of articles describe in detail the science goals of the future Space Infrared telescope for Cosmology and Astrophysics ([[*SPICA*]{}]{}). The [[*SPICA*]{}]{} satellite will employ a 2.5-m telescope, actively cooled to below 8K, and a suite of mid- to far-infrared spectrometers and photometric cameras, equipped with state-of-the-art detectors. In particular, the [[*SPICA*]{}]{} Far Infrared Instrument (SAFARI) will be a grating spectrograph with low ($R$$=$300) and medium ($R$$=$3000–11000) resolution observing modes instantaneously covering the 35–230$\mu$m wavelength range. The [[*SPICA*]{}]{} Mid-Infrared Instrument (SMI) will have three operating modes: a large field of view (12$\times$10) low-resolution 17–36$\mu$m spectroscopic ($R$$=$50–120) and photometric camera at 34$\mu$m, a medium resolution ($R$$=$2000) grating spectrometer covering wavelengths of 18–36 and a high-resolution echelle module ($R$$=$28000) for the 12–18 domain. A large-field-of-view (160$\times$160)[^1], three-channel (100, 200, and 350) polarimetric camera (B-BOP [^2]) will also be part of the instrument complement. These articles will focus on some of the major scientific questions that the [[*SPICA*]{}]{} mission aims to address; more details about the mission and instruments can be found in @Roelfsema+2018.
INTRODUCTION: SPICA AND THE NATURE OF COSMIC MAGNETISM {#sec:intro}
======================================================
Alongside gravity, magnetic fields play a key role in the formation and evolution of a wide range of structures in the Universe, from galaxies to stars and planets. They simultaneously are an actor, an outcome, and a tracer of cosmic evolution. These three facets of cosmic magnetism are intertwined and must be thought of together. On one hand, the role magnetic fields play in the formation of stars and galaxies results from and traces their interplay with gas dynamics. On the other hand, turbulence is central to the dynamo processes that initially amplified cosmic magnetic fields and have since maintained their strength in galaxies across time [@Brandenburg05]. A transfer from gas kinetic to magnetic energy inevitably takes place in turbulent cosmic flows, while magnetic fields act on gas dynamics through the Lorentz force. These physical couplings relate cosmic magnetism to structure formation in the Universe across time and scales, and make the observation of magnetic fields a tracer of cosmic evolution, which is today yet to be disclosed. Improving our observational understanding of cosmic magnetism on a broad range of physical scales is thus at the heart of the “Origins” big question and is an integral part of one of ESA’s four Grand Science Themes (“Cosmic Radiation and Magnetism”) as defined by the ESA High-level Science Policy Advisory Committee (HISPAC) in 2013.
As often in Astrophysics, our understanding of the Universe is rooted in observations of the very local universe: the Milky Way and nearby galaxies. In the interstellar medium (ISM) of these galaxies, the magnetic energy is observed to be in rough equipartition with the kinetic (e.g. turbulent), radiative, and cosmic ray energies, all on the order of $\sim 1\, {\rm eV\, cm^{-3}} $, suggesting that magnetic fields are a key player in the dynamics of the ISM [e.g. @Draine2011]. Their exact role in the formation of molecular clouds and stellar systems is not well understood, however, and remains highly debated [e.g. @Crutcher2012]. Interstellar magnetic fields also hold the key for making headway on other main issues in Astrophysics, including the dynamics and energetics of the multiphase ISM, the acceleration and propagation of cosmic rays, and the physics of stellar and back-hole feedback. Altogether, a broad range of science topics call for progress in our understanding of interstellar magnetic fields, which in turn motivates ambitious efforts to obtain relevant data [cf. @Imagine18]. Observations of Galactic polarization are a highlight and a lasting legacy of the [*Planck*]{} space mission. Spectacular images combining the intensity of dust emission with the texture derived from polarization data have received world-wide attention and have become part of the general scientific culture [@planck2014-a01]. Beyond their popular impact, the [*Planck*]{} polarization maps represented an immense step forward for Galactic astrophysics [@PlanckXII2018]. [*Planck*]{} has paved the way for statistical studies of the structure of the Galactic magnetic field and its coupling with interstellar matter and turbulence, in the diffuse ISM and star-forming molecular clouds. SPICA, the [*Space Infrared Telescope for Cosmology and Astrophysics*]{} proposed to ESA as an M5 mission concept [@Roelfsema+2018], provides one of the best opportunities to take the next big leap forward and gain fundamental insight into the role of magnetic fields in structure formation in the cold Universe, thanks to the unprecedented sensitivity, angular resolution, and dynamic range of its far-infrared (far-IR) imaging polarimeter, $^2$ (previously called SPICA-POL, for “SPICA polarimeter”). The baseline instrument will allow simultaneous imaging observations in three bands, $100\, \mu$m, $200\, \mu$m, and $350\, \mu$m, with an individual pixel ${\rm NEP} < 3 \times 10^{-18}\, {\rm W\, Hz}^{-1/2}$, over an instantaneous field of view of $\sim 2.7' \times 2.7'$ at resolutions of 9, 18, and 32, respectively [@Rodriguez+2018]. Benefiting from a 2.5-m space telescope cooled to $< 8\, $K, will be two to three orders of magnitude more sensitive than current or planned far-IR/submillimeter polarimeters (see § \[subsec:spica-adv\] below) and will produce far-IR dust polarization images at a factor 20–30 higher resolution than the [*Planck*]{} satellite. It will provide wide-field 100–350$\, \mu$m polarimetric images in Stokes Q and U of comparable quality (in terms of resolution, signal-to-noise ratio, and both intensity and spatial dynamic ranges) to [*Herschel*]{} images in Stokes I.
The present paper gives an overview of the main science drivers for the polarimeter and is complementary to the papers by, e.g., @Spinoglio+2017 and @vandertak+2018 which discuss the science questions addressed by the other two instruments of SPICA, SMI [@Kaneda+2016] and SAFARI [@Roelfsema+2014], mainly through highly sensitive spectroscopy. The outline is as follows: Section \[sec:filaments\] describes the prime science driver for , namely high dynamic range polarimetric mapping of Galactic filamentary structures to unravel the role of magnetic fields in the star formation process. Section \[sec:turbulence\] introduces the contribution of to the statistical characterization of magnetized interstellar turbulence. Section \[sec:protostars\] and Section \[sec:massive-sf\] emphasize the importance of polarization observations for our understanding of the physics of protostellar dense cores and high-mass star protoclusters, respectively. Section \[sec:galaxies\] discusses dust polarization observations of galaxies, focusing mainly on nearby galaxies. Section \[sec:dust-physics\] describes how multi-wavelength polarimetry with can constrain dust models and the physics of dust grain alignment. Finally, Sections \[sec:cosmic-rays\], \[sec:disks\], and \[sec:proto-var\] discuss three topics which, although not among the main drivers of the instrument, will significantly benefit from observations, namely the study of the origin of cosmic rays and of their interaction with molecular clouds (Sect. \[sec:cosmic-rays\]), the detection of polarized far-IR dust emission from protoplanetary disks, thereby tracing magnetic fields in the inner layers of the disks (Sect. \[sec:disks\]), and the (non-polarimetric) monitoring of protostars in the far-IR, i.e., close to the peak of their spectral energy distributions (SEDs), to provide direct constraints on the process of episodic protostellar accretion (Sect. \[sec:proto-var\]). Section \[sec:conclusions\] concludes the paper.
[lccc]{} Band & 100 $\mu$m & 200 $\mu$m & 350 $\mu$m
------------------------------------------------------------------------
\
$\lambda$ range & 75-125 $\mu$m & 150-250 $\mu$m & 280-420 $\mu$m
------------------------------------------------------------------------
\
Array size & 32$\times$32 & 16$\times$16 & 8$\times$8\
Pixel size & 5“$\times$5” & 10“$\times$10” & 20“$\times$20”\
FWHM & 9“ & 18” & 32"\
\
Unpol. & 21 $\mu$Jy & 42 $\mu$Jy & 85 $\mu$Jy\
Q, U & 30 $\mu$Jy & 60 $\mu$Jy & 120 $\mu$Jy\
\
Unpol. & 160 $\mu$Jy & 320 $\mu$Jy & 650 $\mu$Jy\
Q, U & 230 $\mu$Jy & 460 $\mu$Jy & 920 $\mu$Jy\
\
Unpol. & 0.09 MJy/sr & 0.045 MJy/sr & 0.025 MJy/sr\
5% Q, U $^\dag$ & 2.5 MJy/sr & 1.25 MJy/sr & 0.7 MJy/sr\
\
& $\geq 100$ & $\geq 100$ & $\geq 100$\
\
& $\geq 20\arcsec$/sec & $\geq 20\arcsec$/sec & $\geq 20\arcsec$/sec\
Surface brightness level in I to map Q, U at 5$\sigma$ over 1 deg$^2$ in 10hr assuming 5% fractional polarization.
Assuming $\geq 1\% $ fractional polarization. \[tab:POL\]
Magnetic fields and star formation in filamentary clouds {#sec:filaments}
========================================================
Understanding how stars form in the cold ISM of galaxies is central in Astrophysics. Star formation is both one of the main factors that drive the evolution of galaxies on global scales and the process that sets the physical conditions for planet formation on local scales. Star formation is also a complex, multi-scale process, involving a subtle interplay between gravity, turbulence, magnetic fields, feedback mechanisms. As a consequence, and despite recent progress, the basic questions of what regulates star formation in galaxies and what determines the mass distribution of forming stars (i,e. the stellar initial mass function or IMF) remain two of the most debated problems in Astronomy. Today, a popular school of thought for understanding star formation and these two big questions is the gravo-turbulent paradigm [e.g. @MacLowKlessen2004; @McKee07; @Padoan+2014], whereby magnetized supersonic turbulence creates structure and seeds in interstellar clouds, which subsequently grow and collapse under the primary influence of gravity. A variation on this scenario is that of dominant magnetic fields in cloud envelopes, and a turbulence-enhanced ambipolar diffusion leading to gravity-dominated subregions [e.g., @Li04; @KudohBasu2008].
Moreover, while the global rate of star formation in galaxies and the positions of galaxies in the Schmidt-Kennicutt diagram [e.g. @Kennicutt+2012] are likely controlled by macroscopic phenomena such as cosmic accretion, large-scale feedback, and large-scale turbulence [@SanchezAlmeida+2014], there is some evidence that the star formation efficiency in the dense molecular gas of galaxies is nearly universal[^3] [e.g. @Gao+2004; @Lada+2012] and primarily governed by the physics of filamentary cloud fragmentation on much smaller scales [e.g. @Andre+2014; @Padoan+2014]. As argued in § \[subsec:fil-paradigm\] and § \[subsec:filaments\] below, magnetic fields are likely a key element of the physics behind the formation and fragmentation of filamentary structures in interstellar clouds.
Often ignored, strong, organized magnetic fields, in rough equipartition with the turbulent and cosmic ray energy densities, have been detected in the ISM of a large number of galaxies out to $z = 2$ [e.g. @Beck2015; @Bernet+2008]. Recent cosmological magneto-hydrodynamic (MHD) simulations of structure formation in the Universe suggest that magnetic-field strengths comparable to those measured in nearby galaxies ($\simlt 10\, \mu$G) can be quickly built up in high-redshift galaxies (in $<< 1\,$ Gyr), through the dynamo amplification of initially weak seed fields [e.g. @RiederTeyssier2017; @Marinacci+2018]. Magnetic fields are therefore expected to play a dynamically important role in the formation of giant molecular clouds (GMCs) on kpc scales within galaxies [e.g. @Inoue+2012] and in the formation of filamentary structures leading to individual star formation on $\sim$1–10$\,$pc scales within GMCs [e.g. @Inutsuka+2015; @Inoue+2018 see § \[subsec:filaments\] below]. On dense core ($\leq 0.1\, $pc) scales, the magnetic field and angular momentum of most protostellar systems are likely inherited from the processes of filament formation and fragmentation (cf. Misugi et al., in prep.). On even smaller ($< 0.01\, $pc or $< 2000\, $au) scales, magnetic fields are essential to solve the angular momentum problem of star formation, generate protostellar outflows, and control the formation of protoplanetary disks [e.g. @Pudritz+2007; @Machida+2008; @LiPPVI]. In this context, will be a unique tool for characterizing the morphology of magnetic fields on scales ranging from $\sim 0.01\,$pc to $\sim 1\,$kpc in Milky Way like galaxies. In particular, a key science driver for is to clarify the role of magnetic fields in shaping the rich web of filamentary structures pervading the cold ISM, from the low-density striations seen in HI clouds and the outskirts of CO clouds [e.g. @Clark+2014; @Kalberla+2016; @Goldsmith+2008] to the denser molecular filaments within which most prestellar cores and protostars are forming according to [*Herschel*]{} results (see Fig. \[taurus\_planck\] and § \[subsec:fil-paradigm\] below).
Dust polarization observations: A probe of magnetic fields in star-forming clouds {#subsec:dustpol}
---------------------------------------------------------------------------------
### Dust grain alignment {#subsubsec:alignment}
Polarization of background starlight from dichroic extinction produced by intervening interstellar dust has been known since the late 1940s [@Hall1949; @Hiltner1949]. The analysis of the extinction data in polarization, in particular its variation with wavelength in the visible to near-UV, has allowed major discoveries regarding dust properties, in particular regarding the size distribution of dust. Like the first large-scale total intensity mapping in the far-IR that was provided by the Infrared Astronomical Satellite (IRAS) satellite data, extensive studies of polarized far-IR emission today bring the prospect of a new revolution in our understanding of dust physics. This endeavor includes pioneering observations with ground-based, balloon-borne, and space-borne facilities such as the very recent all-sky observations by the [*Planck*]{} satellite at $850\, \mu$m and beyond [e.g. @PlanckXII2018]. However, polarimetric imaging of polarized dust continuum emission is still in its infancy and amazing improvements are expected in the next decades from instruments such as the Atacama Large Millimeter Array (ALMA) in the submillimeter and in the far-IR.
The initial discovery that starlight extinguished by intervening dust is polarized led to the conclusion that dust grains must be somewhat elongated and globally aligned in space in order to produce the observed polarized extinction. While the elongation of dust grains was not unexpected, coherent grain alignment over large spatial scales has been more difficult to explain. A very important constraint has come from recent measurements in emission with, e.g., the Archeops balloon-borne experiment [@Benoit+2004] and the [*Planck*]{} satellite [@PlanckXIX2015] which indicated that the polarization degree of dust emission can be as high as $20\%$ in some regions of the diffuse ISM in the solar neighborhood. This requires more efficient dust alignment processes than previously anticipated [@PlanckXII2018].
The most widely accepted dust grain alignment theories, already alluded to by [@Hiltner1949], propose that alignment is with respect to the magnetic field that pervades the ISM. Rapidly spinning grains will naturally align their angular momentum with the magnetic field direction [@Purcell1979; @Lazarian_Draine1999], but the mechanism leading to such rapid spin remains a mystery. The formation of molecular hydrogen at the surface of dust grains could provide the required momentum [@Purcell1979]. Today’s leading grain alignment theory is Radiative Alignment Torques (RATs) [@Dolginov_Mitrofanov1976; @Draine_Weingartner1996; @Lazarian_Hoang2007; @Hoang_Lazarian2016 and references therein], where supra-thermal spinup of irregularly-shaped dust grains results from their irradiation by an anisotropic radiation field [a process experimentally confirmed, see @Abbas+2004].
### Probing magnetic fields with imaging polarimetry {#subsubsec:polarimetry}
In the conventional picture that the minor axis of elongated dust grains is aligned with the local direction of the magnetic field, mapping observations of linearly-polarized continuum emission at far-IR and submillimeter wavelengths are a powerful tool to measure the morphology and structure of magnetic field lines in star-forming clouds and dense cores [cf. @Matthews+2009; @Crutcher+2004; @Crutcher2012]. A key advantage of this technique is that it images the structure of magnetic fields through an emission process that traces the [*mass*]{} of cold interstellar matter, i.e., the reservoir of gas directly involved in star formation. Indirect estimates of the plane-of-sky magnetic field strength $B_{POS}$ can also be obtained using the Davis-Chandrasekhar-Fermi method [@Davis1951; @CF1953]: $B_{POS} = \alpha_{corr}\, \sqrt{4\pi \rho}\, \delta V / \delta \Phi $, where $\rho $ is the gas density (which can be estimated to reasonable accuracy from $Herschel$ column density maps, especially in the case of resolved filaments and cores – cf. [@Palmeirim+2013; @Roy+2014]), $ \delta V $ is the one-dimensional velocity dispersion (which can be estimated from line observations in an appropriate tracer such as N$_2$H$^+$ for star-forming filaments and dense cores– e.g. [@Andre+2007; @Tafalla+2015]), $ \delta \Phi $ is the dispersion in polarization position angles directly measured in a dust polarization map, and $\alpha_{corr} \approx 0.5 $ is a correction factor obtained through numerical simulations [cf. @Ostriker+2001]. Large-scale maps that resolve the above quantities over a large dynamic range of densities can be used to estimate the mass-to-flux ratio in different parts of a molecular cloud. This can test the idea that cloud envelopes may be magnetically supported and have a subcritical mass-to-flux ratio [@MouschoviasCiolek1999; @Shu+1999]. Recent applications of the Davis-Chandrasekhar-Fermi method using SCUBA2-POL $850\, \mu$m data taken as part of the BISTRO survey [@Ward+2017] toward dusty molecular clumps in the Orion and Ophiuchus clouds are presented in @Pattle+2017, @Kwon+2018, and @Soam+2018. Refined estimates of both the mean and the turbulent component of $B_{POS}$ can be derived from an analysis of the second-order angular structure function (or angular dispersion function) of observed polarization position angles $<\Delta\Phi^2 (l)> = \frac{1}{N(l)} \Sigma [\Phi (r) - \Phi (r+l)]^2 $ [@Hildebrand+2009; @Houde+2009]. Alternatively, in localized regions where gravity dominates over MHD turbulence, the polarization-intensity gradient method can be used to obtain maps of the local magnetic field strength from maps of the misalignment angle $\delta $ between the local magnetic field (estimated from observed polarization position angles) and the local column density gradient (estimated from maps of total dust emission). Indeed, such $\delta $ maps provide information on the local ratio between the magnetic field tension force and the gravitational force [@Koch+2012; @Koch+2014]. Additionally, the paradigm of Alfvénic turbulence can be tested in dense regions where gravity dominates, in which the observed angular dispersion $\Delta \Phi$ is expected to decrease in amplitude toward the center of dense cores where $\delta V$ also decreases [@Auddy+2019].
{width="42pc"}
Because the typical degree of polarized dust continuum emission is low ($\sim \, $2%–5% – e.g. [@Matthews+2009]) and the range of relevant column densities spans three orders of magnitude from equivalent visual extinctions[^4] $A_V \sim 0.1$ in the atomic medium to $A_V > 100$ in the densest molecular filaments/cores, a systematic dust polarization study of the rich filamentary networks pervading nearby interstellar clouds and their connection to star formation requires a large improvement in sensitivity, mapping speed, and dynamic range over existing far-IR/submillimeter polarimeters. A big improvement in polarimetric mapping speed is also needed for statistical reasons. As only the plane-of-sky component of the magnetic field is directly accessible to dust continuum polarimetry, a large number of systems must be imaged in various Galactic environments before physically meaningful conclusions can be drawn statistically on the role of magnetic fields. As shown in § \[subsec:spica-adv\] below, the required step forward in performance can be uniquely provided by a large, cryogenically cooled space-borne telescope such as SPICA, which can do in far-IR polarimetric imaging what [*Herschel*]{} achieved in total-power continuum imaging.
Insights from [*Herschel*]{} and [*Planck*]{}: A filamentary paradigm for star formation? {#subsec:fil-paradigm}
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The [*Herschel*]{} mission has led to spectacular advances in our knowledge of the texture of the cold ISM and its link with star formation. While interstellar clouds have been known to be filamentary for a long time [e.g. @Schneider+1979; @Bally+1987; @Myers2009 and references therein], [*Herschel*]{} imaging surveys have established the ubiquity of filaments on almost all length scales ($\sim 0.5\,$pc to $\sim 100\,$pc) in the molecular clouds of the Galaxy and shown that this filamentary structure likely plays a key role in the star formation process [e.g. @Andre+2010; @Henning+2010; @Molinari+2010; @Hill+2011; @Schisano+2014; @Wang+2015]. The interstellar filamentary structures detected with [*Herschel*]{} span broad ranges in length, central column density, and mass per unit length [e.g. @Schisano+2014; @Arzoumanian+2018]. In contrast, detailed analysis of the radial column density profiles indicates that, at least in the nearby molecular clouds of the Gould Belt, [*Herschel*]{} filaments are characterized by a narrow distribution of inner widths with a typical value of $\sim 0.1$ pc and a dispersion of less than a factor of 2, when the data are averaged over the filament crests [@Arzoumanian+2011; @Arzoumanian+2018]. Independent studies of filament widths in nearby clouds have generally confirmed this result when using submillimeter continuum data [e.g. @KochRosolowsky2015; @Salji+2015; @Rivera-Ingraham+2016], even if factor of $\sim \,$2–4 variations around the mean inner width of $\sim 0.1\,$pc have been found along the main axis of a given filament [e.g. @Juvela+2012; @Ysard+2013]. Measurements of filament widths obtained in molecular line tracers [e.g., @Pineda+2011; @FernandezLopez+2014; @Panopoulou+2014; @Hacar+2018] have been less consistent with the [*Herschel*]{} dust continuum results of @Arzoumanian+2011 [2018], but this can be attributed to the lower dynamic range achieved by observations in any given molecular line tracer. @Panopoulou+2017 pointed out an apparent contradiction between the existence of a characteristic filament width and the essentially scale-free nature of the power spectrum of interstellar cloud images (well described by a single power law from $\sim 0.01\,$pc to $\sim 50\,$pc – [@mamd+2010; @mamd+2016]), but @Roy+2018 showed that there is no contradiction given the only modest area filling factors ($\simlt 10\% $) and column density contrasts ($\leq 100\% $ in most cases) derived by @Arzoumanian+2018 for the filaments seen in [*Herschel*]{} images. While further high-resolution submillimeter continuum studies would be required to investigate whether the same result holds beyond the Gould Belt, the median inner width of $\sim 0.1\,$pc measured with [*Herschel*]{} appears to reflect the presence of a true common scale in the filamentary structure of nearby interstellar clouds. If confirmed, this result may have far-reaching consequences as it introduces a characteristic scale in a system generally thought to be chaotic and turbulent (i.e. largely scale-free – cf. [@Guszejnov+2018]). It may thus present a severe challenge in any attempt to interpret all ISM observations in terms of scale-free processes.
Another major result from [*Herschel*]{} studies of nearby clouds is that most ($>75\% $) prestellar cores and protostars are found to lie in dense, “supercritical” filaments above a critical threshold $\sim 16\, M_\odot $/pc in mass per unit length, equivalent to a critical threshold $\sim 160\, M_\odot $/pc$^2$ ($A_V \sim 8)$ in column density or $n_{H_2} \sim 2 \times 10^4\, {\rm cm}^{-3} $ in volume density [@Andre+2010; @Konyves+2015; @Marsh+2016]. A similar column density threshold for the formation of prestellar cores (at $A_V \sim \, $5–10) had been suggested earlier based on ground-based millimeter and submillimeter studies [e.g. @Onishi+1998; @Johnstone+2004; @Kirk+2006], but without clear connection to filaments. Interestingly, a comparable threshold in extinction (at $A_V \sim \, $8) has also been observed in the spatial distribution of young stellar objects (YSOs) with [*Spitzer*]{} [e.g. @Heiderman+2010; @Lada+2010; @Evans+2014].
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Overall, the [*Herschel*]{} results support a filamentary paradigm for star formation in two main steps [e.g. @Andre+2014; @Inutsuka+2015]: First, multiple large-scale compressions of interstellar material in supersonic turbulent MHD flows generate a cobweb of $\sim 0.1$ pc-wide filaments in the cold ISM; second, the densest filaments fragment into prestellar cores (and subsequently protostars) by gravitational instability above the critical mass per unit length $ M_{\rm line,crit} = 2\, c_s^2/G$ of nearly isothermal, cylinder-like filaments (see Fig. \[taurus\_planck\]), where $c_s$ is the sound speed and $G$ the gravitational constant. This paradigm differs from the classical gravo-turbulent picture in that it relies on the unique features of filamentary geometry, such as the existence of a critical line mass for nearly isothermal filaments [e.g. @Inutsuka+1997 and references therein]. The validity and details of the filamentary paradigm are strongly debated, however, and many issues remain open. For instance, according to some numerical simulations, the above two steps may not occur consecutively but simultaneously, in the sense that both filamentary structures and dense cores may grow in mass at the same time [e.g. @Gomez+2014; @ChenOstriker2015]. The physical origin of the typical $\sim 0.1\,$pc inner width of molecular filaments is also poorly understood and remains a challenge for numerical models [e.g. @Padoan+2001; @Hennebelle2013; @Smith+2014; @Federrath2016; @Ntormousi+2016]. @Auddy+2016 point out that magnetized filaments may actually be ribbon-like and quasi-equilibrium structures supported by the magnetic field, and therefore not have cylindrical symmetry. Regardless of any particular scenario, there is nevertheless little doubt after [*Herschel*]{} results that dense molecular filaments represent an integral part of the initial conditions of the bulk of star formation in our Galaxy.
As molecular filaments are known to be present in the Large Magellanic Cloud (LMC – [@Fukui+2015]), the proposed filamentary paradigm may have implications on galaxy-wide scales. Assuming that all filaments have similar inner widths, it has been argued that they may help to regulate the star formation efficiency in dense molecular gas [@Andre+2014], and that they may be responsible for a quasi-universal star formation law in the dense molecular ISM of galaxies [cf. @Lada+2012; @Shimajiri+2017], with possible variations in extreme environments such as the CMZ [@Longmore+2013; @Federrath+2016].
In parallel, the [*Planck*]{} mission has led to major advances in our knowledge of the geometry of the magnetic field on large scales in the Galactic ISM. The first all-sky maps of dust polarization provided by [*Planck*]{} at 850$\, \mu$m have revealed a very organized magnetic field structure on $\simgt \, $1–10 pc scales in Galactic interstellar clouds [@PlanckXXXV2016 see Fig. \[taurus\_planck\]a]. The large-scale magnetic field tends to be aligned with low-density filamentary structures with subcritical line masses such as striations (see Fig. \[taurus\_planck\]b) and perpendicular to dense star-forming filaments with supercritical line masses [@PlanckXXXII2016; @PlanckXXXV2016 see Figs. \[taurus\_planck\]b & \[taurus\_planck\]c], a trend also seen in optical and near-IR polarization observations [@Chapman+2011; @Palmeirim+2013; @Panopoulou+2016; @Soler+2016]. There is also a hint from [*Planck*]{} polarization observations of the nearest clouds that the direction of the magnetic field may change [*within*]{} dense filaments from nearly perpendicular in the ambient cloud to more parallel in the filament interior [cf. @PlanckXXXIII2016]. These findings suggest that magnetic fields are dynamically important and play a key role in the formation and evolution of filamentary structures in interstellar clouds, supporting the view that dense molecular filaments form by accumulation of interstellar matter along field lines.
The low resolution of [*Planck*]{} polarization data (10$\,$at best or 0.4 pc in nearby clouds) is however insufficient to probe the organization of field lines in the $\sim 0.1\,$pc interior of filaments, corresponding both to the characteristic transverse scale of filaments [@Arzoumanian+2011; @Arzoumanian+2018] and to the scale at which fragmentation into prestellar cores occurs [cf. @Tafalla+2015]. Consequently, the geometry of the magnetic field [*within*]{} interstellar filaments and its effects on fragmentation and star formation are essentially unknown today.
Investigating the role of magnetic fields in the formation and evolution of molecular filaments with {#subsec:filaments}
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Improving our understanding of the physics and detailed properties of molecular filaments is of paramount importance as the latter are representative of the initial conditions of star formation in molecular clouds and GMCs[^5] (see § \[subsec:fil-paradigm\] above). In particular, investigating how dense, “supercritical” molecular filaments can maintain a roughly constant $\sim$0.1$\,$pc inner width and fragment into prestellar cores instead of collapsing radially to spindles is crucial to understanding star formation. The topology of magnetic field lines may be one of the key elements here. For instance, a longitudinal magnetic field can support a filament against radial collapse but not against fragmentation along its main axis, while a perpendicular magnetic field works against fragmentation and increases the critical mass per unit length but cannot prevent the radial collapse of a supercritical filament [e.g. @Tomisaka2014; @Hanawa+2017]. The actual topology of the field within molecular filaments is likely more complex and may be a combination of these two extreme configurations.
One plausible evolutionary scenario, consistent with existing observations, is that star-forming filaments accrete ambient cloud material along field lines through a network of magnetically-dominated striations [e.g. @Palmeirim+2013; @Cox+2016; @Shimajiri+2018 see also Figs. \[taurus\_planck\]b & \[taurus\_fibers\]a]. Accretion-driven MHD waves may then generate a system of velocity-coherent fibers within dense filaments [@Hacar+2013; @Hacar+2018; @Arzoumanian+2013; @HennebelleAndre2013 cf. Fig. \[taurus\_fibers\]] and the corresponding organization of magnetic field lines may play a central role in accounting for the roughly constant $\sim 0.1\,$pc inner width of star-forming filaments as measured in [*Herschel*]{} observations (cf. § \[subsec:fil-paradigm\]). Constraining this process further is key to understanding star formation itself, since filaments with supercritical masses per unit length would otherwise undergo rapid radial contraction with time, effectively preventing fragmentation into prestellar cores and the formation of protostars [e.g. @Inutsuka+1997]. Information on the geometry of magnetic field lines [*within*]{} star-forming filaments at $A_V > 8$ is thus crucially needed, and can be obtained through 200–350$\, \mu$m dust polarimetric imaging at high angular resolution with . Large area coverage and both high angular resolution and high spatial dynamic range are needed to resolve the 0.1 pc scale by a factor $\sim \,$ 3–10 on one hand and to probe spatial scales from $> 10\,$pc in the low-density striations of the ambient cloud (see Fig. \[taurus\_planck\]), down to $\sim \, $0.01–0.03 pc for the fibers of dense filaments (see Fig. \[taurus\_fibers\]). In nearby Galactic regions (at $d \sim \, $150–500 pc), this corresponds to angular scales from $> 5\, $deg or more down to $\sim 20$$\,$ or less.
![Simulated striations (from Tritsis and Tassis 2016). Right panel: Volume density image from the simulations. Left panel: Zoomed-in column density view of a single striation, showing the “sausage” instability setting in, with characteristic imprints in both the magnetic-field and the column-density distribution. In both panels, the drapery pattern traces the magnetic field lines and the mean direction of the magnetic field is indicated by a black arrow. The passage of Alfvén waves excites magnetosonic modes that create compressions and rarefactions (colorbar) along field lines, giving rise to striations. The simulated data in both panels have been convolved to an effective spatial resolution of 0.012 pc, corresponding to the $18\arcsec $ HPBW of at $200\, \mu$m.[]{data-label="striations_tritsis_sims"}](atritsis_fig_Version14newSim.jpg){width="\columnwidth"}
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Low-density striations are remarkably ordered structures in an otherwise chaotic-looking turbulent medium. While the exact physical origin of both low-density striations [@Heyer+2016; @Tritsis+2016; @Tritsis+2018; @Chen+2017] and high-density fibers [e.g. @Clarke+2017; @Zamora-Aviles+2017] is not well understood and remains highly debated in the literature, there is little doubt that magnetic fields are involved. For instance, @Tritsis+2016 modeled striations as density fluctuations associated with magnetosonic waves in the linear regime (the column density contrast of observed striations does not exceed $25\%$). These waves are excited as a result of the passage of Alfvén waves, which couple to other MHD modes through phase mixing (see Fig. \[striations\_tritsis\_sims\], right panel). In contrast, @Chen+2017 proposed that striations do not represent real density fluctuations, but are rather a line-of-sight column density effect in a corrugated layer forming in the dense post-shock region of an oblique MHD shock. High-resolution polarimetric imaging data would be of great interest to set direct observational constraints and discriminate between these possible models. Specifically, the magnetosonic wave model predicts that a zoo of MHD wave effects should be observable in these regions. One of them, that linear waves in an isolated cloud should establish standing waves (normal modes) imprinted in the striations pattern, has recently been confirmed in the case of the Musca cloud [@Tritsis+2018]. Other such effects include the “sausage” and “kink” modes (see Fig. \[striations\_tritsis\_sims\], left panel), which are studied extensively in the context of heliophysics [e.g. @Nakariakov+2016], and which could open a new window to probe the local conditions in molecular clouds [@Tritsis+2018b]. A first specific objective of observations will be to test the hypothesis, tentatively suggested by [*Planck*]{} polarization results [cf. @PlanckXXXIII2016] that the magnetic field may become nearly parallel to the long axis of star-forming filaments in their dense interiors at scales $< 0.1$ pc, due to, e.g., gravitational or turbulent compression (see Fig. \[synthetic\_polar\_maps\]) and/or reorientation of oblique shocks in magnetized colliding flows [@Fogerty+2017]. A change of field orientation inside dense star-forming filaments is also predicted by numerical MHD simulations in which gravity dominates and the magnetic field is dragged by gas flowing along the filament axis [@Gomez+2018; @PSLi+2018], as observed in the velocity field of some massive infrared dark filaments [@Peretto+2014]. An alternative topology for the field lines within dense molecular filaments often advocated in the literature is that of helical magnetic fields wrapping around the filament axis [e.g. @Fiege+2000; @StutzGould2016; @SchleicherStutz2018; @Tahani+2018]. As significant degeneracies exist between different models because only the plane-of-sky magnetic field is directly accessible to dust polarimetry [cf. @Reissl+2018; @Tomisaka2015], discriminating between these various magnetic topologies will require sensitive imaging observations of large samples of molecular filaments for which the distribution of viewing angles may be assumed to be essentially random. One advantage of the model of oblique MHD shocks [e.g. @ChenOstriker2014; @Inoue+2018; @Lehmann+2016] is that it could potentially explain both how dense filaments maintain a roughly constant $\sim 0.1\,$pc width while evolving [cf. @Seifried+2015] and why the observed spacing of prestellar cores along the filaments is significantly shorter than the characteristic fragmentation scale of 4$\, \times$ the filament diameter expected in the case of non-magnetized nearly isothermal gas cylinders [e.g. @Inutsuka+1992; @Nakamura+1993; @Kainulainen+2017]. A second specific objective of observations will be to better characterize the transition column density at which a switch occurs between filamentary structures primarily parallel to the magnetic field (at low $N_{H_2}$) and filamentary structures preferentially perpendicular to the magnetic field (at high $N_{H_2}$) [see @PlanckXXXV2016 and § \[subsec:fil-paradigm\] above]. Based on a detailed analysis of numerical MHD simulations, @Soler+2017 postulated that this transition column density depends primarily on the strength of the magnetic field in the parent molecular cloud and therefore constitutes a key observable piece of information. Moreover, @Chen+2016 showed that, in their colliding flow MHD simulations, the transition occurs where the ambient gas is accelerated gravitationally from sub-Alfvénic to super-Alfvénic speeds. They also concluded that the nature of the transition and the 3D magnetic field morphology in the super-Alfvénic region can be constrained from the observed polarization fraction and dispersion of polarization angles in the plane of the sky, which provides information on the tangledness of the field.
As a practical illustration of what could be achieved with , a reference polarimetric imaging survey would map, in Stokes I, Q, U at $100\, \mu$m, $200\, \mu$m, $350\, \mu$m, the same $\sim 500\,$deg$^2$ area in nearby interstellar clouds imaged by [*Herschel*]{} in Stokes I at 70–500 $\, \mu$m as part of the Gould Belt, HOBYS, and Hi-GAL surveys [@Andre+2010; @Motte+2010; @Molinari+2010]. To first order, the gain in sensitivity of over SPIRE & PACS on [*Herschel*]{} would compensate for the low degree of polarization (only a few %) and make it possible to obtain Q and U maps of polarized dust emission with a signal-to-noise ratio similar to the [*Herschel*]{} images in Stokes I. Assuming the performance parameters given in Table \[tab:POL\] (see also Table 4 of [@Roelfsema+2018], and Table 1 of [@Rodriguez+2018]) and an integration time of $\sim 2\,$hr per square degree, such a survey would reach a signal-to-noise ratio of 7 in Q, U intensity at both 200$\, \mu$m and 350$\, \mu$m in low column density areas with $A_V \sim 0.2$ (corresponding to the diffuse, cold ISM), for a typical polarization fraction of 5% and a typical dust temperature of $T_d \sim 15\,$K. The same survey would reach a signal-to-noise ratio of 5 in Q, U at 100$\, \mu$m down to $A_V \sim 1$. The entire survey of $\sim 500\,$deg$^2$ would require $\sim 1500\,$hr of telescope time, including overheads. It would provide key information on the magnetic field geometry for thousands of filamentary structures spanning $\sim 3$ orders of magnitude in column density from low-density subcritical filaments in the atomic (HI) medium at $A_V < 0.5$ to star-forming supercritical filaments in the dense inner parts of molecular clouds at $A_V > 100$.
Key advantages of over other polarimetric facilities {#subsec:spica-adv}
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Far-IR/submillimeter polarimetric imaging from space with will have unique advantages, especially in terms of spatial dynamic range [*and*]{} surface brightness dynamic range. Studying the multi-scale physics of star formation within molecular filaments requires a spatial dynamic range of $\sim 1000$ or more to simultaneously probe scales $> 10\,$pc in the parent clouds down to $\sim 0.01\,$pc in the interior of star-forming filaments (corresponding to angular scales from $\sim 18$$\,$to $> 5 \,$deg in the nearest molecular clouds – see Fig. \[taurus\_planck\]). Such a high spatial dynamic range was routinely achieved with [*Herschel*]{} in non-polarized imaging, but has never been obtained in ground-based submillimeter continuum observations. It will be achieved for the first time with in polarized far-IR imaging.
The angular resolution and surface brightness dynamic range of will make it possible to resolve 0.1$\,$pc-wide filaments out to 350 pc and to image a few % polarized dust emission through the entire extent of nearby cloud complexes (cf. Fig. \[taurus\_planck\]), from the low-density outer parts of molecular clouds ($A_V \simlt 0.5$) all the way to the densest filaments and cores ($A_V > 100$). In comparison, [*Planck*]{} had far too low resolution (10$\,$ at best in polarization) to probe the magnetic field within dense 0.1 pc-wide filaments or detect faint 0.1 pc-wide striations. Near-infrared polarimetry cannot penetrate the dense inner parts of star-forming filaments, and ground-based or air-borne millimeter/submillimeter polarimetric instruments, such as SCUBA2-POL (the polarimeter for the Sub-millimetre Common-User Bolometer Array 2 – [@Friberg+2016]), NIKA2-POL (the polarization channel for the New IRAM KID Arrays 2 – cf. [@Adam+2018]), HAWC$+$ (the High-resolution Airborne Wideband Camera-Plus for SOFIA, the Stratospheric Observatory for Infrared Astronomy – [@Harper+2018]), or BLAST-TNG (the next generation Balloon-borne Large Aperture Submillimeter Telescope for Polarimetry – [@Galitzki+2014]), will lack the required sensitivity and dynamic range in both spatial scales and intensity.
![Surface-brightness sensitivity of for wide-field polarimetric imaging compared to other existing or planned polarimetric facilities. The total surface-brightness level required to detect polarization (i.e., Stokes parameters Q, U) with a signal-to-noise ratio of 7 per resolution element (e.g. $9\arcsec $ pixel at $200\mu $m for ) when mapping 1 deg$^2$ in 2 hr assuming 5% fractional polarization is plotted as a function of wavelength for each instrument (SOFIA-HAWC$+$, , BLAST-TNG, CCAT-p, SCUBA2-POL, CSST, NIKA2-POL). For comparison, the typical surface-brightness level expected in total intensity from the diffuse outer parts of molecular clouds ($A_V = 1$) is shown for two representative dust temperatures ($T_d = 10\,$K and $T_d = 14\,$K, assuming simple modified blackbody emission with a dust emissivity index $\beta = 2$), as well as the SED of the halo of the nearby galaxy M82 [cf. @Galliano+2008; @Roussel+2010]. []{data-label="spica_sensitivity"}](spica_sensitivity.pdf){width="\columnwidth"}
More specifically, BLAST-Pol, the Balloon-borne Large Aperture Submillimeter Telescope for Polarimetry operating at 250, 350, and 500 $\mu$m [@Fissel+2010; @Fissel+2016], has only modest resolution (30–1), sensitivity, and dynamic range. HAWC$+$, the far-infrared camera and polarimeter for SOFIA [@Dowell+2010]; and BLAST-TNG [cf. @Dober+2014], both benefit from a larger 2.5-m primary mirror equivalent to that of SPICA and thus have comparable angular resolution, but are not cooled and therefore are two to three orders of magnitude less sensitive (Noise Equivalent Power ${\rm NEP} > 10^{-16}\, {\rm W\, Hz}^{-1/2}$) than . Stated another way, the mapping speed[^6] of will be four to five orders of magnitude higher than that of HAWC$+$ or BLAST-TNG. Future ground-based submillimeter telescopes on high, dry sites such as CCAT-p (the Cerro Chajnantor Atacama Telescope, prime) and CSST (the Chajnantor Submillimeter Survey Telescope) will benefit from larger aperture sizes (6$\,$m and 30$\,$m, respectively) and will thus achieve higher angular resolution than SPICA at 350$\, \mu$m, but will be limited in sensitivity by the atmospheric background load on the detectors and in spatial dynamic range by the need to remove atmospheric fluctuations. The performance and advantage of over other instruments for wide-field dust polarimetric imaging are illustrated in Fig. \[spica\_sensitivity\].
Dust polarimetric imaging with ALMA at $\lambda \sim \,$0.8–3$\,$mm will provide excellent sensitivity and resolution, but only on small angular scales (from $\sim \,$0.02$\,$ to $\sim 20\arcsec $). Indeed, even with additional observations with ACA (the ALMA Compact Array), the maximum angular scale recoverable by the ALMA interferometer remains smaller than $\sim \,$1$\,$ in total intensity and $\sim \, 20\arcsec $ in polarized emission (see ALMA technical handbook)[^7]. This implies that ALMA polarimetry is intrinsically insensitive to all angular scales $> 20\arcsec $, corresponding to structures larger than 0.015–0.05$\,$pc in nearby clouds. Using multi-configuration imaging, ALMA can achieve a spatial dynamic range of $\sim 1000$, comparable to that of [*Herschel*]{} or , but only for relatively high surface brightness emission. Because ALMA can only image the sky at high resolution, it is indeed $\sim \,$2–3 orders of magnitude less sensitive to low surface brightness emission than a cooled single-dish space telescope such as SPICA. Expressed in terms of column density, this means that ALMA can only produce polarized dust continuum images of compact objects with $N_{H_2} \simgt 10^{23}\, {\rm cm}^{-2}$ (such as substructure in distant, massive supercritical filaments – see [@Beuther+2018]) at significantly higher resolution ($\sim 1$$\,$ or better) than SPICA, while polarimetric imaging of extended, low column density structures down to $N_{H_2} \simgt 5 \times 10^{20}\, {\rm cm}^{-2}$ (such as subcritical filaments and striations) will be possible with . Furthermore, the small size of the primary beam ($\sim \,$0.3-1$\,$at 0.8–3$\,$mm) makes mosaicing of wide ($> 1\, {\rm deg}^2$) fields impractical and prohibitive with ALMA. In practice, ALMA polarimetric studies of star-forming molecular clouds will provide invaluable insight into the role of magnetic fields within individual protostellar cores/disks and will be very complementary to, but will not compete with, the observations discussed here which target the role of magnetic fields in the formation and evolution of filaments on larger scales.
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The turbulent magnetized interstellar medium {#sec:turbulence}
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Magnetic fields and turbulence are central to the dynamics and energetics of gas in galactic disks, but also in their halos and possibly in the cosmic web at large. These two intertwined actors of cosmic evolution coupled to gravity drive the formation of coherent structures from the warm and hot tenuous gas phases to the onset of star formation in molecular clouds.
Reaching a statistical description of turbulence in the magnetized ISM is an outstanding challenge, because its extreme characteristics may not be reproduced in laboratory experiments nor in numerical simulations. This challenge is of fundamental importance to Astrophysics, in particular to understand how galaxies and stars form, as well as the chemical evolution of matter in space. This section summarizes the contribution we expect to bring to this ambitious endeavor.
Interstellar magnetic fields {#subsec:turbulence_polarization}
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Magnetic fields pervade the multi-phase ISM of galaxies. In the Milky Way, and more generally in local universe galaxies, the ordered (mean) and turbulent (random) components of interstellar magnetic fields are comparable and in near equipartition with turbulent kinetic energy [@Heiles05; @Beck2015]. The galactic dynamo amplification is saturated, but exchanges between gas kinetic and magnetic energy still occur and are of major importance to gas dynamics. Magnetic fields are involved in the driving of turbulence and in the turbulent energy cascade [@Subramanian08]. The two facets of interstellar turbulence: gas kinematics and magnetic fields, are dynamically so intertwined that they may not be studied independently of each other.
The multiphase magnetized ISM is far too complex to be described by an analytic theory. Our understanding in this research field follows from observations, MHD simulations and phenomenological models. MHD simulations allow us to quantify the non-linear ISM physics but within numerical constraints that limit their scope. They may guide the interpretation observations but alone they do not provide conclusive answers because they are very far from reproducing the high Reynolds ($R_e$) and magnetic Prandtl numbers ($P_m$)[^8] of interstellar turbulence [@Kritsuk11]. The fluctuation dynamo and shock waves contribute to produce highly intermittent magnetic fields where the field strength is enhanced in localized magnetic structures. The volume filling factor of these structures decreseases for increasing values of the magnetic Reynolds number $R_m = R_e\times P_m$ [@Schekochihin02; @Brandenburg05]. The inhomegeneity in the degree of magnetization of matter associated with intermittency is an essential facet of interstellar turbulence [@Falgarone15; @Nixon19], which simulations miss because they are far from reproducing the interstellar values of $R_m$. In this context, to make headway, we must follow an empirical approach where a statistical model of interstellar turbulence is inferred from observations.
The promise of {#subsec:turbulence_promises}
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will image dust polarization with an unprecedented combination of sensitivity and angular resolution, providing a unique data set (Sect. \[subsec:spica-adv\]) to characterize the magnetic facet of interstellar turbulence. This leap forward will open an immense discovery space, which will revolutionize our understanding of interstellar magnetic fields, and their correlation with matter and gas kinematics.
At $200\,\mu$m, for the SED of the diffuse ISM [@planck2013-XVII], the surface brightness sensitivity of is three orders of magnitude greater than that of the [*Planck*]{} $353\,$ GHz all-sky map for deep imaging (10 hr per square degree) and a few hundred times better for faster mapping (2 hr per square degree). The analysis of dust polarization at high Galactic latitude with the [*Planck*]{} data is limited by sensitivity to an effective angular resolution of $\sim \,1^\circ$ in the diffuse ISM and $10\arcmin$ in molecular clouds where the column density is larger than $10^{22}\,$Hcm$^{-2}$ [@PlanckXII2018], while will map dust polarization with a factor $\sim \,$20–70 better resolution at 100–350$\, \mu$m.
Dust polarization probes the magnetic field orientation in dust-containing regions, i.e., mostly in the cold and warm phases of the ISM, which account for the bulk of the gas mass, and hence of the dust mass. These ISM phases comprise the diffuse ISM and star-forming molecular clouds. They account for most of the gas turbulent kinetic energy in galaxies [@HennebelleFalgarone2012]. Thus, among the various means available to map the structure of interstellar magnetic fields, dust polarization is best suited to trace the dynamical coupling between magnetic fields, turbulence, and gravity in the ISM. This interplay is pivotal to ISM physics and star formation. It is also central to cosmic magnetism because it underlies dynamo processes [@Subramanian08].
Observations have so far taught us that magnetic fields are correlated with the structure of matter in both the diffuse ISM and in molecular clouds [@Clark+2014; @PlanckXXXII2016; @PlanckXXXV2016] but this correlation does not fully describe interstellar magnetism. Data must also be used to characterize the intermittent nature of interstellar magnetic fields.
The magnetic structures identified in MHD simulations (Fig. \[fig:simus\_intermittency\]) may be described as filaments, ribbons or sheets with at least one dimension commensurate with dissipation scales of turbulent and magnetic energy in shocks, current sheets or through ambipolar diffusion [@Momferratos14; @Falgarone15]. While the viscous and Ohmic dissipation scales of turbulence are too small to be resolved by , turbulence dissipation due to ion-neutral friction is expected to occur on typical scales between $\sim 0.03\,$pc and $\sim 0.3\,$pc [cf. @Momferratos14], which is well within the reach of for matter in the local ISM.
Although dust polarization does not measure the field strength, the polarization angle may be used to map these magnetic structures, as illustrated in Fig. \[fig:simus\_intermittency\]. The figure shows that the largest values of the increment of the polarization angle, $\Delta \Phi$, delineate structures that tend to follow those of intense dissipation of turbulent energy. will allow us to identify magnetic structures such as those in Fig. \[fig:simus\_intermittency\] even if their transverse size is unresolved because i) they are highly elongated and ii) their spatial distribution in the ISM is fractal.
Regions of intermittency in interstellar turbulence correspond to rare events. Their finding requires obtaining large data sets combining brightness sensitivity and angular resolution, as illustrated by the CO observations with the Institut de Radioastronomie Millimétrique (IRAM) 30m telescope analyzed by @Hily-Blant08 and @Hily-Blant09. has the unique capability to extend these pioneering studies of the intermittency of gas kinematics to dust polarization observations, tracing the structure of magnetic fields (Sect. \[subsubsec:polarimetry\]), with a comparable angular resolution. will also greatly strengthen their statistical significance by covering a total sky area more than two orders of magnitude larger.
holds promises to reveal a rich array of magnetic structures, characterizing the intermittency of interstellar magnetic fields. [*Planck*]{} data, at a much coarser scale, gives a first insight at the expected outcome of the observations illustrated in Fig. \[fig:planck\_intermittency\]. Magnetic structures will be identified in the data as locations where the probability distributions of the increments of the polarization angle, and the Stokes Q/I and U/I ratios, depart from Gaussian distributions. Compared to [*Planck*]{}, will only map a small fraction of the sky ($\sim 1 \%$ for nearby molecular clouds and diffuse ISM observed away from the Galactic plane) but it will probe the field structure on much smaller scales (by a factor $30$ or more) where the surface density of magnetic structures is expected to be much larger.
![Non-Gaussianity of the magnetic field structure in the [*Planck*]{} dust polarization data. This all-sky image, in Galactic coordinates centered on the Galactic center, presents the modulus of the angular polarization gradient, $|\nabla{\psi}|$, built from the [*Planck*]{} data at 353GHz smoothed to 160 resolution. Figure adapted from Appendix D of @PlanckXII2018. []{data-label="fig:planck_intermittency"}](gradPg-GNILC_160acm_mollview.pdf){width="8cm"}
Observing strategy {#subsec:observations}
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will considerably expand our ability to map the structure of interstellar magnetic fields. These data will be complementary to a diverse array of polarization observations of the Galaxy.
Stellar polarization surveys will be combined with [*Gaia*]{} astrometry [e.g. @Tassis+2018] to build 3D maps of the magnetic fields in the Galaxy but with a rather coarse resolution, comparable to that of the density structure of the local ISM in @Lallement18.
Synchrotron observations at radio wavelengths with the Square Kilometer Array (SKA) and its precursors will probe the structure of magnetic fields [@Dickinson15; @Haverkorn15], in particular in ionized phases through Faraday rotation [@Gaensler11; @Zaroubi15]. SKA will also provide Faraday rotation measurements toward $\sim 10^7$ extragalactic sources [@Johnston-Hollitt15], which will be available for comparison with dust polarization data as illustrated in the pioneering study of @Tahani+2018.
will allow us to study interstellar turbulence over an impressive range of physical scales and astrophysical environments from the warm ISM phases to molecular clouds. Observations of nearby galaxies are best suited to probe the driving of turbulence in relation to galaxy dynamics (spiral structure, bars, galaxy interaction, outflows) and stellar feedback as discussed in Sect. \[sec:galaxies\]. Galactic observations will probe the inertial range of turbulence over 4 to 5 orders of magnitude from the injection scales ($\sim 100\,$pc - $1\,$kpc) down to $0.01\,$pc. The smallest physical scales will be reached by observing interstellar matter nearest the Sun, away from the Galactic plane: the diffuse ISM at high Galactic latitudes and star-forming molecular clouds in the Gould Belt. These sky regions are best suited for the study of turbulence because the overlap of structures along the line of sight is minimized. The Gould Belt clouds are already part of the filament science case in Sect. \[sec:filaments\]. This survey will include star-forming clouds and diffuse clouds representative of the cold neutral medium. Deeper polarimetric imaging of high Galactic latitude fields (10 hr per deg$^2$), sampling regions of low gas column density ($\rm A_V \sim 0.1$ to 0.3), will allow us to probe turbulence in the warm ISM phases. These deep imaging observations could potentially share the same fields as those used to carry out a SPICA-SMI cosmological survey. The size of the area that may be mapped to that depth (of order $\sim 100\,$deg$^2$) will be optimized with the needs of this survey. Altogether, we estimate that will cover a total area of about 500 deg$^2$ away from the Galactic Plane including diffuse ISM fields at high Galactic latitudes, which will be available to study turbulence in diverse interstellar environments. At the angular resolution of at $200\,\mu$m, these data correspond to a total of $2 \times 10^7$ polarization measurements. This number is 20 times larger than the statistics offered by the [*Planck*]{} polarization data. The gain in angular resolution and sensitivity is so large, that will supersede [*Planck*]{} in terms of data statistics, even if the maps used cover only $\sim 1\%$ of the sky.
A wealth of spectroscopic observations of HI and molecular gas species, tracing the gas density, column density and kinematics, will become available before the launch of with SKA and its precursors [@McClure-Griffiths15], and the advent of powerful heterodyne arrays on millimeter ground-based telescopes, e.g. the Large Millimeter Telescope and the IRAM 30m telescope. Furthermore, we will be able to investigate the link between coherent magnetic structures and turbulent energy dissipation observing main ISM cooling lines from H$_2$, CII, and OI with the SPICA mid and far-IR spectrometers SMI and SAFARI. These complementary data from SPICA and ground-based observatories will be combined to characterize the turbulent magnetized ISM statistically. The data analysis will rely on on-going progress in the development of statistical methods [e.g. @Makarenko18], which we will use to characterize the structure of interstellar magnetic fields and their correlation with gas density and velocity. This process will converge toward an empirical model of interstellar turbulence, which will be related to ISM physics comparing data and MHD simulations.
Magnetic fields in protostellar dense cores {#sec:protostars}
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Current state of the art
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In molecular clouds, protostellar dense cores are the “seeds” where the gravitational force proceeds to form stars. Class 0 objects are the youngest known accreting protostars: most of their mass is still in the form of a dense core/envelope ($M_{\rm env} \gg M_\star $) and this phase is characterized by high accretion rates of gas from the dense core onto the central stellar object, accompanied by ejection of powerful highly collimated flows [@Andre00; @Dunham+2014].
How many stars can be formed out of a typical molecular cloud depends not only on the physical conditions in filamentary structures, but also on the detailed manner the gravitational collapse proceeds within individual protostellar cores (i.e. would it form a single/binary stellar system, one or several low-mass stars or high-mass stars?).
By the end of the protostellar phase, the star has gained most of its final mass: understanding the role of magnetic fields during the protostellar stage is therefore crucial to clarify how they affect some of the most remarkable features of the star formation process, such as the distribution of stellar masses, the stellar multiplicity, or the ability to host planet-forming disks [@McKee07; @LiPPVI]. The development of numerical magneto-hydrodynamical models describing the collapse of protostellar cores and the formation of low-mass stars, has opened new ways to explore in more details the physical processes responsible for the formation of solar-type stars. MHD models suggest that protostellar collapse proceeding with initially strong and well aligned magnetic field produces significantly different outcomes than hydrodynamical or weakly magnetized models [@Fiedler93; @Hennebelle08a; @Masson16]. For example, if the field is strong enough and well coupled to the core material, magnetic braking will regulate the formation of disks and multiple systems during the Class 0 phase. This has been the focus of recent studies [@Hennebelle16; @Krasnopolsky11; @Machida11] because it could potentially explain the low-end of the size distribution of protostellar disks [e.g. @Maury10; @SeguraCox18; @Maury18b].
![Composite images of the G14.225-0.506 massive star forming region [@Busquet13; @Busquet16; @Santos16]. [*Top left panel:*]{} R band optical polarization vectors (red segments) overlaid on [*Herschel*]{} 250 $\mu$m image overlapped (from Santos et al. 2016). [*Central panels:*]{} SOFIA/HAWC+ 200 $\mu$m images (beam $14\arcsec$) of the Northern (top) and Southern (bottom) hubs, with black segments showing the magnetic field direction (F. Santos, private communication). [*Right panels:*]{} Submillimeter Array (SMA) images of the 1.2 mm emission toward the center of the Northern (top) and Southern (bottom) hubs [@Busquet16], with orange segments showing the magnetic field direction (Añez et al. in preparation). []{data-label="Fig_G14"}](G14_Figure4spicapol.jpg){width="\linewidth"}
All protostellar cores are magnetized to some level and current observations suggest that at least in some cases the magnetic field at core scale is remarkably well organized, pointing toward scenarii with strong field even at the high column densities typical of protostellar cores [e.g., IRAS 4A, G31.41, G240.31, NGC 6334, L1157, B335: @Girart06; @Girart09; @Qiu14; @Li-HB15; @Stephens13; @Galametz18; @Maury18a], while in other cases (e.g., @Girart13 [@Hull17b; @Ching17], and Fig. \[Fig\_G14\] for an example in the G14 massive star forming region) the core-scale magnetic field shows very complex morphology. In Fig. \[Fig\_G14\], for instance, it is noteworthy that the northern hub of the G14 region, with a more uniform magnetic field, has a lower level of fragmentation than the southern hub (that shows a more perturbed magnetic field). These observations suggest the field may remain organized at scales where collapse occurs in most solar-type progenitors, and also at least some of the massive cores [@Zhang14; @Beuther+2018]. Current results may be biased, however, because present single-dish facilities selectively trace magnetic fields from the brightest regions within star-forming cores (dust polarization is only detected at the highest column densities, see Fig. \[Fig\_G14\]).
Role of magnetic fields in controlling the typical outcome of protostellar collapse
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can perform statistical studies in unprecedentedly large samples of protostellar cores, testing for example whether the magnetic field in cores is directly inherited from their environment (if the magnetic field in low-density filamentary structures, see § \[subsec:filaments\], connects to the magnetic field in high-density cores, which is expected in the strong field case), or if the field in cores is disconnected from the local field in the progenitor cloud (weak field case). While the ALMA interferometer can only provide constraints on the magnetic field topology at the smallest scales ($\sim \,$0.02$\,$ to $\sim \,$20, i.e. $<5000$ au in Gould Belt clouds), the polarization capabilities of other facilities probing larger spatial scales (SMA, NIKA2, and HAWC+) are severely sensitivity limited. Accordingly, studies linking cores and filaments can currently be carried out in bright, massive star-forming regions mostly (see Fig. \[Fig\_G14\]), and only in a handful of nearby solar-type protostellar cores.
![Potential role of the magnetic field topology at core scales in the formation of disks and multiple systems. [*Top:*]{} Magnetic field (red/orange line segments, from dust polarization observations with the SMA at 850$\,\mu$m) in two solar-type Class 0 protostellar cores [@Galametz18]. The blue arrows indicate the jet/rotation axis of these cores, aligned with the core-scale magnetic field in L1157 (left), and mostly orthogonal to it in L1448N (right). [*Bottom:*]{} Level of core rotation (from kinematic observations at core scales, @Yen15a and Gaudel et al. in prep) as a function of the misalignment between the rotation axis and the magnetic field (observed at core scale with the SMA – @Galametz18) in a sample of nearby Class 0 sources. There is a hint that large misalignments of the magnetic field at core scales lead to sources with large rotational gradients and multiple systems at smaller scales (red symbols). []{data-label="Fig_SMA"}](StokesI_Borientation-models-3.jpg){width="0.9\linewidth"}
Some indications have been found, in small samples ($<20$ objects), that the topology of the magnetic field at core scales may be linked to the distribution of angular momentum in solar-type cores, and hence that the magnetic field may be of paramount importance to set the initial conditions for the formation of protoplanetary disks and multiple systems (see Fig. \[Fig\_SMA\] and @Galametz18 [@SeguraCox18]). An example of the type of studies that could extend to the full mass function of protostellar cores in a statistical fashion is shown in Fig. \[Fig\_G14\] and Fig. \[Fig\_SMA\]: these two figures illustrate the tentative link between the magnetic field topology at core scale and the disks and multiplicity fraction found [*within*]{} protostellar cores at smaller scale. The magnetic field properties found with at dense core scales can be compared with the protostellar properties observed with interferometers at smaller scales to build correlation diagrams similar to the one shown in Fig. \[Fig\_SMA\]. In this way, observations can test the hypothesis, tentatively suggested by current studies of the brightest protostars, that magnetic fields regulate the formation of disks and multiple systems during the main accretion phase. Observations of large samples of protostars could be carried out thanks to the sensitivity and spatial resolution of , which is crucially needed not only to populate diagrams such as Fig. \[Fig\_SMA\], but also because only statistics will allow us to solve the degeneracy induced by projection effects intrinsically linked to dust polarization (tracing only the magnetic field component in the plane of the sky). Moreover, will provide information on the geometry of magnetic field lines across the full protostellar core mass distribution, probing different behaviors in different mass regimes, and potentially as a function of environment in different star-forming regions. The angular resolution and surface brightness dynamic range of will make it possible to resolve most $\sim \,$2000–20000 au protostellar cores in nearby star-forming regions out to 250 pc. A wide-field survey of all nearby clouds as envisaged in § \[subsec:filaments\] ($\sim 2\,$hr per square degree) will map dust polarization (fraction $>1\%$) at core scales with signal-to-noise ratio $>7$, in complete populations of $\simgt$1000 protostars (Class 0 and Class I) and their parent cloud/environment, from massive protostellar cores down to the low-mass progenitors of solar-type stars. In contrast, current millimeter/submillimeter polarimetric instruments, such as SCUBA2-POL, NIKA2-POL, SOFIA/HAWC$+$, or BLAST-TNG (cf. § \[subsec:spica-adv\]) are limited to the subset of the $\sim$50-100 brightest cores, and without the important context provided by the magnetic field information in the parental clouds.
Role of magnetic fields in high-mass star and cluster formation {#sec:massive-sf}
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As mentioned in § \[subsec:fil-paradigm\], supercritical molecular filaments are believed to be the preferred birthplaces of solar-type stars. It is however unclear whether the filamentary paradigm – or an extension of it, based on unusually high line masses or levels of turbulence [e.g. @Roy+2015] – also applies to high-mass star formation and can lead to the quasi-static formation and monolithic collapse of massive prestellar cores, then forming high-mass protostars. The most recent observational results, partly obtained with [*Herschel*]{}, suggest that high-mass stars and stellar clusters form in denser, more dynamical filamentary structures called ridges[^9] which exceed the critical line mass of an isothermal filament by up to $\sim $ two orders of magnitude (see, e.g., the review by [@Motte+2018a]). Such massive structures may originate from highly dynamical events at large scales like converging flows and cloud-cloud collisions, continuing on median scales through the global collapse and filament feeding of ridges. The role of magnetic fields in this scheme is poorly known and may be as crucial as for low-mass star formation.
In the hypothesis of large-scale cloud collapse, gravity overcomes the magnetic field support and the magnetic field follows the infall gas streams from cloud scales ($\sim$100 pc) to accumulation points at scales between $\sim$1 pc to $\sim$0.1 pc, with a typical hourglass geometry toward these accumulation points [@Girart09; @Cortes+2016]. Large-scale collapse leads to very dense, massive structures at pc scales, which are either spherical (hubs) or elongated (ridges) [@HartmannBurkert2007; @Schneider+2010; @Hill+2011; @Peretto+2013]. Pilot works with ground-based facilities (SMA) toward the DR21 ridge in Cygnus X [@Zhang14; @Ching17] show that the magnetic field is ordered at the scale of the ridge and mostly perpendicular to its main axis, as for low-mass supercritical filaments (cf. § \[subsec:fil-paradigm\] and Fig. \[taurus\_planck\]), suggesting mass accumulation along field lines. However, while large-scale collapse and strong ordered magnetic fields are probably a key ingredient, the detailed physical processes at the origin of ridges remain, for now, a mystery. At some point, ridges fragment into hundreds of protostellar cores in local, short, but violent bursts of star formation, leading to exceptionally large instantaneous star formation rates [@NguyenLuong+2011; @Louvet+2014]. This clustered mode of fragmentation in ridges may differ in nature from the filamentary mode of fragmentation leading to low-mass star formation at significantly lower average densities. As a matter of fact, top-heavy core mass functions, overpopulated with high-mass protostellar cores, begin to be found with ALMA in the massive, young ridges of the Galactic plane [@Csengeri+2017; @Motte+2018b; @YCheng+2018]. The magnetic-field configuration (field topology and field strength) inside the hubs and ridges, at scales of a few 0.1 pc, may limit fragmentation (see [@Commercon+2011] for MHD simulations and Fig. \[Fig\_G14\] for recent observations) and favor the formation of massive protostellar cores against their low-mass counterparts. Dynamical processes associated with local accretion streams and global collapse may also favor the growth of high-mass protostellar cores due to competitive accretion [@Smith+2009]. Elucidating the relative roles of – and coupling between – magnetic fields and dynamics, is therefore of crucial importance for understanding the origin of high-mass stars and their associated clusters. For example, if cluster-scale global collapse is required to form massive stars near the bottom of the gravitational potential well, the collapsing flow should drag the magnetic field on the cluster scale into a more or less radial configuration. If, on the other hand, only localized collapse of a pre-existing massive core is required to produce a massive star [@McKee03], the collapse-induced field distortion is expected to be limited to the smaller, core region.
Observationally, this requires probing the magnetic field configuration from cloud scales ($\sim$100 pc) to protostellar scales ($\sim$0.01 pc) within massive dense ridges/hubs (down to so-called “massive dense cores” or MDCs; 0.1–0.3pc; [@Motte07; @Bontemps+2010b]). Ultimately, a spatial dynamic range[^10] as high as $10^4$ (from 100 pc to 0.01 pc) is thus needed. In nearby high-mass star-forming regions, located at $\sim 1$ to 3 kpc, this translates to angular scales from a few ($\sim \,$2–6) degrees down to $\sim 0.1\arcsec$. While the $0.1\arcsec$ scale of individual pre-/protostellar cores in these regions is only reachable with SMA or ALMA, the inner scale of massive dense ridges/hubs or MDCs ($\sim 0.2\,$pc, or $\sim14\arcsec-40\arcsec$ at $\sim$1–3 kpc distance) can be reached with the angular resolution of . This MDC scale is of particular importance for high-mass star and cluster formation since high-mass protostellar cores appear to form in only a subset of MDCs, possibly those with a high level of magnetization (see, e.g., [@Motte+2018a; @TCChing+2018] and Fig. \[Fig\_G14\]). With a typical spatial resolution of $10\arcmin $, [*Planck*]{} polarization data already provide some indications on the magnetic-field geometry at scales between 100 pc and $\sim \, $3–9$\,$pc, but [*Planck*]{} maps are strongly limited by the confusion arising from several layers of dust emission along the line of sight within the large beam. The spatial resolution of is required to separate the contributions of these layers and focus on the polarization signal from high-mass star-forming ridges, hubs, and MDCs. The high sensitivity of will also be crucial to trace the magnetic field topology all the way to the outer environment of ridges and hubs, where the column density of dust reaches values below $A_V \sim 1$–2.
Magnetic fields in galaxies {#sec:galaxies}
===========================
Magnetic fields are an important agent that influences the structure and evolution of galaxies [e.g. @Tabatabaei+16]. The magnetic pressure in the ISM is comparable in magnitude to the thermal, turbulent, and cosmic ray pressures [e.g. @Ferriere01; @BoularesCox1990; @Beck2007], so the magnetic field contributes significantly to the total pressure which supports a galactic gas disk against gravity. The interplay between the magnetic field, gravity, and turbulence is central to the process of star formation [see, e.g., @McKee07; @HennebelleFalgarone2012; @Crutcher2012], both on the scale of individual stars and filaments (cf. § \[sec:filaments\] and § \[sec:protostars\]), and for the formation of molecular clouds out of the magnetized diffuse interstellar gas [e.g. @Kortgen+2018]. On even larger scales in galaxies, magnetic fields control the density and distribution of cosmic rays [e.g. @KoteraOlinto2011], mediate the spiral arm shock strength [e.g. @ShettyOstriker2006], and may even modulate rotation of galaxy gas disks [e.g. @Elstner+2014] and quench high-mass star formation [e.g. @Tabatabaei+18]. Magnetic fields play an important role in launching galactic winds and outflows [@Heesen+2011], regulate gas kinematics at the disk-halo interface [@HenriksenIrwin2016], and ultimately connect galaxies to the intergalactic medium [@Bernet+2013].
Current observational status
----------------------------
Significant progress has been made in recent decades to characterize the interstellar magnetic fields of external galaxies, with measurements of the magnetic field strength and orientation obtained for about one hundred nearby galaxies [see the appendix of @BeckWielebinski2013][^11]. This effort has established a broadly consistent picture of the large-scale ($>1$kpc) properties of galactic magnetic fields. As in the Milky Way, the interstellar magnetic field in external galaxies can be described as a combination of large-scale regular fields and small-scale turbulent fields. Observations of face-on spiral galaxies demonstrate that galaxies typically host spiral fields in the disk, with the observed large-scale magnetic field orientations appearing similar to the material spiral arms. The vertical structure of the field is more easily probed via observations of edge-on systems, which typically show an X-shaped structure such that the field tends to become more inclined (and eventually perpendicular) with increasing distance from the midplane. Field strengths vary, but are usually in the range of several to tens of $\mu$G, with roughly similar contributions from ordered and random field components [@BeckWielebinski2013].\
To date, magnetic field properties in external galaxies have mostly been investigated via observations of synchrotron emission at GHz frequencies [for a review, see @BeckWielebinski2013; @Beck2015]. At the same frequencies, Faraday rotation of background radio sources provides an alternative, direct determination of the direction and strength of magnetic field within galaxies. This technique has been used for detailed studies of the Milky Way’s magnetic field [e.g. @TerralFerriere17; @Mao+10], but its application to external galaxies has thus far been limited to the most nearby galaxies [which have a large angular size and thus a sufficient number of bright background sources, e.g. M31, LMC, @Han+98; @Gaensler+05]. Faraday rotation measurements for a much larger sample of nearby galaxies is an important science driver for SKA [see e.g. @Beck+15].\
At other wavelengths, the magnetic field structure of a much smaller number of external galaxies has been surveyed using optical polarization e.g. the Magellanic Clouds (LMC and SMC; Mathewson & Ford 1970), NGC1068, and M51 [@Scarrott+1987]. Wide-field imaging of polarized extinction at infrared (IR) wavelengths is a newer capability that has been used to probe the magnetic field structure in nearby edge-on galaxies, with results that are generally in good agreement with radio observations [e.g. @Clemens+2013; @MontgomeryClemens2014]. The IR extinction technique is less suited to observations of face-on galaxies, due to the relatively short path length that can produce internal extinction [see e.g. @PavelClemens2012 for the case of M51].\
will probe the magnetic field structure via observations of dust polarization in emission. An important advantage of this technique compared to radio synchrotron observations is that it traces the magnetic field structure in the cold gas where star formation occurs, with minimal contamination from the warm ionized gas in the halo [e.g. @Mao+2015]. Studies in nearby galaxies further suggest that the magnetic field is coupled to the interstellar gas independently of the star formation rate [e.g. @Schinnerer+13; @Tabatabaei+18], highlighting the importance of tracing the field in the neutral ISM. The SCUBA-POL camera on the James Clerk Maxwell Telescope (JCMT), operating at 850$\mu$m, obtained the first such dust polarization observations of external galaxies [e.g. @Matthews+2009], but was only able to access extremely bright extragalactic regions, such as the centre of the nearby starburst system M82 [e.g. @Greaves+2000]. The [*Planck*]{} mission has recently provided all-sky measurements of the polarized submillimeter dust emission, but with an angular resolution ($\sim\,$10 at 353GHz) sufficient to resolve sub-kiloparsec scales only in the closest Local Group galaxies ($<1$Mpc). With $\sim$arcsecond resolution, a key opportunity for ALMA will be targeted imaging of the detailed magnetic field structure in extragalactic molecular clouds, but wide-field polarization surveys of nearby galaxies will remain impractical, due to prohibitive integration times for fields larger than a few square arcminutes.
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Key opportunities for on nearby galaxies
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Mapping the structure of interstellar magnetic fields in the cold ISM of nearby galaxies is crucial to understand how magnetic fields influence gas dynamics in galaxies, and in particular the role of the field in regulating star formation, driving galactic outflows, and fuelling galactic nuclei. Observational studies of these processes in nearby galaxies complement Milky Way studies, which typically have superior spatial resolution, but may be limited by distance ambiguity and line-of-sight confusion. Among current and near-future facilities, only will be able to make these measurements across a representative sample of external galaxies, probing a much wider range of ISM conditions than those encountered in the Milky Way, and to conduct spatially complete mapping of the field structure in Local Group targets.\
To highlight ’s unique capabilities for such an effort, Fig. \[fig:ng\_polsnr\] shows the estimated signal-to-noise ratio in polarized intensity for several iconic nearby galaxies at $100\,\mu$m (top row) and $200\,\mu$m (bottom row) after 2 hr on-source integration with . These maps are constructed assuming the performance parameters given in Table \[tab:POL\] and a conservative polarization fraction of 1%. At the intrinsic resolution of the 100$\mu$m band, the sensitivity is sufficient for tracing the detailed polarization structure of bright features such as spiral arms, bar dust lanes, and galaxy centers. For the galaxies in Fig. \[fig:ng\_polsnr\], several hundred independent 100$\mu$m polarization vectors would be obtained. Measurements in the more sensitive 200$\mu$m and 350$\mu$m bands would essentially cover the entire galactic disk within $0.6\,R_{25}$ (where $R_{25}$ is the optical radius). The excellent signal-to-noise ratio that can be achieved with a modest integration time per galaxy means that could conduct the first systematic survey of the polarized far-IR dust emission – and hence magnetic field structure – in $\sim100$ nearby galaxies. This is slightly larger than the combined sample of galaxies targeted by the VNGS and KINGFISH [*Herschel*]{} nearby galaxy projects, and would require only $200$hr of on-source observing time. In the remainder of this section, we highlight some of the potential science drivers for such a nearby galaxy survey.
### Testing and refining Galactic dynamo models
The currently favored paradigm for interstellar magnetic fields is that they are amplified by dynamo action. In this scenario, weak primordial fields in young galaxies are quickly amplified by a small-scale turbulent dynamo, which continuously supplies turbulent fields to the ISM after the formation of a galactic disk in $\lesssim10^{9}$yr [@Schleicher+2010]. The large-scale field is then amplified by the mean-field $\alpha - \Omega$ dynamo effect [e.g. @Ruzmaikin+88], whereby the combination of differential rotation of the galactic disk ($\Omega$ - effect) and helical turbulence ($\alpha$ - effect) presumably driven by supernova explosions [@FerriereSchmitt2000], produce small-scale turbulent and organize some fraction of them into regular large-scale patterns.\
The mean field dynamo is expected to generate a regular magnetic field with both poloidal and azimuthal components, and nearly all polarized synchrotron observations of face-on disk galaxies show a large-scale spiral pattern. To date, the magnetic field pitch angles $p_{B}$ and azimuthal structure that have been observed in nearby disk galaxies via observations of polarized radio synchrotron emission are broadly compatible with the predictions of mean field dynamo theory [@Fletcher2010; @vanEck+15]. Yet the precise nature of the magnetic field generated via the mean-field dynamo depends on properties of the host galaxy. For example, the rotation curve determines the shear strength in a differentially rotating galaxy disk, and hence how the azimuthal field component is generated from the poloidal field. The $\alpha$-effect – by which a poloidal field component is generated from the azimuthal field – is thought to be powered by supernova explosions, which depend on a galaxy’s star formation rate. As our knowledge of external galaxies grows, the logical next step is to refine dynamo models for specific galaxies to include all relevant observed galaxy properties – e.g. the ionized and molecular gas density distributions, rotation curve, star formation rate, gas inflow and outflow rates – and test the model predictions for individual galaxies against the observed properties of the magnetic field. The first attempt to do this systematically for a sample of galaxies [@vanEck+15] was hampered by inconsistencies in the available radio observations. A sample of galaxies observed with the same instrument at the same resolution and sensitivity is necessary to allow the details of dynamo theory, such as how the dynamo saturates, to be tested against data. ’s moderate resolution, full-disk sampling of the magnetic field structure across a statistically significant sample of nearby galaxies would provide precisely this test.
### Magnetic fields and gas flows in barred and spiral galaxies
Mapping the structure of the magnetic field across a sample of nearby galaxies is needed to understand the typical dynamical importance of the field on galactic scales. Of particular interest is how gas flows in galaxies – e.g. gas streaming along spiral arms, inflow along bar dust lanes, and starburst-driven outflows – interact with the field. While independent estimates of the field strength will still be required, observations at sub-kiloparsec resolution of the magnetic field structure and complementary spectral line data for tracing interstellar gas kinematics will be extremely valuable for studying the interplay between the field and motions within the cold gas reservoir across the local galaxy population.\
In face-on disk galaxies, the large-scale field traced by radio polarization observations tends to follow a spiral pattern. This pattern is expected from mean field dynamo theory, and is not directly connected to a galaxy’s baryonic (i.e. gas/stellar) spiral structure. Observations of spiral galaxies indeed show that the field pattern is not always spatially coincident with the spiral arms, and in several cases [most famously NGC6496 @Beck2007] the ordered field pattern is most pronounced in the interarm region. Some of the large-scale field patterns in galaxies may be due to the combined action of shear and compression in the interstellar gas, which renders the turbulent field anisotropic (and hence ordered) over large scales [e.g. in M51, @Fletcher+2011; @Mulcahy+14; @Mulcahy+16]. Current observations also suggest that the average pitch angle of the regular spiral field pattern is often similar to the pitch angle of the local spiral arm $p_{M}$. This is not a direct prediction of mean-field dynamo theory, but would be expected if spiral shocks amplify the magnetic field component parallel to the shock. Significant discrepancies between $p_{B}$ and $p_{M}$ in the inter-arm region, as well as large azimuthal and radial variations in $p_{B}$, are also observed, the origin of which are not yet well understood.\
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The central regions of barred galaxies are the site of fast radial gas inflow, strong shocks, and intense star formation. Barred galaxies often show strong gas streaming along the shock fronts at the edge of bars, which develop because the gas is rotating faster than the bar pattern. Radio polarization observations of the prototypical barred galaxy NGC1097 [@Beck+2005] reveal strongly polarized emission along the bar with field orientations parallel to the gas streamlines. The observed polarization pattern suggests that the field is amplified and stretched by shear in the compression region, and that the field is frozen into the gas and aligned with the gas flow over a large part of the bar. If this result holds generally in barred galaxies, the polarization pattern in bars – especially using a tracer that preferentially probes the dense interstellar gas – would provide important complementary information on the plane-of-sky gas flows to the line-of-sight kinematic information obtained from molecular emission lines. In combination with estimates for the magnetic field strength, information about the magnetic field structure in the central regions of barred galaxies would also provide useful constraints for models of AGN fuelling. One of the main problems in this area is to generate mass inflow rates that are compatible with the observed nuclear activity. Magnetic stress in circumnuclear rings [e.g. @Beck+1999] and fast MHD density waves [e.g. @Lou+2001] have been proposed as potential mechanisms, but current observational data for the field strength and structure in the inner regions of barred galaxies is not sufficient for a rigorous test of these models.
### Magnetism in dwarf galaxies
Due to their slow rotation, the amplification of magnetic fields should be less efficient in dwarf galaxies. Yet observations of radio polarized intensity show that several nearby low-mass galaxies host large-scale ordered fields, e.g. the Magellanic Clouds, NGC4449 and IC10 [@Chyzy+2003; @Gaensler+05; @Mao+12; @Chyzy+16; @Heesen+18]. Dwarf galaxies are also more likely to exhibit star formation powered outflows and galactic winds, due to their shallow gravitational potential. The magnetized nature of these outflows has been observed in some dwarf systems [@Chyzy+2000; @Kepley+2010], consistent with some models of a cosmic ray driven dynamo [@Siejkowski+2014; @DuboisTeyssier2010]. To date, all dwarf galaxies with detected ordered magnetic fields are star-bursting, participating in a galaxy-galaxy interaction, and/or experiencing significant gas infall, suggesting the importance of enhanced turbulence for the magnetic field properties and evolution of these systems. observations of a sample of local dwarf galaxies with a range of masses, interaction properties and star formation histories, would provide valuable input for theories for the amplification of magnetic fields in such systems, and their role in magnetizing the intergalactic medium (IGM).
### The Magellanic Clouds
The Large and the Small Magellanic Cloud (LMC, SMC) are the closest gas-rich galaxies to the Milky Way. A survey of the Magellanic Clouds would for the first time probe the magnetic field structure in the cold ISM across all spatial scales between the clumpy sub-structure within GMCs ($\sim2$pc) and the galactic disk (several kpc). Observations across such a large range of spatial scales are needed to decipher the dynamical importance of the magnetic field for the inherently hierarchical process of star formation, i.e. from the formation of GMCs out of the diffuse ISM, down to the formation of individual stars. Spatially complete surveys of dust emission in the Magellanic Clouds with ALMA are unfeasible due to their large angular size ($\sim50$ and $\sim10$ deg$^2$ for the LMC and SMC respectively).\
As an example of what could be achieved with , Fig. \[fig:mc\_polsnr\] shows the estimated signal-to-noise ratio for a 50hr polarimetric imaging survey of the LMC at $100\, \mu$m, $200\,\mu$m, and $350\, \mu$m. This hypothetical survey would achieve a signal-to-noise ratio of $3$ for the polarized intensity at 100$\,\mu$m for interstellar gas with column densities above $\sim2.5 \times 10^{21}$cm$^{-2}$ [equivalent to $A_{V}\sim0.4$ in the LMC– @WeingartnerDraine2001]. This sensitivity would yield $\sim$0.5 million independent measurements of the magnetic field orientation in the interstellar gas on spatial scales of 2pc, including in the column density regime of the atomic-to-molecular phase transition. At 200$\mu$m and 350$\mu$m, a similar number of significant detections of the magnetic field orientation would be achieved in even more diffuse gas ($\sim1 \times 10^{21}$cm$^{-2} \approx 0.15$mag). This represents $\sim$ two orders of magnitude increase in detail over measurements with [*Planck*]{}’s 353GHz channel in the Magellanic Clouds, and would provide the first spatially complete view of the parsec-scale magnetic field structure in the molecular gas reservoir of any galaxy.
### Wavelength dependence of polarization in U/LIRGs and AGNs
The polarization of luminous external galaxies such as Luminous Infrared Galaxies (LIRGs), Ultraluminous Infrared Galaxies (ULIRGs) and active galactic nuclei (AGNs), in the far-IR and submillimeter can arise from synchrotron emission but also from emission or absorption by aligned dust grains in the optically thick clouds that surround young stars and AGN tori [e.g. @Efstathiou+97; @Aitken+02]. Information in the far-IR and submillimeter can be combined with information at 10$\mu$m and 18$\mu$m as well as near-IR data from the ground to study the switch in position angle by about 90 deg that is predicted as polarization changes from dichroic absorption at shorter wavelengths to dichroic emission at longer ones. Several highly polarized galaxies in the mid-IR were found by @SiebenmorgenEfstathiou01 with Infrared Space Observatory (ISO) and more recently by @LopezRodriguez+18a with CanariCam on the 10.4-m Gran Telescopio Canarias (GTC).\
This is a science area where significant progress can be achieved with , which will provide sensitive polarization measurements at 100–350$\mu$m for a large sample of luminous external galaxies. Such information is currently available for very few objects. In a recent study of the nearby radio galaxy Cygnus A using data from HAWC+ onboard SOFIA, @LopezRodriguez+18b showed that this approach can be very useful for unravelling the polarization mechanisms in the infrared and submillimeter and providing an independent method of estimating the contributions of AGN tori and starbursts to the SEDs. Exploring the role of AGNs and star formation in galaxies is a scientific objective of wide interest. The opportunity to study multi-wavelength polarization with will be complementary to other methods such as spectroscopy [e.g. @GonzalezAlfonso+17] and traditional SED fitting of the total emission [e.g. @Gruppioni+17].
Distant galaxies and the potential detection of the Cosmic Infrared Background polarization {#subsec:cib}
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The build-up of coherent magnetic fields in galaxies and their persistence along cosmic evolution is being investigated with analytical models of the galactic dynamo [e.g. @Rodrigues19] and numerical simulations of galaxy formation [e.g. @Martin-Alvarez18]. These studies suggest that the mean-field dynamo is effective early in the evolution of galaxies but, today, polarization data available to trace the redshift evolution of galactic magnetic fields are very scarce [@Mao17]. While SKA holds exciting promises to extend observations of cosmic magnetism to the distant universe [e.g. @Basu+2018; @Mao18], we argue here that can also uniquely contribute providing the first polarimetric extragalactic survey at far-IR wavelengths.
To quantify what could be achieved with , we consider the point source sensitivity of a polarimetric extragalactic survey for an integration time of 10 hr per deg$^2$ (Table \[tab:POL\]). At the detection limit of [*Herschel*]{} imaging surveys in total intensity, the signal-to-noise ratio of in Stokes Q and U is $\sim 200$ at $200\, \mu$m, and $\sim 100$ at $100\, \mu$m and $350\, \mu$m. This sensitivity needs to be compared to the few existing values of the far-IR polarization fraction for galaxies as a whole.
The net polarization fraction resulting from the integrated emission of galaxies depends on the existence of a coherent mean magnetic field and on viewing angle. In disk galaxies, the polarization angle is aligned with the projection of the galaxy angular momentum vector on the plane-of-the-sky, and the polarization fraction increases from a face-on to an edge-on view. Integrating the [*Planck*]{} dust polarization maps at 353GHz over a $20^\circ$ wide band centered on the Galactic plane, @deZotti18 found a polarization fraction $p= 2.7$%. Within a simple model, $p$ is expected to scale as sin$\,i$, where $i$ is the inclination angle of the galaxy axis to the line of sight. For this scaling, the mean $p$ fraction averaged over inclinations is $1.4\%$. This may be taken as a reference value for spiral galaxies like the Milky Way, but $p$ is likely to be on average lower for distant infrared galaxies. Indeed, SOFIA polarization imaging of the two template starburst galaxies, M82 and NGC253, revealed regions with different polarization orientations, which tend to average out when computing the integrated polarized emission yielding an overall mean $p \sim 0.1$% [@Jones19].
Even if only detects polarized emission from only a small fraction of [*Herschel*]{} galaxies, the number of detections will be significant given the present dearth of such measurements. If the detections are numerous, the emission from galaxies could even limit the polarization sensitivity of deep surveys. This is a possibility that needs to be assessed. Beyond the study of individual galaxies, we anticipate that the main outcome of a deep polarimetric extragalactic survey with could follow from a statistical analysis of the data.
Statistical analysis is the reference in cosmology and much can be learned without detecting sources individually. In particular, the cross-correlation of surveys across the electromagnetic spectrum is a powerful means commonly used. In the far-IR, this is illustrated by the results obtained stacking [*Herschel*]{} data on positions of extragalactic sources in the near- and mid-IR. This approach has been successfully used to statistically identify sources accounting for the the bulk of Cosmic Infrared Background (CIB), although they were too faint to be detected individually [@Bethermin12; @Viero13_stacking]. , which will extend these studies to polarization, is uniquely suited to detect – or set tight constraints on – the CIB polarization. We note that the analysis of point sources circumvents the difficulty of separating the CIB from the foreground polarized emission of the diffuse Galactic ISM.
For polarization, data stacking needs to be oriented to align polarization vectors. This can be achieved by using, e.g., galaxy shapes measured from near-IR surveys at the appropriate angular resolution. Another interesting path will be the study of correlations between extragalactic survey data and maps of the cosmic web inferred from weak-lensing surveys. The angular momenta of galaxies are not randomly oriented on the sky. The cosmic web environment has a strong influence on galaxy formation and evolution, and tidal gravitational fields tend to locally align the spins of dark-matter halos and galaxies. Such alignments, observed in dark-matter simulations, bear information on galaxy formation and evolution, as well as on the growth of structure in the Universe [@Kirk15]. In this picture, low-mass haloes tend to acquire a spin parallel to cosmic web filaments, while the most massive haloes, which are typically the products of later mergers, have a spin perpendicular to filaments [e.g. @Codis12; @Dubois14]. Quasar observations provide observational evidence of a correlation between the polarization orientation of galaxies and the large-scale structure of the Universe [@Hutsemekers14], but only for a small number of sources. can uniquely contribute to characterizing this correlation for infrared-luminous galaxies.
The scientific goals outlined here are new and promising but still qualitative. Modelling is required to assess the scientific outcome of a deep polarimetric extragalactic survey with and decide on the best observing strategy in terms of survey depth and sky coverage.
Constraining Dust Physics {#sec:dust-physics}
=========================
The polarization of thermal dust emission depends on the shape, size, composition, and alignment efficiency of dust grains, and also on the 3D structure of the Galactic magnetic field on the line of sight and within the instrument beam. Dust polarized emission can therefore bring specific constraints on the alignment mechanism of dust grains and possibly on the grain shape. Despite this complex nature, it can also be used to constrain the optical properties of aligned dust grains (emissivity, spectral index), which are large grains at thermal equilibrium [@PlanckXXI2015; @PlanckXXII2015].
As described below, , with its high angular resolution and good wavelength coverage of the polarized dust SED, will be a key instrument to provide new constraints on grain alignment theories and inform the evolution of aligned grain properties from the diffuse ISM to the densest cloud cores.
Probing the grain alignment mechanism
-------------------------------------
Grain alignment is subordinate to various processes. First, grains must rotate supra-thermally[^12] to be well aligned. Radiative torques, or chemical torques resulting from the formation of H$_2$ molecules on the surface of grains are good candidates for grain spin-up. Second, alignment torques of magnetic, radiative, or mechanical origin [@Hoang_Lazarian2016] are needed to align the supra-thermally rotating dust grains. Compared to the time evolution of molecular clouds and cores, the alignment of grains along magnetic field lines by radiative torques is a fast process, with a timescale on the order of $10^3$–$10^4\,$ yr [@HL14][^13], so that situations where grain alignment would be out-of-equilibrium can be safely ignored. While the mechanisms of dust grain alignment are still debated (see, e.g., Sect. \[subsubsec:alignment\]), polarization measurements at UV to optical wavelengths imply that the alignment efficiency of dust grains is sensitive to grain size. Such a behavior, well observed in the Mie regime where absorption and scattering are size-dependent [@BH83], is however more difficult to extract from the polarized thermal SED because the dust temperature and dust spectral index depend more on the grain shape, internal structure, and composition (through its emissivity) than on the exact grain size.
![\[Andersson\_geometry\] Sketch of the geometry around a single star dominating the heating of the local ISM. The magnetic field direction is represented by the horizontal dashed lines. The aligned dust grains are sketched as prolate rotating parallelograms. If RATs dominate, dust alignment will be more efficient in regions with low values of $\psi$, the angle between the stellar radiation and magnetic field directions. Figure adapted from [@Andersson_Potter_2010] and [@Andersson+2011].](Andersson_fig.pdf){width="\columnwidth"}
As already mentioned in Sect. \[subsubsec:alignment\], the leading grain alignment theory is Radiative Alignment Torques (RATs) [e.g. @Lazarian_Hoang2007 and references therein]. The RATs alignment mechanism, if present, will lead to characteristic signatures in observations. For instance, in this theory, the alignment efficiency is directly dependent on the angle between the incident radiation field and the magnetic field direction ($\psi$). When dust is heated by a single nearby star or in starless dense cores where the field is strongly attenuated and anisotropic, the incident radiation field direction is well characterized. Since the magnetic field orientation projected on the plane of the sky can be determined from the polarized signal, mapping dust polarization in such regions can in principle be used to test alignment by RATs. This is illustrated in Fig.\[Andersson\_geometry\] which sketches the relative geometry of the radiation field and magnetic field in a region of the ISM where the radiation field is dominated by a single star. In such regions, the RATs alignment theory predicts a stronger alignment and therefore a higher polarization fraction where $\psi$ is close to 0 deg. Despite an expected clear signature, direct observational evidence of this angular effect has been scarce. The only positive detection reported in emission is by [@Vaillancourt_Andersson2015] who detected a periodic modulation of the dust polarization fraction around the Becklin-Neugebauer Kleinmann-Low (BNKL) object in Orion OMC-1. Some authors also claim to have evidenced a correlation between dust temperature and polarization fraction, as expected for dust grains aligned through the RATs mechanism [@Andersson+2011; @Mats2011]. There are also a few reports that indicate a possible influence of H$_2$ formation [e.g. @Andersson+2013 in IC63]. In any case, only a handful of cases have been investigated so far and there is certainly a bias in the literature for publishing detections rather than non-detections. Given the intrinsically tangled nature of magnetic field geometry, chance coincidences are very difficult or even impossible to exclude for these few isolated studies and a statistically representative study is clearly needed.
Such a study has not been carried out so far using [*Planck*]{} all-sky data, essentially because the number of interstellar regions where dust is directly and predominantly heated by a single star is very low at the [*Planck*]{} angular resolution. Attempts to unambiguously detect a statistical increase of the polarization fraction with dust temperature in the [*Planck*]{} data, which would also be attributable to radiation-enhanced spin up and alignment of dust grains, have not led to a strong conclusion. [@PlanckXII2018] showed that it is possible to disentangle, statistically, between what can be attributed to variations in grain alignment efficiency or grain properties, and what is due to line-of-sight and beam averaging of magnetic field structures. This study demonstrated that there is no strong variations in grain alignment efficiency in the diffuse ISM (up to a column density $N_{\rm H} \sim 2\times10^{22}$ cm$^{-2}$), but, due to the low angular resolution of the [*Planck*]{} data, could not conclude in the case of the high-density ISM. [@PlanckXII2018] did not find any correlation either between dust temperature and polarization fraction in the diffuse ISM. In conclusion, the analysis of the [*Planck*]{} all-sky data have so far not allowed to strongly confirm or rule out any specific grain alignment theory, but have provided an upper limit to the drop of alignement in the diffuse ISM. Owing to its much higher angular resolution and sensitivity, will allow us to systematically map the polarization of dust emission around thousands of individual stars heating the nearby ISM locally. The good coverage of the polarized SED will allow us to measure the temperature of aligned dust grains responsible for the polarized emission. Analysis of the polarization fraction as a function of the angle between the known radiation field and the magnetic field direction derived from polarization will allow us to test, for the first time, the RATs alignment theory with statistical significance, in a way that is not affected by local variations and the complexity of individual objects. At the same time, we will be able to detect, if present, the polarized emission resulting from the small temperature difference between grains heated face-on and edge-on immersed in an anisotropic radiation field [@Onaka+1995]. This process, which is only efficient at short wavelength [$\lambda \le 100\,\mu$m, @Onaka2000], would appear in data as a characteristic difference between the polarization fraction and angle measured at $100\,\mu$m and the ones measured by unaffected channels at longer wavelengths.
Altogether, the high resolution, sensitivity, and spectral coverage of will set unprecedented, probably unexpected constraints on the physics of grain alignment in star-forming regions, a topic which [*Planck*]{} observations could hardly address.
Dust polarization as a proxy for dust evolution
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![Polarization fraction as a function of wavelength predicted using the `DustEM` (<http://www.ias.u-psud.fr/DUSTEM>) numerical tool [@Compiegne2011; @Guillet+2018]. The vertical bands show the photometric channels. The dashed band shows a suggested shifted location for the short-wavelength band of at $70\,\mu$m, which would better cover the Wien part of the polarized dust SED. In model A, only silicate grains are aligned, while carbon grains are randomly aligned. In model D, both silicate and carbon are aligned, with carbon inclusions incorporated in the silicate matrix (6% in volume). Figure adapted from [@Guillet+2018].[]{data-label="model_dustem"}](PsI_G0_spica_bands.pdf){width="\columnwidth"}
The wavelength range covered by will allow us to disentangle between various dust models. The Wien part of the polarized dust emission is currently not constrained. It is in this wavelength range that dust models present the strongest differences in spectral variations of $P/I$ (Fig. \[model\_dustem\]), in particular between those where carbon grains are aligned and those where they are not [@Draine_Hensley2013; @Guillet+2018]. Moreover, as dust models now predict both emission and absorption properties of dust grain populations in polarization [@Siebenmorgen+2014; @Draine_Hensley2017; @Guillet+2018], joint observations of common targets with and survey experiments targeting extinction polarization of background stars, such as PASIPHAE [@Tassis+2018], can be used to further test such models. High-resolution polarization observations with will allow us to probe dust properties in dense environments and to further characterize dust evolution between diffuse and dense media [e.g. @Kohler+2015]. Observationally, a number of studies based on emission and extinction data have provided evidence of grain growth within dense clouds. One of the main results is an increase of the far-IR/submillimeter emissivity by a typical factor of 2–3 compared to standard grains in the diffuse medium [cf. @Stepnik+2003; @PlanckCollaboration25_2011; @Ysard+2013; @Roy+2013; @Juvela+2015], as predicted by calculations of aggregate optical properties [@Ossenkopf+1994; @Kohler+2012; @Kohler+2015]. Another striking result is the so-called [*coreshine*]{} effect, i.e., enhanced mid-IR light scattering detected with [*Spitzer*]{} toward a number of dense cores [@Pagani+2010; @Steinacker+2010], implying the presence of larger [@Steinacker+2015] or, taking into account the change in dust optical properties, only moderately larger [@Ysard+2016], dust grains.
Stochastic emission by very small grains ($a\ll 10$nm) is known to contribute significantly to the 60$\mu$m and 100$\mu$m emission bands. This contribution is estimated to be on the order of 13% at 100$\mu$m, and 45% at 60$\mu$m [e.g. @Jones+2013]. When the 100$\mu$m band is used to derive the dust temperature[^14], this contamination affects the accuracy of mass determinations using dust continuum measurements. The formation of dust aggregates first removes the very small grains from the gas phase, as suggested by observations showing a significant decrease in the 60$\mu$m emission [@Laureijs+1991; @Bernard+1999; @Stepnik+2003; @Ysard+2013] and as predicted by dust evolution models [@Ossenkopf+1994; @Kohler+2015]. Because small grains are not aligned with the magnetic field, such contamination is absent from the polarized thermal emission SED[^15]. As a consequence, the dust temperature derived[^16] from the polarized dust SED that can observe will only reflect the temperature of large aligned dust grains in the transition from the diffuse to the dense ISM, a constraint that will be used in addition to that inferred from the total intensity SED to study dust evolution processes.
Just like unpolarized emission, polarized dust emission will also probe variations in dust emissivity as expected from dust evolution in dense clouds. Dense environments could not be properly characterized in polarization at the low resolution of the [*Planck*]{} data. Unlike unpolarized emission, the anisotropic nature of polarized emission makes it sensitive to the grain shape. The formation of dust aggregates by grain-grain coagulation must have its counterpart in polarization, and will detect signatures which will have to be analyzed through detailed modeling of the coagulation process. Here again, the combination of data with the increasing amount of high-resolution polarization observations in the optical and the near-IR will provide strong constraints on dust evolution [@PlanckXXI2015; @PlanckXII2018].
Observations of total dust emission intensity at far-infrared and submillimeter wavelengths with [*Planck*]{} and [*Herschel*]{} have also brought surprises. One of them is evidence that the logarithmic slope of the dust emission SED at long wavelengths, often referred to as the dust emissivity index, $\beta$, exhibits significant variations at large scales across the Galaxy. The [*Planck*]{} all-sky data clearly show variations of $\beta$ along the Galactic Plane, from very steep SEDs toward inner regions of the Milky Way to much flatter SEDs ($\beta \simeq1.5$) toward the Milky Way anticenter [@PlanckXI+2014]. This has also been confirmed in the far-infrared by the analysis of Hi-GAL data [@Paradis+2012]. Even larger variations have been found in observations of external galaxies, with the SMC and LMC having $\beta\simeq1.3$ and $\beta\simeq1.0$, respectively [@PlanckXVII2011]. Such variations are observed in the [*Herschel*]{} data within individual nearby galaxies such as M31 [@Smith+2012] and M33 [@Tabatabaei+2014]. The origin of these variations is currently unclear and three main classes of dust models have been proposed to explain them. The first type involves the mixing of different materials during the dust life-cycle [@Kohler+2015; @Ysard+2015]. The second type of models invokes Two-Level-System (TLS) low-energy transitions in the amorphous material composing dust grains [@Meny+2007] as the cause for the flattening of the SED. The third type of models proposes that magnetic inclusions in dust grains [@Draine_Hensley2013] could produce the observed variations [@Draine_Hensley2012]. Determining the origin of these variations is critical in many respects, not only to understand the dust cycle in the ISM, but also for accurate mass determinations from dust continuum measurements (which require good knowledge of the dust emissivity, its wavelength dependence, and its spatial variations).
In this domain again, extensive polarimetric imaging at far-IR wavelengths is likely to play a critical role in the future. Dust models based on dust evolution have not yet presented their predictions in polarization, but the other two classes of models mentioned above predict significantly different behaviors for the polarization fraction as a function of wavelength. TLS-based models essentially predict a flat spectrum for the polarization fraction, a prediction compatible with [*Planck*]{} observations. In contrast, metallic-inclusion models predict variations of the polarization fraction in the submillimeter [@Draine_Hensley2013], which are not observed. It will be possible to evidence those distortions of the polarization SED by comparing far-IR observations with existing submillimeter [*Planck*]{} data for the Magellanic clouds and polarization data obtained with new ground-based polarimetric facilities such as NIKA2-POL and SCUBA2-POL for Milky Way regions/sources and nearby galaxies. Correlating changes in the polarized SED with variations of $\beta$ will allow us to constrain models of the submillimeter dust emissivity in a very unique way.
Toward a tentative detection of polarization by dust self-scattering in the densest cores
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In the past decade, interesting constraints on grain sizes in dense clouds have come from the detection of the “coreshine” effect [@Pagani+2010], which results from scattering of near-IR stellar photons by dust grains present in the cloud. This has been interpreted as evidence of grain growth ($a=1\, \mu$m, [@Steinacker+2010]) or grain compositional and structural evolution with a modest size increase ($a<0.5\, \mu$m, [@Ysard+2016]). More recently, it has been demonstrated that very large ($a > 10\, \mu$m) dust grains are able to produce polarization by scattering thermal dust emission, a process that is called ’self-scattering’. This was first predicted to be observable in protoplanetary disks [@Kataoka+2015] and then confirmed by numerous ALMA observations [@Kataoka+2016; @HYang+2017; @Girart+2018]. , with its $100\,\mu$m polarized channel, would be able to detect and characterize the spectral dependence of polarization by scattering due to $\sim 15\,\mu$m dust grains [@Kataoka+2015], if present. For the effect to be observable, the thermal emission of dust must first present a quadrupolar anisotropy: scattering grains must receive more far-IR irradiation along one direction in the plane of the sky than along the perpendicular direction. Such a condition is naturally met in a dense protostellar core, or in the presence of density gradients. Second, high local densities ($>10^6\, {\rm cm}^{-3}$) must be present along the line of sight so that dust grains can have grown to the very large sizes ($a \sim \lambda/2\pi$) needed for scattering to occur in the far-IR. Observing such a high-density medium should be feasible at $100\,\mu$m at the resolution of SPICA, but simulations of grain growth and polarization by scattering are needed to confirm this idea.
Polarization by self-scattering at $100\,\mu$m with will most likely concern only a few lines of sight through the densest cores. Because polarization due to scattering sharply declines at wavelengths larger than the grain size, it will not alter the polarized emission from aligned dust grains at $200\,\mu$m and $350\,\mu$m used to trace the local magnetic field orientation in molecular clouds (§ \[sec:filaments\] to §\[sec:massive-sf\]).
Molecular clouds and the origin of cosmic rays {#sec:cosmic-rays}
==============================================
Polarimetric imaging of molecular clouds (MCs)[^17] with will also be very useful for the study of the origin of cosmic rays (CRs). Indeed, CRs pervade the whole galaxy, and their interaction with the dense gas of MCs has two important consequences. First, the interactions of high energy CRs (kinetic energy larger than a few hundred MeV) with the gas make MCs bright $\gamma$-ray sources. Second, CRs of low energy ($\approx 1-100$ MeV) are the only ionizing agents able to penetrate MCs and regulate the ionization fraction of MC dense cores. For these reasons, observations of enhanced levels of $\gamma$-ray emission or ionization rates from MCs reveal the presence of a CR accelerator in their vicinity (see review by [@GabiciMontmerle2015]).
A first problem arises, that of the *propagation* of cosmic rays inside molecular clouds, for which the strength, and above all, the topology of the magnetic field around and inside them plays a central role. At high energies, CRs are unaffected by the magnetic field, so that they interact with all the gas (atomic as well as molecular): the $\gamma$-ray emissivity is simply proportional to the product (CR flux $\times$ cloud mass). In other words, for a given cloud mass, determining the $\gamma$-ray luminosity of a molecular cloud allows the local CR flux to be measured, irrespective of the magnetic field. Over galactic scales, it is well established that the CR flux deduced in this way is essentially uniform [@Ackermann+2011], which means that the CR diffusion away from their sources is efficient enough not to be sensitive to large-scale spatial features like the spiral arms. However, there may be $\gamma$-ray “hot spots” close to CR sources, and this is precisely what happens when a supernova remnant (SNR) collides or is located in the vicinity of a molecular cloud. Many examples of such SNR-MC associations are known [@GabiciMontmerle2015]. The reason for the enhanced $\gamma$-ray emission is that SNR shock waves accelerate CR in situ, via the so-called Diffuse Shock Acceleration, or DSA, mechanism [@Drury1983], so this particular configuration can be considered as a “CR laboratory”: If the shock-accelerated CRs are insensitive to the local magnetic fields when they reach high energies, the process by which they do, in other words the *acceleration* mechanism itself depends very much on it, and again on its topology close to the SNR shock. For instance, it is well known from theoretical DSA models that the acceleration efficiency depends strongly on the angle between the shock front and the local magnetic field lines [@CaprioliSpitkovsky2014]. So, again, knowing the magnetic field topology on small spatial scales in MCs impacted by SNRs would allow a detailed study of the CR acceleration process in the vicinity of the shock front.
In particular, a new picture of the ISM in star-forming regions has now to be taken into account: in the standard picture summarized above, the gaseous medium in which the SNR shock propagates is assumed uniform. But as recent work has shown (see Sect. \[subsec:fil-paradigm\] and Fig. \[taurus\_planck\]), the structure of MCs is not uniform, but *filamentary*, down to scales of parsecs (in length) and $\sim 0.1$ pc (in width), i.e., precisely those that are accessible to at distances $\approx 1$ kpc. One of the key changes is then that the shock would cross the ambient magnetic field lines at all angles, and likely perturb them and enhance the turbulent component of the magnetic field, on spatial scales comparable to that of the filaments: for not-too-distant sources, would then act as a “magnifying-glass” to study the shock-ambient gas interactions at unprecedented small spatial scales, and put entirely new, perhaps even unexpected, constraints on DSA models (e.g., see the CR escape issues raised by @Malkov+2013).
At low energies, the situation is markedly different, because the transport properties of CRs become very sensitive to a variety of processes governed by the magnetic field properties which may hamper the penetration of CRs into MCs, and reduce the rate at which the gas is ionized by these particles [@Phan+2018]. Like $\gamma$-ray production at high CR energies, the ionization by low-energy CR can also be measured by way of infrared and millimeter-wave observations, which detect lines of various molecules and radicals (like H$^+_3$, HCO$^+$, DCO$^+$, etc. – cf. [@Indriolo+2015]). This has been done for many MCs in the Galaxy, but more recently also for SNR-MC collision regions [@GabiciMontmerle2015]: here again an enhancement of MC ionization has been observed. The results tentatively suggest a proportionality between the SNR-accelerated high-energy and low-energy CR fluxes, constraining the acceleration mechanism, or a proportionality between the partially irradiated, ionized gas, and the fully irradiated, $\gamma$-ray emitting gas, or both.
More generally, both low- and high-energy CRs are affected in their propagation in the diffuse ISM by *diffusion* effects, which are still poorly known –and directly influenced by magnetic fields. The spatially average diffusion coefficient of CRs in the Galaxy is constrained by a number of observations, and is often assumed to be isotropic on large Galactic scales ($\gg 100$ pc). On the other hand, in order to explain a number of $\gamma$-ray observations of SNR-MC associations (characterized by spatial scales of $\sim 10-100$ pc), a diffusion coefficient about two orders of magnitudes smaller (i.e., slower bulk propagation) than the average Galactic one needs to be assumed. However, such a discrepancy could be reconciled if CR diffusion is in fact anisotropic on such small scales [@NavaGabici2013]. An anisotropic diffusion is indeed expected for spatial scales smaller than the magnetic field coherence length [@Malkov+2013]. Knowing the topology of the magnetic field in such regions is thus of paramount importance in order to interpret $\gamma$-ray observations correctly.
![The region surrounding the W28 SNR ($d \sim 2$ kpc; shock approximated by the dashed white circle), as seen in cold dust emission at $353\, $GHz by *Planck* (color image with background $B$-field “drapery” from polarization data), in TeV-GeV $\gamma$-rays (white contours), and CO (green areas, well correlated with the $\gamma$-ray sources – [@Aharonian+2008],). The labels highlight the various CR processes discussed in the text, at high energies (HECR) and low energies (LECR). The *Planck* and beams are indicated by a light green circle (label “P”) and a dot (label “S”), respectively. []{data-label="w28"}](spica_wp_W28_CR_3.jpg){width="\columnwidth"}
All of these issues can be illustrated by a recent study of the W28 SNR (cf. Fig. \[w28\]), a well-known example of an SNR-MC collision [@Vaupre+2014]. This SNR is located in the Galactic plane, at $d \approx 2$ kpc from the Sun, with an estimated (very uncertain) age $\approx 10^4$ yr. At this distance, the SNR apparent diameter ($\sim 30'$) gives a linear diameter $D \approx 20$ pc. An observation by the High Energy Stereoscopic System (*HESS*) Čerenkov telescope, in the TeV $\gamma$-ray range [@Aharonian+2008], covering a large field-of-view of $\sim 1.5^\circ \times 1.5^\circ$ (with a resolution of $\sim 0.1^\circ$), has revealed a complex of several resolved $\gamma$-ray sources. One of the sources, which is spatially correlated with a part of the SNR shock outline, was also detected as a bright GeV source by the *Fermi* satellite, contrary to the other sources, which are either dimmer or undetected [@Abdo+2010]. This multiple source was soon correlated with a complex of molecular clouds mapped in CO by the *NANTEN*[^18] telescope, showing that the SNR was in fact colliding with the molecular cloud associated with the GeV-TeV source (a physical contact being confirmed by the existence of several OH masers), the other sources being away, far upstream of the SNR shock.
Calculations indicated a factor $\approx 100$ enhancement of the local high-energy CR flux, qualitatively consistent with a local CR acceleration by the SNR shock. Using the IRAM 30-m telescope to observe various molecular and radical tracers (H$^{13}$CO$^+$, C$^{18}$O, etc.) in the millimeter range, @Vaupre+2014 were also able to calculate the MC ionization rate $\zeta$ at several locations. They found $\sim 2-3$ order of magnitude enhancements of $\zeta$ (or lower limits) over its average Galactic value ($\zeta_0 \approx 4-5 \times 10^{-17} \rm{erg~s}^{-1}$ – e.g. [@Indriolo+2015]), for the cloud correlated with GeV-TeV emission, i.e., indirect evidence for a similar enhancement of the low-energy CR flux, but no such enhancements for the clouds far upstream of the SNR shock. Within the “GeV-TeV bright” cloud, the measurements were separated by the IRAM telescope resolution, $\sim 12 ''$, i.e., comparable to (only 1.5 times better than) the resolution at $200\, \mu$m (or a linear scale $\sim 0.15$ pc). The (projected) distance to the other clouds is $\sim 10$ pc, and this is seen as the diffusion length for high-energy CRs (TeV CRs reaching the distant clouds before the GeV CRs).
Thus, the W28 SNR and its environment provide us with a case study with all the ingredients needed to improve our understanding of the origin of CRs, and their relation with magnetic fields down to scales $\sim 0.15$ pc, i.e., [*much smaller than observable before*]{}: $(i)$ CR *acceleration* by the SNR shock; $(ii)$ *diffusion* of CRs between clouds as a function of energy; $(iii)$ *penetration* of low-energy CRs in “average” clouds, irradiated only by ambient, galactic CRs (cf. Fig. \[w28\]).
For more distant sources, the “magnifying-glass” effect of on small spatial scales would of course decrease, but an interesting link could then be established with the *Čerenkov Telescope Array* (*CTA*, presently under construction; @Actis+2011), which operates in the 20 GeV–300 TeV regime. Until *CTA* actually observes, it is difficult to make accurate predictions on how far SNR-MC systems such as W28 could be detected in $\gamma$-rays. For CR studies, the main point is not simply the detection, but the location of the emission with respect to the shock. For the moment, only two cases are known in which the $\gamma$-rays are clearly upstream of the shock: W28 (detected by *CGRO*, *Fermi* and *HESS*, so from GeV to TeV energies), and W44 (detected by *Fermi* only, so at GeV energies only). About 20% of *HESS* sources, and most of the SNRs detected by *Fermi* are SNR-MC systems (@HESS2018; @Ackermann+2011), but apart from W28 and W44 it is difficult to distinguish between upstream and downstream $\gamma$-ray emissions (or both). Taking into account that *CTA* is $\sim$ 10 times more sensitive than *HESS*, and taking W28 as a template, representative of a Galactic disk SNR-MC population, we estimate that $\approx$ ten W28-like sources could be possibly detected and be sufficiently resolved by *CTA*, hence be good targets for future observations, which would in turn spur new theoretical work on CR acceleration on very small spatial scales not considered at present.
[.4]{} {width="\linewidth"}
[.42]{} {width="\linewidth"}
Polarized dust emission from protoplanetary disks {#sec:disks}
=================================================
As already mentioned in Sect. \[sec:filaments\], magnetic fields may regulate the gravitational collapse and fragmentation of prestellar dense cores, thereby influencing the overall star formation efficiency [@MouschoviasCiolek1999; @Dullemond+2007; @Crutcher2012]. It is thus natural to expect that, during core collapse, magnetic fields can be dragged inward, leaving a remnant field in the protoplanetary disk formed subsequently.
If protoplanetary disks are indeed (weakly) magnetized, then the MHD turbulence arising from magneto-rotational instability (MRI) is thought to be the primary source of disk viscosity, a crucial driving force for disk evolution (e.g. disk accretion) and planet formation [@Balbus+1998; @Turner+2014]. Despite this general consensus, our knowledge about magnetic fields in disks is actually very limited and incomplete at this stage, largely due to the lack of observational constraints on magnetic field properties (geometry and strength) in protoplanetary disks. Polarimetric observations of thermal dust emission at centimeter or millimeter wavelengths with single-dish telescopes, such as the Caltech Submillimeter Observatory (CSO) and JCMT, or interferometric arrays, such as the Karl G. Jansky Very Large Array (JVLA), the SMA, the Berkeley-Illinois-Maryland Array (BIMA), and the Combined Array for Research in Millimeter-wave Astronomy (CARMA), have been used extensively to map magnetic field structure in YSOs at scales from $\sim 50\,
$AU to thousands of AU (see [@Crutcher2012] for a review). However, due to the limited sensitivity and angular resolution offered by current facilities and the nature of centimeter/millimeter observations, most of these studies have been focused on magnetic fields in molecular clumps/cores (cf. § \[sec:protostars\]), or Class 0/Class I objects [e.g. @Qiu+2013; @Rao+2014; @Liu+2016], rather than classical protoplanetary disks around Class II objects. Using CARMA, @Stephens+2014 spatially resolved the HL Tau disk in polarized light at 1.3 mm, and their best-fit model suggested that the observation was consistent with a highly tilted (by $\sim$50$^\circ$ from the disk plane), toroidal magnetic field threading the disk. This conclusion was challenged by follow-up studies, which showed that the 1.3 mm polarization of HL Tau could also arise solely from dust scattering as opposed to dichroic emission from elongated grains aligned with the magnetic field [@Kataoka+2015; @Yang+2016]. However, more recent ALMA observations at 0.87, 1.3, and 3.1 mm indicated that dust scattering alone may not be able to explain all of the multi-wavelength polarization data [@Kataoka+2017; @Stephens+2017].
Recently, @Li+2016 have been able to highlight the signature of a magnetic field in the AB Aur protoplanetary disk at mid-IR wavelengths. Using observations of the AB Aur protoplanetary disk at 10.5 $\mu$m with the GTC/CanariCam imager and polarimeter, they detected a polarization pattern in the inner regions of the disk compatible with dichroic emission polarization produced by elongated grains aligned by a tilted poloidal magnetic field. The observed polarization level (2-3 %) was somewhat lower than that predicted by theory [@Cho_Lazarian2007], although this is something naturally expected since the modeling assumes alignment efficiencies and intrinsic particles polarizability which are probably overestimated. At a wavelength of 10.5 $\mu$m where protoplanetary disks are optically thick, the observations are probing the disk properties down to depths corresponding to the $\tau$=1 optical depth surface. This depth is relatively small (less than $\sim$10%) compared to the disk scale height. Longer wavelengths, up to about 200 $\mu$m where the disk can still be moderately optically thick up to large distances from the star, are emitted by cooler material located deeper within the disk. Thus, by measuring the level of polarization at wavelengths in the range 100–300 $\mu$m, we expect to be able to compare the levels of polarization as a function of optical thickness, thereby getting an indirect signature of the magnetic field at different depths within protoplanetary disks. This will provide constraints on the importance of MRI-induced turbulence. Together with 10 $\mu$m and ALMA similar types of observations, this will allow us to build a tomographic view of the magnetic field along the vertical profile of the disks. Such measurements would also have large impacts on our understanding of planet formation processes.
For the purpose of this paper, we used the same modeling approach as described in @Li+2016, with also the same disk parameters based on the example of AB Aur, in order to predict expected polarization levels levels at 100 $\mu$m, 200 $\mu$m, and 350 $\mu$m if observed by SPICA. Given the angular resolution of SPICA at these wavelengths, we do not expect, apart from exceptional cases, to angularly resolve the polarized emission, therefore we computed an integrated value, considering the object as unresolved.
In Fig. \[diskspol\], we show the predictions of the model, integrated over the full spatial extent of the disk, for different disk inclinations with respect to the line of sight and for observing wavelengths within the range. Two types of magnetic field configuration are considered, poloidal and toroidal, which are the simplest ones, and those also widely discussed in the literature. We can see from the simulations that the poloidal magnetic field configuration produces stronger integrated polarization signatures compared to the toroidal configuration.
As mentioned earlier, the origin of the dust continuum polarization on the disk scale is still uncertain, with potential contributions from scattering by large grains in addition to that from emission by magnetically (or radiatively) aligned grains. The instrument will generally not (or barely be able to) resolve protoplanetary disks, which poses the problem of disentangling these various mechanisms. Fortunately, polarization by aligned dust grains and polarization by dust self-scattering have different dependences on wavelength and optical depth [@HYang+2017]. SPICA will greatly extend the wavelength coverage of ALMA (from 870$\, \mu$m to 100$\, \mu$m), which will make it easier to disentangle the contributions from the competing mechanisms. Such an effort is a pre-requisite for using dust continuum polarization to probe both disk magnetic fields and grain growth, the crucial first step toward the formation of planetesimals and ultimately planets. Moreover, given the plan to image the whole extent of nearby star-forming regions with (cf. end of § \[subsec:filaments\]), several tens of protoplanetary disks will be detected in Stokes I, Q, U. It will therefore be possible to derive statistical trends about the presence of magnetic fields, and any bias can be controlled providing that the inclination and position angles of the disks are known.
Variability studies of protostars in the far-infrared {#sec:proto-var}
=====================================================
At the core ($\leq 0.1\,$pc) scale, the formation of a solar-type star is well understood as a continual mass assembly process whereby material in the protostellar envelope is accreted onto a circumstellar disk and then transported inward and onto the protostar via accretion columns [@Hartmann+2016]. The observational evidence for the mass assembly rate of low-mass stars is provided by the lifetimes of the various stages and the bolometric luminosities, which are dominated by accretion energy at early times [e.g. @Dunham+2014 and references therein]. These two quantifiable measures are in significant disagreement and circumstantial evidence exists for the episodic nature of mass assembly – bullets in outflows [e.g. @Plunkett+2015], FU and EX Ori phenomenon [@Hartmann+1996; @Herbig2008], numerical calculations of disk transport [e.g. @Armitage2015]. Moreover, high spatial resolution images of young disks reveal macroscopic structure including rings [@ALMA+2015], spirals [@Perez+2016], and fragmentation [@Tobin+2016], suggesting that the transport of material through the disk is not a smooth and steady process.
Recently, significant variability has been detected in the submillimeter continuum emission of several nearby protostars [see, e.g., @Mairs+2017b; @Yoo+2017; @Johnstone+2018], through an ongoing multi-year monitoring survey of eight nearby star-forming regions with JCMT at $850\, \mu$m [@Herczeg+2017]. While uncertain due to small number statistics, it appears that roughly 10% of deeply embedded protostars vary over year timescales by around 10% at submillimeter wavelengths [@Johnstone+2018]. The dominant mode of variability uncovered by the survey is quasi-secular, with the protostellar brightness increasing or decreasing for extended - multi-year - periods [@Mairs+2017b; @Johnstone+2018] and suggesting a link to non-steady accretion processes taking place within the circumstellar disk where the orbital timescales match those of the observed variability. These long timescales also allow for significant amplification of the overall change in submillimeter brightness after many years. One source, EC 53 in Serpens Main, has an eighteen-month quasi-periodic light curve [@Yoo+2017 and Fig. \[proto\_var\_fig\]], previously identified through near-IR observations [@Hodapp+2012], which is likely due to periodic forcing by a long-lived structure within the inner several AU region of the disk.
![Time variation observed over 27 epochs at $850\, \mu$m for the Class I protostar EC 53 in the Serpens Main star-forming region as part of the “JCMT Transient Survey” [@Yoo+2017]. The typical uncertainty in a single measurement is $\sim 20\,$mJy (S/N $\sim$ 50) and the peak to peak brightness variation is almost 500mJy. Figure adapted from [@Johnstone+2018b]. []{data-label="proto_var_fig"}](EC53_850microns_Atel.pdf){width="\columnwidth"}
Stronger variability is expected at far-IR wavelengths where Class 0 and Class I YSOs have the peak of their SEDs and the envelope emission directly scales with the internal (accretion) luminosity of the underlying protostar [@Dunham+2008; @Johnstone+2013]. At submillimeter wavelengths the emission typically scales with the envelope temperature and thus the far-IR signal is expected to be around four times larger such that a 10% variability in the submillimeter relates to a 40% variability in the far-IR. Contemporaneous monitoring of EC 53 in the near-IR and at 850/450$\, \mu$m has confirmed that the longer submillimeter wavelength shows just such a diminished response to the underlying change in internal luminosity as proxied by near-IR observations (Yoo et al., in prep.). Thus, carefully calibrated monitoring of protostars with SPICA should uncover a significantly larger fraction of variables than the 10% obtained by the JCMT survey.
Thanks to its high continuum sensitivity and mapping speed at wavelengths around the peak of protostellar SEDs, , used as a total-power imager, will be ideal for monitoring hundreds of forming stars over multi-year epochs, allowing an unprecedented statistical determination of the variation in accretion on these timescales. Typical nearby deeply embedded protostars have far-IR brightnesses greater than $\sim10\, $mJy and thus will be observed to a S/N $\simgt $ 100 by in a fast scanning mode. As demonstrated for ground-based submillimeter observations [@Mairs+2017a], instrument stability will need to be carefully monitored in order to achieve precise relative flux calibration of a few percent between epochs. Additional critical requirements for will be a large, $10^5$ or higher, dynamic range and instrument robustness against extremely bright sources within the field.
Three interconnected monitoring surveys are envisioned. First, the bulk of the $\sim1000$ nearby, Gould Belt, deeply embedded protostars will be observed every six months while SPICA is in orbit, requiring coverage of $\sim 20$ deg$^2$ (a modest twenty hours of observing per epoch). This will allow for a detailed statistical characterization of variability across multiple years. Additionally, a few carefully chosen nearby star formation fields, each roughly a square degree, will be observed weekly during their expected few month continuous observing window [for information on observing strategies for SPICA, see @Roelfsema+2018]. For both of these nearby samples, an even larger number of Class II YSOs will be observable within each field. While these sources will be fainter at far-IR wavelengths as the emission probes the disks directly, the enhanced numbers will allow for a determination of the importance of variability throughout the evolution of a protostar. Finally, a sample of more distant high-mass star-forming regions should be observed yearly to search for rare, but extremely bright, bursts such as FU Ori events. While SPICA will not have the spatial resolution to separate individual protostars within these regions, evidence of a significant brightening can be easily followed-up with ground-based telescopes such as ALMA (see [@Hunter+2017] for an example of a brightening in a high-mass star-forming region).
Concluding remarks {#sec:conclusions}
==================
Magnetic fields are a largely unexplored “dimension” of the cold Universe. While they are believed to be a key dark ingredient of the star formation process through most of Cosmic time, they remain very poorly constrained observationally, especially in the cold ISM of galaxies [e.g. @Crutcher2012].
Benefiting from a cryogenic telescope, SPICA-POL or will be two to three orders of magnitude more sensitive than existing or planned far-IR/submillimeter polarimeters (cf. Fig. \[spica\_sensitivity\]) and will therefore lead to a quantum step forward in the area of far-IR dust polarimetric imaging, one of the prime observational techniques to probe the topology of magnetic fields in cold, mostly neutral environments. In particular, systematic polarimetric imaging surveys of Galactic molecular clouds and nearby galaxies with have the potential to revolutionize our understanding of the origin and role of magnetic fields in the cold ISM of Milky-Way-like galaxies on scales from $\sim 0.01\, $pc to a few kpc. The three main science drivers for are 1) probing how magnetic fields control the formation, evolution, and fragmentation of dusty molecular filaments (Sect. \[sec:filaments\]), thereby setting the initial conditions for individual protostellar collapse (Sects. \[sec:protostars\] and \[sec:massive-sf\]); 2) characterizing the structure of both turbulent and regular magnetic fields in the cold ISM of nearby galaxies, including the Milky Way, and constraining galactic dynamo models (Sect. \[sec:turbulence\] and Sect. \[sec:galaxies\]); and 3) testing models of dust grain alignment and informing dust physics (Sect. \[sec:dust-physics\]). Other science areas can be tackled with, or uniquely informed by, observations, including the problem of the interaction of cosmic rays with molecular clouds (Sect. \[sec:cosmic-rays\]), the study of the magnetization of protoplanetary disks (Sect. \[sec:disks\]), and the characterization of variable accretion in embedded protostars (Sect. \[sec:proto-var\]). Last, but not least, the leap forward provided by in far-IR imaging polarimetry will undoubtedly lead to unexpected discoveries, such as the potential detection of polarization from the CIB (Sect. \[subsec:cib\]).
This paper is dedicated to the memory of Bruce Swinyard, who initiated the SPICA project in Europe, but unfortunately died on 2015 May 22 at the age of 52. He was ISO-LWS calibration scientist, Herschel-SPIRE instrument scientist, first European PI of SPICA and first design lead of SAFARI. This work has received support from the European Research Council under the European Union’s Seventh Framework Programme (ERC Advanced Grant Agreement no. 291294 – ‘ORISTARS’, and Starting Grant Agreement no. 679937 – ‘MagneticYSOs’). D.$\, $J. is supported by the National Research Council of Canada and by an NSERC Discovery Grant. H.$\, $B. acknowledges support from the ERC Consolidator Grant CSF-648505 and financial help from the DFG via the SFB881 “The Milky Way System” (subproject B1). The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
Affiliations {#affiliations .unnumbered}
============
[^1]: Some other [[*SPICA*]{}]{} papers refer to this field of view as 80$\times$80, but it is 160$\times$160 according to the latest design.
[^2]: B-BOP stands for “B-fields with BOlometers and Polarizers”.
[^3]: With the possible exception of extreme star-forming environments like the central molecular zone (CMZ) of our Milky Way [@Longmore+2013] or extreme starburst galaxies [e.g. @Garcia-Burillo+2012]. See other caveats for galaxies in @Bigiel+2016.
[^4]: In Galactic molecular clouds, a visual extinction $A_V = 1$ roughly corresponds to a column density of H$_2$ molecules $N_{H_2} \sim 10^{21}\, {\rm cm}^{-2}$ [cf. @Bohlin+1978].
[^5]: There is a whole spectrum of molecular clouds in the Galaxy, ranging from individual clouds $\sim \,$2–10$\,$pc in size and $\sim 10^{2-4}\, M_\odot$ in mass to GMCs $\sim \,$50$\,$pc or more in diameter and $\sim 10^{5-6}\, M_\odot $ in mass ([@Williams00a], [@Heyer+2015], and references therein).
[^6]: The mapping speed is defined as the surface area that can be imaged to a given sensitivity level in a given observing time.
[^7]: Note that total power ALMA data can only be obtained for spectral line observations and are not possible for continuum observations.
[^8]: The magnetic Prandtl number is the ratio between the kinetic viscosity and the magnetic diffusivity.
[^9]: By ridge, we do not mean the crest of a given filament but a massive elongated structure ($> 1000\, M_\odot $ of dense molecular gas with $n_{H_2} > 10^5\, {\rm cm}^{-3}$), consisting of a dominant highly supercritical filament and an accompanying network of sub-filaments, often themselves supercritical.
[^10]: The spatial dynamic range is defined as the ratio of the largest to the smallest spatial scale accessible to an instrument or an observation (see also Sect. \[subsec:spica-adv\]).
[^11]: The list is continually updated in the arXiv version at [https://arxiv.org/abs/1302.5663]{}
[^12]: The grain thermal energy, which is radiatively balanced, is not in equipartition with the grain rotational energy, allowing for the suprathermal rotation of large grains under specific torques [@Purcell1979].
[^13]: More precisely, the grain alignment timescale by the RATs mechanism is about three orders of magnitude shorter than the local free-fall time.
[^14]: This is the case for [*Planck*]{} studies using 100$\mu$m IRAS data, but not for [*Herschel*]{} results based on SED fitting between 160$\mu$m and 500$\mu$m.
[^15]: This is probably also valid for the zodiacal light emission, which severely contaminates the 60$\mu$m band near the Ecliptic plane in total intensity, but should not in polarization because the large warm grains responsible for this emission are not known to be aligned. This will however have to be checked.
[^16]: In the Rayleigh regime ($a\ll \lambda$) that characterizes thermal dust emission, the influence of the magnetic field and alignment efficiency on polarization observables is achromatic, and therefore does not affect the spectral dependence of the SED, but only its amplitude. The spectral index of the polarized SED will therefore characterize the optical properties of aligned grains.
[^17]: In this section, “molecular clouds” are used in a broad sense, ranging from small individual dark clouds $\sim \,$2–10$\,$pc in size to GMCs $\sim \,$50$\,$pc in diameter (see Sect. \[sec:filaments\]). There is no clear cut-off size for the physical effects discussed here.
[^18]: NANTEN means “Southern Sky” in Japanese.
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